Non-dispersive wave packets in periodically driven quantum systems

More documents

Recommendations

Info

436 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 “Floquet eigenstates” of the system—multiplied by oscillatory functions: | (t)〉 = cj exp −i Ejt |Ej(t)〉 ; (73) ˝ with j |Ej(t + T )〉 = |Ej(t)〉 : (74) The Ej are the “quasi-energies” of the system. Floquet states and quasi-energies are eigenstates and eigenenergies of the Floquet Hamiltonian H = H0 + V cos !t − i˝ 9 : (75) 9t Note that, because of the time-periodicity with period T =2 =!, the quasi-energies are de ned modulo ˝! [92]. The Floquet Hamiltonian (75) is nothing but the quantum analog of the classical Hamiltonian (58) in extended phase space. Indeed, −i˝9=9t is the quantum version of the canonical momentum Pt conjugate to time t. In strict analogy with the classical discussion of the previous section, it is the Floquet Hamiltonian in extended phase space which will be the central object of our discussion. It contains all the relevant information on the system, encoded in its eigenstates. In a quantum optics or atomic physics context—with the external perturbation given by quantized modes of the electromagnetic eld—the concept of “dressed atom” is widely used [18]. There, a given eld mode and the atom are treated on an equal footing, as a composite quantum system, leading to a time-independent Hamiltonian (energy is conserved for the entire system comprising atom and eld). This picture is indeed very close to the Floquet picture. If the eld mode is in a coherent state [18] with a large average number of photons, the electromagnetic eld can be treated (semi)classically—i.e., replaced by a cos time dependence and a xed amplitude F—and the energy spectrum of the dressed atom exactly coincides with the spectrum of the Floquet Hamiltonian [92]. By its mere de nition, Eq. (74), each Floquet eigenstate is associated with a strictly time-periodic probability density in con guration space. Due to this periodicity with the period of the driving eld, the probability density of a Floquet eigenstate in general changes its shape as time evolves, but recovers its initial shape after each period. Hence, the Floquet picture provides clearly the simplest approach to non-dispersive wave packets. Given the ability to build a Floquet state which is well localized at a given phase of the driving eld, it will automatically represent a non-dispersive wave packet. In our opinion, this is a much simpler approach than the attempt to build an a priori localized wave packet and try to minimize its spreading during the subsequent evolution [44,45,34,47]. Note that also the reverse property holds true. Any state with T-periodic probability density (and, in particular, any localized wave packet propagating along a T-periodic classical orbit) has to be a single Floquet eigenstate: Such a state can be expanded into the Floquet eigenbasis, and during one period, the various components of the expansion accumulate phase factors exp(−iEiT=˝). Hence, the only solution which allows for a T -periodic density is a one-component expansion, i.e., a single Floquet eigenstate. 10 10 One might argue that Floquet states di ering in energy by an integer multiple of ˝! could be used. However, the Floquet spectrum is ˝!-periodic by construction [92], and two such states represent the same physical state.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 437 To summarize, the construction of non-dispersive wave packets in a time-periodic system is equivalent to nding localized Floquet eigenstates. The existence of such states is far from obvious, as the Floquet spectrum is usually very complex, composed of quasi-bound states, resonances and continua. This is why a semiclassical analysis can be very helpful in nding these objects. 3.1.3. Semiclassical approximation Dealing with highly excited states, a semiclassical approximation can be used to determine quasienergies and Floquet eigenstates [80]. If the driving perturbation is su ciently weak, we have shown in Section 3.1.1 that the classical dynamics close to a non-linear resonance is essentially regular and accurately described by the pendulum Hamiltonian (66). It describes a system with two degrees of freedom (along t and ˆ , with their conjugate momenta ˆPt and Î, respectively) which is essentially regular. For semiclassical quantization, we may then use the standard EBK rules, Eq. (45), introduced in Section 2. The momentum ˆPt is a constant of the motion, and the isovalue curves of ˆPt; Î; ˆ lying on the invariant tori can be used for the EBK quantization scheme. Along such a curve, t evolves from 0 to 2 =!, with ˆ = − !t kept constant. Thus, itself is changed by 2 , what implies that the Maslov index of the unperturbed (I; ) motion has to be included, leading to the following quantization condition for ˆPt, 1 2 T 0 ˆPt dt = ˆPtT 2 = k + 4 ˝ (76) with integer k. Since T =2 =! is just the period of the resonant driving, we get the quantized values of ˆPt : ˆPt = k + ˝! ; (77) 4 which are equally spaced by ˝!. Thus, we recover semiclassically the !-periodicity of the Floquet spectrum. For the motion in the (Î; ˆ ) plane, we can use the isocontour lines of the pendulum Hamiltonian Hpend, Eq. (66), as closed paths, keeping ˆPt and t constant. Depending on the nature of the pendulum motion (librational or rotational), the topology of the closed paths is di erent, leading to distinct expressions: • For trapped librational motion, inside the resonance island, the path is isomorphic to a circle in the (Î; ˆ ) plane, with a Maslov index equal to two. The quantization condition is 1 Î d 2 ˆ = N + 1 ˝ (librational motion) 2 (78) with N a non-negative integer. Of special interest is the “fundamental” state, N =0, which exhibits maximum localization within the resonance island and is therefore expected to represent the optimal non-dispersive wave packet. • For unbounded rotational motion, outside the resonance island, the path includes a 2 phase change for and acquires the Maslov index of the unperturbed motion: 2 1 Î d 2 ˆ = N + ˝ (rotational motion) : 4 (79) 0
Page 1 and 2: Abstract Physics Reports 368 (2002)
Page 3 and 4: A. Buchleitner et al. / Physics Rep
Page 27: A. Buchleitner et al. / Physics Rep
Page 43 and 44: → ; → ; A. Buchleitner et al. /
Page 79 and 80:
A. Buchleitner et al. / Physics Rep
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
|ψ (z)| 2 |ψ (z)| 2 0.3 0.2 0.1 0
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
γ /∆ σ/∆ 10 -1 10 -2 10 -3 0.
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139:
show all

Non-dispersive wave packets in periodically driven quantum systems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?