Reeta Vyas - Physics - University of Arkansas

PHYSICAL REVIEW A 76, 052317 (2007)Generation and evolution of entanglement in coupled quantum dots interactingwith a quantized cavity fieldAmab Mitra and Reeta VyasDepartment of Physics, University of Arkansas, Fayetteville, Arkarlsas 72701. USADaniel ErensoDepartment of Physics & Astmnomv, Middle Tennessee State University, Murfeesboro, Tennessee 37132, USA(Received 14 August 2007; published 20 November 2007)The generation of entanglement between two identical, interacting quantum dots-initially in groundstates-by a coherent field and the subsequent time evolution of the entanglement are studied by calculatingthe concurrence between the two dots. The results predict that while it is possible to generate entanglement (orentanglement of formation, as defined for a mixed state) between the two dots: at no time do the dots becomcfully entangled to each other or is a maximally entangled Bell state ever achieved. We also observe that thedegree of entanglement increases with an increase in the photon number inside the cavity and a decrease in thedot-photon coupling. The behavior of the two-dot system, initially prepared in an entangled state and interactingwith thermal light, is also studied.DOI: 10.llO3/PhysRevA.76.0523 17 PACS number(s): 03.67.Mn, 78.67.Hc, 03.65.Ud. 42.55.SaI. INTRODUCTIONThe coherent evolution of two quantum bits (qbits) in anentangled state of the Bell type is at the heart of the study ofquantum entanglement and is fundamental to both quantumcryptography [l] and quantum teleportation [2]. Differentsystems and methods for the preparation and measurement ofmaximally entangled states are being investigated intensively.Many of these investigations have centered on atomicand quantum-optical systems [3-51. However, generation ofentangled states in solid-state systems such as semiconductorquantum dots (QDs) has also received a lot of attention[6-91. Recent experiments have shown that QDs can showmany interesting quantum features such as entanglement[lo], photon antibunching [I 11, Rabi oscillations [12], andmodification of spontaneous emission decay rate [13].In our present work, we study the generation and evolutionof entanglement between two coupled, identical quantumdots interacting with a quantized cavity field. Similarstudies have been reported by several previous authors.Quiroga and Johnson [6,7] have shown that two andthree equally coupled QDs, interacting with a suitably chosenfield, can generate maximally entangled Bell andGreenberger-Home-Zeilinger (GHZ) states respectively.However, they used a classical field instead of a quantizedone and measured the degree of entanglement between thedots by calculating the overlap of the state vector with acompletely entangled state. Yi et al. [8] extended this workfor two coupled QDs interacting with a quantized coherentfield; however, they do not give any explicit expression forthe entanglement between the dots as well, rather take theratio of the probabilities of the QDs population being foundin two fully entangled Bell states as an indirect measure ofentanglement. Wang et al. [9] have considered a fully quantizedfield, and they have also given an exact mathematicalexpression for the entanglement between the two dots, butthey did not take into consideration any interaction betweenthe dots. In our work we consider the QDs to be coupled toeach other, taking into account the possibility of their mutualinteraction, and we measure the degree of entanglement betweenthe two dots by calculating the concurrence betweenthe two dots, after taking partial trace over the field states.We assume the dots to be initially in the ground state andstudy the creation of entanglement as a result of the interactionwith the coherent cavity field and between thcn~selves.Several interesting results come out which diflcs from thqpredictions where the field is treated as classical [h,7] or thedot-dot interaction is neglected [9] or entanglemen( is calculatedonly qualitatively by finding the probability of the systemstate to be in a Bell state [6-81. Wc ti:.: t1:it the dctscould never be made fully entangled, even when we negl:i:tall kinds of losses, and in general the entanglemen! functionis not periodic in time. We also study the effect of the fieidstrength and the relative strengths of the coupling parameterson entanglement. Next we consider the interaction with ther- ,ma1 light and obtain an interesting observatio~i that if anentangled state is left to interact with the thermal light; itreduces the amount of entanglement, but does no1 completelydestroy it.The paper is organized as follows. Section I1 sutnmarizesthe model Hamiltonian describing N coupled QDs interactingwith a quantized electromagnetic field. In Sec. Ill wc use thisHarniltonian for two QDs to determine the state vector of thecoupled QD-field system. We then study the rvolution o'fentanglement for the cavity field, initially in coherent ari'dthermal states. Section IV summarizes the main ~.csults cii'this paper.U. TWO COUPLED QDs INTERACTING ,.WITH A QUANTIZED CAVITY kit,%. 2We consider N identical semiconductor quantym dots tha:are equally coupled to each other via Coulombic interaction.The QDs interact with a quantized field (dipole interactiod)in a high-Q cavity. Then the coupled QD-field system is .described by the Hamiltonian [6]1050-2947/2007n6(5)/05231 7(7) 052317- 1 02007 The American Physical socle,t;

Conditional homodyne detection of light with squeezed quadrature fluctuationsJustin Vines, Recta Vyas, and Surcndra Singhllepartrne~it o/'P/ysirs, Universiiy of Arkanscls, Fayettevrlle, Arkclnsa,~ 72701, USA(Received 16 June 2006; published 24 August 2006)We discuss the detection of field quadrature fluctuations in conditional homodyne detection experiments andpossible sources of error in such an experiment. We also present modifications to these expeiiments to helpeliminate such errors and extend their range of applicability.I. INTRODUCTIONFluctuations of light provide a window on the underlyingquantum dynamical evolution of a light-emitting source.Einission of a photon by a light source signals quantumfluctuations in progress, and a measurement that is conditionedon a photodetection allows us to study the timeevolution of the fluctuations [I]. For example, measurementof light intensity conditioned on a photodetection, alsoknown as two-time intensity correlation, reveals informationregarding bunching and antibunching that is not available inunconditional intensity measurements [2-61.The conditional measurement of quadrature fluctuations(CMQF) proposed by Cannichael er ul. [7] reveals in a novelway the nonclassical nature of light from a cavity containingtwo-level atoms; this has been experimentally observed byFoster et al. [XI. Nonclassical effects in conditional intensityand squeezing in light from a degenerate parametric oscillatorhave been studied [6]. The conventioilal methods of dctectingquadrature squeezing involve unconditional measurementsthat are degraded by detection inefficiencies and donot explore the time evolution of quadrature fluctuations[1,2,9,10]. The CMQF, on the other hand, is essentially independentof detection efficiency and provides a sensitiveprobe of the fluctuations' development over time. It has beenshown that the conditional measurement can reveal remarkablenonclassical behavior of not only squeezed but also ofunsqueezed quadrature fluctuations [6-81.111 Sec. 11, we briefly summarize-the theoretical conceptsunderlying a CMQF experiment [6,7]. We then consider intracavitysecond harmonic generation (ISHG) [ll-131 andpresent a theoretical analysis of the nonclassical features ofISHG quadrature fluctuations.The CMQF technique achieves a measurement of thequadrature fluctuations of a given source ficld by crosscorrelalinga photon count with a balanced homodyne detection.This technique requires the use of auxiliary coherentoscillators (coherent laser sources), and the amplitudesor intensities of the coherent oscillator fields must be setto values that depend on the properties of the source field. Itcan be shown that the accuracy of the measurement'sfinal results is extremely sensitive to the precision of theseadjustments. In some cases, a very small error in theseadjustments may give rise to incorrect conclusions about thestate of the source field's quadrature fluctuations. We demonstratethis in Sec. III by developing a theoretical model ofsuch an crror in the CMQF mcasurement and exploring itseffccts on conclusions that might be drawn about the state ofthe ISHG field.Weak fields with approximately Gaussian fluctuations areideal candidates for accurate quadrature fluctuation measurementsusing the CMQF method, provided that coherent fieldadjustments can be made sufficiently precise. However, thetechnique is limited in its applicability to a source field withnonzero third-order moments [I 4,351, which may obscurethe results of the mcasurement, even with perfect control ofcoherent light parameters. In Sec. IV, we propose an extensionof the CMQF method that can achieve a measurementof the quadrature fluctuations of a completely generic sourcefield while eliminating the effccts of third-order fluctuationmoments. Additionally, this extended CMQF measurementavoids the need for precise adjustments of coherent laserfields, thus perhaps averting the kinds of error discussed inSec. III. In Sec. V, we summarize our findings.II. CONDITIONAL MEASUREMENT OF QUADRATIJREFLUCTUATIONS FOR LSHGThe quadrature variables for an optical ficld with annihilationand creation operators 6, and 6: are defined bywhere C#J is an arbitrary phase [2]. It follows from thisquadrature definition that the variances (:(Aig)?) and(:(A?,)':) are related lo (he intensity of the field fluctuations(A~~AG,) bywhere colons denote time and normal ordering of theoperators enclosed by them. For classical fields, both quadraturevariances are always greater than or equal to zero,being equal to zero only in the classical coherent state.For quantum fields, however, the nornlally ordered varianceof a quadrature Xg can become negative as long as the normallyordered variance of ?@ increases in such a way thatEq. (3) is still satisfied. In such a case, thc quadrature i4 issaid to be squeezed and the field 6 is said to be in a squeezedstate [2]. This fact and the fact that the fluctuation intensity isnonnegative lead to the inequality [6,7]1050-2947/2006/74(2)/023817(7) 023817-1 02006 The American Physical Society

Eur. Phys. J. D 29, 95-103 (2004)DOI: 10.1140/epjd/e2004-00018-2On tlie bichromatic excitation of a two-level atomwith squeezed lightAmitabh Joshia, Reeta Vyas, Surendra Singh, and Min XiaoDepartment of Physics, University of Arkansas, Fayetteville AR 72701, USAReceived 3 September 2003 / Received in final form 11 November 2003Published online 17 February 2004 - @ EDP Sciences, Societk Italiana di Fisica, Springer-Verlag 2004Abstract. Analytical results for the dynamical evolution of a single two-level atom coupled to an ordinaryheat bath and bichromatically excited with two finite bandwidth squeezed fields are presented bysolving the Heisenberg equations of motion. Photon statistics of the system in terms of the second-orderintensity-intensity correlation function are also discussed. Transient fluorescent intensity as well as intensitycorrelation function exhibit oscillatory phenomenon even in the weak field limit. The latter also showsenhanced delayed bunching effect. All these effects are sensitive to the bandwidth of squeezed light.PACS. 42.50.Ct Quantum description of interaction of light and matter; related experiments -42.50.D~ Nonclassical states of the electromagnetic field, including entangled photon states; quantum stateengineering and measurements1 IntroductionStudies related to the dynamical evolution, the photonstatistics, spectral and radiative properties of one andmanv two-level atoms embedded in a broadband saueezedbath have been the topics of keen interest in quantumoptics. Gardiner [I] studied the interaction of a two-levelatom with a broadband squeezed bath and predicted unequalpolarization quadrature-decay rates. Carmichael,Lane and Walls [2] were able to discover a significantphenomenon of sub-natural linewidth in the fluorescencespectrum of a driven two-level atom, in the presenceof squeezed light. Atomic absorption spectrum has discussedby Ritsch and Zoller [3] in the presence of coloredsqueezed vacuum. Since then many interesting results inatom-squeezed field interaction have been reported whichinclude both two and three-level atoms interacting withbroad bandwidth or narrow bandwidth squeezed baths [4,51. In a recent study the interaction of a two-level atomwith the squeezed vacuum of bandwidth smaller than thenatural atomic linewidth was considered and the holeburning and the three-peaked structure in spectra of fluorescenceand transmitted field were predicted [6]. Theseresults essentially show that squeezed fields having pairwisecorrelations and anisotropic noise distribution cangive rise to interesting phenomena including novel featuresin spectral properties of atoms, formation of purestates and photon statistics. A more realistic model of finitebandwidth squeezed light interacting with a singletwo-level atom has been studied by Vyas and Singh (71and Lyublinskaya and Vyas [8] where the source of thea e-mail: ajoshiouark. edusqueezed light employed was a degenerated parametricoscillator (DPO) operating below threshold and a homodynedDPO [9]. In another work, a two-level atom insidean optical parametric oscillator has been considered andhole and dips in the fluorescence and transmitted lighthas been observed [lo]. The interest in other sources ofsqueezed light as well as its applications in a wide varietyof areas has continued unabated [ll-131.The interaction of a single two-level atom with bichromaticdriving field has also been studied extensively boththeoretically and experimentally [14]. These studies weremotivated by the observations that the bichromatic natureof the driving field can lead to a number of novel featureswhich are different from the monochromatic case. For example,the fluorescence intensity exhibits resonances atsubharmonics of the Rabi frequency and different spec-tral characteristics when com~ared with the usual Mollowtriplet. Recently, some new calculations for resonance fluorescenceand absorption spectra of a two-level atom drivenby bichromatic field have been reported [15]. Also, reportedare the effects of broadband squeezed reservoir onthe second order intensity correlation function and squeezingin the resonance fluorescence for a bichromaticallydriven two-level atom [16]. Coherent population trappingand Sisyphus cooling under bichromatic illumination havealso been studied [17]. In another recent work the electromagneticallyinduced transparency (which normally occursin three-level atoms) has been demonstrated in a twolevelatom excited by a bichromatic field (one strong andone weak field) and possibility of squeezed-light generationhas also been discussed 1181. . .In this work, we study the interaction of a single twolevelatom with a bichromatic electromagnetic field that is

PHYSICAL REVIEW A 67, 063808 (2003)Conditional measurements as probes of quantum dynamicsShabnanl Siddiqui, Daniel Erenso, Reeta Vyas, and Surcndra SinghDepurtmet~t q/'Pl~):sics. Universii): ofAt.kutans, &vetfe\~ille, Arkamcus 72701, USA(Rcccivcd 5 December 2002; published 17 June 2003)Wc discuss conditional rneasuremcnts as probcs of quantum dynamics and show that thcy provide differentways to characterize quantum fluctuations. \Ve illustrate this by considering thc light from a subthresholddcgcncratc parametric oscillator. Analytic rcsults and curvcs arc prcscntcd to illustrate thc behavior.DOI: 10.1103/PhysRcvA.67.063808PACS numbcr(s): 42.50.Dv, 42.50.Ar, 42.65.K~1. INTRODUCTIONFluctuations are an integral feature of quantum dynamicalevolution. For dissipative quantum optical sources, thesefluctuations are reflected in the photoemissions fYom thesesources. In this paper, we will use the terms photoemissionsand photodetections interchangeably because photodetectionis sin~ply a matter of casting a net to catch the photons emittedby the source. A photoemission signals a fluctuation inprogress. Hence, a conditional measurement that commenceswhen a photoenlission has occurred catches a fluctuation inthe act and, in fact, allows us to observe the time evolutionof the fluctuation [I]. This sort of information is not availablein unconditioned measurements. Thus, conditional measurenientsallow us to probe quantum dynamics at a deeperlevel. Perhaps the best known example of conditional measurementis the measurement of the second-order intensitycorrelation that measures fluctuations of light intensity followinga conditioning photodetcction [2]. This is the basis ofthe observation of photon bunching or antibunching. Recently,measurements of quadrature squeezing conditionedon a photodetection have also been proposed and reported incavity quantum electrodynamics [3,4]. In this papel; we considerconditional measurements in the context of parametricoscillators and show that such measurements provide sensitiveprobes of quantum dynamics and the language of conditionalmeasurements provides powerhl conceptual toolsfor unraveling and understanding nonclassical features ofquantum dynamics [5- 81.Optical parametric oscillators (OPOs) and an~plifiers [9]based on down-conversion have played a central role in thestudies of nonclassical photon correlations and variousschemes for quantum coinmunication and computation [lo].The fitndamental process in optical parametric oscillators isconversion of a pump photon of frequency w,, into a pair ofphotons (signal and idler) of lower frequencies w, and wi ina nonlinear medium inside an optical cavity. The processconserves energy and momentum. Conservation of energyrequires the pump and down-converted frequencies to be relatedby wp= us+ wi and conservation of momentum, alsoknown as phase matching, requires the use of certain anisotropicmaterial media and specific states of polarization forthe pump, signal, and idler photons. If the down-convertedphotons have the same frequency (ws= wi=wd), polarization,and direction of propagation, the process is called degenerateotherwise it is called nondegenerate. In the fonnercase, we speak of a degenerate parametric oscillator (DPO)and in the latter of a nondegeneratc parametric oscillator.We begin by considering a degenerate paranlet~ic oscillatorin Sec. I1 and discuss conditional measurements of intensityand amplitude. Conditional measurements of quadraturefluctuations are discussed in Sec. 111. We restrict our considerationsto its operation below the threshold of sustainedoscillations. This allows us to obtain simple analytic cxpressionsfor various quantities of interest. The results are summarizedin Sec. IV.11. CONDITIONAL MEASUREMENTS OF A DPOThe field from the DPO is governed by the interactionHamiltonian for phase-matched down-conversion inside anoptical cavity driven by a classical injected signal [5,6]. Theequation of motion for the density operator bd of the DPOfield is thenwhere K is the mode-coupling constant, E is the dimensionlessamplitude of the classical pump field, y is the cavitylinewidth, and id and 6; are the annihilation and creationoperators for the DPO. In writing the equation of motion forthe density matrix, we have neglected pump depletion that isa reasonable approxin~ation for low down-conversion efficienciesand subthreshold operation of the DPO consideredhere. The combination KE has been chosen to be real by asuitable definition of the phases of id and 61.Using the positive-P representation for the density matrix,we can map the equation of motion for the annihilation andcreation operators for the DPO field onto a set of stochasticequations for the c-number amplitudes cud and ad, corresponding,respectively, to the annihilation and creation operators6 and Cid. These equations read [5,6]where [,it) and &(t) are two statistically independent realGaussian white-noise processes with zero means and unitintensity. Nonnally ordered averages of ri$ and rid can thenbe calculated by using the mapping1 0S0-2947/2003/67(6)/063808(9)/$20.00 67 063808-1 Q2003 The American Physical Society

PHYSICAL REVIEW A 67, 013818 (2003)Quantum well in a microcavity with injected squeezed vacuumDaniel Brenso, Rceta Vyas, and Surcndra SinghI)epczrtmenI of Ph.vsics, Universiw of Ar/can.sas, Fayetteville, Arkansas 72 701(Received 27 August 2002; published 3 1 January 2003)A quantum wcll with a singlc cxciton mode in a microcavity driven by squcczed vacuum is studicd in thelow exciton density regime. By solving the quantum Langevin equations, we study the intensity, spectrum, andintensity correlation function for the fluorcsccnt light. An cxprcssion for thc Q function of thc ficld insidc thccavity is derived from the solutions of the quantum Langevin cquations. Using the Q function, the intracavityphoton number distribution and thc quadrature fluctuations for both thc cavity and fluorcsccnt fields arcstudied. Several interesting and new el'fects due to squeezed vacuum are found.DOI: 10.1 103/PhysRevA.67.013818PACS number(s): 42.55.Sa. 78.67.De, 42.50.Dv, 42.50.L~1. INTRODUCTIONWith the development of semicond~ictor optical microcavities,there has been considerable interest in excitoncavitycoupled systems [1,2]. These systcms have revealedsome interesting phenomena that are similar to those observedin the interaction of a two-level atom with light [3-91.The exciton-cavity system gives rise to the so-called polaritons,which are the no~nlal inodes of a coupled excitonphotonsystem. The excitation spectrum of the compositeexciton-cavity system is characterized by two well-resolvedpolariton resonances (or nornlal mode resonances) when g>(ye, yc), where g is the dipole coupling between the excitonand the cavity mode, and ye and y, are the exciton andcavity mode damping rates, respectively. In this limit, anexcitation of the cavity mode can lead to a coherent oscillatoryenergy exchange (or normal mode oscillation) betweenthe exciton and the cavity due to the vacuum Rabi oscillations.The vacuum Rabi oscillations in a coupled exciton-photonsystem in sen~iconductor microcavity lasers have been observedby Weisbuch et al. [2]. Following this observation,extcnsive experimental and theoretical studics have been carriedout [lo- 171. These studies have confirmed normal modesplitting and oscillatory emission from exciton microcavities.Theoretical investigations in the linear regime, where the excitonscan be approximated as bosons, have bccn carricd outby Pa11 et 01. [I 51. Wang et al. [I 61 investigated the effects ofinhomogeneous broadening of excitons on normal mode oscillationsin semiconductor microcavities using the coupledoscillator model. Their results show that inhomogen~ousbroadening can drastically alter the coherent oscillatory energyexchange process cven in regimes whcre normal modesplitting remains nearly unchanged.In this paper, we study the excitonic system in a microcavitywhere the cavity is driven by squeezed vacuum. Anoutline of the system is shown in Fig. 1. A sen~iconductorquantum well is embedded between two Bragg reflectingmirrors. One of these mirrors acts as an input port throughwhich light in a squeezed vacuum state is injected into thecavity. We includc dissipation of both thc cavity and excitonmodes. In Sec. 11, we derive the quantum Langevin equationsfor the exciton and cavity modcs. We solve these equationfor the case in which the damping constants are equal. These~.csults are used in Scc. I11 to study the cffects of initial cavityphoton number as wcll as squeezed-vacuum photon numberon the intensity, spectrum, and the second-order intensityco~relation of the fluorescent light. In Sec. IV, we obtain theQ-distribution function and use it to study the intracavityphoton number distribution and squeezing of thc cavitymode and the fluorescent light. We summarize the principalresults of the paper in Sec. V.11. QUANTUM LANGEVIN EQUATIONWe consider a semiconductor quantum well fQW) in thelinear excitation regime where the density of excitons issmall so that exciton-exciton interaction can be ignored. Theexcitons can then be approximated as a dilute boson gas [IS].In this approximation, the microscopic Hamiltonian in theinteraction picture describing the exciton-cavity system isgiven by [17,19]The Hamiltonian of Eq. (1) is written in the rotating-waveapproximation and in the dipole approximation. Here and6 are the annihilation operators for the cavity and excitonmodes, respcctivcly, in a frame rotating at frequencyw, , f, (I? ,) is the reservoir operator responsible for cavityfield (exciton) damping, g is the coupling constant characterizingthe strength of interaction between the exciton and thecavity field, and detuning A w = (we- w,), where w, and w,.are the frequencies of the exciton and cavity modes, respectively.Normally, the exciton and cavity inodcs are coupled toa continuum of thennal reservoir modes. This leads to theirMirro@fl-MirrorpmpFIG. 1. An outline of the physical system.1050-2947/2003/67(1)/0138 18(10)/$20.00 67 013818-1 Q2003 The American Physical Society

PHYSICAL REVIEW A, VOLUME 65, 063808Two-level atom coupled to a squeezed vacuum inside a coherently driven cavityDaniel Erenso and Reeta VyasDepartment of Phj~sics, Univer:~ir), of Arkansas. F'czyetteville, Arkaizsns 72701(Rcceivcd 8 Septembcr 2001 ; published 5 June 2002)A single two-lcvcl atom in a cohcrcntly driven cavity and damped by a broadband squcczcd vacuumcentered about the atomic transition liequency is studied. A second-order Fokker-Planck equation for thissystem is obtained without using system size cxpansion. Effects of dctunings and cavity decays arc alsoincorporated in the Fokker-Planck equation. This equation is used lo study atomic inversion, fluorescentspectrum, and the intcnsity corrclations of the transmitted and fluorcsccnt photons in thc bad-cavity limit.Several interesting effects in the atomic inversion, spectrum, and intensity correlations due to the squeezedvacuum arc presented. Thesc results are also compared with an atom that is dampcd by a thermal rcscrvoir.DOI: 10.1103/PhysKcvA.65.063808PACS nu~nbcr(s): 42.50.Lc, 42.50.Ct, 42.50.Dv, 42.50.ArI. INTRODUCTIONWith the gcncration and dctcction of sqiieczed light [I-41increasing attention is being given to the study of interactionof squcezed light with a two-level atoin [5-131. Gardinerstudied a single two-level atom embedded in a broadbandsqueezed vacuum and showcd that tlie two quadratures of thcatomic polarization decay at two distinct decay rates that aresensitive to the phase correlations of the squcezed vacuum[IS]. Carmichael rt al. [6] studied the fluorescent spectrum ofan aton1 immersed in a broadband squeezed vacuum whenthe atom is driven by a coherent field. They predicted that forwcak driving fields the incoherent spectrum would narrow asthe amount of squeezing is increased. In this linlit the spectminis insensitive to the relative phase between tlie drivingfield and the squeezed vacuum. For strong driving fields, onthe other hand, the central pcak of the Mollow spectrum canbroaden or narrow, depending on the relative phase betweenthe squcezed vacuum and the driving field. The photon numberdistribution for this system has been calculated by Jagatapand Lawande [7]. Vyas and Singh considered resonancefluorescence in the weak-field limit when the atoin isdriven by squeezed liglit from an optical parametric oscillator[8], Lyublinskaya and Vyas considered when the atom isdriven by nonclassical light from intracavity second harmonicgeneration and a homodyne degenerate parametric oscillator[9].Parkins and Gardiner [lo] considered a single two-levelatom in a cavity whcn thc squeezed light is incidcnt uponone of the output mirrors. Rice and Pedrotti [I I.] placed atwo-level atom coupled to an ordinary vacuum inside a coherentlydriven optical cavity coupled to a broadbandsqueezcd reservoir through the output mirror. They foundthat it was possible to overcoine the cavity enhancement partof tlie linewidth. This work was extended by Rice and Baird[12] to calculate the second order intensity co~~elation functiong(2)(7-) and spectra of the fluorescent light.In this paper we study a single two-level atom placedinside a coherently driven cavity where the atom is directlycoupled to a squeezed vacuum but the cavity mode decaysinto an ordinary vacuum. The squeezed vacuum spectrum isconsidcrcd to bc broadband ccntcred at the atomic transitionfrequency. The model proposed here can be implemented byassuming a short cavity, which subtends a largc solid anglc atthe atom. Squeezed modes of the short cavity are directlycoupled to the atom. Another cavity with its axis perpendicularto the short cavity is driven by a coherent driving field.We derive an exact Fokker-Planck equation following theapproach of Wang and Vyas [14-161. An appealing featureof the Fokker-Planck equation approach is that it allowsquantum-operator averages to be calculated as classical-likeaverages. Thus analogies between classical and quantumfluctuations can be drawn that help in developing an intuitivefeeling for quantum fluctuations. The Folcker-~lanck equationcan be converted into a set of stochastic differentialequations which can be solved numerically and in manycases analytically. We study the effects of a squeezedvacuum on atomic inversion, the fluorescent spectrum, andthe second-order intensity col~elation function of the transmittedand fluorescent light in the bad-cavity limit. We findthat in the presence of squeezing the threshold value of thecooperativity parameter for seeing vacuum Rabi splitting canbe lowered. This bchavior is phase sensitivc and cannot beseen if squeezed light is replaced by thermal light. We findthat the transmitted light can show antibunching even for alarge cooperativity parameter. We explain the behavior ofantibunching in ternls of self-hoinodying of coherent andincoherent components. We also find that, for large values ofsqueezing, antibunching results due to a reduction in the intensityfluctuations of the incoherent component. This differsfrom the case for small squeezing. For sinall squeezing antibunchingresults from an interference of the coherent componentwith the incoherent component.Squeezed VacuumFIG. 1. Physical scheme.1050-2947/2002/65(6)/063808( 1 1)/$20.00 65 063808-1 02002 The American Physical Society

Erenso et al. Vol. 19, No. 6IJune 2002lJ. Opt. Soc. Am. B 1471Higher-order sub-Poissonian photon statisticsin terms of factorial momentsDaniel Erenso, Reeta Vyas, and Surendra SinghDepclrtment of Physics, University of Arkansas, Fayetteville, Arkansas 72701Received September 10, 2001We introduce the concept of higher-order super-Poissonian and sub-Poissonian statistics and show that higherordersub-Poissonian statistics is a signature of a nonclassical field. Fields generated in intracavity secondharmonicgeneration and single-atom resonance fluorescence are shown to exhibit higher-order sub-Poissonianstatistics. 0 2002 Optical Society ofAmericaOCIS codes: 270.0270, 270.5290, 190.4970, 270.2500.1. INTRODUCTIONNonclassical properties of electromagnetic fields receive agreat deal of attention, as these properties provide a testingground for the predictions of quantum electrodynamics.'~'Sub-Poissonian Photon statistics based onsecond-order intensity correlations provide one way tocharacterize the nonclassical nature of a light beam.3.4Nonclassical effects as they relate to higher-order momentshave also been discussed in the literat~re.~-'Aganval and Tara discussed higher-order nonclassical effectsfor single-mode fields in terms of normally orderedmoments."erina and co-workers studied noilclassicalbehavior in optical parametric processes, Raman andBrillouin scattering6 and nonlinear optical couplers7 interms of moments of integrated intensity. Lee consideredsingle-mode fields and defined higher-order noi~classicaleffects in terms of factorial moments of the photon distributionby using majorization theory.8 Vyas and Singhconsidered higher-order nonclassical effects in terms offactorial moments of the photocount distribution.' Inthis paper we introduce criteria for evaluating higherordersub-Poissonian statistics in terms of factorial momentsand show that higher-order sub-Poissonian statisticsare indicative of nonclassical fields. The criteria thatwe introduce are independent of the efficiency of detection.We then show that the light from intracavitysecond-harmonic generation (ISHG) and light fromsingle-atom resonance fluorescence exhibit higher-ordersub-Poissonian statistics.2. HIGHER-ORDER SUB-POISSONIANSTATISTICSFor second-order sub-Poissonian statistics variance((Am)') = (m2)- (m)' of the photon-counting distributionis less than the meail of the distribution, (m). Bynoting that second-order factorial moment (m(")= (m(m - 1)) - (m') - (m), we can write the criterionfor the second order sub-Poisson statistics asNote that for a Poissonian distribution (m(") = (m)', soinequality (1) holds as an equality. The departures fromPoisson statistics are then characterized in terms of theFano factor (F = ((~m)~)l(m)) or the Q parameter {Q= [(m'") - (m)2]l(m)}.3~4 For a sub-Poissonian distributionthe Q parameter is negative. We now extend thiscriterion to higher-order factorial moments. The lth (1 isa positive integer) order factorial moment of the photocountdistribution is defined by(m(')) = m(m - 1) ...( m - 1 + l)p(m, T), (2)m=lwhere p(m, T) is the probability of detecting m photoi~sin the counting interval [0-TI. Here we have suppressedthe time argument in the factorial moments. Toextend the criteria for sub-Poissonian statistics to higherordermoments we introduce a parameter S1 :It is easily proved that, for a Poisson distribution, S1= 0 for all 1. Parameter Sl for 1 3 2 provides a measureof the deviation of the I th factorial moment from that fora Poisson distribution with the same mean. Sl > 0 definesa super-Poissonian distribution, and S1 < 0 definesa sub-Poissonian distribution. Note that parameter Sz isnot equal to the Q parameter but is related to it byS, = Ql(m). As the values of higher-order factorial momentscan be large, it is convenient to use normalized factorialmoments rather than the analogs of Q to extend theconcept of sub- and super-Poissonian statistics to higherordermoments. Another advantage of using the SL parametersis that they are independent of the efficiency ofdetection.We now show that negative values oF parameter S1 forI > 2 (higher-order sub-Poissonian statistics) indicate thenonclassical nature of light. To establish this we notethat for a classical field the factorial moments must satisfythe inequalitys3'0740-3224/2002/061471-05$15.00 O 2002 Optical Society of America

PHYSICAL REVIEW A, VOLUME 64, 043806Nonclassical effects in photon statistics of atomic optical bistabilityDaniel Erenso, Ron Adan~s, Hua Deng, Rceta Vyas, and Surendra Sing11Department of Physics, University oJArkunsuv, F~ryetfeville, Arkansas 72701(Received 30 Miiy 2001; published 13 September 2001)Homodync statistics of light generated by an atomic system cxhibiting optical bistability arc analyzed. Usingthe dynamical equations of lnotion for a single atom in a coherently driven cavity in the good cavity limit, weshow that thc homodync ficld can bc dcscribcd in terms of two iltdepcndcnt rcal Gaussian stochastic processcsand a coherent component. By making a Karhuneil-Loeve expansion of thc field variables we derive thegcl~crating function for the photoclcctron statistics. From this gc~~crating function photoclcctron-counting distribution,factorial moments, and waiting-time distribution are obtained analytically. These quantities are directlymeasurablc in photon-counting expcrimcnts. We show that thc homodync ficld cxhibits many interestingnonclassical features including nonclassical etrects in higher-order factorial moments.DOI: 10.1 103iPhysRevA.64.043806PACS number(s): 42.50.Ar, 42.50.Dv, 42.50.CtTnteraction of a single two-level atom with a quantumficld insidc a coherently drivcn cavity in the good cavitylimit, is known to show optical bistability [1,2]. We will referto this system as single atom optical bistability (SAOB).Similarly, N two-level atoms placed inside a coherentlydriven cavity also exhibit optical bistability that we shallrefer to as multiatom optical bistability (MAOB) [3,4]. Thesesystems are also known to show antibunching, although thesize of antibunching is small. In order to enhance antibunchingand other nonclassical effects, several schcmcs based oninterference [5], passive filter cavities [6], or homodyne detection[7] have been proposed.Homodyning a field with a coherent local oscillator (LO)provides one way of cnhancing nonclassical effects. The homodynefield can exhibit strong nonclassical featurcs, whichare not shown by the original ficld. The homodyile statisticsare sensitive to the phase difference betwecn the signal andthe LO. An example of this behavior is provided by the lightfrom the degenerate parametric oscillator, which is highlybunched and super-Poissonian. When this field is homodynedwith a LO, the homodync field shows a variety ofnonclassical cffccts such as antibunching, sub-Poissonianstatistics, and violation of other classical inequalities [8-101.In this paper we consider homodyning of the light from asystem that exhibits SAOB with the light beam from a LO ata losslcss beam splitter as shown in Fig. 1. A detector ofefficicncy 77 placed at one of the output ports of the beamsplitter detects the homodyne field and generates photoelectricpulses, which are measured by suitable electronics. Westudy photoelectron statistics measured by the detector. InSec. I1 we start from the equations of motion derived byWang and Vyas for a single two-level atom in the good cavitylimit [2] and show that the ficld from the SAOB can beexpressed in terms of two Gaussian random variables. Wethen derive the equations that govern the dynamics of thehomodyne field. Using these equations and applying theKarhunen-Lokve expansion for the field variables, we calculatethe moment generating function for the photocount distribution.We also show that a system exhibiting MAOB canalso be described by similar expressions with an appropriatechange in parameters. In Scc. TI1 we present an analytic expressionfor thc moment gencrating function. Photon statisticsof the homodync field are then analyzed with the help ofthe moment generating function. The photocount distribution,its moments, and the waiting-time distribution for thehomodyne field are calculated. Finally, in Sec. VI, a summaryof the main results of the paper is presented.11. DYNAMICS OF THE HOMODYNE FIELD ANDTHE GENERATING FUXCTIONIn this section we derive equations of motion describingthe dynamics of the homodync field when the signal is fromthe SAOB. We will see that similar equations are obtainedwhen the signal is from thc MAOB.Consider a single damped two-level atom with transitionfiequcncy w,, interacting with a single mode of a cavitywith resonance frequency w,. The cavity is driven by a coherentexternal field of frequency w, and amplitude E. In theelectric dipole and rotating-wave approximation, the Hamiltonianfor this system call be written asHereand dt are the annihilation and creation operators forthe cavity mode, a+, 6.- , and 6, are the Pauli spin matricesdescribing the two-level atom, g is the atom-field couplingconstant, and H,,,, describes atomic losses due to spontaneousdecay and field losses at the cavity mirrors.Two-level atom+FIG. I. System for homodyning the SAOB field with the LOfield. BS denotes the beam splitter and D denotes a detector.1050-2947/2001/64(4)/043806(7)1$20.00 64 043806-1 02001 The American Physical Society

VOLUME 86, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARCH 2001Entanglement, Interference, and Measurementin a Degenerate Parametric OscillatorHua Deng, Daniel Erenso, Reeta Vyas, and Surendra SinghDepartment of Physics, University of Arkansas, Fayetteville, Arkansas 72701(Received 25 August 2000)Quantum dynamical equations of motion for homodyne detection of the degenerate optical parametricoscillator are solved exactly. Nonclassical photon statistics are shown to be a consequence of interferenceof probability amplitudes, entanglement of photon pairs from such an oscillator, and the role ofmeasurement in quantum evolution.DOI: 10.1 103/PhysRevLett.86.2770PACS numbers: 42.50.Dv, 42.50.A~ 42.65.K~Fluctuations of photon beams reflect the quantum dynamicsof photoemissive sources. In quantum mechanics,probabilities for observed events are derived from an underlyingwave function that can interfere and collapse as itevolves. A consequence of this is that quantum mechanicscan lead to correlations between observed events whicha classical stochastic theory may not. Examples of thesenonclassical correlations include squeezing, antibunching,and violations of Bell's inequalities [1,2].The subthreshold degenerate parametric oscillator(DPO) has played a central role in the study of nonclassicalphoton correlations, particularly, squeezing [1,2]. TheDPO radiates a highly bunched light beam that exhibits alarge degree of squeezing. Interestingly, the squeezed andhighly bunched light from the DPO when combined with acoherent light field, as in homodyne detection, is expectedto display a rich variety of nonclassical photon correlationsincluding antibunching [3]. It is intriguing that a highlybunched entangled photon beam from the DPO whenmixed with a coherent field will exhibit correlations similarto those exhibited by a single-atom resonance fluorescencein free space or in cavity quantum electrodynamics (QED)[4,5]. Antibunching of light emitted by a single two-levelatomic system can be eventually traced to the atomic deadtime that a two-level atom cannot emit a second photonimmediately after the emission of a first photon. The situationis not so simple for homodyne detection of the lightfrom the DPO because there is no obvious mechanismfor a dead time. By solving the equations of motion forhomodyne detection exactly, we show that nonclassicalphoton correlations in homodyne detection of the DPO area consequence of the interference of probability amplitudes,entangled nature of photon pairs generated by theDPO, and measurement. These are the features that mostdistinguish quantum mechanics from classical mechanics.An outline of the experimental setup for homodyne detectionof the DPO light is shown in Fig. 1. The DPO andlocal oscillator (LO) fields are combined by a beam splitterto produce the source field at the detector. The field fromthe DPO is governed by the interaction Hamiltonian for aphase matched DPO driven by a classical injected signalof amplitude E [6]:ihK&H = (&J2 - 6;) + Aloss . (1)LHere K is the mode-coupling constant and and adtarethe annihilation and creation operators, respectively, forthe DPO. H~,,, describes the loss suffered by the DPOfield. The combination KE can be chosen to be real by asuitable definition of phases.The equation of motion for the density matrix bd of theDPO field is thent ..t.. t+ ~(2adbdad - adadbd - bdadad)? (2)where 2 y is the cavity decay rate. The steady-state solutionto this equation in positive-P representation is given bywhere -a < x, y < are both real variables and In) is acoherent state of ad with adlx) = xlx). From this expressionfor the density matrix, the steady-state expectationvalue of an operator 0 can be calculated as (o),, =T~(o$,,). This leads to the following expectation valuesFrom DPOFrom LODetectorFIG. 1. Schematic experimental setup for the homodyne detectionof the light from a degenerate parametric oscillator.2770 0031 -9007/01/ 86(13)/2770(4)$15.00 O 2001 The American Physical Society

PHYSICAL REVIEW A, VOLUME 62, 033803Higher-order nonclassical effects in a parametric oscillatorReeta Vyas and Surendra SinghPhysics Deportment, Lr/ziversity of Arkcmsas, Fayetleville, Arkansas 72701(Received 10 January 2000; published 8 Aubwst 2000)Factorial ~noments for thc light from a degencratc paramctlic oscillator mixed with a cohcrcnt local oscillatorare calculated and shown to reveal novel nonclassical features of light. We have round novel regimeswhcrc the nonclassical charactcr of thc ficld is rcflccted not in thc violations of incqualitics bascd on thcsecond-order moments but in those based on higher ordcr moments. These violations can be observed mphotoclcctric mcasurclnents of light.PACS numbcr(s): 42.50.Dv, 42.50.Ar, 42.65.K~Nonclassical properties of light such as squeezing [I],sub-Poissonian statistics [2], and antibunching [3] have beendiscussed in terms of quadratic field or intensity correlations.With the developn~ent of techniques for making higher ordercorrelation measurements. the interest in nonclassical effectsnaturally extends to the higher order correlations. Of particularinterest are physical systems where nonclassical featuresof light are revealed in higher order correlations but not inthe lower order correlations. Higher order squeezing was introducedby Hong and Mandel [4]. Aganval and Tara introducednonclassical behavior in terms of normally ordercdmoments of a distribution [5]. Lee [6] considered a singlemode field and defined higher order nonclassical effects interms of factorial moments of the photon distribution by usingmajorization theory.Jn this paper we present a simple criterion for higher ordernonclassical behavior in tcrms of factorial moments of thephotoelectron counting distribution. We then show that thelight fi.om a subthreshold degenerate parametric oscillator(DPO), when superposed on the light from a coherent localoscillator (LO), satisfies the criteria for highcr order nonclassicalbehavior in terms of factorial moments. These violationscan be measured in photon counting experiments.Let us first recall that the probability of detecting m photoelectronsin the time interval [O - T] is [7]where, for simplicity, we have suppresscd the dependence ofthe moments on the counting interval T. The second factorialmoment (rn(l)) is related to the variancc ((dm)') of photonnumber distribution byFor a coherent field (Poissonian distribution) ((Am12)= (m) is independent of the value of T. Expressing p(m, 71in Eq. (3) in terms of 0 and using thc Glauber-Sudarshan Prepresentation for the density matrix of the field, the factorialmoments can be written aswhere U is a positive real number and for a classical state theprobability distribution P(U) must be a positive and nonsingularfunction (no more singular than a 6 function). Nowdefine a positive fbnction in tams of two positive real variablesU and Waswhere the colons : . . . : denote time ordering and normal orderingof the operator product between the colons. In wi-itingEq. (I), we have assumed stationary light fields. The operatorfi is given byOn expanding and simplifying this, we find thatTherefore for a classical probability density P(U,W)[= P(U)P(W)], which is positive and no more singular thana 6 function,where i(t) is the photon flux operator (number of photonsper second) and r] is the detection efficiency. The angularbrackets denote averaging with respect to the state of thefield.The /-th order factorial moment (/ is a positive integer)of the photoelectron counting distribution is defined byThus the factorial inoments of photon counting distributionfor a classical field must satisfy the inequalities1050-2947/2000/62(3)1033803(5)15 15.00 62 033803-1 Q2000 The American Physical Society

JOURNAL OF APPLIED PHYSICS VOLUME 84, NUMBER 7 1 OCTOBER 1998Thermal diffusivity measurement of solid materials by the pulsedphotothermal displacenient techniqueG. L. Bennis, R. Vyas, and R. GuptaDepartment of Physics, University of Arkansas, FayetteviNe, Arkansas 72701S. Ang and W. D. BrownDepartment of Electrical Engineering, University of Arkansas, FayefteviNe, Arkansas 72701(Received 27 February 1998; accepted for publication 2 July 1998)A simple, noncontact technique for the measurement of thermal diffusivity of solids isexperimentally demonstrated. The technique is based on the photothermal displacement effect.Excellent agreement between the quasistatic theory of photothermal displacement and theexperiment has been obtained. The technique has been demonstrated by measuring the thermaldiffusivities of GaAs and InGaAsIAlGaAs multiple quantum wells. O 1998 American Institute ofPhysics. [SOO21-8979(98)04519-81I. INTRODUCTIONThermal diffusivity of a material is an important property,a knowledge of which is required for a variety of applications.Of the many methods available for making diffusivitymeasurements, noncontact methods are generallypreferable over the more traditional methods in which probesmust be inserted into the sample. One of the popular noncontactmethods is the photothermal technique.'!2 In this technique,local heating of the sample is created by a laser beam(pump beam), and the effects of the heating are observed,generally by another, weaker laser beam (probe beam). Theexperiments can be performed by using either a cw modulatedpump beam or a pulsed pump beam. The former areknown as the frequency-domain experiments, whereas thelatter are known as the time-domain experiments. The followingvariations of the photothermal technique have beenused to measure thermal diffusivity in solids: (i) Phototherma1radiometry. In this method, the heating is produced by apulsed laser and the time evolution of the heating is measuredby observing the infrared emission from the samplesurface. Thermal diffusivity is obtained from the timeevolutionof the radiometric Chen and ande el is'have used a variation of this technique, called rate-windowspectrometry, to enhance the signal-to-noise ratio. (ii) Photothermalrefraction: The heating of the sample is observedby a probe laser passing through the sample, which is transparentto the probe beam. The heating produces refractiveindex gradients in the sample, which cause the probe beam todeflect. The thermal diffusivity is obtained by measurementsof this deflection either with cw-modulated excitation6 orwith pulsed e~citation.~ (iii) Photothermally induced mirageeffect: In this technique, the measurement is made in a fluidmedium surrounding the sample. The heat from the solidsample is conducted to the fluid, which produces refractiveindex gradients in the fluid. Deflection of a probe beam passingthrough the fluid is measured. Generally, a cw-modulatedpump is used and deflection is measured as a function ofpump-probe distance. Thermal diffusivity is obtained eitherby an analysis of the phase information contained in thesignal839 or by a multiparameter fit to the data.'' (iv) Probebeam reflection: The temperature change of the sample canalso be monitored by observing the change in the reflectivityof the sample with(v) Photothermal displacement:In 1983, Olmstead et al.I3 showed that the heatingof the sample by the pump beam produces a thermoelasticdeformation of the sample which can be detected bydeflection of the probe beam reflected from the sample surface,interferometric methods, or an attenuated total reflectionscheme. Analysis of the signal, either in the frequencyor time domain, in principle, would yield the thermal diffusivity.However, to the best of our knowledge, this techniquehas not been fully exploited and is the subject of this article.(vi) Transient thermal grating: In this technique, two coherentbeams are made to interfere on the surface of the sample.This produces a thermal grating. In one of the schemes, thisgrating is detected by observing the temperature inducedchanges in reflectivity.14 The thermal grating also produces adeformation grating on the sample surface by thermoelasticdeformation, which can be observed by the diffraction of aprobe beam,15 or by the deflection of a well-focused probebeam from one undulation of the deformation grating.'6 Jaureguiand welsch17 have reviewed the theory of thermal gratingsin both the frequency and time domains. The above listof various photothermal techniques for thermal diffusivitymeasurements is by no means exhaustive, as the literature onthis subject is vast. A good recent review of the field is givenby Park et a/.In this article we demonstrate that the thermal diffusivitycan be measured very simply by the temporal evolution ofthe thermoelastic deformation following a short laser pulse.As mentioned in (v) above, to the best of our knowledge, thissimple technique has not been fully exploited previously.The feasibility of thermal diffusivity measurement by thismethod was reported by Karner et a/." and by Vintsents and~andornirskii.~' However, no analyses of the data were performedto obtain actual thermal diffusivity values. We findthat our data fit extremely well to the relatively simple theoryof thermoelastic deformation given by ~i;' and we obtain0021 -8979/98/84(7)/3602/9/$15.00 3602 O 1998 American Institute of PhysicsDownloaded 26 Feb 2008 to 130.284.237.6, Redistribution subject to AIP license or capyrlght; see http:l1jap.aip.orgljaplcopyright.jy1

Photothermal lensingdetection: theory and experimentQifang He, Reeta Vyas, and Rajendra GuptaPhotothermal lensing signal shapes are experimentally investigated and compared with those predictedtheoretically in our earlier paper. The investigation included flowing and stationary media and pulsedand cw excitations. Good qualitative agreement between theory and experiment is found. Since thelensing signal is almost always accompanied by a deflection signal, the influence of the deflection signalon the detection of lensing signal is investigated. For a perfectly aligned detection geometry theinfluence of the deflection signal on the lensing signal is negligible, but in the presence of misalignmentsa significant amount of deflection signal could be superimposed on the lensing signal. The effect oflensing on the deflection signal is also been considered. The effect of the finite size of the probe beam onthe lensing signals is also investigated. O 1997 Optical Society of AmericaKey words: Photothermal lensing, photothermal deflection, thermal blooming.1. IntroductionThere is currently an intense interest in the techniqueof photothermal spectroscopy1 because thetechnique has found many applications in diversefields.2 Sometime ago we gave the complete theoryof photothermal lensing spectroscopy (PTLS) in aflowing fluid medium.3 A general treatment wasgiven that consisted of pulsed (of arbitrary pulselength) or cw (modulated or unmodulated) excitation,flowing or stationary media, and transverse or collineargeometry. To our knowledge no systematicexperimental investigation of the signal shapes presentedin our previous paper has yet been published.In the present paper we present the experimentalinvestigation of the lensing signals and their comparisonwith the theoretical results presented in Ref. 3,which we will hereafter refer to as paper I. Duringthis investigation we realized that the photothermaldeflection spectroscopy (PTDS) signal, which almostalways accompanies the PTLS signal, has the potentialfor complicating the shapes of the PTLS signals.Therefore in this paper we also present both the the-When this research was performed all the authors were with theDepartment of Physics, University of Arkansas, Fayetteville, Arkansas72701-1201. Q. He is now with the Arkansas State University,Beebe Branch, Beebe, Arkansas 72012.Received 24 October 1996; revised manuscript received 26 February1997.0003-6935/97/277046-13$10.00/0O 1997 Optical Society of Americaoretical and the experimental investigation of the effectof PTDS on the shapes of the PTLS signals. Forcompleteness, in Appendix A we also discuss the effectof PTLS on the shapes of PTDS signals. Theeffect of the finite size of the probe beam on the detectionof the lensing signal has also been investigated.The basic idea underlying dual-beam PTLS isshown in Fig. 1. A laser beam (pump beam) propagatesthrough a medium, and it is tuned to one of theabsorption frequencies of the medium. The mediumabsorbs some of the optical energy from the laserbeam. If the collision rate in the medium is sufficientlyhigh compared with the radiative decay rates,most of the energy appears in the translationalrotationalmodes of the medium within a short periodof time. In other words, the laser-irradiated regionbecomes slightly heated. The refractive index of themedium is thus modified. The refractive-indexchange can be monitored in several different ways.4In this paper we are concerned with a technique thatrelies on the lensing effect of the medium to monitorthe refractive-index change. A weak probe beampasses through the pump-irradiated region, as shownin Fig. 1. Owing to the curvature of the refractiveindex, the probe beam diverges, which can be detectedas a change in the intensity of the probe beampassing through a pinhole. In other words, underthe influence of the pump beam the medium acts as adiverging lens. In certain circumstances the mediumacts as a converging lens also. If a pulsedpump laser is used, a transient lens is formed; the7046 APPLIED OPTICS / Vol. 36, No. 27 / 20 September 1997

Theory of photothermalspectroscopy in an optically dense fluidQifang He, Reeta Vyas, and R. GuptaThe theory of photothermal spectroscopy in an optically dense fluid is presented. The general case isconsidered in which the fluid may be flowing or stationary, and the excitation could be cw (modulated orunmodulated) or pulsed (arbitrary pulse length). All three detection schemes, deflection, phase shift,and lensing, are considered. This is the most complete theory of photothermal spectroscopy in fluids todate. O 1997 Optical Society of AmericaKey words: Photothermal deflection, photothermal phase shift, photothermal lensing, photothermalinterference, thermal blooming.1. IntroductionThere is currently a great interest in the technique ofphotothermal spectroscopy. This is because thetechnique has numerous applications in many diversefields.1 For example, the technique can beused for the measurement of species concentration,thermal diffusivity, laser spatial profile, relaxationtime, local temperature, and flow velocity. Thesemeasurements are nonintrusive and can be performedwith high degrees of spatial and temporalresolution and are particularly useful in hostile environmentssuch as combustion.2The principle of the technique is simple. A laserbeam (pump beam) passes through the medium ofinterest. The laser is tuned to an absorption line ofthe medium and the optical energy is absorbed by themedium. In a fluid medium, if the collisionalquenching rate is much higher than the radiativerate, most of this energy appears in the rotationaltranslational(thermal) modes of the molecules. Theheating of the medium modifies its refractive index.The change in the refractive index of the medium isdetected by a second (generally a He-Ne) laser beam(probe beam).Three distinct methods of monitoring the change inthe refractive index are in use. 1. Photothermalphase-shift spectroscopy (PTPS). In this techniqueThe authors are with the Department of Physics, University ofArkansas, Fayetteville, Arkansas 72701.Received 26 March 1996; revised manuscript received 29 July1996.0003-6935/97/091841-19$10.00/0O 1997 Optical Society of Americaone can measure the refractive-index change directlyby placing the sample inside a Fabry-Perot cavity orin one arm of an interferometer. The refractiveindexchange produces a change in the optical pathlength that is detected as a fringe shift.3j4 2. Photothermaldeflection spectroscopy (PTDS). Therefractive-index change produced by the absorption ofthe pump beam is, in general, nonuniform. The nonuniformrefractive index causes the probe beam todeflect; this is detected by a position-sensitive opticaldetector.5 The PTDS signal thus is proportional tothe gradient of the refractive index anlax. 3. Photothermallensing spectroscopy (PTLS). The nonuniformrefractive index also causes a focusing ordefocusing of the probe beam, which is detected as anintensity change of the probe beam as it passesthrough a pinhole.6 The PTLS signal is proportionalto the curvature of the refractive index a2n/dx2.In a series of papers Gupta, Vyas, and collaboratorshave provided a coherent theoretical treatment of thethree techniques in fluid media.7-10 Both pulsedand cw optical excitation were considered. For thecw case the theory allowed for both modulated andunmodulated excitation. For the pulsed case bothshort and long excitation pulses compared with thediffusion and forced convection times were considered.Both the stationary and flowing media weretreated. In other words, that theoretical developmentis valid for general conditions and has the furtheradvantage that all three techniques (PTPS,PTDS, and PTLS) and the various conditions (cw orpulsed excitation, stationary or flowing medium, etc.)are treated in a coherent fashion.Here we generalize the theory further to includeoptically thick media. Therefore the theory pre-20 March 1997 / Vol. 36, No. 9 / APPLIED OPTICS 1841

PHYSICAL REVIEW A VOLUME 55, NUMBER 1 JANUARY 1997Cavity-modified Maxwell-Bloch equations for the vacuum Rabi splittingChangxin Wang and Reeta VyasPhysics Department, University of Arkansas, Fayetteville, Arkansas 72701(Received 5 March 1996; revised manuscript received 29 August 1996)We derive a set of cavity-modified Maxwell-Bloch equations for a two-level atom in a cavity driven by acoherent field from the single-atom Fokker-Planck equation in the bad-cavity limit. These equations have thesame form as the Maxwell-Bloch equations for a two-level atom in free space interacting with a coherent field.The presence of a cavity is reflected in three cavity-modified parameters: decay rate ye//, Rabi frequencya, and detuning A,//. We show that the cavity-modified Maxwell-Bloch equations provide an easy way tostudy cavity-modified spontaneous emission, cavity-induced radiative energy level shifts, and vacuum Rabisplitting. [S1050-2947(96)075 12-91PACS number(s): 42.50.Ct, 32.80.-t, 42.50.L~The radiative properties of atoms in a cavity have beeninvestigated by many authors in the framework of cavityquantum electrodynamics [I]. Fields inside a cavity are subjectto boundary conditions and when the dimensions of thecavity are comparable to the wavelength of the radiation, thespectrum and spatial distribution of the electromagnetic fieldmodes are modified. As a consequence. atoms inside such acavity exhibit many different radiative properties from thosein free space, such as cavity-modified spontaneous emission[2], cavity-induced radiative frequency shifts [3], andvacuum Rabi splitting [4]. In the laboratory, several experimentshave verified the enhancement or inhibition of thespontaneous emission rate of an atom in a cavity [S]. Thephenomena of vacuum Rabi splitting in the strong couplingregime have been observed experimentally, for cavities witha large number of atoms, as well as for a small number ofatoms [6,7]. In particular, the case of a single atom on theaverage inside an optical cavity has been studied experimentally[7].To describe a single atom inside a cavity we derived ageneralized second-order Fokker-Planck equation for asingle two-level atom interacting with a cavity mode, withoutusing the system size expansion and any truncation [8].In this paper, we show that this exact single-atom Fokker-Planck equation in the bad-cavity limit leads to a set ofcavity-modified Maxwell-Bloch equations. We use theseequations to study the cavity-modified spontaneous emission,radiative level shifts, and vacuum Rabi splitting.Our quantum dissipative system consists of a singledamped two-level atom with transition frequency w, interactingwith a damped cavity mode of resonance frequencyw,. The cavity is driven by a coherent external field of frequencyw0 and amplitude E . In a frame rotating at the frequencyw0 of the external field, the behavior of the combinedatom-field system is governed by the master equation for thedensity operator p(t) [8],Here 6 and it are the field annihilation and creation operators;&+, &- , and &, are the Pauli spin operators describingthe two-level atom; y is the atomic spontaneous emissionrate; and 2 K is the rate at which the cavity is losing photons.Our model incorporates both the atomic detuningA,(= w,- wo) and the cavity detuning A,(= w,- wo).The master equation (1) can be transformed into a secondorderFokker-Planck equation without using system size expansionor any truncation in the bad-cavity limit [8]. Byeliminating the field variables adiabatically, we obtained asecond-order Fokker-Planck equation containing only theatomic variables. The drift terms of the Fokker-Planck equationlead to the differential equations for the mean values ofthe atomic operators [see Eqs. (28) in Ref. [8])], which canbe expressed in the following form:where cavity-modified parameters aredb-= -iAc[6t6,p]-iAll[&, ,p]+g[6t&--6&+ ,p]dtHere c = ~ ~ / is K the ~ single-atom version of the cooperat-ivity parameter, Y = 2 fig el^ is the dimensionless drivingYfield amplitude, 6, = 2 A, 1 y is the atomic detuning param-+~[i~-a^,bl+ -[2&-6&+-&+&-p-p&+&-]2eter, and aC= A, IK is the cavity detuning parameter. Equations(2) have the same form as the well-known Maxwell-+ ~[26p6~-ii~iip-p6~;]. (1) Bloch equations for a single two-level atom interacting with1050-2947/97/55(1)/823(4)/$10.00 - 55 823 O 1997 The American Physical Society

Modern Physics Letters B, Vol. 10, No. 14 (1996) 661-669@World Scientific Publishing CompanyPARTICLE TRAPPING BY OSCILLATING FIELDS:CONNECTING SQUEEZING WITH COOLINGB. BASEIA'Instituto de Fisica, Universidade de Sio Paulo, Caiza Postal 66318,05389-970, Slo Paulo (SP), BmzilREETA VYASPhysics Department, University of Arkansas, Fayetteville, AR 72701, USAV. S. BAGNATOInstifufo de Fisica e ~u:mica de Sa'o Carlos, Uniuersidode de Sio Paulo,CP 369, 13560-970, Sio Carlo$ (Sf), BmtilReceived 6 November 1995Revised 15 June 1996The first observation of the squeezing effect outside the optical domain was reported vervrecently for trapped atoms [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)l.Putsuing this line and a sequel of previous works of ours we employ a model by Glauberand a method of invariants by Lewis and Riesenfeld to establish a connection betweenthe squeezing and the cooling effects in the system. Rom this connection the occumnceof squeezing could be detected through the measurement of the dispersion in the variableP (or i) and of the temperature.Laser cooling and trapping techniques have played an important role in newadvances of atomic physics. hiany techniques of cooling atoms or ions down totheir zero point vibrational energy have emerged from the successful application ofradiation pressure and the Paul and Penning traps.'?* These techniques are appliedto reduce relative motion of the trapped particle and such systems, with extremelylow center of mass energy, must be treated quantum me&anically. Squeezed statesof electromagnetic field are weII known in Quantum O~tics.~ For squeezed statesfluctuations in one of the quadrature is less than the mcuurn fluctuations. Recentlythe idea of squeezed states has been extended to mechanical motion ofa trapped particle and schemes for preparation of squeezed state of motion hasbeen pr~~osed.~*~ Heizen and Wineland4 proposed generation of squeezed state of'Email: hasilioOif.usp.br; On leave from Departamento de Fisica, Universidade Federal daPara'ba (PB), Brazil.PACS Nus.: 32.80.Pj; 32.80.P~; 42.50.D~661

Physica A 232 (1996) 273-303PHYSIGA BMultiphoton interaction of a phased atom with asingle mode fieldA.N. Chabaay*, B. Baseiaa, Changxin wangb, Reeta vyasba Unicersidade Federal da Paraiba. Departamento de Fisica, CCEN, Caixa Postal 5008, CEP58.051-970. Joao Pessoa, Paraiba, BrazilPhysfcs Department, Unicersiry of Arkansas, Fayettecille, AAR 72701, USA-AbstractReceived 29 November 1995Multiphoton interaction of a phased atom prepared in a coherent superposition of its ground andexcited states with a general single mode optical field is studied. Calculations are done by usinga state vector of the atom-field composite system. Dynamic behavior of atomic inversion, fieldstatistics, and squeezing are studied for the multiphoton interaction of a phased atom includingthe effects of detuning. Results are discussed for various special cases of initial superposed atomicstate and field states such as number state, coherent state and squeezed states.1. IntroductionIn this paper, we discuss the dynamics of a two-level atom initially in a state whichis a coherent superposition of its ground and excited states (called phased atom), witha field initialIy in a pure state in a lossless cavity. We have used a generalizationof the Jaynes-Cummings model [I] (JCM), proposed by Buck and S~k~mar [2], thatinvolves a multiphoton interaction between the atom and the field. The Hamiltonianfor this system in the dipole and the rotating wave approximation isWe shall refer to this as p-photon JCM. The standard model, which corresponds toP = I, will be called simply the JCM. Here w is the mode frequency of the field andA is the detuning, A = wo - po, coo being the transition frequency of the atom, and9 is the coupling strength for the ndiatiowatom interaction. The Pauli matrices c+,O-, and 03 represent the mising, lowering, and the inversion operators for the atom' Corresponding author."--.. ..-.,. . . #-.-- ,a . tnnr clr-.n.nr Crimr- R v AII riht~ resenled

PHYSICAL REVIEW A VOLUME 54, NUMBER 5 NOVEMBER 1996Fokker-Planck equation in the good-cavity limit and single-atom optical bistabilityChangxin Wang and Reeta VyasPhysics Department, University of Arkansas, Fayetteville. Arkansas 72701(Received 23 January 1996; revised manuscript received 11 July 1996)A generalized Fokker-Planck equation is used to study the problem of single-atom optical bistability in thegood-cavity limit. The effects of quantum fluctuations are investigated by linearizing fluctuations around thesteady-state value. It is shown that in the good-cavity limit, quantum fluctuations are relatively small, andhence there exist both absorptive and dispersive single-atom bistability. By comparing our single-atom Fokker-Planck equation with the N-atom Fokker-Planck equation, we find that in the good-cavity limit, these twodifferent Fokker-Planck equations reduce to identical equations. We therefore come to an important conclusionthat the results obtained from the many-atom Fokker-Planck equation in the good-cavity limit are also valid fora single atom. [S1050-2947(96)00411-81PACS number(s): 42.50.Lc, 42.50.Ct, 42.50.Dv, 42.50.ArI. INTRODUCTIONThe simplest quantum mechanical model for radiationmatterinteraction is the Jaynes-Cummings model [I]. It consistsof a single two-level atom interacting with a singlequantized electromagnetic field mode. The Jaynes-Cummings model is exactly solvable under a variety of conditions.These analytical solutions provide important insightsinto the quantum characteristics of the electromagnetic field.As a result, this model has become a cornerstone in the modemquantum optics [2]. On the other hand, the simplicity ofthe Jaynes-Cummings model brings some limitations. Onelimitation results from the neglect of dissipation. At the opticalfrequencies dissipation is present both as spontaneousemission and as cavity loss through finite reflectivity mirrors.Consequently, an extension of the Jaynes-Cummings modelthat includes dissipation and an external driving field hasrecently drawn considerable interest. This driven Jaynes-Cummings model with dissipation exhibits many novel effects,including photon antibunching [3], cavity-enhancedspontaneous emission [4], and nonclassical steady-stateatomic inversion [5].Of particular interest is whether or not optical bistabilityexists for this single-atom system. McCall and Gibbs [6]proved that the semiclassical criteria for absorptive bistabilitycould be met for a single atom in a cavity. However, theysuggested that quantum fluctuations would strongly affectbistability and no reasonable hysteresis could be preservedfor the single-atom system.In the bad-cavity limit, it has been shown that large quantumfluctuations do indeed destroy bistability predicted bythe semiclassical theory. Hence there is no single-atom bistabilityin the bad-cavity limit [3,7]. In the good-cavity limit,Sarkar and Satchel1 [8] have used the master equation approachto study optical bistability with small numbers ofatoms. However, they failed to find any evidence for singleatombistability because they chose a very small saturationphoton number n, in their numerical calculations. Using anumerical approach, Savage and Carmichael [9] have shownthat single-atom optical absorptive bistability does exist for aparameter regime at the interface between the quantum limitand the classical limit.In Ref. [7] we have derived a generalized Fokker-Planckequation for a single-atom in an optical cavity and studied itsbehavior in the bad-cavity limit. In this paper we study thegeneralized Fokker-Planck equation in the good-cavity limitby adiabatically eliminating atomic variables. We derive ananalytical solution to the problem of single-atom optical bistability.We use linearized theory to investigate the effectsof quantum fluctuations on single-atom absorptive and dispersivebistability. We show that for a given cooperativityparameter as the saturation photon number n, increasesquantum fluctuations decrease. In this limit the condition forthe good-cavity is better satisfied. In the good-cavity limit,therefore, quantum fluctuations are too small to destroysingle-atom bistability. We compare our single-atom Fokker-Planck equation with an N-atom Fokker-Planck equation(FPE). The N-atom FPE was thought to be valid only for alarge number of atoms (N* 1) [lo-141. In the good-cavitylimit, we find that our single-atom Fokker-Planck equation isidentical to the N-atom Fokker-Planck equation if onechooses N= 1. This leads to an important conclusion that inthe good-cavity limit, all the results obtained from theFokker-Planck equation for many-atoms are also valid forthe single-atom system.11. SINGLE-ATOM FOKKER-PLANCK EQUATIONIN THE POSITIVE-P REPRESENTATIONThe extension of the Jaynes-Cummings model, as describedabove, consists of a single damped two-level atomwith transition frequency o, interacting with a singledamped cavity mode with resonance frequency o, . The cavityis driven by a coherent external field of frequency oo andamplitude E. In a frame rotating at the frequency oo of theexternal field the behavior of the combined atom-field systemis governed by the master equation for the density operator&t),4453 O 1996 The American Physical Society

PHYSICAL REVIEW A VOLUME 54, NUMBER 3 SEPTEMBER 1996Homodyne detection for the enhancement of antibunchingReeta Vyas, Changxin Wang, and Surendra SinghPhysics Department, University of Arkansas, Fayetteville, AR 72701(Received 15 February 1996; revised manuscript received 22 April 1996)We propose a scheme based on homodyne detection for enhancing antibunching in second-harmonic generationand multiatom optical bistability. We show that depending on the reflectivity of the beam splitter,relative field strengths, and relative phase it is possible to achieve perfect antibunching in the superposed field.We also discuss other nonclassical effects exhibited by the superposed field and present curves to illustrate thebehavior. [S 1050-2947(96)09008-71PACS number(s): 42.50.Dv, 42.50.Ar, 42.65.K~I. INTRODUCTIONSqueezing [I], antibunching, and sub-Possonian statistics[2,3] are nonclassical features of the electromagnetic field.These nonclassical features have been of considerable interestas they provide testing grounds for the prediction ofquantum electrodynamics. Squeezing is related to the wavelikecharacter of the electromagnetic field. It is measured ininterference experiments. Antibunching and sub-Poissonianstatistics, however, reflect the particlelike behavior of thefield and are measured in photon counting experiments. Asdiscussed in Ref. [4] squeezing, antibunching, and sub-Poissonian statistics are, in general, distinct nonclassical effectsin the sense that an electromagnetic field may exhibitone but not the other.The antibunching effect has been predicted in intracavitysecond-harmonic generation (ISHG) [5,6] and multiatom opticalbistability (MAOB) [7,8]. However, the predicted sizeof antibunching is small and would be difficult to detectexperimentally, as it occurs against a large coherent background.The predicted antibunching in these systems is inverselyproportional to the saturation photon number n,,which is of the order of lo6- 10' for the ISHG, andlo3 - lo4 for the MAOB. Several schemes based on interference[9] or passive filter cavities [lo-121 have been proposedto enhance the antibunching effect.We propose a scheme based on homodyne detection[13-151 for enhancing antibunching in these systems. Homodynedetection experiments have been used for measuringphase-sensitive properties of squeezed light [I]. It has beenshown that the light from a degenerate parametric oscillator,which is highly bunched and super-Poissonian [16,17], canexhibit many nonclassical effects using a similar detectionscheme [13]. In the homodyne detection experiment we considerthe interference of the signal beam from the ISHG orthe MAOB with a coherent local oscillator (LO) at a losslessbeam splitter as shown in Fig. 1. A detector of efficiency 7 isplaced at one of the output ports of the beam splitter. Thestatistics measured at the detector is sensitive to the relativephase between the signal and the LO. Thus particlelike properties(photon statistics) are intimately connected to wavelike(phase) property of the field. Because of this phase dependence,the homodyne field can exhibit enhanced antibunchingand violation of various classical inequalities. Since inthis scheme one can readily adjust various parameters suchas the strength of the local oscillator, transmittance, and relativephase, this scheme may provide a better way of enhancingantibunching.In Sec. I1 we briefly describe the homodyne detectionscheme. In Sec. 111 we apply this technique to the ISHG. InSec. IV we discuss the enhancement of antibunching for theMAOB. Finally, a summary and main conclusions of thepaper are presented in Sec. V.11. HOMODYNE DETECTIONFigure 1 shows a schematic diagram for the homodynedetection experiment. For the ISHG, a nonlinear crystal isplaced inside the cavity, whereas for the MAOB, N two-levelatoms are placed inside the cavity. The light from the ISHGor the MAOB is superimposed with the light from a LO at alossless beam splitter. The annihilation operators b, and b2at the output ports are related to those at the input ports by[13,141withNLC or N atomsIIIIFIG. 1. System for homodyning the ISHG or MAOB field withthe LO field. For the ISHG a nonlinear crystal (NLC) is placedinside the cavity and for the MAOB N two-level atoms are placedinside the cavity. BS denotes a beam splitter and D denotes a detector.1050-2947/96/54(3)/2391(6)/$10.00 3 2391 O 1996 The American Physical Society

VOLUME 74, NUMBER 12 PHYSICAL REVIEW LETTERS 20 MARCH 1995Exact Quantum Distribution for Parametric OscillatorsReeta Vyas and Surendra SinghDepartment of Physics, University of Arkansas. Fayetteville, Arkansas 72701(Received 15 August 1994)An exact quantum distribution for the nondegenerate parametric oscillators is presented and used todiscuss their coherence properties. It is found that while each mode individually approaches a classicalstate, many quantum features exhibited by their combination survive even in the semiclassical limit.PACS numbers: 42.50.Dv, 42.50.Ar. 42.65.K~Optical parametric oscillators (OPOs) are quantummechanical devices with a definite threshold for selfsustainedoscillations [ 1 -41. They have played a centralrole in squeezing and twin-beam noise reduction experiments[5,6]. Theoretical understanding of these propertiesof the OPOs has been based mostly on linearized treatmentsabove and below the threshold of oscillation [7-91.In the region of threshold where linearization fails, thecomplex-P distribution has been used [lo]. The complex-P distribution, unfortunately, does not have the characterof a probability density and is of limited use for gaininginsights into the coherence properties of the OPOs.Another distribution, closely related to the complex P,is the positive-P distribution which is a true probabilitydensity [ll]. In this paper we present an analyticsolution for the positive-P function for the optical parametricoscillators. With the help of analytic solutionstremendous insights into the coherence properties of otheroscillators have been gained [12- 141. Our analytic treatmentis based on the observation that the quantum dynamicsof OPOs is naturally confined to a bounded regionin an eight-dimensional phase space. The solution presentedhere provides us with an elegant picture of howthe coherence properties of the OPOs are transformed inthe threshold region. It also allows us to discuss quantumfeatures that survive even as the field amplitudes grow upto macroscopic values above threshold.We model the parametric oscillator by two quantizedfield modes of frequencies w, and w2 interacting witha third mode of frequency w3 = w, + w2 inside anoptical cavity via a X(2) nonlinearity. Modes o, and0 2 experience linear losses characterized by the decayrates yl = y2 = y, and mode w3 suffers linear lossescharacterized by the decay y3. The cavity is excited by aclassical pump at frequency w3. In the interaction picturethe microscopic Hamiltonian takes the form(1)where aj and 6,tare the annihilation and creation operatorsfor the modes, K is the mode coupling constant, e isthe pump field amplitude, and ffl,,, describes mode losses.This nonlinear quantum mechanical problem can bemapped into a classical stochastic process by using thepositive-P representation [Ill. Eliminating the pumpmode adiabatically (y3 >> y) we obtain the following setof Ito stochastic differential equations:where vi are white noise Gaussian random processeswith zero mean and correlation functions given by{q,(t)v,(t')) = Sij8(t - t'). Here time is measured inunits of ye', no = 2-yy3/~' is parameter that sets thescale for the number of photons necessary to explore thenonlinearity of interaction, and u = 2y3e/~is a dimensionlessmeasure of the pump field amplitude scaled togive a = no as the threshold condition. In the absenceof mode losses adiabatic elimination of the pump mode isnot justified [15]. Note that the adiabatic approximationdoes not amount to a neglect of the entanglement ofpump and down-converted modes. Complex variablestai and ai, correspond to 2; and ai, respectively. In thepositive-P representation, ai and ai, are not complexconjugates of each other.Equations (2)-(5) describe trajectories in an eightdimensionalphase space. An examination of theseequations reveals that the eight-dimensional phase spaceis naturally divided into two subspaces. If we considerthe four-dimensional subspace a2 = (cYI)*, a2* = (a~*)*,and < a/2, Ia2*J2 < u/2, we notice that thetrajectories starting in this subspace initially remainconfined to this subspace. In other words, the conditiona2 = (a,)f, aZr = (alr)* is preserved for all times ifinitially we start out in this subspace. The initial state,2208 0031 -9007/95/74(12)/2208(4)$06.00 O 1995 The American Physical Society

PHYSICAL REVIEW A VOLUME 52, NUMBER 2 AUGUST 1995'Statistical properties of a charged oscillator in the presence of a time-dependentelectromagnetic fieldA. L. de Brito, A. N. Chaba, and B. BaseiaDepartamento de Fisica-CCEN, Universidade Federal da Paraiba, Caixa Postal 5008, 58.051-970 Joao Pessoa, Paraiba, Brazil..Reeta VyasDepartment of Physics, University of Arkansas, Fayetteville, Arkansas 72701(Received 29 August 1994; revised manuscript received 13 January 1995)The statistical properties of a charged oscillator in the presence of a uniform magnetic field are investigatedfor the case of a time-dependent electromagnetic field. Quite general results are obtained whenthe time dependence of the external field is not specified. As an example, an oscillating external field isconsidered. It is found that, depending upon the parameters, this system can show many nonclassicalfeatures, such as sub-Poissonian statistics and squeezing.PACS number(s): 42.50.Dv, 84.30.Ng, 03.65. - wI. INTRODUCTION 11. CHARGED PARTICLEIN A TIME-DEPENDENT ELECTROMAGNETICIn both quantum mechanics and electromagneticFIELD AND TIME-DEPENDENTtheory one can find extensive literature on charged parti-HARMONIC OSCILLATORcles and charged oscillators in the presence of a constant[I] or time-dependent fields [2]. People mainly concen-Let us consider a particle, which may be an isotopic ostratetheir attention on calculating propagators, invaricillator,of charge e and mass M moving in gn axiallyants, eigenvalues and eigenvectors, coherent andsymmetric magnetic field defined by the vector potentialsqueezed states, etc. There is also considerable interest in(cf., e.g., Lewis and Riesenfeld [2])the production of quantum states of light, such assqueezed states [3], sub-Poissonian states [4], andphoton-number states [5]. Agarwal and Arun Kumar [6] and a scalar potentialstudied statistical properties of a one-dimensional oscillatorwith a time-dependent frequency and showed that anonadiabatic change of frequency may exhibit nonclassicalbehavior such as squeezing and sub-Poissonian statis- where (2.1) is valid if B (t) is uniform. We assum: that'tics. Abdalla [l] investigated statistical properties of a the magnetic field B is switched on at time t =O. k is acharged oscillator in the presence of a spatially constant unit vector along the symmetry axis; r2=x2+y *, where xbut time-independent magnetic field. He considered sta- and y are two Cartesian components perpendicular to thetistical aspects that, in the realm of quantum optics, re- symmetry axis. B (t) and q( t) are arbitrary piecewise',late photon-number fluctuations, bunching and anti- continuous functions of time and c is the speed of light.bunching, quasiprobability distribution function, etc. The Hamiltonian for the present system is given bySince a charged oscillator in a magnetic field is not thesame as a mode of a radiation field in a magnetic field, wewill prefer the term "excitation" instead of "photon" tobe consistent with a realm that is not that of quantum optics.whereIn the present paper we study statistical properties ofan excitation for a time-dependent (TD) electromagneticfield. Among the motivations one can cite the problem ofparticle trapping by oscillating fields (cf. Glauber [2]) andothers involving TD Hamiltonians [7].The present paper is arranged as follows In Sec. I1 wedefine the system Hamiltonian. In Sec. I11 we quantizethe system and obiain the Heisenberg equations ofmotion. In Sec. IV we solve the Heisenberg equations for where LZ is the z component of the angular momentumour TD system and calculate the variances of the quadra- L,=(rXp),=xp,,-ypx andture phase amplitudes. In Sec. V we consider some statisticalproperties for our system and present results for atime-dependent frequency.I1050-2947/95/52(2)/15 18(7)/$06.00 - 52 1518 a1995 The American Physical Society ',

PHYSICAL REVIEW A VOLUME 5 1, NUMBER 3 MARCH 1995Fokker-Planck equation for a single two-level atom:Applications in the bad-cavity limitChangxin Wang and Reeta VyasPhysics Department, University of Arkansas, Fa~etteville, Arkansas 72701(Received 14 October 1994)A generalized second-order Fokker-Planck equation is derived for a single two-level atom in acavity driven by an external classical field without using system size expansion or any truncation.Effects of detuning, as well as atomic and cavity decays, are incorporated in this Fokker-Planckequation. This equation is used to study atomic inversion and the second-order intensity correlationfunction of light in the bad-cavity limit by adiabatically eliminating the field variables. Several novelfeatures in atomic inversion and intensity correlations that arise due to atomic and cavity detuningare discussed. Curves are presented to illustrate the behavior.PACS number(s): 42.50.Lc, 42.50.Ct, 42.50.Dv, 42.50.ArI. INTRODUCTIONThe Fokker-Planck equation has played an importantrole in quantum optics. For the problem of N two-levelatoms interacting with a cavity field, the Fokker-Planckequation approach is well developed and it has beenwidely used to study optical bistability [I-71, photon antibunching[I-31, and squeezing [8,9]. Haken [lo] used theSudarshan-Glauber P representation [11,12] based on thenormally ordered characteristic function to map an operatormaster equation into a c-number Fokker-Planckequation. This equation, when applied to study opticalbistability, results in a nonpositive diffusion matrix. Thescaling argument, used by Haken to drop the offendingterms for the laser near threshold, does not work for opticalbistability, photon antibunching, and squeezing. Animportant step towards resolving this problem was takenby Drummond and Gardiner [13] who introduced theso called positive-P representation. The Fokker-Planckequation based on the positive-P representation has beenused to study quantum effects in an atomic system consistingof N two-level atoms in the good-cavity limit [I]and the bad-cavity limit [2,3]. Another approach basedon the symmetrically ordered Wigner representation hasalso been used for optical bistability, which for opticalbistability leads to a positive diffusion matrix [4,5].The Fokker-Planck equation approach offers some appealingfeatures [6,7]. First, the analogies between classicaland quantum fluctuations can be drawn. This helpsin develodne - intuition for auantum fluctuations. Secuond, all quantum-operator averages can be calculated asclassical-like averages by integrating the correspondingc-number variables multiplied by the distribution function.Third, the Fokker-Planck equation has been studiedextensively in classical statistical physics and mathematicaltechniques developed for analyzing the Fokker-Planck equations for classical systems can be applied tosolve quantum-mechanical problems. Finally, a Fokker-Planck equation can be converted into an equivalent setof first-order stochastic differential equations, which canbe solved analytically or numerically.The Fokker-Planck equations derived for N two-levelatomic systems are, unfortunately, not applicable to asingle-atom system. Various representations, such as theSudarshan-Glauber-Haken, Wigner, or positive-P representations,used for mapping a density-operator masterequation into a c-number differential equation alllead to a generalized Fokker-Planck equation which containsderiktives of all orders. To obtain a second-orderFokker-Planck equation the system-size expansion is usedto justify the truncation of series and retain only terms upto the second-order derivatives. The system-size expansion,however, breaks down when the number of atomsN becomes small. That is why most of the studies onquantum fluctuations by using the Fokker-Planck equationassume the system size (here, the number of atoms)to be large. On the other hand, small systems like a singletwo-level atom are of particular interest because in thesmall systems quantum effects are expected to be mostdramatic. So far, to study the quantum effects of a singletwo-level atom inside a cavity, the density-operator masterequations have been used by most authors [14-181.In this paper we derive an exact second-order Fokker-Planck equation for a single two-level atom interactingwith a quantized cavity mode. The cavity is drivenby an external classical field. This Fokker-Planck equationincludes the effects of atomic detuning, cavity detuning,atomic decay, and cavity decay. Since no partialderivatives beyond second-order exist in our Fokker-Planck equation no system-size expansion or truncationis needed. Our work may offer an alternative for studyingcoupled single-atom field. systems.Section I1 describes our model and the phase space representationused to arrive at the Fokker-Planck equation.In Sec. 111, a generalized second-order Fokker-Planckequation for both field and atomic variables is derived.In Sec. IV, field variables are adiabatically eliminatedfrom the generalized Fokker-Planck equation by usingthe bad-cavity limit. A generalized set of Bloch equationsdescribing the dynamics of a single atom coupledto the field inside the cavity are derived. These equationsare used to study the steady-state inversion. In Sec.105@2947/95/5 1(3)/25 16( 14)/$06.00 - 51 2516 01995 The American Physical Society

3 1 October 1994Physics Letters A 194 (1994) 153-158PHYSICS LETTERS AI!i. .i:' !/1Scattering of atoms by light:probing a quantum state and the variance of the phase operatorB. Baseia "*I, Reeta Vyas b2, Cklia M.A. Dantas c3, V.S. Bagnato c*3 Ia Departamento de Fisica, CCEN, Universidade Federal ah Parah. 58059-970 Joio fissoa (PB), BrazilDepamnt of Physics, University of Arkansas. Fayettcville, AR 72701, USAlnstituto de Fisica de SCo Carlos, Universidade de Sdo Poulo, Cx. Postal 369, 13560-970 Sdo Carlos (SP), BrazilReceived 22 June 1994; revised manuscript received 6 September 194; accepted for publication 12 September 1994Communicated by F?R. Hollandi:f;,FGrI*:-7';AbstractAn experiment is proposed to study the quantum state and the variance of the phase operator by measuring the momentumdistribution of the atoms resulting from the interaction of an atomic beam with a single mode electromagnetic field via atwo-photon interaction. A comparison of the results with those obtained for a one-photon interaction is made. Adwt~gesin using the two-photon interaction are pointed outIn the last decade there has been great interest in the deflection of atoms by light fields [I]. The momentumtransferred to an atomic beam by a standing wave has been shown to be an important tool in the determinationof the light field properties [Z-41.Recently Freyberger and Herkornmer [4] proposed a simple and interesting setup to obtain complete informationabout the quantum state of a single mode of the radiation field. The proposed method uses a barnof two-level atoms as a meter for the quantized mode radiation field. Atoms are prepared in a supemsitionState (Jg) + eie Ie)), where Ig) and le) denote ground and excited states of the atom respectively, and 6r 18 therelative phase between them. The prepared atomic beam passes through a region near the node of a standingelectromagnetic wave of wavelength A in a cavity. Atoms interact with the field via a one-phoron interactiondffcribed by the resonant Jaynes-Cumming model. It is assumed that the time of interaction is very short sothat spontaneous decay of the atom is negligible. The atomic diffraction pattern resulting from the interactionshows asymmetries providing information about the state of the field and also about the expectation value ofthe phase operator exp(-$) [4].The proposed one-photon interaction method (OPIM) is interesting, however, for some special fields thecannot provide complete information about the field. As-in the OPIM, we consider the standing lightwave to be a quantum state IP(x, 0)) = C Cmlrn) with the coefficients C,,, of a number state (m). For the case' E-mail: basilio@ifqsc.sc.usp.br.2Email: w@24533uafsysb.uark.edu.3E-mail: vandcr@ifqsc.sc.usp.br.0375-9601/94/$f)7.00 @ 1994 Elsevier Science B.V. All rights nservcd0375-9601 (94)00755-1

PHYSICAL REVIEW A VOLUME 47, NUMBER 4 APRIL 1993Homodyne photon statistics of the subthreshold degenerate parametric oscillatorA. B. Dodson and Reeta VyasPhysics Department, University of Arkansas, Fayetteville, Arkansas 72701(Received 17 August 1992)Homodyne statistics when the light from a degenerate parametric oscillator (DPO) is mixed withcoherent light from a local oscillator are discussed. By using dynamical models of these beamswe discuss the photon sequences underlying the superposed beam in terms of photoelectric pulsesequences recorded by a detector. The field produced by the parametric oscillator is expressed interms of two independent real Gaussian random variables. Using this property of the field variables,we derive the generating function for the photoelectron statistics analytically. From this generatingfunction, expreseions for the photoelectron-counting distribution, factorial moments, and thewaiting-time distribution are derived. These quantities are directly measurable in photon-countingexperiments. The results reported here are applicable to arbitrary strengths of the signal and thelocal-oscillator fields. We also show that, depending on the relative strength and relative phasebetween coherent light from the local oscillator and squeezed light from the DPO, the superposedlight beam may exhibit many interesting nonclassical features. These and other effects are describedand curves are presented to illustrate the behavior.PACS nurnber(s): 42.50.Dv, 42.50.Ar, 42.65.K~I. INTRODUCTIONInterest in the optical degenerate parametric oscillator(DPO) as a vehicle for studying the photon statisticsof squeezed light [I, 2) has motivated the publication ofseveral papers describing the statistics of the DPO [3-61.Vyas and Singh obtained exact analytical expressions forthe statistics of the DPO [3, 41. In this paper we applythe same approach to study the statistics of the fieldproduced by superposing the light from the DPO withthe light from a coherent local oscillator (LO) in homedyne detection experiments. In homodyne detection experimentsthe LO frequency is the same as that of thesignal. Such experiments have been used for measuringphase-sensitive properties of squeezed light [I, 2, 61.Detailed discussions of homodyne detection and beamsplittermodeling may be found in the literature 16-17].The light from the DPO produces a large amount ofsqueezing 121. Inside the DPO cavity a pump photonis down converted into two photons at the subharmonicfrequency. These pairs of photons are highly correlated,and it is this correlation that results in various interestingnonclassical features that we describe in this paper.In the homodyne detection experiments usually astrong LO is used. This assumption is also implicit in thetheoretical treatments 16-17]. In this limit many effectsdiscussed in this paper are either too small or the techniqueof photoelectron counting cannot be used as thephoton fluxes involved are too high. In our approach wedo not make this strong-local-oscillator approximation.This allows us to treat the case of even small photonnumbers and interestingly we find that many quantumeffects that are otherwise small are more pronounced inthis regime. Since low photon numbers are involved photoelectroncounting experiments can be used to measurethese effects.The theoretical investigations of the fluctuat~on prop-;erties of the light from the DPO operating belaw thresholdshow that this light may exhibit pronounced photonbunching and super-Poissonian statistics 13-81. Thesefeatures by themselves do not reflect the unusual natureof the light from the DPO. In this paper we show thatwhen the output light from the DPO is homodyned witha coherent LO field, the superposed field exhibits a vanetyof interesting statistical features, ranging froq super-Poissonian and highly bunched photon sequences to sub-Poissonian and antibunched photon sequon--. Betweenthese limits the superposed field can show photo11 bun, 11-",ing accompanied by sub-Poissonian statistics, and antibunchingaccompanied by super-Poissonian statistics, indicatingclearly that sub-Poissonian statistics and photonantibunching are distinct effects [8,18-211. Other interestingfeatures are reflected in the waiting-time distribution[22]. The waiting-time distribution for the photonsshows both a minimum and a maximum. The minimum,is found at the time of the order of the cavity lifetime.This minimum corresponds to a reduced probability ofdetecting two consecutive photons separated by a timeof the order of the cavity lifetime.In addition to the effects described in the precedingparagraph the second-order intensity correlation functiog'shows new violations of Schwartz's ineaualities, indicat-.ing the quantum nature of the light from the DPO. ~11-.;of these features depend on the relative strengths of thetwo fields and their phase differences. The superpose$Ifield thus offers a system that allows us to generate a va- ., ,riety of nonclassical features simply by turning a virtual"knob" of parameters.It is interesting to note that thermd L;;b'r is aisubunched and super-Poissonian, but homodyning the ~1.e::-ma1 field with a local oscillator does not show these non:.classical features [14]. This clearly indicates that the n6n- .01993 The American Physical Society

PHYSICAL REVIEW A VOLUME 48, NUMBER 5 NOVEMBER 1993Single-atom fluorescence with nonclassical lightIrina E. Lyublinskaya and Reeta VyasPhysics Department, University of Arkansas, Fayetteville, Arkansas 72701(Received 20 May 1993)The statistical properties of fluorescent light from a single two-level atom driven by nonclassicallight beams that exhibit antibunching and squeezing are discussed in the weak-field limit. The fieldsproduced in intracavity second-harmonic generation and those produced by superposing the lightfrom a degenerate parametric oscillator with a coherent field are considered to be the models ofnonclassical light beams. These fields can be described in terms of real Gaussian random processeswith nonzero mean coherent components. Analytical expressions for the fluorescent light intensity,the second-order intensity correlation function, and the variance of the photon number in the steadystate are obtained. The effects of driving-field correlations, detuning, and other parameters characterizingthe atom-field interaction on the statistics of scattered light are investigated. It is foundthat for a certain range of parameters the second-order intensity correlation function may vanishat nonzero times. Under certain other circumstances the antibunching of the fluorescent photons isenhanced. These and other interesting features that appear in the scattered light are discussed.PACS number(s): 42.50.Lc, 32.80.-t, 42.50.Dv, 32.50.+dI. INTRODUCTIONThe phenomenon of resonance fluorescence, when asingle two-level atom is illuminated by a light beamwhose frequency is nearly resonant with the transitionfrequency of the atom, has been treated by a numberof workers [I-101. Most treatments assume the incidentlight beam to be in a coherent state. This leads to mathematicallysolvable expressions for photon statistics. Acomplete solution to photon statistics in resonance fluorescenceunder coherent illumination has been given inRef. [5]. Recently, two-level atoms driven by nonclassicalstates of light have also been considered [7-101.Carmichael, Lane, and Walls [9] investigated the phenomenaof subnatural linewidth in the presence of broadbandsqueezed light. The interaction of a single two-levelatom with finite-bandwidth squeezed light was studied byVyas and Singh [lo]. They considered a degenerate parametricoscillator (DPO) operating below threshold as thesource of squeezed light.In this paper we study the interaction of a single twolevelatom with an electromagnetic field that is producedby superposing the light from the DPO with the lightfrom a coherent local oscillator (LO) at a beam splitterin a homodyne-detection-type setup. We will referto this light source as HMDPO. Another source of lightwe consider is the intracavity second-harmonic generation(SHG). The fundamental beam derived from thissource is used for illuminating the atom. We also studythe effects of detuning on the statistical properties of thefluorescent photons.The fundamental process underlying the DPO is theconversion of a pump photon into two photons at thesubharmonic frequency inside an optical cavity resonantat the subharmonic frequency. Due to the strong correlationsbetween the pairs of photons produced in the processof down-conversion the light from the DPO is highlybunched and super-Poissonian [ll]. When the light fromthe DPO is superposed with the light from a coherent LOat a beam splitter, the superposed field exhibits a varietyof quantum features [12], ranging from super-Poissonianand highly bunched photon sequences to sub-Poissonianand antibunched photon sequences. Thus the HMDPOfield allows us to control the statistical properties of thedriving field simply by changing parameters such as relativephase and the relative intensities of the two superposedbeams.The SHG produces antibunched and sub-Poissonianphoton sequences at the fundamental wavelength. Thenonclassical effect of photon antibunching in the secondharmonicgeneration was predicted theoretically [13-151.We assume that both the DPO and the SHG are operatingbelow threshold and consider a fully quantummechanicaltreatment of noise in both sources. In bothcases the field can be modeled by two real Gaussian randomprocesses with different variances and correlationtimes [Ill. Because of the finite correlation time of theincident field, atomic states and field states develop correlationsduring their dynamical evolution. These correlationsrender the problem of photon statistics morecomplex for these fields, in general, than in the casefor coherent excitation. For weak driving fields, such asthose produced by the light sources under consideration,it is possible to apply perturbative approach to understandthe behavior of a two-level atom when the excitingfield has finite correlation time [lo]. This is the approachtaken in this paper.The paper is organized as follows. In Sec. I1 we summarizethe Heisenberg equations of motion governing thetime evolution of atomic and field operators for finitedetuning between the atomic transition frequency andthe mean-field frequency. Using the formal solutions ofthese equations we write down general expressions for thescattered light intensity, normalized second-order intensitycorrelation function, and the variance of the photoncounting distribution. In Sec. 111 these quantities are cal-1050-2947/93/48(5)/3966( 14)/$06.00 3966 01993 The American Physical Society

PHYSICAL REVIEW A VOLUME 46, NUMBER 1 1 JULY 1992Photon-counting statistics of the subthreshold nondegenerate parametric oscillatorReeta VyasDepartment of Physics, Uniuersity ofArkunsos, Foyetteuille, Arkansas 72701(Received 9 September 1991)Statistics of photons emitted by a nondegenerate parametric oscillator (NDPO) operating belowthreshold are discussed. Analytical expressions for the generating function are obtained when two nondegeneratemodes of the NDPO are mixed on a photocathode or when the intensities of the two modesare added. From the generating function, photon-counting distributions and waiting-time distributionsas measured by a detector placed outside the cavity are derived. Similarities and differences between thedistributions for the NDPO, the degenerate parametric oscillator, degenerate four-wave mixing, nondegeneratefour-wave mixing, and thermal light are also discussed.PACS number(s): 42.50.Dv, 42.50.Ar, 42.65.K~I. INTRODUCTIONThere has been considerable interest in the fluctuationproperties of the nondegenerate parametric oscillator(NDPO) in recent years. In the NDPO a pump photon isdown-converted into two nondegenerate photons insidean optical cavity. These photons come out of the cavity,independently, in a lifetime of the order of the cavity lifetime.Since two nondegenerate photons are producedsimultaneously inside the cavity, the NDPO generateshighly correlated twin beams [I]. This feature of theNDPO has been used in reducing the noise in thedifference intensity I, -I, below the shot-noise limit [2].The nondegenerate modes of the parametric oscillator arepresent even when the parametric oscillator is operated inthe degenerate mode. It is of interest to investigate howfluctuation properties of the parametric oscillator changewhen nondegenerate modes are also taken into account.The degenerate parametric oscillator (DPO) is known toproduce large amounts of squeezing [3-51. Photon statisticsfor the DPO have been studied by several differentgroups [6,7] by using different techniques. For theNDPO, the mean and the variance of the photons comingout of the cavity have been calculated by Collett andLoudon [8]. Photon-counting statistics for an idealizedmodel of squeezed light have also been studied [9]. Inthis paper a realistic system, an open system with a pump- -and dissipation, for the NDPO is considered, and statisticsof the photons emitted by the NDPO are discussedwhen the two nondegenerate photons have the same frequency.Here nondegeneracy refers to the direction ofpropagation or polarization. Two cases are considered,one in which the amplitudes of the two nondegeneratemodes are homodyne detected, and the second in whichthe intensities of the two modes are added together. Inboth cases, analytic expressions for the generating functionsfor photoelectric-counting statistics are obtained byfollowing our earlier approach to the DPO [6]. In thenext section, we show that the field for the NDPO can bedescribed in terms of four real Gaussian random processes.In Sec. 111 the generating function for the NDPO isobtained, and photon statistics in terms of photoncountingdistribution, factorial moments, and waitingtimedistribution are discussed. These results are comparedwith those for the DPO, degenerate four-wave mixing(DFWM), nondegenerate four-wave mixing (NFWM),and thermal light.11. GAUSSIAN RANDOM VARIABLESOF THE NDPO FIELDThe Hamiltonian, in the interaction picture, for theNDPO where one pump photon is down-converted intotwo nondegenerate photons can be written as [lo]Here pump depletion is assumed to be negligible, which isa good approximation below threshold. Parameter K is amode-coupling constant, E is the dimensionless amplitudeof the pump beam incident on the cavitthe annihilation operators, and at and a, ?'* are the and creation areoperators for the two down-converted fields of theNDPO. a,,,, describes dissipation suffered by thesubharmonic modes due to linear absorption, scattering,and transmission.In order to obtain the equations of motion correspondingto the annihilation and creation operators, the masterequation with the interaction Hamiltonian [Eq. (111 is expressedin the positive-P representation [I I]. This leadsto a Fokker-Planck equation for probability distributionof the subharmonic mode with a positive diffusion constant.The distribution is defined over a four-dimensionalcomplex space. The four Langevin equations obtainedfrom this Fokker-Planck equation are [I21- 46 395@ 1992 The American Physical Society

,n of atomic clocks, wherenegligible [2]. In such de.pathways can couple amofmicrowave fieldt results this would causet frequency. Consequentl,sitic pathway would respility.optics Communications 9 1 ( 1992) 347-353~~flh-HollandTWO-photon interaction of a phased atom with squeezed vacuumC. Wang and Reeta VyasDeparlmenf of Physics, Llniversily ofArkansas, Fayeffeville, AR 72701. USAp. ~ambropoulos for:garding the effects Received 3 February 1992; revised manuscript received 17 March 1992t processes. This w;pace Corporation':h program.Two-photon interaction of a squeezed-vacuum state with a single atom prepared in a coherent superposition of its ground andexcited states is studied. Long time and short time evolution of quadrature squeezing is discussed and compared with the correspondingbehavior in the conventional Jaynes-Cummings model. It is found that the two-photon interaction of a phased atomwith field in the vacuum slate can yield upto 451 squeezing below the shot-noise limit. With a suitably prepared phased atom thevacuum state of the field can evolve periodically into a pure state that is a coherent superposition of the vacuum state ando-photon Fock state.cs of a two-level atom interacting with a single mode of the radiation field in a lossless cavity caned by the Jaynes-Cummings model (JCM) ". This model is perhaps the simplest solvable modelarchi, IEEE Trans. lnstexamples clearly indicate the essential role of quantum mechanics in the dynamical evolution ofnd fluctuations because these features are absent in the semiclassical version of this model.atom with a single mode squeezed state within the framework of the JCM. The possibility of gen-sted by Meystre and Zubairy [3]. Knight and co-workers [ 10,11] have discussed the JCM with astate that is a coherent superposition of the vacuum state and the number state. They found that af 25qo squeezing can occur in the JCM if the atom is prepared initially in a coherent superpositionstate or in the excited state.is also examined as a special case and it is found that a maximum squeezing of 45Yo below the347

Green's function soluSion to the tissue bioheat equationReeta VyasDepartment of Phprics, Unimity of Arkansas, Fayetteuille, Arkansas 72701M. L. RustgiDepartment of Phpsics, StateU.'niuersi@ of New York at Buffalo, Buffalo, New York 14260[Received 18 October 1991;; accepted for publication 13 May 1992)A Green's function solutiw to the tissue bioheat equation including blood flow in cylindricalgeometry is obtained. Nlmerical results for temperature variation in the bovine muscle arereported when @he tissue ias exposed to neodymium-yttrium-aluminum garnett (Nd:YAG)hers with Gaussian prae and a comparison with recent measurements is made. A strongdependence of the tissue mperature on the beam radius and pulse time is found.I. INTRODUCTIONThe influence of lasers in biologicdamd medical research isbecoming increasingly important"-8 The neodymiumyttrium-duminumgarnett (Nd:YLLG) laser, which is capableof being operated as eithar pulsed or continuouswave (cw), with pulse widths vaqkag, with mode of operation,from :lanoseconds to picosamnds and power to theorder of 10" W has eclipsed the ruse of ruby laser. TheNd:YAG laser finds most of its -litxition in the utilizationof hs 1064-nm light. The k tissue interaction ismainly determined by two param-, the interaction timeof the radiation with the tissue a~ld the effective energydensity which brings about an eEcct whereby the tissuespecificabsorption must be taken $into consideration. Atlow qgy density with a long expsure the absorption oflighL a .manly leads to photochedcal processes. With decreasinginteraction time and higher energy densities is thedomain of the photothermal-indd effects. In this paper,a Green% function solution to thehue bioheat equation isobtained to describe the tempera distribution due to alaser beam with Gaussian pr ~2- k The effect of thermalconduction on the tewrd$re k y curve within theframework of the bioheat equationhas been investigated by~andhu,~ who pointed out that inbnited cases, the decaycurve may be used to measure Wood flow. Though thesolution of the bioheat equation OPH excluding blood flowhas beem obtained by a number d ;authors?-' the Green'sfunction approach used here is qifte general and includesthe blood flow in the sdution of dne bioheat equation.In Sec. I1 simple analytical expressions for Green'shnction and temperature distribtiion using a model withcylindrical symmetry are derived The heat source is assumedbo be a pulsed laser beam with a Gaussian profile.The method is sufficiently general tto enable calculation oftemperature distribution for other sources with cylindricalsymmetry such as cw laser usingthis Green's function. InSec. 111 results are compared with available experimentaldata.'is derived. The derivation is based on a cylindrically symmetricmodel in which the laser beam is traveling in the zdirection. The temperature distribution for this system isdescribed by a differential equation which is essentially anequation of energy conservation. The Green's function approachis followed to solve the differential equation for thetemperature distribution. Since the Green's function obtainedfor the differential equation is independent of thesource term, the same Green's function can be used tocalculate temperature distribution for various sources withdifferent spatial and temporal profiles. Even though thisproblem is solved for a cylindrically symmetric system, thisapproach can be extended to calculate the temperature distributionsfor systems that do not possess cylindrical symmetry.The bioheat equation was first proposed by pennesg andthe inherent assumptions in this equation are outlined inHodson et aL lo The differential equation describing thetemperature distribution can be written as9.''where the first and the second terms on the right-hand sideof Eq. (1) are due to the thermal diffusion and blood flow,the temperature above the baseline temperature of the tissueis T(r,z,t), S(r,z,t) is the power deposited per unitvolume due to the absorption of the laser energy, the ther-mal diffusivity D is equal to K/~C, where K, p, and C arethermal conductivity, density, and specific heat of the tissue,respectively. The parameter b is given by opbCJC.Here o is volumetric blood flow, Cb is the specific heat ofthe blood, and pb is density of the blood. The boundaryconditions imposed on the temperature and its derivativesare that they vanish as r and z go to infinity. These boundaryconditions require that the source energy must vanishat infinity. It is also assumed that at time t=0, the temperatureis zero. The solution of Eq. ( 1 ) can be expressedin terms of Green's function G(r,z,t; r1,z',t') and thesource term S(r,z,t) as"11. P"F&N'S FUNCTION AND TEMPERATUREDl: t6BUTION S(r1,z',t') G( r,z,t;r1,z',t' )In this section, an expression fot the temperature distributionT(r,z,t) in a tissue that is a~osed tothe laser beam x r'dr'dz' dt'. (2)1319 Hed. Phys. 19 (5). Sep/Oct lXB2 0994-2405/91/5131406S01.20 @ 1992 Am. Assoc. Phys. Med. 1319

PHYSICAL REVIEW A VOLUME 45, NUMBER 11 I JUNE 1992Resonance fluorescence with squeezed-light excitationReeta Vyas and Surendra SinghDepartment of Physics, University of Arkansas, Fayetteuille, Arkansas 72701(Received 30 August 199 1)Resonance fluorescence from a single two-level atom driven by a beam of squeezed light is studied inthe weak-field limit. We consider the situation where the atom is coupled to the ordinary vacuum andonly a few field modes corresponding to the driving field are squeezed. The field produced by the degenerateoptical parametric oscillator is used as the driving field. Heisenberg equations of motion are solvedin the steady-state and analytic expressions for the fluorescent-light intensity and the spectrum offluorescent light are derived. We also consider photon statistics of fluorescent light. In particular,squeezing, antibunching, and sub-Poissonian statistics of fluorescent photons are discussed, and analyticexpressions for the quadrature variance and the two-time intensity correlation function are presented.Contrary to the case of coherent excitation, the second-order intensity correlation function does not factorize.This and other differences are discussed, and curves are presented to illustrate the behavior ofvarious quantities. We also present results for thermal excitation of the atom.PACS number(s): 42.50.Lc, 32.80. - t, 42.50.Dv, 32.50. +dA single two-level atom interacting with an electromagneticfield is a fundamental model of quantummechanics. This simple dissipative quantum system liesat the heart of the physics of light-atom interaction. Aninteresting aspect of this problem is the phenomenon ofresonance fluorescence when the atom is illuminated bv alight beam whose frequency is nearly resonant with thetransition frequency of the atom. The fluorescent lightunder these conditions displays many purely quantummechanicalfeatures [l]. These features are most clearlyreflected in the photon statistics of fluorescent light [2,3].The problem of photon statistics in resonance fluorescencehas been treated by a number of workers usingseveral different techniques [4]. Most treatments assumethe incident light beam to be in a coherent state. A completesolution to photon statistics in resonance fluorescenceunder coherent illumination has been given recently[5]. Other models of the incident light beam that takeinto account the fluctuations of incident light have alsobeen considered. These include the phase-diffusion model[6], the chaotic-field model [7], and the jump models [8]of phase and amplitude fluctuations. More recently, nonclassicalstates of the driving field have also been considered.Gardiner considered the ~roblem of radiativedecay in the presence of broadband (white-noise)squeezed light [9]. The possibility of a subnaturallinewidth in resonance fluorescence under similar conditionshas been investigated by Carmichael, Lane, andWalls [lo]. More realistic models of squeezed light,where only a few modes are saueezed (colored saueezedlight), have been considered b; Ritsch and ~olle; in thediscussion of the atomic absorption spectrum 1111. Inthreshold [12,13]. This field can be modeled by two realGaussian processes with different variances and correlationtimes [14]. Because of the finite correlation time ofthe incident field, atomic states and field states developcorrelations during their dynamical evolution. Thus, unlikethe case of coherent excitation, where the factorizationof correlation functions [2,3,5,15] leads to asimplified description, the problem of photon statisticsbecomes complex in the present case. Nevertheless, forweak driving fields appropriate to subthreshold degenerateparametric oscillators, it is still possible to gainsome insight into the behavior of a two-level atom whenthe exciting field is squeezed.In Sec. I1 we derive the equations of motion governingthe time evolution of atomic and field operators. Solutionsto these equations are used to discuss the time evolutionof fluorescent-light intensity in Sec. 111. The spectrumof scattered light is calculated in Sec. IV. Photonstatistics and the two-time intensity correlation functionof scattered light are discussed in Sec. V, and Sec. VIpresents results for thermal beam excitation of the atom.Finally, the principal results of this paper are summarizedin Sec. VII.11. EQUATIONS OF MOTIONConsider a two-level atom having energy states 11) and12) separated by an energy gap fh, and interacting withan electromagnetic field via a dipole interaction. TheHamiltonian for this system, in the rotating-wave approximation,isB=tio$,+io,+-[A'-'(~,t)a-ct)-A(+)co,t)a+ct)]this paper we discuss the interaition ofa single ~wo~level +& . (1)atom with a finite-bandwidth squeezed light. The modelof squeezed light that we adopt corresponds to light froma degenerate parametric oscillator operating belowHere f3F represents the energy of the electromagneticfield. R,(t), b,(t), and f7,(t) are three dynamical spin45 8095 @ 1992 The American Physical Society

volume 79, number 1,2 OPTICS COMMUNICATIONSI October 1990Two-mode ring laser with a saturable absorberDepartment of Physics. University ofArkansas, Faye1teville. AR 72701. USAReceived 9 November 1989Deterministic as well as fluctuation properties of the mode intensities in a two-mode bidirectional ring laser with a saturableabsorber (TRSA) are studied. Both homogeneously and inhomogeneously broadened media are considered. Equations of motionfor the field amplitudes are derived by using Lamb's semiclassical approach to laser theory. Stability of various steady states inthe multidimensional parameter space is discussed. It is found that for a certain range of operating parameters a homogeneouslybroadened laser may exhibit tetrastability and an inhomogeneously broadened laser may exhibit tristability. In the former casethe intensity probability density for mode shows a three peak structure.a saturable absorber. We will see that the addition ofa saturable absorber can lead to many new phenom-We consider both the deterministic and statisticalith a absorber can exhibit many oms. In case of similar atoms the level of excitation,distinguish the two cells. The amplifying and absorbingmedia are physically separated from one anotherso that they interact only through the cavityfield. As a result the total polarization driving thelaser field can be written as the sum of the polarivityabsorbers have also been used inzations of the amplifying and absorbing media. Forilization and ~ - ~ ~ i ~ ~ h an i absorbing[31. ~ ~ medium that saturates faster than thesers with saturable absorbers have been gain medium polarization terms Up to fifth order inentally [7]. Much less attention hasultimode lasers with a saturable abafirst step in this direction we con-bility of steady states. We have followed Lamb'ssemiclassical approach [ 10,11] to the laser theory toobtain polarization terms up to fifth order. We con-

PHYSICAL REVIEW A VOLUME 42, NUMBER 1 1 JULY 1990Photon-counting statistics of the subthreshold transient degenerate parametric oscillatorReeta Vyas and A. L. DeBrito*Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701(Received 13 February 1990)Transient photoelectron-counting statistics of a degenerate parametric oscillator (DPO) operatingbelow threshold are studied. A generating-function technique has been used and analytical expressionsfor photon-counting distribution, factorial moments, and waiting-time distribution have beenderived. These results are compared with the steady-state results for the DPO and thermal light.Recently there has been considerable interest in thesqueezed light generated by a degenerate parametric oscillator(DPO) below threshold.' This system is known toproduce the largest amount ofThe fundamentalprocess in the DPO consists of the downconversionof a pump photon of frequency 20 into twophotons at the subharmonic frequency o. Quantum statisticalproperties for an idealized squeezed state havebeen discussed by many author^.^,^ For the DPO meanzero and variances of photons escaping the cavity havebeen calculated by Collett and ~oudon~ and photoncountingstatistics for small counting times have been calculatedby Agarwal and dam.' Analytical results forphoton statistics including dissipation for the DPO belowthreshold have been calculated by Vyas and Singh.' Wolinskyand Carmichae19 have followed a numerical approachto obtain photon-counting statistics of the DPO.Photon-counting distribution for the subthreshold DPOis found to exhibit even-odd oscillations for countingtimes greater than a few cavity life times. However, it isinteresting to note that even though photons are createdin pairs inside the cavity, intracavity photon number distributionin the steady state does not show even-odd oscillation~.~~'~On the other hand, the transient intracavityphoton number distribution does exhibit even-odd oscillations,"suggesting that transient DPO can exhibitdifferent features than those exhibited by the DPO in thesteady state.In this paper we discuss quantum-statistical propertiesof the DPO in the transient regime, that is, during itsevolution from the vacuum state towards the steady state.We concentrate on the statistical properties of photonsemitted by the cavity. These properties can be studied bya detector placed outside the cavity.In Sec. I1 we obtain the generating function for theDPO below threshold in the transient regime. From thisgenerating function we obtain exact analytical expressionsfor the photon-counting distribution, factorial moments,and waiting-time distributions in Sec. 111. Theseresults for the DPO in the transient regime are then comparedwith those for thermal light and the DPO in thesteady state. A summary of the results of this paper ispresented in Sec. IV.11. EQUATION OF MOTIONAND GENERATING FUNCTIONIn the interaction picture the Hamiltonian for theDPO operating below threshold can be written asI2Here pump depletion is assumed to be negligible. K is themode coupling constant and E is the dimensionless amplitudeof the pump beam incident on the cavity. a and a tare the annihilation and creation operators for thesubharmonic field and file,, describes losses suffered bythe subharmonic mode due to linear absorption, scattering,and transmission. It has been shown that in the positiveP representationI3 subharmonic field produced by theDPO can be described in terms of two independent, realGaussian random variables u , and u These randomvariables satisfy the following equations:ui=-hiui+~~2qi(t) , (2)where i can take values 1 or 2 and q (t) and q2( t ) are 6-correlated, real Gaussian white-noise processes withThe decay constants h, and A2 are given byHere (2y )- ' is the cavity lifetime at the subharmonic frequency.Below threshold (A > I K E ~ ) both the decay constantsare positive and the solutions of Eqs. (2) can bewritten asIn deriving this solution we have assumed that at timet =O the oscillator starts in the vacuum state with u ,(O).=O=u,(O). From Eqs. (3) and (5) we find592 01990 The American Physical Society

PHYSICAL REVIEW A VOLUME 40, NUMBER 9 NOVEMBER 1, 1989Photon-counting statistics of the degenerate optical parametric oscillatorReeta Vyas and Surendra SinghDepartment of Physics, University ofdrkansas, Fayetteville, Arkansas 72701(Received 22 May 1989)Nonclassical light beams generated by the degenerate optical parametric oscillator operatingbelow threshold are analyzed in terms of photoelectron-counting sequences. The positive-P representationis used to calculate the generating function for photoelectron statistics in a closed form.This generating function is used to derive expressions for the photoelectron-counting and waitingtimedistributions. The dependence of these distributions on mean photon number inside the cavityand efficiency of detection is studied. The relationship between photoelectron-counting sequenceand the photon emission sequence is used to present a simple physical picture of light beams producedby the degenerate parametric oscillator.I. INTRODUCTIONSqueezed states of light have been observed in a varietyof physical systems.'-) These states do not admit a positivenonsingular diagonal representation in terms ofcoherent states and are, therefore, an example of nonclassicalstates of the electromagnetic field. Since squeezingonly refers to the variance of the two quadrature componentsof the electric field, it does not filly characterizethese states. With experimental realization of thesestates, increasing attention is being paid to theirquantum-statistical properties.435 These properties foridealized squeezed states are well kn~wn.~-~The systemsin which squeezed states have been observed experimentallyare dissipative nonlinear systems, and photon statisticalproperties of squeezed states produced by thesesystems have received much less attention.The largest amount of squeezing has been observed inan optical parametric oscillator (OPO) operating belowthre~hold.~.' This simple dissipative quantum system hasplayed an important role in recent studies of squeezing.In an OPO (Ref. 10) a strong pump beam interacts with anonlinear crystal and is frequency down-converted intotwo beams of smaller frequencies inside an optical cavity.If the two beams produced in down conversion have thesame frequency, then the oscillator is termed a degenerateparametric oscillator (DPO); otherwise it is termeda nondegenerate parametric oscillator (NDPO). Aquantum-mechanical treatment of the OPO is of courseessential since it generates light with nonclassical properties.For an oscillator a distinction must be made betweenintracavity photon statistics and the statistics of photonsemitted by the cavity. Intracavity statistics are notdirectly observable. The statistics of photons emitted bythe cavity can be measured in photon-counting experiments.The statistics of the field inside and outside thecavity are, of course, related. Many recent studies of thequantum-statistical properties of the DPO have centeredaround the calculation of the spectrum of squeezing"*'2inside and outside the cavity because of the subtleties involvedin the detection of squeezed light. Intracavityfield statistics were discussed by Drumrnond, McNeil,and ~alls" by using the complex-P representation andby Graham by using the Wigner function'). More recently,Wolinsky and Carrni~hael~*'~ have provided a completedescription of the quantum-statistical properties ofthe intracavity field by using the positive-P representation.For the photons escaping the cavity, the mean andvariance of photon counts have also been calculated byCollett and ~oudon.'~In this paper we discuss the quantum-statistical propertiesof photon beams generated by an OPO as measuredby a detector placed outside the cavity. These propertiescan be studied in photoelectric-counting and correlationexperiments with low-intensity light beams appropriatefor an OPO below threshold. From the measured photoelectronstatistics, photon statistics of the incident lightbeam can be derived. For a detector of unit efficiencyeach photodetection corresponds to an emission of a photonby the cavity. In this case, the photoelectric-countingsequence and the photon emission sequence areequivalent. We begin by expressing the photoelectroncountingstatistics in terms of a generating function inSec. 11. The statistics of the waiting time between successivephotoelectric counts can also be derived from thesame generating function. In Sec. I11 the c number equationsof motion for the DPO operating below thresholdare presented. This is done by using the positive-P representation.The solutions to these c-number equations areused to obtain a closed form expression for the generatingfunction. From this generating function exact expressionsfor the photoelectron-counting distribution and thewaiting-time distribution are derived in Sec. IV. Intracavityphotonstatistics are discussed in Sec. V. We concludeby summarizing the principal results of the paper inSec. VI.11. THE GENERATING FUNCTIONConsider a photoelectric detector illuminated by a stationaryweak beam of light. The probability p(m, T) ofdetecting m photoelectric counts at the output of thedetector in a time interval T is given by1'5147 @ 1989 The American Physical Society

1110 OPTICS LETTERS / Vol. 14, No. 20 / October 15,1989Quantum statistics of broadband squeezed lightReeta Vyas and Surendra SinghDepartment of Physics. University of Arkansas. Fayetteville, Arkansas 72701Received May 8,1989; accepted August 2.1989Photon-emission sequences in nonclassical light beams generated by an optical parametric oscillator are studied.Exact photon-counting and waiting time distributions are derived by using a generating function technique basedon the positive-P representation. This approach is used to present a simple physical picture of photon emissionsfrom this nonclassical light source.Squeezed states of light are nonclassical states of lightthat have been experimentally realized recently in avariety of physical systems.ll2 Since squeezing refersonly to the variance of the quadrature components ofthe electric field, the need to characterize these statesin terms of their photon-counting distributions hasbecome urgent. These distributions for idealizedsqueezed states are well known.3 As a practical devicefor producing squeezed light, the performance of thedegenerate optical parametric oscillator (DPO) hasbeen un~urpassed.~In this Letter we present an analytic treatment forthe counting statistics of photon beams generated bythe DPO. Our approach leads to exact expressions forboth the counting and waiting time distributions.When these distributions are analyzed they lead to aclear picture of photon beams generated by the DPO.The DPO is modeled by two resonant modes of anoptical cavity, with frequencies 20 and 0, interactingwith each other through an intracavity nonlinear crystal.The pump mode 20 is excited by an injectedclassical signal of amplitude C. In the interaction picturethe Hamiltonian with perfect phase matchingreadswhere b and at and 6 and 6t are the annihilation andcreation operators for the subharmonic mode w andthe pump mode 20, respectively, K is the mode-cou-pling constant, r is the cavity linewidth at the pumpfrequency, and BIo8, describes linear absorption andscattering losses suffered by the modes inside the crystaland the cavity mirrors. By a suitable combinationof phases, K and E have been chosen to be real.The Hamiltonian in Eq. (I) is used to derive anoperator master equation for the density matrix of thefield. This operator equation is converted into anequivalent set of classical Langevin equations for thecomplex field amplitudes by using the positive9repre~entation.~~~ The resulting equations for thesubharmonic mode below threshold, where pump depletionis negligible, arewhere 7 is the cavity linewidth at the eubharmonicfrequency, [l(t) and 52(t) are two real statistically independentGaussian white-noise processes with zeromean and unit intensity, and a! and a!* are two complexvariables associated with operators b and bt in thepositive-P representation. Unlike in the diagonal-Prepresentation, a! and a!* are not complex conjugates.The oscillator threshold is defined by KE = y. In thisLetter we restrict our analysis to the case KE < y correspondingto the experiments of Ref. 2. Equations (2)ensure that the variabIes a and a* are real in the steadystate because any imaginary parts to a! and a!* decayaway. It follows that the realvariables ul = (a! + a!*)/2and u2 = (a - a!*)/2 are statistically independentGaussian stochastic processes with zero mean and correlationfunctionswhere Xlnz = y r KC are positive decay constants. Thisequation completely determines the statistical propertiesof a! and a* and ul and uz.The generating function with parameter s for photon-countingstatistics as measured by a photodetectorplaced outside the cavity is given bywhere T is the counting interval, 0 I q I I is theefficiency of detection, f(t) is the photon number fluxoperator (number per second) for the Iight beam generatedby the oscillator, and T-: denote the time andnormal ordering of operators inside the colons. Onusing the positive-P representation the normally orderedoperator average in Eq. (4) becomes1 10-X exp 2sqy u?(t)dt 9 (5)where we have used the correspondence f(t) -. 2ya!a!*.The factorization in Eq. (5) is a consequence of thestatistical independence of the variables ul and uzexpressed by Eq. (3). Since ul and ui are real Gaussianvariables with an exponential correlation function,the averages in Eq. (5) may be evaluated by making a1)0146-9592/89/201110-03$2.00/0 O I989 Optical Society of America

Pulsed and cw photothermal phase shift spectroscopy ina fluid medium: theoryB. Monson, Reeta Vyas, and R. GuptaA theoretical treatment of photothermal phase shift spectroscopy in a fluid medium for the most generalconditions is given. The medium is assumed to be flowing. Results for a stationary medium appear as aspecial case. Both pulsed and cw excitation are considered. For pulsed excitation, the results are valid forexcitation pulses of arbitrary length. For the cw case, modulated excitation is explicitly considered, and theresults for unmodulated excitation appear as a special case.I. IntroductionPhotothermal spectroscopy is currently an activearea of research, and a wide range of applications ofthis technique has been developed.' In this technique,a laser beam (pump beam) is partially or fullyabsorbed by a medium of interest. In the presence offast quenching collisions, most of this energy appearsas heat, and the refractive index of the laser-irradiatedregion is modified. This change in the refractive indexcan be monitored by a second and weaker laser beam(probe beam) in several different ways. For example,the nonuniform refractive index produced by the absorptionof the pump beam can be detected by thedeflection of the probe laser beam passing through themedium. This technique is called photothermal deflectionspectrscopy (PTDS).%3 If the refractive indexof the medium has a nonzero curvature, the mediumalso acts like a lens, and a probe beam passing throughthe medium changes shape. This can be detected as achange in the intensity of the probe beam passingthrough a pinhole. This technique of monitoring thephotothermal effect is called photothermal lensingspectroscopy (PTLS).4-6 One may also detect thephotothermal effect by placing the medium in onebeam of an interferometer (or inside a Fabry-Perotcavity). The change in refractive index causes a fringeshift which can conveniently be detected as an intensitychange of the central fringe. We shall refer to thisThe authors are with University of Arkansas, Physics Department,Fayetteville, Arkansas 72701.Received 28 September 1988.0003-6935/89/132554-08$02.00/0.O 1989 Optical Society of America.technique as photothermal phase shift spectroscopy(PTPS), and it is the subject of this paper. Thistechnique was discovered by Stone7 and by Davis andPetuchow~ki,~.~ who refer to this technique as phasefluctuation optical heterodyne (PFLOH) spectroscopy.Many applications of this technique have beendiscussed in theIn this paper we give a theoretical treatment ofPTPS in fluids for the most general conditions, that is,for a flowing medium. Results for a stationary mediumappear as a special case. Moreover, both pulsedand cw excitation are considered. For the pulsed case,the results are valid for excitation pulses of arbitrarylength. For the cw case, modulated excitation is consideredexplicitly, and the results for unmodulatedexcitation appear as a special case. Both collinear andtransverse geometries are considered. Davis and Petuchowskighave given a theoretical treatment of thissubject previously. However, their results are validonly for a stationary medium, collinear geometry, andin the pulsed case, for delta function excitation only.Therefore, our results have a much more general applicability.Davis and Petuchowski have considered,however, the effect of nonzero relaxation times, whereaswe have assumed that all relaxation times are negligiblecompared to the thermal diffusion and convectiontimes. In this aspect, Davis and Petuchowski'sresults are more general than ours.Expressions for the temperature distribution due tocw and pulsed excitation of a flowing medium arederived in Sec. 11. The explicit case of heating by apump laser beam is considered, which is generally theexperimental situation. However, the heating of themedium may be produced by any suitable radiation,e.g., microwaves.14 Expressions for the PTPS signalsare derived in Sec. 111. The explicit case of a MichelsonInterferometer is considered. Expressions forother types of interferometer either are identical (e.g.,2554 APPLIED OPTICS I Vol. 28, No. 13 1 1 July 1989

PHYSICAL REVIEW A VOLUME 39, NUMBER 3 FEBRUARY 1, 1989Photoelectron waiting times and atomic state reduction in resonance fluorescenceH. J. Carmichael, Surendra Singh, Reeta Vyas, and P. R. Rice*Department of Physics, Uniuersity of Arkansas, Fayetteuille, Arkansas 72701(Received 12 September 1988)Photoelectron counting sequences for single-atom resonance fluorescence are studied. The distributionof waiting times between photoelectric counts is calculated, and its dependence on drivingfieldintensity and detection efficiency is discussed. The photoelectron-counting distribution is derivedfrom the waiting-time distribution. The relationship between photoelectron counting sequencesand photon emission sequences is discussed and used to obtain an expression for the reducedstate of the atom during the waiting times between photoelectric counts. The roles of irreversibilityand the observer in atomic state reduction are delineated.I. INTRODUCTIONThe fluorescent photons emitted by a single coherentlydriven two-level atom exhibit the nonclassical property ofphoton antib~nchin~.'-~ The antibunching of fluorescentphotons is seen in temporal correlations betweenphotoelectric counts; the detection of one photon makesthe detection of a second, after just a short delay, improbable.Photon antibunching is traditionally defined interms of the degree of second-order temporal coherenceg'2'(r,r +T). This is the joint probability for recordingphotoelectric counts in the intervals [r,t +At) and[r +~,t +r+At), normalized by the probability for twoindependent photoelectric counts. For antibunched lightthe joint probability for recording photoelectric countsclosely spaced in time falls below the probability for statisticallyindependent counts (separated by a time longerthan the coherence time); thus, gt2'(t,t) < 1.The antibunching of fluorescent photons is alsoreflected in the sub-Poissonian character of the probabilitydensity p(n,r,t +T) for recording n photoelectriccounts in the interval [t, t + T).~ p (n,t,t + T) can be derivedfrom g'2'(t,t +T), although the detailed algebraicrelationship is quite complicated. Both g(')(t,t +T) andp (n, t, t + T) have been calculated for single-atom resonancefluorescence by a number of workers.'-lo Becauseof the complexity of general expressions in the timedomain, some workers only give the Laplace transformfor the photoelectron counting distribution, or give explicittime-dependent expressions only for limiting cases,such as short and long counting times.Recent theoretical work on "quantum jumps""-'2 hasdrawn attention to the distribution of waiting times betweenphoton emissions as another useful quantity forcharacterizing photon statistics-in terms of measuredquantities, the distribution of waiting times between photoelectrons.By "waiting time" we mean the time T betweena photoelectric count recorded at time t, and thenext, recorded at time t +T. If photoelectron sequencescan be described by a Markov birth process, a single conditionalprobability density w ( T I < ) specifies the distributionof waiting times between every pair of photoelectrons.We call this the photoelectron waiting-time distri-bution. Photoelectron waiting times for coherent lightare exponentially distributed.13 Antibunching impliesthat photons tend to be separated in time. The distributionof waiting times should then tend to peak around theaverage time between photoelectric counts.Photoelectron waiting times are certainly not new tothe field of photon statistics. Indeed, when a time-toamplitudeconverter is used for a delayed coincidencemeasurement, the raw data provide the distribution ofwaiting times between photoelectric counts. However,when the count rate is sufficiently low, this distribution isproportional (aside from dead-time corrections) tog'2'(t,t +T). l4 This relationship provides the techniqueused to measure g'2)(t,t +T) in the experiments of Kimbleef & on photon antibunching in resonance fluorescence.Thus, the waiting-time distribution and its relationshipto g'2)(t,t +T) are known. But the waiting-timedistribution has not been mentioned until re~entl~'~''"'~in the large theoretical literature on resonance fluorescence.This is a deficiency, since it provides a clearerphysical picture of photon emission sequences, correspondingphotoelectron counting sequences, and theirnonclassical properties, than g'"(t,t +TI. In this paperwe revisit the problem of single-atom resonance fluorescenceand focus attention on the waiting-time distribution.(We will discuss waiting times between photonemissions as well as between photoelectrons. When thedistinction is not important we simply refer to "the waitingtimes" or "the waiting-time distribution.")There are probably two main reasons for the lack of at-tention paid to w (TI t) in early work on resonance fluorescence.The first is that g(2)(r,r +T), not w(T(~), is thequantity accessible to measurement. It might be asked,why not use a time-to-amplitude converter to measurethe quantity it gives directly, the photoelectron waitingtimedistribution w (?It)? The problem is that photoelectricdetection is very inefficient. The average time betweenphotoelectric counts is unavoidably much longerthan the correlation time of the fluorescent light. Thenw ( ~(t) is proportional to g'2'(t, t +T); w (rlt) can be measured,but only when it effectively reduces to g'2'(t,t +TI.Actually, the proportionality between these quantitiesdoes not hold for all T, but it holds over many correlation1200 @ 1989 The American Physical Society

228 J. Opt. Soc. Am. BNol. 6, No. 21February 1989 Satyanarayana et al.Ringing revivals in the interaction of a two-level atom withsqueezed lightM. Venkata Satyanarayana, P. Rice, Reeta Vyas, and H. J. CarmichaelDepartment of Physics, University of Arkansas, Fayetteville, Arkansas 72701Received May 12,1988: accepted October 4.1988The Jaynes-Cummings interaction of a two-level atom with the radiation field is studied when the radiation isinitially in a strongly squeezed coherent state. The dynamic response of the atomic inversion shows echoes aftereach revival when the squeezed coherent state exhibits an oscillatory photon-counting distribution due to the phasespaceinterference effect. The sensitivity of the dynamic behavior to approximations used in computing the atomicinversion is discussed. Comparison is made with the intensity-dependent interaction model of Buck and Sukumar[Phys. Lett. BlA, 132 (1981)lf this model does not exhibit echoes. he mean, variance, and entropy for the photonnumberdistribution are calculated and found to show behavior similar to that of the atomic inversion.1. INTRODUCTION sum is not possible even for. an initial coherent state, althoughapproximate expressions that reproduce the generalcharacter of the revivals have been obtained.3s4 We comparethe exact numerical evaluation of Eq. (1) with variousapproximations when P(n) corresponds to a squeezed coherentstate.The plan of the paper is as follows: In Section 2 we brieflyreview the oscillatory nature of P(n) for squeezed coherentstates. We study the corresponding dynamical response ofthe atomic inversion in Section 3. The exact numericalevaluation of w(t) is presented, demonstrating the occurenceof ringing revivals, and the origin of the ringing behavior isdiscussed. In Section 4 we obtain a closed analytical expressionfor w(t) in the harmonic approximation, in which thesquare root in the argument of the cosine in Eq. (1) is expandedto first order. The ringing of the revivals is lost inthis approximation. Expansion of the square root to secondorder recovers the ringing behavior. The summation formulathat yields an analytical result in the harmonic approximationmay also be used to derive an exact integralrepresentation for w(t) when the radiation field is initially ina squeezed coherent state. This integral representation isgiven in Appendix A. We study the photon statistics andentropy for the field in Section 5. Our results are summarizedin Section 6.The Jaynes-Cummings model of optical resonance1 describingthe interaction of a single two-level atom with a singlemode of the radiation field has predicted a number of interestingfeatures in the dynamical behavior of the atomic inversion.24Much attention has focused on the collapse andrevival of Rabi oscillations because this effect provides evidencefor the granularity of the radiation field.3.4 Experimentalrealizations of the Jaynes-Cummings model havebeen obtained by using Rydberg atoms interacting with theradiation field in a high-Q microwave cavityS6 Recentlyobservations on the collapse and revival of Rabi oscillationswere reported.6 In this paper a new feature in the dynamicalbehavior of the atomic inversion is studied, with theradiation field prepared in a strongly squeezed coherentstate whose photon-counting distribution is oscillatory.7Under these conditions, the collapse following each revivalhas an oscillatory envelope (echoes), a phenomenon that wecall ringing revivals.Milburn has studied the interaction of a two-level atomand a single mode of the radiation field with the field preparedin a squeezed coherent state.8 He showed that thecollapse time depends on the direction of the squeezing andfound that for certain squeezed states the response of theatom is similar to that for chaotic radiation. However, Milburnrestricted his study to states for which the coherentcontribution to the photon-number variance is dominant.The new behavior described in this paper is obtained withsqueezed coherent states for which the squeezed contributionto the photon-number variance is dominant.Our results for the atomic inversion are based on thenumerical evaluation of the series1Squeezed coherent states are now quite familiar in quantumoptics; we simply state their definitiong and refer the readerto two recent collections of papers1° for further details and areview of current activity regarding these states.The squeezed coherent states for a single-mode radiationfield can be obtained from the vacuum (0) aswhere P(n) is the photon-number distribution for the initialstate of the radiation field and X is the coupling constant forthe atom-field interaction. The atom is assumed to be in itsground state initially. An exact analytical evaluation of thiswhere S(t) is the squeezing operatorS(S) = exp[%(t*a2 - Sat2)1 (3)and D(a) is the displacement operatorQ 1989 Optical Society of America

Photothermal lensing spectroscopy in aflowing medium: theoryReeta Vyas and R. GuptaA complete and general theoretical deecription of dual-beam photothermal lensing spectroscopy is given.The results are valid for the moat general conditions, that is, for flowing as well as stationary media, and for cwas well as pulsed excitation. For pulsed excitation, the results are valid for arbitrary pulse length. The cwresults apply to both modulated as wellas unmodulated excitation. Both transverse and collinear geometriesare considered.I. IntroductionIn this paper we present the theory of dual-beamphotothermal lensing spectroscopy (PTLS) in a fluidmedium valid for the most general conditions, that is,for flowing as well as stationary media and for cw aswell as pulsed excitation. For pulsed excitation, thepulse length is arbitrary, and for cw excitation both themodulated and the unmodulated sources are considered.Both the transverse and the collinear geometriesare considered. A unified treatment of all cases ispresented. This is the first comprehensive treatmentof this important subject.The basic idea underlying dual-beam PTLS isshown in Fig. 1. A laser beam (pump beam) propagatesthrough a medium, and it is tuned to one of theabsorption frequencies of the medium. The mediumabsorbs some of the optical energy from the laserbeam. If the collision rate in the medium is sufficientlyhigh compared to the radiative rates, most of theenergy appears in the translational-rotational modesof the medium within a short period of time. In otherwords, the laser-irradiated region gets slightly heated.The refractive index of the medium is thus modified.The refractive-index change can be monitored in severaldifferent ways.' In this paper, we are concernedwith a technique that relies on the lensing effect of themedium to monitor the .refractive-index change. Aweak probe beam passes through the pump-irradiatedThe authors are with University of Arkansas, Physics Department,Fayetteville, Arkansas 72701.Received 6 June 1988.0003-6935/88/224701-11$02.00/0.O 1988 Optical Society of America.region, as shown in Fig. 1. Due to the curvature of therefractive index, the probe beam diverges, which canbe detected as a change in the intensity of the probebeam passing through a pinhole. In other words, underthe influence of the pump beam, the medium adslike a diverging lens. In certain circumstances, themedium acts like a converging lens also. If a pulsedpump laser is used, a transient lens is formed; theprobe beam changes shape shortly after the pumpbeam is fired and returns to its original shape on thetime scale of the diffusion of heat from a probe region.If a cw laser is used, it is generally convenient to amplitudemodulate its intensity, and the PTLS signal consistsof oscillations in the intensity of the probe beampassing through the pinhole. The PTLS takes onmore interesting dimensions if a flowing medium isused. The PTLS technique has been discussed extensivelyin the literature for trace detection of chemicals?and the use of PTLS in a flowing medium hasrecently been demonstrated for flow velocity measurement~.~Although the thermal lensing effect may alsobe observed by monitoring the pump beam itself, inthis paper we only consider the dual-beam techniquein which the thermal lens is created by the pump beamand monitored by the probe beam.The photothermal lensing effect was observed accidentallyby Gordon et a1.4 in 1964 when they placed acell filled with a liquid sample inside a He-Ne lasercavity. These authors correctly identified the effectand gave a theoretical description of it. In 1973, Huand Whinnery5 gave a detailed theoretical descriptionof the effect for an extracavity sample and determinedthat the maximum signal occurred when the samplecell was placed one confocal distance away from thewaist of the beam. They also demonstrated the usefulnessof this technique for measurements of smallabsorptivities. A cw laser was used, and a shutter wasemployed to effect the change necessary to observe the15 November 1988 / Vol. 27. No. 22 / APPLIED OPTICS 4701

Continuous wave photothermal deflection spectroscopy ina flowing mediumReeta Vyas, 6. Monson, Y-X. Nie, and R. GuptaA complete and rigorous theoretical treatment of the continuous wave photothermal deflection spectroscopyin a flowing medium is given. The theoretical resulta have been verified experimentally.I. IntroductionThere is presently extensive interest in the techniqueof photothermal deflection spectroscopy(PTDS),1-3 and this technique has found a wide rangeof applications. In this paper we present a comprehensiveand rigorous theoretical treatment of the cwPTDS technique for the most general conditions, i.e.,for a flowing medium. The results for a stationarymedium appear as a special case. Very interestingresults have been found, and these results will extendthe range of applications of cw PTDS in fluids. Resultsof an experiment used toverify the theory are alsopresented.The principle of the PTDS technique is as follows:A dye laser beam (pump beam) passes through themedium, and it is tuned to an absorption line of themolecules or atoms of interest. The molecules absorbthe laser energy, and in the presence of quenchingcollisions some of this energy appears in the rotational-translationalmodes (heating) of the molecules ofthe medium. If the quenching rates are high comparedwith the radiative rates, almost all the absorbedenergy appears in the heating of the laser irradiatedregion. The temperature change of the laser irradiatedregion results in a change of the refractive index ofthat region. This change in the refractive index can beprobed by a second, and weaker, laser (probe beam).In general, the refractive index is nonuniform, and therefractive-index gradient deflects the probe beam.The deflection of the probe beam may be detectedThe authors are with University of Arkansas, Physics Department,Fayetteville, Arkansas 72701.Received 22 December 1987.0003-6935/88/183914-07$02.00/0.63 1988 Optical Society of America.conveniently using a position sensitive optical detector.In the case of a cw pump beam, it is generallyconvenient to intensity modulate the pump beam atsome frequency f. In this way, the deflection is alsomodulated at frequency f, which can then be detectedconveniently using a phase-sensitive detector.The only previous theoretical study of the cw PTDS,to our knowledge, is that of Jackson et aL4 Theseauthors, however, consider only a stationary medium.Our study is more general (i.e., a flowing medium), andJackson et ale's results appear as a special case (flowvelocity = 0) in our solutions. Moreover, this treatmentof the cw PTDS unifies the theory of cw PTDSwith that of pulsed PTDS in a flowing medium, publishedearlier by Rose et aL5The theory of cw PTDS is given in Sec. 11. Thetheoretical results are discussed in Sec. 111. The apparatusis described in Sec. IV, and the experimentalresults are presented in Sec. V.II. TheoryFigure 1 shows the basic geometry of the pump andprobe beams. Two cases are considered: TransversePTDS, in which the probe beam is perpendicular to thepump beam, and the collinear PTDS, in which thepump and probe beams are parallel. We assume thepump beam to be propagating along the z axis. Theorigin of the coordinate system lies on the axis of thepump beam. For the transverse PTDS, the probebeam propagates along they axis, and for the collinearPTDS, the probe beam propagates in the z direction.In both cases, the medium flows with velocity v, in thex direction. The pump and probe beams do not necessarilyintersect, and they may be separated by a variabledistance x in the x direction. For the collinearcase, they may also be separated by a distance y in theydirection (not shown in Fig. 1). The general case,where the pump and probe beams make an arbitraryangle 8 will not be considered here. This case has beentreated by Rose et al.5 previously in connection with3914 APPLIED OPTICS I Vol. 27, No. 18 I 15 September 1988

PHYSICAL REVIEW A VOLUME 38, NUMBER 5 SEPTEMBER 1, 1988Waiting-time distributions in the photodetection of squeezed lightReeta Vyas and Surendra SinghDepartment of Physics, University of Arkansas, Fayetteville, Arkansas 72701(Received 24 March 1988)Distribution of waiting-time intervals between the arrivals of successive photons on a photocathodeilluminated by a beam of light is discussed. Analytic expressions for the conditional andunconditional distributions for squeezed light are derived in the high degeneracy limit. Results forbinomial and thermocoherent states are also given. Curves are presented to illustrate the behavior.I. INTRODUCTIONSqueezed states of light have been observed in a numberof experiments.'-4 Similar to the states exhibitingphoton antib~nchin~,~ squeezed states6 are nonclassicalstates. For these states variance in one of the quadraturecomponents of the electric field is smaller than the correspondingvariance for a coherent state. The nonclassicalnature of these states is also reflected in the fact that thecorresponding phase space density in the coherent statediagonal representation7.' does not exist as an ordinaryprobability density. Photon number distributions forsqueezed states have been discussed by a number of authors9-l4and they reveal several interesting features ofthese states. Depending on the parameters characterizinga squeezed state, such a state may exhibit sub-Poissonian or super-Poissonian photon statistics.I4 Since,for short times at least, sub-~oiisonian (super-Poissonian)photon statistics reflect antibunching (bunching),I5squeezed states are capable of exhibiting antibunching orbunching under suitable circumstances. The propertiesof photon antibunching or bunching are best visualized interms of the theory of photoelectric detection.16-" Considera photoelectric detector illuminated by a beam oflight. For an ideal photodetector of unit detectionefficiency and zero dead time, the photoelectric pulses appearingat the output of the photodetector are in one-toonecorrespondence with the arrival of photons on thephotocathode. In terms of this sequence of photoelectricpulses, photon antibunching implies that the detection ofa photon at a certain time t renders the detection ofanother photon immediately following the first less probable.The opposite is implied by photon bunching. Thismeans that for an antibunched beam of light, successivephotoelectric pulses will be separated, on the average, bylarge time intervals. We may consider this to be areflection of the tendency of the photons in the lightbeam to be separated in time. This physically appealingpicture of the behavior of photons in a light beamemerges from photodetection theory. Photons themselvesdo not lend directly to such an interpretation. Thistime-dependent behavior of photons will be reflected inthe distribution of the time interval between successiveph~todetections.''~~~ It is the purpose of this investigationto derive the time interval distribution for photons ina squeezed state.In Sec. I1 we present an outline of the photodetectiontheory leading to a general expression for the time intervaldistribution. This expression is used to discuss the behaviorof photons in a squeezed beam in Sec. 111. Wealso discuss the behavior of waiting time distributions forbinomial and thermocoherent states of the field in Sec.IV.11. TIME INTERVAL DISTRIBUTION FUNCTIONIn order to discuss the time interval distribution for alight beam we need the threefold joint probability of photodetection,for the time interval T between successivephotodetections is defined in terms of three events: aphotodetection event at some time t,, no photodetectionevents in the interval [t , , t2( = t, + T )], and one photodetectionat time t2. First let us recall that the probabilityof detecting n photoelectric events in [t ,,t2] isI7whereand ?(t) is the photon flux operator (number of photonsper unit time) and : . . : denotes the time-ordered normalproduct of operators. The detection efficiency 77 dependson the characteristics of the detector. The angularbrackets denote averaging with respect to the state of thefield. From Eqs. (1) and (2) the probability of detectingone photon in the interval [t -At, t] is found to bewhere At is some small time interval. We also find fromEq. (1) that the probability that no photodetection occursin the interval [t t2] isEquations (3) and (4) will be used later in the paper. Thesingle-fold photoelectric counting formula (1) is wellknown.16-" The threefold photoelectric counting formula(to

Pulsed photothermal deflection spectroscopy in a flowingmedium: a quantitative investigationA. Rose, Reeta Vyas, and R. GuptaA comprehensive investigation of pulsed photothermal deflection spectroscopy in a flowing medium has beencarried out. A rigorous solution of the appropriate diffusion equation has been obtained, and experimentshave been conducted to verify the theoretical predictions. Absolute measurements of the photothermaldeflection were made and no adjustable parameters were used in the theory. Very good agreement betweenthe theory and the experiment was obtained.I. IntroductionRecently there has been extensive interest in thetechnique of photothermal deflection spectroscopy(PTDS). Since the initial work of Davis,' and of Boccara,Fournier, Amer and co-~orkers,2-~ extensive newapplications have been developed by Tam and coworker~~-~and by others.1° Gupta and co-workershave demonstrated the usefulness of this technique forcombustion diagnostic^.^^-^^ Motivated by this application,here we present results of a comprehensive andquantitative investigation of pulsed PTDS in a flowingmedium.The principle of the PTDS technique is quite simple:A dye laser beam (pump beam) passes throughthe medium of interest and is tuned to one of theabsorption lines of the molecules that are to be detected.The molecules absorb optical energy from thelaser beam and, if the pressure is sufficiently high (i.e.,if the quenching rates are sufficiently fast compared tothe radiative rates), most of the energy quickly appearsin the rotational-translational modes of the medium.The dye laser irradiated region thus gets slightly heated,leading to changes in the refractive index of themedium in that region. If the density of the absorbingmolecules is uniform over the width of the dye laserbeam, the refractive index acquires the same spatialprofile as the dye laser beam (e.g., a Gaussian profile ifthe dye laser is working in the TEMoo mode). Now ifanother laser beam (called the probe beam) overlapsthe pump beam, it is deflected due to the changes inThe authors are with University of Arkansas, Physics Department,Fayetteville, ~rkansas 72701.Received 25 July 1986.0003-6935/86/244626-18$02.00/0.O 1986 Optical Society of America.the refractive index created by the absorption of thepump beam. This deflection is easily measured by aposition-sensitive optical detector. If a pulsed dyelaser is used, a transient deflection of the probe beam isobtained. The deflection of the probe beam is proportionalto the concentration of the absorbing molecules.Therefore the technique can be used to measure major-ity and minority species con~entrations.~l-~~ If themedium is flowing, the heat pulse produced by theabsorption of the dye laser travels downstream withthe medium. The heat pulse is, of course, accompaniedby changes in the refractive index and can bemeasured by the deflection of a suitably placed probebeam. The flow velocity of the medium can be measuredfrom a measurement of the transit time of theheat pulse between two probe beams downstream fromthe pump beam.7J5J8J9 The heat pulse broadens dueto thermal diffusion as it travels downstream. Thethermal diffusion coefficient of the medium can bemeasured from the broadening of the signal which, inturn, yields the local temperature of the medium.l4*6Therefore photothermal deflection spectroscopy is avaluable optical diagnostic technique for varied applications(combustion diagnostics is just one of the applications).Quantitative measurements of speciesconcentrations, temperature, and velocity are moredifficult, however, than one might assume from theabove discussion. The PTDS signal in a flowing mediumdepends in a complicated manner on all threeparameters (concentration, velocity, and temperature).For example, the amplitude of the signal dependsnot only on the concentration of the absorbingspecies but also on the temperature and the flow velocity.The width of the signal depends on both thetemperature and the flow velocity. The measurementof the transit time is affected by the broadening due tothermal diffusion. Therefore, a good theoretical andexperimental understanding of the size and shape of4626 APPLIED OPTICS 1 Vol. 25, No. 24 1 15 December 1986

PHYSICAL REVIEW A VOLUME 33, NUMBER 1 JANUARY 1986Laser theory without the rotating-wave approximationReeta Vyas and Surendra SinghDepartment of Physics, University of Arkansas, Fayetteville, Arkansas 72701(Received 17 June 1985)The effect of counter-rotating terms in the interaction Hamiltonian on the photon statistics is investigatedfor a single-mode laser. The treatment is based on the Scully-Lamb model. A perturbationexpansion is used to derive an equation of motion for the field-density matrix. The counterrotatingterms are found to lead to both transient and secular effects. With some reasonable approximationsthe steady-state photon-number distribution is derived and it is shown that the counterrotatingterms tend to raise the threshold of laser action and broaden the photon-number distribution.The relative size of these effects is found to be very small and depends on the square of the ratioof the atomic linewidth and the transition frequency.I. INTRODUCTIONIn quantum optics, NMR, quantum electronics, andother branches of resonance phenomena it is customary tomake the rotating-wave approximation (RWA).' Underthis approximation certain terms which oscillate at twicethe resonance frequency are dropped from the interactionHamiltonian. The resulting Hamiltonian describes thedynamics of the system quite adequately. However, therapidly oscillating terms (also called the counter-rotatingor energy-nonconserving terms) can give rise to physicaleffects even if small. For example, in the Rabi problemthey give rise to the Bloch-Siegert shift of the resonancefrequency and small high-frequency amplitude modulationof the Rabi oscillation^.^ In laser theory the role ofthese counter-rotating terms has not been investigated. Inthis paper we wish to consider the effect of such terms inlaser theory. We find that the counter-rotating terms giverise to both secular and time-dependent effects. In particular,their effect is to raise the threshold of laser oscillationand broaden the photon-number distribution. However,the magnitude of these corrections is very small asexpected. In our investigations we adopt the model ofScully and Lamb3 for a single-mode laser. Unlike them,however, we use a perturbative approach.3*4 In Sec. I1 wedescribe the details of the model and derive a masterequation for the density matrix of the laser field for atraveling-wave mode. In Sec. 111, this master equation issolved in the steady state and changes produced by thepresence of counter-rotating terms are discussed. We alsoconsider the standing-wave case and discuss the differencesthat arise in the equations of motion.11. EQUATIONS OF MOTIONOF THE DENSITY MATRIXWe consider a single-mode electromagnetic field of frequencyo interacting with a group of N identical twolevelatoms. The energy separation between the upperatomic level ( a ) and the lower atomic level / b ) is fbo,where oo is equal or close to the field mode frequency.The atoms may decay nonradiatively out of the two levels/ a ) and ( b ) to various other levels at a rate y. FollowingScully and Lamb3 we suppose that the laser gain isprovided by atoms introduced uniformly at an averagerate NP throughout the cavity. These excited atoms interactwith the field inside the cavity and make their contributionsto the field one at a time. The total rate ofchange of the optical field may be calculated as a coarsegrainedderivative by multiplying the change produced byone atom by the rate NP at which atoms are introducedin the excited state. Similarly. laser loss is simulated byconsidering another set of fictitious two-level atoms withbroad levels introduced into the cavity in their lower stateat a certain uniform rate.3 These atoms absorb radiationand model the loss suffered by the laser field which is ultimatelydue to photons escaping from the cavity eitherbecause of the finite transmittivity of the mirrors or becauseof scattering from various elements inside the cavity.The Hamiltonian for an interacting atom and asingle-mode electromagnetic field is given bywhere s,,R^,,R^, are the three Pauli spin-f operators, pis the transition dipole moment between the levels / a )and / b) which we take to be real, and r is the position ofthe atom. The single-mode electric field operator @(r,t) isgiven by (in mks units)where E is a unit polarization vector which we take to bereal corresponding to linear polarization, a^ and ii ' are theannihilation and creation operators for the field, u (r) isthe cavity mode function, and V is the cavity volume(quantization volume). The third term in Eq. (1)represents the interaction Hamiltonian d. This term inthe interaction picture can be written aswhere01986 The American Physical Society

-==--d WOSRb d ~ i ,J of the Optical Society of Amer ,ica A - Optics Image Science and Vision, v 2, Issue 13, p25-p25, 1985me llw Intensities of the two modes. thadiierenca. and the autocarelatlon and crossarrelationhulctlons were calculated. The kind ofbehavla exhibited by the laser Is found to dependoitlcally on the phases Of the backscatlcmd fiilds.Sams values lead to posltive crou conelatlms InUm rqim where the laser ordinarily exhlblts zema negatlve correlations, and othen to nsgathremRNOOn 1-modes are not locked togalhr they undergo randcmjumps In Integer multiples of 2r. the resubof the urlcuktions are mmpared with expalmen-I results from whlm some conclusions about themopmaa, / backscattering can be &awn. (12 mln)IlUW4 Paper rllhbraunNWS likasuremenl of flnt-pasasgttlm dleblkrllarrfor a Os~nchsd lawMARVIN R. YOUNG and SURENORA SINW. U.Arkansas. Physics Departmnt. Fayetbvllls, AR72701.I The dlstributlons of first-passage tlm for Ihoi bulld up 01 hser radlatlon hom zero IntemHy toSome predetermined value are measured for a CTJwiidmd her. Experiments were performed on ai sing~ernade He-Ne laser operating nearthnshold.aswltching was performed by applying a voItsge,Pulse to an intracavlly awustwptlc modulator.Rwu)ts are compared to thaoretld calculatlms.(12 mln)%€fA VYAS and !SURENCRA SINGH. U. Arkansas.Physics Depamnent. Fayeneville. AR 72701.W. R. M. G PAYNE. and W. R. GARRm,Oak Ridge National Laboratory. Heam and SafelyResearch Dlvkion. Oak Ridge. TN 37831.A preWoadenlng study of t h r mand me-phG+gn resonances was perfofmeti mingcounterpropagatlng, linearly polarized laserbeam. Self-bmadening. & well .as Wm collislO+Induced etfects due to Uw presenur Ol a sscondmble gas. was investigated fa tJm 6s. 6s'. Sdand7sofxenonandn Sdof -ton. A muniphotonionlzatlon scheme In a calibratd proportionalcounter is the basls of a new tectmlque by whichmllislonal processes can be lnvestlgated tor h epresscae regimes whlch have previously been inaaxssibleto hvestlgation, psrtlcularly for selfbroaddlwhere the mean (ree path of the @OmmboflheOrknof10'3cmfapeosures~kwas 1 Ton. Using ihis tschnlqus. we have rsmdedwlllsiorAnduced wldmP and shifts of the re-~laser~ando(herbnvldres0lQuenNnelectronics certain t m oscilhtlng at twh tharesonance freq~~cyarsdopped from me lntaac- wnce for pr~ssues up b 1000 Tm. In the hmfmHamlltonh. he ettm of mese rmntenotatdreda of tur range. asymnebic. sell-broadsniwIO tanu on the operauon of a single-made laser ISllne da%c8s were o b d wlh full wldms andWnsidereduslng therodelof ScuIfyandLamb. &isshim vrhlch were greater than five times Iho laserfaad that these t m lead to a shin of the laser bandwklth [0.002-0.005 M1). (12mm)thesholdandabmadeningof~photonnunb~6sbbutlm.(12mSn)A. S. MARATHAY. 8. A. CAPRON. and M. SAR-GRJr Ifl. U. Arlmrw OpUal S c i i Cents. T*~n. AZ 85721.Refweme 1 &rive the twO+hOton15 October 1985suption coofliclent for a hanogeneourly bmadmedimsubjecbd toJEmRSONan arbmarib Intsn?w,sabra@ wave. We have gsneralked this mWVto allow fw bppk broedeninq with a ccuntnlngprobs and r a m waves. We have cardeda~ Integral3 over s Lorentrian Doppler dlsbibutlcnadytlcally and-have specialized m a number ofmX1 Absoldw two-phalon abwrptlon and Ilmlts. such as emar.m Doppler W i n g . In%hr.epMon knlratlon crcm sedans la atdc this limit, we llnd the simple wok, ebsapti~norlmcoetRclent a, 2ndyDIh - Iw.Tt). wlwre ao Isthe rnsatuatedabsorptirn coeMdentbWOLAS J. BAMFORD. LEONARD E. &lSIHSKI, r,Istheseh.amr~.ykmh*0photonancl wUlAfw4 K. BISG-EL. SRI I-. Chm coherence decay rate comtanS us k the St* 3N'ftleal mi- Laboratay. Menlo PYk. CA 04025. parmeter. T, ia the popdatran difference decayBefas r photon dstectian Ichsma?i for at- tkne, and a, L I/[? + r(o + WA - PZ- ~111.~ e r r d m o l ~ c a n b s p w e d m a Fa ~ van- Stark shifts (0. 0). thb is the~ooli~aoss~la~armnl~bwi- rtandsdtwophotarFmbs~-~tiom must-be know. We hllua rma~crad bu,forhw,mecom~lexLaMltLianf)tbb~e~mcm~sectiaata~40a".~~wrlam speck b, Reme dlam*. T bnlgus of tw-~QI OXCI~~~ thaesoaoolaedtDmefwre~heams~tame2p~h~;,,$q1985 Annual Meeting OpUcal Society of America P25 ! 'il3~?~,,.0 trarultlon at 226 nm. Excltatlon was 1. M. Sargent Ill. S. Ovadii. and M c 111 Phys.tXmltored by Ob~(*~lng fluorescence at 845 nrn Rev A (1985) to be published.from the 3p SPzt,o + 3s3S, transition. The fluerescence collection system was callbraled absnlutely us[yl spa- scanelm hm TUX4 Saluratlon MKb In mukiphalon lonlzamalecularhydrogen. The spatial and toqua1 tbnpmfiles of the exclting dye laser pulses were carefullymeaswedln agently focused excltatlon peom JOHN A YEAZEU. MlCW S. MALCUIT. CAReby(peak IntensHy -10 MWl& allwlng the LOS R. STROLD. Js., end ROBERT W. BOYD, U.t w w cross section to be detmlntd. The Rochastw. Institute of OpUcr. Rochester, NYaos section fa absorption of a third, identical14627.photon (2 + 1 photoionization) was Umn beter- We have sMled two-photon rescfwl Wv*mined by rneasurlng the absolute nun- of Ions photon lonlzatlon of atomic rodium vapor. A meproduced per laser pulse. The meanred cmss laser tuned to elther the 46 or 5s twmtw alseclloluarea= 2.5 X 10-~cm4/Wandopl= 6.1 bwed transnlon was used to excite the atun ud aX 10-lS d. Wng a square pulse In spaw and second m e Intense her (A = 7000 A) yaa usedtime, we calculate that - 1 % IoniaUon Is poorlble tor lonizatlon. Tha Ion yleld was measwed 8s aforal~s~n~at226nmof 1 J/c~Inal(kri hnction of remtlaser lntwity aml detwlng.long pulsa.(12 mh) the nonrsMMnt Laser IntsmiRy, and the temporaloverlap of the exciting ard lonlzlng 1- pldses.We have found' Umt Bt high laser Intenaltlm theNX2 IWrdmt) prrrwrre bmadmhg of throe- photolmlzdtiar yield decreases as he nwyescphotonresonances fn gascrWR kzw mlemity Is increased. When tuwd tome 5s level a value of --0.6 was obtained !u thsexponentla1 index avan with the resonant lasatuned to rnaxlrnize the ion ykld. A IheoretlCdlmodel lncludlng the ettw of Stark shln QI IMgumdstale, power broadening 01 me n~-~~-"elelevel due to 6atwation of the WWhotm WaWlthand by rapid trw~tiOnrr to the Icnizatbl cerltlwasused to predict the experlmsnbl resub.Good agreement between lf=W and emirequiredthe use 01 a photolonii WOW Kcamfor me 5s bvel which was -20 times Weator rtranme acwptad value. Redts for photo~onbtbban the 4dlevel are also pressnted. (12 mEn)1 L AIM, R. W. Boyd. J KrasWl. M. S. Malcun.and c R. s~aud. Jr.. mys. Rev Len. 51. 309(1985).TUX5 Canp.tltlar b.trrwn cohnn( nnd Lnc*Mlmtmnllmu~poccrt..DAMEL J GAUMmL M I W S MA1 CUP. andROBERT W. BOYD. U Rocbsler, 1-1- of Dptlu. Rorhsstn. NY 11827Canp3tMn ha. besn obrsrred beeen qlCfledemtulon (ASQ and fw-wavemlxhg (FWH) Revlously, -Ion betweenan k&~.m$ and coharent proceu was o~xmedm ths rumof suppasion of muniphoton bnizatkmby m~ m i c ganaatta, I,# xr exoerCmcrn.un~npnbsef~bnedprecac)yto~?- jd- allowed transmon h amlc sodC~~pagnjUm&uia,wa,WredInb~the fprward backward d~rectiom. In CheWcmMom. arther FWM a ASE can cUX Igm-,ng the p~~~ibillty of canpetltlon betyscn~opmcs~~,checakulaWpaln fuASE~smuchman mat (a FWM Ho*erer. we have ch--d expmimentally that FWM and not ASE oc--. TO ~~nnure mat H IS the pewncs d emFW w- AS€, we have exmsdthe atanlc vapor lmtsad With WWKW'Wpvnp waves of muem tregoency t.ra case.FWM. be~ng a pr-9 ~~ cG-ands~ngASEnobserved Themtlwof*ccmpet~lan is erpbned by 60hnnQfoupledh&xwell and demy mablx equam 'mwsy~yshmr nisfandmthesuppr-b&JO to me desrmctive nderterence Wweenpetmws d e x ~ ~ CA ~ (he o n 3a WAwtuchresult.l~n~Lqofme~'="~~qand me. (12 m)