The previous section showed that the tensor Hamiltonian can be useful for aligning the atomic and optical polarizations. In this section, we show that, as a direct consequence of the nonlinearity in atomic spin operators, the tensor term can also be used to measure the atomic pumping efficiency.
Optical pumping is the incoherent process [31] by which an atom absorbs angular mo- mentum from an optical beam, thus aligning the spin of the atom. We discuss simulations of this process for a real multilevel cesium atom in section 12.5.9. When it comes to measuring the spin-state of a single atom, there are various state tomography procedures available. In section 12.5.5, we discuss a standard Stern-Gerlach technique by which the internal sub- levels are separated with a sufficiently strong magnetic field gradient and measured via fluorescence. This technique is used for state tomography in experiments similar to ours, for example [105]. There are many reasons for not wanting to use this technique however, including experimental complexity and a limited measurement resolution. In principle, one can use the continuous measurement as discussed in this work to determine the initial spin- state. Extended notions on tomography with continuous quantum measurement and control are discussed in [106, 107].
Mostly we are interested in a very restricted version of the general tomography problem and we would like to know with what efficiency we have prepared the particular coherent spin-state |f = 4, mf = 4i. Here we present a simple example showing how we can do this by aligning the spin-state along a certain direction, e.g., with a adiabatic holding field, sending in a beam polarized along x, and measuring a particular output Stokes component.
By following this procedure for two different configurations, the pumping efficiency can be determined as follows.
Consider first the atomic alignment with θ = φ = 0 and measured Stokes component hSˆy0i. Referring to equation (6.77) and making the replacement f → ηf where η is repre-
sentative of the pumping efficiency, we have
γz,1≈γ0ηfX
f0
α(1)f,f0
α0∆f,f0 (6.99)
with theγx,1=γy,1 = 0. Then by specializing equation (6.84) to this case we see that hSˆy,10 i ≈ −hSˆxiγz,1. (6.100) Next, consider the state with θ=π/2 andφ=π/4 such that
γy,2≈γ0η2f(f−1/2)X
f0
αf,f(2)0
α0∆f,f0 (6.101)
with the others again zero. The measurement of the circular polarization then gives hSˆz,20 i ≈ hSˆxiγy,2. (6.102) Now by considering the measured ratio of the measurements we have hSˆz,20 i/hSˆy,10 i ∝ η, with some model dependent terms canceling in the ratio. Clearly, this scheme uses the fact that one interaction is quadratic while the other is only linear.
Of course, mapping the measured ratio to the pumping efficiency will depend on the model being used for the state population. Our goal is only to prepare a measurement that is approximately correct near unity pumping efficiency. Consider two models of the state distribution. In one, given a “temperature parameter” t we populate the states according to
ρ=
f
X
i=−f
tm|f = 4, m=iihf = 4, m=i| (6.103) then normalize. In another model, we allow only two levels to have any population and define the state as
ρ=t|f = 4, m= 4ihf = 4, m= 4|+ (1−t)|f = 4, m= 3ihf = 4, m= 3|. (6.104) As seen in figure 6.6, the models give nearly the same result for the measured ratio for nearly perfect preparation efficiencies greater than 90%. Experimental measurements using
Figure 6.6: Measurement of pumping efficiency as described in the text. On the left, the upper plot corresponds to dragging the spins in the xz-plane and measuring hSˆy0i. The lower plot corresponds to dragging the spins in the xy-plane and measuring hSˆz0i. The measurements referred to in the text are the peaks of these curves. On the right the ratio is mapped to the pumping efficiency via two different models.
this technique are presented in section 13.6.
Chapter 7
The Unconditional Collective Master Equation
Thus far we have considered only the semiclassical descriptions of the atom-light interaction.
In chapter 5, we used a classical description of the light to derive a single atom master equation with spontaneous emission. After that, we discussed the decomposition of the polarizability Hamiltonian in chapter 6, where we treated the light quantum mechanically, but only considered one of two semiclassical limits where either the optical moments or spin moments were considered as classical.
In this chapter, we start from the fully quantum Hamiltonian of the last chapter, with quantum light and quantum atoms, and derive the unconditional master equation for the collective state of the atoms. Under these circumstances, we can begin using some of the quantum measurement language that was introduced in chapter 2. In the case that the Hamiltonian only consists of the QND vector term, this will result in no entanglement un- conditionally, only antisqueezing. On the other hand, the conditional solution will produce spin-squeezing as discussed in the next chapter. However, if we include the tensor terms, and the assumption that all of the atoms interact with the same optical mode symmetri- cally, we will see that collective spin-squeezed states can be prepared via the unconditional master equation described here.
The latter assumption is a significant one, which will not be completely true under current experimental circumstances, but we continue with it for pedagogical reasons. A more realistic, and complicated, examination of the effect of the tensor term in a spatially extended atomic cloud is analyzed in [26]. For simplicity, we also continue to neglect the spontaneous emission, and other three-dimensional effects, discussed in chapter 5.