ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
168 9. Two-mode Gaussian states in the lab Frequency-Doubled Nd:YAG Laser Isolator Servo ⊗12 MHz λ/2 Servo KTP Q 0 Q 1 H 1 Q 2 { PZT H2 Q3 H3 Q4 PZT Ramp - H5 - H4 Figure 9.5. Experimental setup. A continuous-wave frequency-doubled Nd:YAG laser pumps below threshold a type II OPO with a λ/4 plate inserted inside the cavity (Q0). The generated two-mode vacuum state is characterized by two simultaneous homodyne detections. The infrared output of the laser is used as local oscillator after filtering by a high-finesse cavity. SA1,2,3: spectrum analyzers. Q1,...,4 and H1,...,5: respectively quarter and half waveplates. PD Lock: FND-100 photodiode for locking of the OPO. PD Split: split two-element InGaAs photodiode for tilt-locking of the filtering cavity. We make no assumption on the form of the CM which is written in the general case σ+ σ = γ γ ± T ± σ− ⎛ a ⎜ = ⎜ b ⎝ c d b e f g c f h i d g i j ⎞ ⎟ ⎠ . When the LO phase is chosen so that zero corresponds to the long axis of the noise ellipse of the first mode, the CM is written in the form ⎛ ⎞ ⎜ ⎟ σ = ⎜ ⎟ ⎝ ⎠ = ′ σ + γ ′ ± , a ′ 0 c ′ d ′ 0 e ′ f ′ g ′ c ′ f ′ h ′ i ′ d ′ g ′ i ′ j ′ where a ′ and c ′ correspond respectively to the maximum and minimum noise levels measured in a linear scale on the spectrum analyzer for the first mode, which we will choose arbitrarily to be A+. One can also easily determine h ′ , i ′ and j ′ from the spectrum analyzer traces for A−: when the LO phase is chosen so that zero corresponds to the long axis of the noise ellipse of the second mode, the CM is written in the form σ ′′ ′′ h 0 − = 0 j ′′ and σ ′ − can be easily deduced from σ ′′ − by applying a rotation. The angle of this rotation is given by the phase shift ϕ between the two traces (see Fig. 9.6). This operation is performed numerically. We have now measured both diagonal blocks. In order to measure the non-diagonal blocks, one records on an additional spectrum analyzer a third signal, the difference between the two homodyne detection γ ′T ± σ ′ − - SA1 IN Trig SA2 SA2 IN Trig IN Trig
9.2. Experimental production and manipulation of two-mode entanglement 169 Figure 9.6. Spectrum analyzer traces as a function of the local oscillator phase. signals (these signals being themselves the difference between their respective photodiodes photocurrents, see Fig. 9.5). Let us consider the case where the waveplate Q3 is not present; for a given LO phase ψ1, the homodyne detections will give photocurrents which are proportional to the amplitude noise for the A+ beam, ˆq+, and to the phase noise for the A− beam, ˆp−. The signal recorded on spectrum analyzer SA2 is, in this case, proportional to ˆs = ˆq+ − ˆp− whose variance is 〈î 2 〉 = 〈ˆq 2 +〉 + 〈ˆp 2 −〉 − 2〈ˆq+ ˆp−〉 = a ′ + j ′ − 2d ′ . a ′ and j ′ being already known, it is easy to extract d ′ from this measurement. For a LO phase ψ1 + π/2, one will get using a similar procedure e ′ , h ′ and f ′ . Let us now add the wave plate Q3. For a LO phase ψ1, one will get a ′ , h ′ and c ′ and for ψ1 + π/2 e ′ , j ′ and g ′ thus completing the measurement of the CM. 9.2.4. Experimental measures of entanglement by the negativity As all experimental measurements, the measurement of the CM is subject to noise. It is thus critical to evaluate the influence of this noise on the entanglement. A quantitative analysis, relating the errors on the measured CM entries (in the A± basis) to the resulting error in the determination of the logarithmic negativity (the latter quantifying entanglement between the corresponding A1 and A2 modes) has been carried out and is summarized in Fig. 9.7 in absence of mode coupling. In general, the determination of the logarithmic negativity is much more sensitive to the errors on the diagonal 2×2 blocks α and β (referring to the reduced states of each mode) of the CM σ [see Eq. (2.53)] than to those on the off-diagonal ones (γ, and its transpose γ T , whose expectations are assumed to be null). Let us remark that the relative stability of the logarithmic negativity with respect to the uncertainties on the off-diagonal block adds to the reliability of our experimental estimates of the entanglement. Notice also that, concerning the diagonal blocks, the errors on diagonal (standard form) entries turn out to affect the precision of the logarithmic negativity more than the errors on off-diagonal (non standard form) entries. This behavior is reversed in the off-diagonal block, for which errors on the off-diagonal (non standard form) entries affect the uncertainty on the entanglement more than errors on the diagonal (standard form) entries.
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168 9. Two-mode Gaussian states in the lab<br />
Frequency-Doubled<br />
Nd:YAG Laser<br />
Isolator<br />
Servo<br />
⊗12 MHz<br />
λ/2<br />
Servo<br />
KTP<br />
Q 0<br />
Q 1 H 1 Q 2<br />
{<br />
PZT<br />
H2<br />
Q3<br />
H3<br />
Q4<br />
PZT<br />
Ramp<br />
-<br />
H5 -<br />
H4<br />
Figure 9.5. Experimental setup. A continuous-wave frequency-doubled<br />
Nd:YAG laser pumps below threshold a type II OPO with a λ/4 plate inserted<br />
inside the cavity (Q0). The generated two-mode vacuum state is characterized<br />
by two simultaneous homodyne detections. The infrared output of the<br />
laser is used as local oscillator after filtering by a high-finesse cavity. SA1,2,3:<br />
spectrum analyzers. Q1,...,4 and H1,...,5: respectively quarter and half waveplates.<br />
PD Lock: FND-100 photodiode for locking of the OPO. PD Split: split<br />
two-element InGaAs photodiode for tilt-locking of the filtering cavity.<br />
We make no assumption on the form of the CM which is written in the general<br />
case<br />
<br />
σ+<br />
σ =<br />
γ<br />
γ ±<br />
T ± σ−<br />
⎛<br />
a<br />
⎜<br />
= ⎜ b<br />
⎝ c<br />
d<br />
b<br />
e<br />
f<br />
g<br />
c<br />
f<br />
h<br />
i<br />
d<br />
g<br />
i<br />
j<br />
⎞<br />
⎟<br />
⎠ .<br />
When the LO phase is chosen so that zero corresponds to the long axis of the noise<br />
ellipse of the first mode, the CM is written in the form<br />
⎛<br />
⎞<br />
⎜<br />
⎟<br />
σ = ⎜<br />
⎟<br />
⎝<br />
⎠ =<br />
<br />
′ σ + γ ′ <br />
±<br />
,<br />
a ′ 0 c ′ d ′<br />
0 e ′ f ′ g ′<br />
c ′ f ′ h ′ i ′<br />
d ′ g ′ i ′ j ′<br />
where a ′ and c ′ correspond respectively to the maximum and minimum noise levels<br />
measured in a linear scale on the spectrum analyzer for the first mode, which we<br />
will choose arbitrarily to be A+. One can also easily determine h ′ , i ′ and j ′ from<br />
the spectrum analyzer traces for A−: when the LO phase is chosen so that zero<br />
corresponds to the long axis of the noise ellipse of the second mode, the CM is<br />
written in the form<br />
σ ′′ <br />
′′ h 0<br />
− =<br />
0 j ′′<br />
<br />
and σ ′ − can be easily deduced from σ ′′ − by applying a rotation. The angle of this<br />
rotation is given by the phase shift ϕ between the two traces (see Fig. 9.6). This<br />
operation is performed numerically. We have now measured both diagonal blocks.<br />
In order to measure the non-diagonal blocks, one records on an additional spectrum<br />
analyzer a third signal, the difference between the two homodyne detection<br />
γ ′T ±<br />
σ ′ −<br />
-<br />
SA1<br />
IN Trig<br />
SA2<br />
SA2<br />
IN Trig<br />
IN Trig
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