1.3 Spin squeezing in cavity-quantum electrodynamics
1.3.4 Squeezing by cavity feedback
pcκ/κ0 (1.73)
In practice, the measurable anti-squeezing is usually higher than this prediction due to technical noise. The consequence of the excess squeezing can be twofold: first, the anti-squeezing leads to an reduction in total spin length (coherence) due to the finite Bloch sphere curvature. However, this effect is negligible for a reasonably large atom number. Second, also due to the Bloch sphere curvature, the uncertainty in Sy can leak into Sz if the final state is not precisely at Sz = 0. After all, a clock is meant to measure such a deviation. In fact, the record demonstration of 20 dB squeezing has an increase in the uncertainty area of 19 dB, which can completely lose its metrological advantage in a realistic context of clocks [83].
1.3.3.4 Prospective squeezing in TACC-2
Here I give an estimate of achievable squeezing in the TACC-2 experiment based on typi-cal parameters that are detailed in the following chapters. For a typitypi-cal thermal cloud at 200 nK, the average cooperativity is about C = 0.42. We currently probe the cavity through transmission with the probe laser detuned at δp = κ/2 and a total detection efficiency of 0.5 is realistic. Fig. 1.5(a) shows the calculated squeezing parameter as a function of the measurement strength for 2 × 104 atoms (a typical number for clock measurements), using Eq. 1.69. The situation of optimum detection is also plotted for comparison. The squeezing reaches a maximum of 18 dB for the realistic case, while the ideal detection can give 23 dB.
We also plot this optimum squeezing as a function of atom number (Fig. 1.5(b)), showing that it is not limited by the atomic structure for the current setup.
1.3.4 Squeezing by cavity feedback
Recalling the effective Hamiltonian of the system (Eq. 1.42), we note that the interaction terms ∼ ˆc†ˆcˆSz can result in the OAT Hamiltonian if the intra-cavity photon number ˆc†ˆc is engineered to be correlated with Sz. In fact, as Sz determines the cavity frequency, probing the cavity with a detuning correlates the transmitted photon number (which determines ˆc†ˆc) with the cavity frequency, hence with Sz.
10 1 100 101 102 103
Figure 1.5 Achievable squeezing in TACC-2. In both panels, we assume an average cooperativity Ceff = 0.42, which is typical for the experiment. We assume, for the solid curves, measurements by cavity transmission with probe detuning κ/2 (ηd = 1/2) and a detector efficiency of qd= 0.5 which is close to the current setting, while the dashed curves assume perfect detection with homodyne method (ηd = qd = 1, still collecting half of the exiting light), representing almost the theoretical limit of the setup. In (a) we plot the squeezing parameter as a function of φac (and nd) for 2 × 104 atoms. We identify on the left the PSN limit and on the right the Raman flip limit (we set p = 1/40 for our clock states). (b) The optimum squeezing as a function of atom number. We see that ultimate limit NC ∼ (ωat/Γ)2 is rather remote for the current experiment.
Formally, the system Hamiltonian is treated in an open quantum system with an input coherent field |βi and decay channels into free space. The evolution of the spin distribution can be described through the evolution of the moments of spin operators in the Heisenberg picture, with the cavity field traced out [43]. We will assume a detuning of κ/2 to have a near maximum correlation between Sz and cavity transmission.
Since Szis a constant of motion, hSziand ∆Szare conserved. But the cavity field induces correlation between Sy and Sz. Most practically, one evaluates the evolution of the raising operator ˆS+ = ˆSx+ i ˆSy. Following [43], it can be shown that for small φac per photon, the lowest moments read
h ˜S+iβ = e−2SQeiQSSzS+(0) (1.74) h ˜S+2iβ = e−2QS eiQS(2Sz−1)S+2(0) (1.75) where h iβ denotes the partial trace over the cavity field, ˜S+denotes ˆS+(t) after the evolution time t, and we have defined a dimensionless shearing strength
Q ≡ Snt 4g2
∆cκ
!2
(1.76) with nt = |β|2κt/2 = nd/qd the average transmitted photon number during the evolution.
We notice that the precession rate of ˆS+ is indeed proportional to Sz. The correlation between Sy and Sz is given by
and the increased uncertainty in ˆSy due to the OAT reads:
(∆ ˜Sy)2 = S
4(2S + 1) − S
4(2S − 1)e−2QS cos2S−1Q S
(1.78)
≈ S
2(1 + 2Q + Q2) (1.79)
where the approximation is S 1 and |QSz/S| 1. We identify the initial noise of the CSS in the first term, the noise from the cavity photon shot noise (Q ∝ nt∝ t) in the second, and the cavity feedback (∝ t2) in the third. It is the last term that allows the OAT squeezing in the open quantum system [43], at a price of stretching the uncertainty region beyond the minimum uncertainty area for Q > 1.
We also have the spin variance after a rotation along S with an angle −α to verify the squeezing
∆Sα2 = 1 2
V+−qV−2+ W2cos(2α − 2α0) (1.80) where V±= (∆Sy)2±(∆Sz)2, W = h ˜SySz+ SzS˜yi and tan 2α0 = W/V−.
The minimum (at angle α0) and maximum (at α0+ π/2) uncertainties, in terms of the squeezing parameter, approximately scale as ξα0 ∼ 2/Q and ξα00+π/2 ∼ Q2. The noise reduction is limited by PSN as for squeezing by QND measurement. But the anti-squeezing grows more quickly – the back-action of measuring Sz through the cavity feedback. This increase of the total uncertainty area (√
2Q) can in principle be recovered by detecting the light leaking out of the cavity, basically performing a cavity measurement.
Limits The anti-squeezing imposes a severer limit due to the curvature of the Bloch sphere.
More precisely, the squeezing scales as [43,104]
ξα20 ≈ 2
Q+ Q4
24S2 (1.81)
where the second term is attributed to the Bloch sphere curvature. This gives an optimum squeezing ξcurv2 ≈ 1.5 · S−2/5 (see also [106]). Nevertheless, it is shown more recently that the excess anti-squeezing can be suppressed by far detuning the probe laser with respect to the cavity resonance, essentially performing a weak measurement [84].
On the other hand, photon scattering into free space is always detrimental. Scattering induced Raman flip leads to similar squeezing limit as for QND-measurement based squeezing (Eq. 1.72). However, these are insignificant compared to the curvature limit, already for a moderate cooperativity C ∼ 1.