noise floors, then this should not be a problem, but in practice the electronic noise floor is never going to be zero.
master equation
dˆρ(t) =−i[ ˆHS,ρ(t)]dtˆ +D[√
MFˆz] ˆρ(t)dt +√
ηH[√
MFˆz] ˆρ(t)
2p
M η[y(t)dt− hFˆzidt]
(8.31) where ˆHS is the system Hamiltonian,η is the quantum detection efficiency, and M is the measurement strength, which is physically derived in the next section for our experimental scheme, but left general here. The superoperators, Dand H, are defined as
D[ˆc] ˆρ ≡ ˆcˆρˆc†−(ˆc†cˆρˆ+ ˆρˆc†ˆc)/2 (8.32) H[ˆc] ˆρ ≡ ˆcˆρ+ ˆρˆc†−Tr[(ˆc+ ˆc†) ˆρ] ˆρ, (8.33) and the photocurrent is represented as
y(t)dt=hFˆzi(t)dt+dW(t)/2p
M η. (8.34)
The stochastic quantity dW(t) ≡ 2√
M η(y(t)dt − hFˆzi(t)dt) is a Wiener increment and dW(t)/dt is a Gaussian white noise associated with the shotnoise of the homodyne local oscillator.
As mentioned in section 2.4, there are several intuitive features to notice about this equation. First, if M = 0 we recover the usual closed-system equation of motion for the state. If M 6= 0, but the quantum efficiency of the detector η = 0, then we get the unconditional evolution of the state due to the measurement. If M 6= 0 and η 6= 0, then the update rule is an explicit function of the photocurrent y(t). This conditional part of the equation is reminiscent of classical estimators that take the difference between what the photocurrent is and what the photocurrent is expected to be, to get aninnovation dW(t) by which the system is updated. In fact, this stochastic master equation (or stochastic Schrodinger equation, SSE) is one of the most general mathematical examples of continuous quantum measurement and is discussed in many different contexts [71, 72, 23]. We discuss its properties in more detail in chapter 9.
8.2.2 Deriving the Measurement Strength from the Polarimeter SNR The SME above is the filter that essentially describes our polarimetry experiment (far-off resonance) although we have not derived it in this context. We avoid the derivation, and identify the physical measurement strength with our experimental parameters, by using results from previous chapters. In the end, it is the form of M we are interested in, but we are also interested in the validity of the SME as a model. There are two reasons that it will not be an exact model. First, if the detuning is not large enough, the tensor terms will destroy the description, as discussed in the unconditional case in chapter 7. Second, we only consider the SME as a valid description for times less than the spontaneous emission time as derived in chapter 5. We end this chapter with a discussion related to these crudely modeled, but real, effects, although in the next few chapters we largely ignore them for pedagogical reasons.
(Update: Led by Luc Bouten, our group has recently finished a paper in which the polarimetry stochastic master equation is derived and shown to be as assumed in this thesis [111]. The paper uses a formal quantum stochastic differential equation (QSDE) formalism and also shows that the scheme is equivalent to performing homodyne detection on the y channel when the probe beam is initially polarized along x. This imbalanced homodyne detection case is as opposed to the balanced homodyne case considered earlier in this chapter.)
In chapter 7, equation (7.32), we showed that the (ignoring the tensor Hamiltonian) the unconditional master equation is simply
dˆρ = hSˆ0i
2dt D[ˆγz] ˆρdt (8.35)
= D[√
MFˆz] ˆρdt. (8.36)
Here, we have now physically identified the measurement strength M in terms of our ex- perimental parameters as
M = 1
8~2τss−1σ0 A
(8.37)
where we have used the definition, from equation (7.48), of the spin-squeezing time as
τss−1 = 2Iσ0
~ω
Γ 4
X
f0
α(1)f,f0
α0∆f,f0
2
. (8.38)
Thus, for the η = 0 case we can identify this measurement strength as having the same function as the measurement strength in equation (8.31). Notice that the measurement strength is a rate when ~is assumed unitless. Also notice that the ratio of areas, the cross section divided by the beam area,σ0/A∝λ2/Acan never be greater than one in free-space due to the diffraction limit.
Rather than this indirect approach for deriving the physical measurement strength, we can also identify it through a direct expression for the physical photocurrent in units of power. As shown in section 6.4, the measurement of ˆSy will result in equation (6.87) when atoms are present. Now we wish to include measurement noise in order to calculate the signal-to-noise ratio. It is readily shown that all terms not linear in ˆFz vanish in equation (6.87) provided that the state is aligned along x with θ = π/2 and φ = 0. That is, a pure Faraday rotation Hamiltonian is recovered when the atomic magnetization vector is oriented along the x-axis. However, rotating ˆFin the xy-plane results in elliptically polarized scattered probe light, and moving out of this plane results in nonlinear atomic dephasing due to scattering terms, which are quadratic in the single-particle spin operators, ˆfz. These adverse effects are avoided for the experimental geometry where ˆF is collinear with the x-axis.
Taking the input probe field to be in an x-polarized optical coherent state, and consid- ering the small γ limit, equation (6.87) leads to a semiclassical photocurrent (with units of optical power) of the form,
yt=η√
ShFˆzi+√
η ζt, (8.39)
where we have made the substitution,~N fcosθ→ hFˆzi(see equation (6.69)), and included the photodetector quantum efficiency, η. We have introduced the scattering strength S defined as
S = 1
~2
Ipσ0
Γ 4
X
f0
α(1)f,f0
α0∆f,f0
2
, (8.40)
which depends upon the probe intensity, Ip = P/A, determined by the coherent state amplitude, P = 2~ω|β|2 and cross sectional area, A = πr2 (for a mode-matched probe laser). It is useful to note that the scattering strength has units of W2/~2 (power squared per~2) and characterizes the degree of coupling between the atoms and the probe field;√
S quantifies the polarimeter optical power imbalance per unit spin (as ˆFz has units of~). As discussed above, we have represented the optical shotnoise byζt=√
~ωP dWt/dtsuch that
∆ζ2 = ~ωP. To convert the above power units to photocurrent units, we simply use the conversion e/~ω, or the responsivity if the detection efficiency has not been included.
Our expressions are similar to previous results [15, 104, 45, 24]. However, our specific expressions for γx, γy and γz, and hence M, account for the detailed hyperfine structure of the atomic excited states, including the fact that the oscillator strengths and signs of the contributions from different participating excited states are not equal, and doing so is required for quantitative agreement between theory and experiment.
Finally, we can identify the physical photocurrent equation (8.39) with the photocurrent from equation (8.34). The measurement strength is then seen to be given by
M = S
4∆ζ2, (8.41)
which again matches the above expression forM, in equation (8.37), when expanded.
Now that we have identified the measurement rate, we spend a considerable amount of time in the next two chapters investigating the SME under ideal circumstances without spontaneous emission or tensor complications. There it will be clear exactly how the SME conditionally projects an initial coherent spin-state into a spin-squeezed state. In the next section, we consider the expected degree of squeezing from a much simpler perspective.
8.2.3 Squeezing by Averaging
Now consider a measurement of ˆFz as described above (and displayed in figure 14.1). In the small time limit where probe induced decoherence can be neglected, the full quantum filter describing this measurement is equivalent a classical model in which ˆFz is simply a random constant on every trial drawn from a distribution with variance equal to the quantum variance ofh∆ ˆFz2i0[20]. Then the generally complicated full quantum filter [23] is equivalent to linear regression, or fitting a constant to the noisy measurement record in real
time. In essence, the optimal filter serves to average away the optical shotnoise to reveal the underlying value of ˆFz. Under these statistical assumptions, with the signal-to-noise ratios of the previous section, the quantum uncertainty at small times is given by
h∆ ˆFz2iτ = h∆ ˆFz2i0
1 + 4M ηh∆ ˆFz2i0τ. (8.42) This can be shown either with the full quantum filter or by using the equivalent classical model combined with Bayesian estimation (from which a Kalman filter or linear regression can be derived). In the filtering terminology, the above expression is seen to be the solution to a Riccati equation as discussed in chapter 10 [49].
We now define the signal-to-noise ratio as
SNR2≡4M ηh∆ ˆFz2i0τ (8.43)
and express the degree of squeezing (ignoring decay of the ˆFx) as
W ≡ h∆ ˆFz2it
h∆ ˆFz2i0 (8.44)
= 1
1 + SNR2. (8.45)
Usingh∆ ˆFz2i0 =~2N f /2, we can express the signal-to-noise ratio as SNR2 =ηODf
4 τ
τss (8.46)
where the optical depth is OD =N σ0/A=ρLσ0, ifA is the area andL the length of the cloud. (This SNR is implied throughout the rest of this thesis, as opposed to the SN R definitions used earlier.)
To get the most spin-squeezing in an experiment, we want to make the SNR as large as possible. The quantum efficiency η is typically of order unity for good photodiodes, thus one cannot gain orders of magnitude by considering better detectors. The optical depth should be made as large as possible, but typically cold atom densities and cloud sizes are limited by technical considerations as discussing in chapter 12. Finally, one wants to average for a long time to make the fit to a random offset buried better resolved, thus the ratio τ /τss should be as large as possible. Unfortunately, the averaging time is limited both by
technical considerations (e.g., atoms falling out of the trap) and fundamental considerations, because the averaging time needs to be less than the spontaneous emission time, τ τsc for this entire story to remain valid. With a cavity, the measurement timeτsscan be made much smaller than theτsc and more squeezing can be obtained, but in free-space these two quantities are fixed relative to each other as discussed in the next section.
In appendix E, we consider a filtering model with spontaneous emission, where the ideal filter isnot equivalent to the averaging filter discussed here. There we analyze exactly how poorly the nonoptimal averaging filter performs in the presence of decay.
8.2.4 Squeezing versus Decay Timescales
Ignoring the tensor Hamiltonian temporarily, it is interesting to consider the ratio of the timescale τsc from equation (5.46) describing the incoherent decay and the timescale τss from above, which is related to the rate at which spin-squeezing occurs. The larger the ratio τsc/τss is, the more squeezing one expects to get. In the following, we assume either an optical polarization parallel or perpendicular to the atomic spin alignment. In figure 8.1, these ratios are plotted as a function of detuning.
First consider the parallel polarization case. When plotting the timescale ratio we see that close to resonance (less than 300 MHz as discussed in chapter 6) the ratio for the cesium D2 line is nearly an order of magnitude higher than far-off resonance. (The perpendicular polarization case displays similar behavior but exhibits a higher overall ratio.) The reason for this behavior is that all of the excited level contributions to τsc are the same sign (positive), while the contributions to τss are alternating in sign. So na¨ıvely one wants to be close to the 4–5’ transition, where the negative effect of the 4–4’ transition on the measurement strength can be neglected, giving a higher expected ratio of times and hence more spin-squeezing. However, this logic tells us to be close to resonance where all of the tensor terms are large and, as we have seen in chapter 7, the simple spin-squeezing story gets considerably more complicated. Thus one has two options: either go to a large detuning and take the lower ratio of timescales for the sake of simplicity, or try to work close to resonance and either understand or cancel the tensor terms in some way, e.g., [112]. This strategy is discussed more in chapter 14.
In the next chapter, we numerically and analytically investigate the behavior of the ideal SME of equation (8.31). For convenience, we temporarily neglect the spontaneous emission
Figure 8.1: Comparisons of the of the spontaneous emission timescales (perpendicular po- larizations τsc,⊥, parallel polarizations τsc,k) and the spin-squeezing timescale (τss) as a function of detuning from the f = 4 to f0 = 5 transition.
and tensor effects.
Chapter 9
Eigenstate Preparation with Measurement and Control
In this chapter, we start with the idealized stochastic master equation of equation (8.31) and analyze it in full detail, with the addition of feedback to make the preparation of Fˆz eigenstates deterministic. This chapter is adapted directly from [21] and is somewhat repetitive of previous chapters in order to be self-contained. However, the two sections section 9.7 and section 9.8 are new and present original analytic results. The idea of this chapter is made considerably more rigorous, albeit for a spin system of only one spin, in [22] where notions of stochastic stability with state-based control laws are introduced. That work and the review [23] also discuss the constructive derivation of adequate control laws rather than the intuitive process used in this chapter.