Chapter 8
Polarimetry and Quantum Measurement
In previous chapters, we have described general quantum measurement concepts (Chapters 1–2) and unconditional descriptions of the probe-atom interaction without measurement (Chapters 3–7). Now we begin to put the two together to consider how the measurement of the probe beam allows us to conditionally evolve the collective quantum spin-state of the atom cloud. A proper description of this quantum estimation process will allow us to understand the back-action of the measurement and conditional preparation of spin- squeezed states.
We begin by describing some of the basic concepts behind the detection of optical polarization rotations with an emphasis on signal-to-noise ratios (SN R). The concept of shotnoise in a polarimeter is introduced prior to the conditioning section because it is the fundamental source of noise that the quantum filter estimating the atomic state must fight against. Then we describe the derivation of the stochastic master equation telling us how to conditionally update the collective state of the atoms via the inherently noisy measurement record.
quarter-waveplate and one half-waveplate), which together can enact a general rotation on the Stokes sphere as described in section 3.2.5. After passing through the waveplates, the beam is split with a polarizing beamsplitter (PBS) and each beam is then detected by a photodiode. The photodiode currents are subtracted and amplified to give the output of the polarimeter, which is proportional to the measured Stokes component. See section 11.6 for a description of the actual components used in the experiment.
8.1.1 Optical Shotnoise
Consider a photodiode absorbing a continuous powerP. The power leads directly to a mean photocurrent of value
I =RP. (8.1)
The responsivity of a photodiode is defined as R= ηe
~ω[A/W] (8.2)
whereηis the quantum efficiency of the detector,eis the electron charge, and~ωis the single photon energy. If the quantum efficiency is equal to unity, one can think of each photon as being converted into an electron that contributes to the current. In a measurement of the current, there will be random fluctuations in the signal due to the random uncorrelated arrival times of the electrons, described by a Poisson process. In a given time τ, one can show via the properties of the Poisson process (see http://qwiki.caltech.edu/wiki/
Shot_Noiseor [108]) that the RMS fluctuation is related to the mean current by i2rms= eI
τ . (8.3)
For an averaging filter with time constantτ, the frequency interval is defined as ∆f = 1/2τ (note the factor of two) [108]. The optical shotnoise floor is then
irms =
√ 2eI
h A/
√ Hzi p
∆f (8.4)
= p
2e2ηP/~ω h
A/
√ Hzi p
∆f . (8.5)
This is consistent with the noise one expects to get from performing photon detection on an optical coherent state [109, 68]. The goal in the design of the photodiode detector is to get the optical shotnoise floor significantly larger than the electronic noise floor of the detector for the power the experiment demands. See chapter 11 for a discussion of the detector used in our experiment. Note that when representing the optical shotnoise as a Wiener process we use
I(t)dt=Idt+
√
eIdW(t) (8.6)
where I is a constant representing the mean current. This expression reproduces the rela- tions above when calculating the RMS current using Itˆo’s rule [52].
8.1.2 Polarimeter Unravellings
After the the probe beam interacts with the atoms, one can imagine measuring the probe beam in many different ways, each of which leads to a differentunravelling of the conditional dynamics as discussed in chapter 2. Note that here we consider the case of an input linearly polarized probe beam of constant power and only change the measurement basis, although we could also imagine adaptively changing the probe beam state for some purpose.
As discussed in section 3.2.7, one can measure any Stokes component with a polarimeter (two input waveplates, a PBS, and a subtracting photodetector). The standard balanced polarimetry measurement is where the linearly polarized probe beam is placed at 45 degrees with respect to the axes of a polarizing beamsplitter (PBS) and half of the light is put on each detector. Any rotations of the linear polarized light that keep it linear will then be measured by the difference output of the photodetectors. In terms of Stokes components, we say that the input state maximizes ˆSx and we measure ˆSy.
As another example, the polarimeter can be operated in a completely “imbalanced”
configuration, with the input state at ˆSx and measuring ˆSx. In this case the analyzer PBS is oriented at 45 degrees to the balanced configuration and all of the noninteracted light goes to one of the detectors. The other detector with less light can then be monitored, possibly in photon-counting configuration, for a signal from the atoms. Note that in this configuration, the detector cannot possibly distinguish which way (cw or ccw) the linear polarization was rotated when a nonzero signal is detected. As a result of this indistinguishability, the conditional dynamics under this unravelling must respect this lack of knowledge, leading to
the possibility of conditional cat state production [90].
As a final possible detection scheme, we mention that the rotations of a linearly po- larized probe beam can considered as coming from a differential phase shift between the circular basis components composing the linearly polarized light. Thus one could also do a measurement by physically separating these two components (e.g., with a quarter-wave plate and a PBS) and then doing a homodyne measurement on each of the components.
The signal-to-noise ratio from this scheme is compared to the balanced polarimetry scheme in the next section.
8.1.3 SN R Comparisons
Consider two separate schemes for measuring the polarization rotation of a single beam. In both, we initialize the input beam polarization along ˆSy on the Stokes sphere such that we can represent its electric field as
Ei = √E0
2(ex+ey). (8.7)
We temporarily ignore some constant prefactors and represent the power simply as P =
|Ei|2 =E02. Now suppose that something rotates the polarization by a small angle φ 1 such that the output field is
E= E0
√2(ex(1 +φ) +ey(1−φ)). (8.8) With this output light, we now consider two different measurement schemes and compare their signal-to-noise ratios.
8.1.3.1 Balanced Polarimetry
In the balanced polarimetry configuration, we measure the two powersPx and Py and then subtract to get the signal
Px = E02
2 (1 +φ)2 (8.9)
Py = E02
2 (1−φ)2 (8.10)
∆PP = Px−Py = 2P φ. (8.11)
We then represent the optical shotnoise as being the square root of total power measured (again ignoring constant prefactors)
ζP =√
P =|E0|. (8.12)
8.1.3.2 Double Homodyne
Now we decompose the output light into its circular components
E=E+e++E−e−, (8.13)
which can be represented as
E±= E0
2 (i(1−φ)∓(1 +φ)). (8.14)
After splitting these components apart, with the quarter-wave plate and PBS, we perform a homodyne measurement on each of the components, i.e., we mix one component with a strong local oscillatorELO at a 50/50 beamsplitter and measure the power difference of the outputs. After mixing the local oscillator with E− we have
EA = (ELO+E−)/√
2 (8.15)
EB = (ELO−E−)/√
2 (8.16)
and
∆P−=PA−PB=ELOE0(1 +φ), (8.17) which gives the total signal. When we take the PLO P limit we have for the overall shotnoise
ζP−=ELO. (8.18)
We also get the same signal from the other homodyne setup (measuring +)
∆P+=ELOE0(1−φ) (8.19)
and the same noise
ζP+ =ELO (8.20)
assuming we used the same power local oscillator. To extract the total signal we subtract the two homodyne outputs to get
∆PH = ∆P−−∆P+=ELOE02φ (8.21)
and add the noises in quadrature to get ζH =√
2|E0|. (8.22)
8.1.3.3 Polarimetry versus Double Homodyne
Comparing the signal-to-noise ratios of the above two analyses we get that the polarimetry signal-to-noise is a factor of√
2 greater than the double homodyne measurement
SN RP = ∆PP/ζP = 2|E0|φ (8.23)
SN RH = ∆PH/ζH =√
2|E0|φ. (8.24)
This difference of √
2 can be explained by the fact that the local oscillator adds unneces- sary noise to the measurement. If squeezed local oscillators were used, the SN R of the polarimetry measurement could be recovered. Also note that if heterodyne measurements were performed on the±state, then these would be another √
2 worse than the homodyne performance [11, 12].
8.1.4 Polarimeter “Amplification”
We end this section by considering a possibly convenient trick, which can be used to reduce the total power detected by the polarimeter while keeping the same signal-to-noise ratio. A major source of technical noise in a balanced polarimeter is offset fluctuations, which are proportional to the total power into the polarimeter. These can be caused, for example, by imperfect balancing. Here we briefly analyze an “amplification” scheme that involves selectively damping the polarization component that does not get measured, as discussed in [110].
Consider a polarimeter that measures the power along axesxandyand with input light of power P aligned mostly along ex0 = (ex+ey)/√
2. Define θ as the angle of the state away from ex0. The polarimeter output is then
Px−Py =Psin(2θ) +ζ(P) (8.25)
where the shotnoise of power P is represented as ζ( ˆP) = √
~ωP dW/dt. Now imagine selectively damping theex0-component by a factor ofχ <1 such that
E = E(cos(θ)ex0+ sinθey0)→ (8.26) E˜ = E(χcos(θ)ex0+ sin(θ)ey0). (8.27)
Now the new polarimeter output is
P˜x−P˜y =χPsin(2θ) +ζ( ˜P) (8.28)
where ˜P =P(χ2cos2(θ) + sin2(θ)). Comparing the signal-to-noise ratios assuming θ 1 we have
SN R = Px−Py
ζ(P) ≈2θ r P
2~ω (8.29)
SN R^ = P˜x−P˜y
ζ( ˜P) ≈2θ r P
2~ω s
χ2
χ2+θ2(1−χ2). (8.30) So if we haveχ2 θ2 then we haveSN R ≈SN R. Given that the polarization rotation is^ going to be small, this technique can be practically useful if, for example, the polarimeter subtraction is not perfect and there are unwanted power fluctuations. This erroneous signal would then go down as χ2 while the desired signal only goes down as χ. This effect can also be useful because in practice detectors often become nonlinear if they absorb too much power.
One large caveat to this approach is that the electronic noise floor does not get decreased at all, thus the relative noise-equivalent-power (NEP) of the detector will go up by a factor of 1/χ2and the detector becomes less optical shotnoise limited compared to the power of the total probe light. If there is enough of an initial cushion between optical and the electronic
noise floors, then this should not be a problem, but in practice the electronic noise floor is never going to be zero.