The second and third lines of the above master equation represent the polarizability Hamiltonian and describe the coherent, dispersive interaction of the beam with the atoms.
In the next chapter, we start from this point to analyze the evolution of the optical polar- ization due to the atomic spin-state. In the remaining part of this chapter, however, we continue to analyze the state dependence of the incoherent spontaneous emission terms.
Figure 5.1: (A) Schematic showing the relative orientation of the spin, magnetic field, and optical polarization. (B) Simulation of ˆfz for an initial state along the x-axis, magnetic field along the y-axis, probe propagation along the z-axis, probe polarization along the y- axis. (C) Simulation for an initial state along x-axis, magnetic field along the y-axis, probe propagation along the z-axis, probe polarization 54.7 degrees from the y-axis. The probe detuning used is 500 MHz.
When considering the following simulation configurations, keep in mind that we continue to consider only the evolution of a single atom, but that evolution consists of many parts, including the polarizability Hamiltonian (which can cause coherent “dephasing”) and the incoherent spontaneous emission. We do not consider the collective evolution of the spin- state until chapter 7.
5.2.1 Decay with Larmor Precession
First we consider Larmor precession dynamics and produce results similar to those from [25]. Figure 5.1A shows the relative orientation of the spin, magnetic field, and optical polarization directions. Figure 5.1B depicts a situation with an initial state polarized along the x-axis, a DC magnetic field along the y-axis, and a probe beam propagating along the z-axis with linear optical polarization along the y-axis. The decay and revivals, or echoes, are from the tensor Hamiltonian part of the master equation, which is shown to be nonlinear in the angular momentum operators in the next chapter. Without spontaneous emission the revival would be complete. Figure 5.1C depicts the same situation, but with the probe polarization aligned along a “magic” angle (54.7 degrees from the y-axis) where the nonlinear Hamiltonian term disappears and a more exponential decay is observed. The
rotating frame analysis used to derive this magic angle effect is presented in section 6.5.
5.2.2 Decay without Larmor Precession
Now we consider the evolution under configurations where either there is no magnetic field or the field and spin-state remain parallel, hence dynamics at the Larmor frequency are not observed. We consider two cases, in one the linear optical polarization is parallel to the initial spin-state, and in the other it is perpendicular.
The simulated evolution under the four possible configurations (two optical polariza- tions, with and without field) is presented in figure 5.2. For parallel polarizations, the field and no field cases are not any different because the evolution due to the field does not break the symmetry of the no field case. For perpendicular polarizations, however, there is a large difference between the field and no field case. Without a field the length of the vector dephases quickly (blue curve in figure 5.2A) and displays echoes similar to the ef- fects demonstrated in the last section. Not surprisingly the perpendicular uncertainties (in the yz-plane) also behave dramatically and asymmetrically in figure 5.2B–C although there is no single-atom spin-squeezing in this configuration. In contrast, the parallel spin case shows simple exponential decay and the uncertainty variances initially grow in time (this is purely a result of the decay and spontaneous emission,not the tensor Hamiltonian). When a field is applied to the spins in the perpendicular field case, the dephasing is eliminated and the resulting spontaneous emission time becomes longer than the parallel field case (as demonstrated below).
figure 5.2B displays the loss of population to the f = 3 ground state where it is clear that the spontaneous emission rate is state (or optical polarization) dependent. A more extreme example of the state dependent pumping could be demonstrated by putting the atoms in a state that was dark to the probe light, e.g., the state|4,4i withσ+ probe light.
Now we go about calculating the expected spontaneous emission decay rates. In analyz- ing the decay, we can consider many quantities, including the moment ˆfz or the population among all of the f = 4 sublevels. Here we consider the damping of ˆfz. Assume we start with the atom aligned along the +z direction and consider the quantity dhfˆzi/dtat time zero calculated from equation (5.44). If we convert this derivative into a scattering rate by
Figure 5.2: As described in the text, simulations of decay and decoherence via linearly polarized probe light for an initial coherent spin-state. The blue (b) and cyan (c) curves represent optical polarizations (along y) perpendicular to the initial spin direction (along x). The red (r) and magenta (m) curves represent parallel optical polarization (along x).
The blue and red curves are without any field, and the cyan and magenta curves are with a field along the spin direction x. The momentshfˆzi and hfˆyi are zero. The probe detuning used is 500 MHz.
assuming initial exponential decay dhfˆzi
dt t=0
= −hfˆzi τsc
t=0
(5.45)
we get the decay rate
τsc−1 = Iσ0
2~ω
Γ2/4
Γ2/4 + ∆245 |0|20.16 +|−|20.04 +|+|20 + Γ2/4
Γ2/4 + ∆244 |0|20.83 +|−|20.41 + Γ2/4
Γ2/4 + ∆243 |−|20.60
. (5.46)
Clearly, we see that this expression respects some expected selection rules by, for example, not allowing positive circular polarization to produce any decay. This is because such light either does not couple to a level (4–3’ or 4–4’) or it couples to a cycling transition that only returns the population to where it started (4–5’). For an optical polarization parallel to the spins we use: |+|2 = |+|2 = 0 and |0|2 = 1. Under the simulation, we see that the Hamiltonian terms alone do not cause decay so the calculated exponential from above works with or without a holding field. For an optical polarization perpendicular to the spins we use: |+|2 =|+|2 = 1/2 and |0|2= 0. Under the simulation, we see that there is substantial Hamiltonian “dephasing” (with echoes) of the moment, but applying a holding field makes equation (5.46) a valid expression at all times.
When the field is applied in figure 5.2, the coherent dephasing is removed and the re- sulting decay curves are exponential with time constants given by equation (5.46). Even though the above expression was only derived for the first timestep, it is seen by simulation to remain valid at all times with such a holding field. Both equation (5.46) and the sim- ulations of figure 5.2 show the feature that spontaneous emission occurs faster for parallel polarizations than perpendicular, which can also be expressed in terms of selection rules and Clebsch-Gordan coefficients.
Again, we have said nothing about the consequences of this decay or non-QND Hamilto- nian evolution on the collective spin-squeezed state preparation and we save this discussion for chapter 7 where both single-atom and collective effects are discussed. In chapter 8, we also refer back to equation (5.46) to provide an estimate of the decay time in relation to the
time that it takes to spin-squeeze the atoms via the measurement. The potential for a less ad hoc combination of spin-squeezing in the presence of decay is discussed in both chapter 15 and appendix E.
Chapter 6
The Irreducible Representation of the Polarizability Hamiltonian
In the last chapter, we demonstrated that adiabatic elimination of the dipole Hamiltonian describing the atom-light interaction produces the polarizability form of the Hamiltonian in terms of only the ground states. In this chapter, we rewrite the polarizability Hamiltonian in terms of the angular momentum operators for the ground state, rather than the dipole operators, and the Stokes operators describing the polarization of the probe beam. There is an irreducible decomposition of the polarizability tensor into purely scalar, vector, and tensor terms, and the physics of each corresponding Hamiltonian is discussed. The scalar term determines the state-independent light shift, which is important in, for example, the trapping of atoms with an optical lattice. The vector term leads to the QND Hamiltonian we desire and provides the basis for much of this work. The tensor term arises because the spin has more than two sublevels and is important at detunings comparable to the hyperfine splittings. After introducing these terms, and investigating their spectral properties, we describe the semiclassical evolution of the probe polarization due to the atomic state via the Hamiltonian components. We then consider the evolution of the spin-state due to the tensor term in special, experimentally relevant, configurations. While the tensor term complicates the ideal description of the quantum measurement, it also proves technically useful because of its nonlinearity. We end by showing how the tensor term provides us with techniques for aligning the optical and atomic polarizations, as well as measuring the degree of pumping of the atomic state.
Much of this chapter is adapted from [29], however many sections are new, including section 6.3, section 6.5, section 6.7, and section 6.8.