Spin-velocity correlations of optically pumped atoms

We present efficient theoretical tools for describing the optical pumping of atoms by light propagating at arbitrary directions with respect to an external magnetic field, at buffer-gas pressures that are small enough for velocity-selective optical pumping (VSOP) but large enough to cause substantial collisional relaxation of the velocities and the spin. These are the conditions for the sodium atoms at an altitude of about 100 km that are used as guidestars for adaptive optics in modern ground-based telescopes to correct for aberrations due to atmospheric turbulence. We use spin and velocity relaxation modes to describe the distribution of atoms in spin space (including both populations and coherences) and velocity space. Cusp kernels are used to describe velocity-changing collisions. Optical pumping operators are represented as a sum of poles in the complex velocity plane. Signals simulated with these methods are in excellent agreement with previous experiments and with preliminary experiments in our laboratory.

Figure 1

Keilson-Storer kernels $W_a$ of (57) with representative memory parameters $a$, compared to cusp kernels $C_s$ of (66) with sharpness parameters $s$ that have the same expectation value, $\langle a\rangle ={\int }_0^1{P}_s\left(a\right)ada=s/(s+1)$, of the memory parameter. As $a\to 0$ and $s\to 0$, the KS and cusp kernels generate Maxwellian distributions.

Figure 2

Resolvent kernels $\overline{W}$ versus buffer-gas pressure $p$ for the initial velocity $y=1.37$. Here, the collision kernel $W_{\mathrm{vd}}$ of (69) is a three-cusp fit from Ref. [7] to a measured kernel for Rb in He from Ref. [14]. The superposition parameters are $f_j=[0.13,0.37,0.50]$ and $s_j=[7.8,27.2,500]$. (a) As described by the expressions of Sec. 8a, with increasing buffer-gas pressure the resolvent kernel $\overline{W}$ evolves from a low-pressure, single-collision limit, $\overline{W}=W_{\mathrm{vd}}$, to a high-pressure limit, which is a Maxwellian distribution $\overline{W}=|v_0\left)\right((v_0|$. (b) With increasing pressure the weights $g_k$ of the resolvent kernel tend to the limit $g_k=[1,0,0]$. (c) The sharpnesses $r_k$ of the resolvent kernel diminish with pressure, and in the high pressure limit, $r_1\to 0$, give a Maxwellian distribution for the only resolvent cusp with appreciable weight, $g_1\approx 1$. (d) The pressure-dependent rates ${\gamma }_{\mathrm{vd}}$ of (A1) and ${\gamma }_{\mathrm{w}}$ of (79) are for $a=1$ mm, $b=5$ mm, $T=60{}^{\circ }$C, and $D_0=0.15$ cm${}^2$ s${}^{-1}$ at 760 Torr. The transition from low-pressure to high-pressure conditions occurs when ${\gamma }_{\mathrm{w}}\approx {\gamma }_{\mathrm{vd}}$.

Figure 3

Energy sublevels of an alkali-metal atom, resonant frequencies ${\nu }_{\overline{f}f}$, and representative transitions for absorption of light (depopulation pumping), or spontaneous emission of light (repopulation pumping). In this example the pumping light is linearly polarized parallel to a small, externally applied magnetic field. Because of the Doppler shift, the resonance frequencies depend on the atomic velocity and on the direction of propagation of the light.

Figure 4

Resonant velocities $\pm x_{\overline{\mu }\mu }$ of (92) for atoms moving along a pump beam (the solid lines sloping up to the right) and for a retroreflected and attenuated probe beam (the dotted lines sloping downward to the right) as a function of the detuning of the laser frequency from the center of the 770 nm D1 line of ${}^{39}$K atoms in a magnetic field of $B=1$ G. Both the pump and probe light are linearly polarized parallel to the magnetic field, as indicated in Fig. 3. The absorption frequencies from Fig. 3 to which each group of resonant velocities correspond are denoted by ${\nu }_{\overline{f}f}$. For a group of unpolarized atoms with dimensionless velocity $x=0.75$, we display the absorption cross sections ${\sigma }_{+}(\nu ,x)$ of a pump beam (the solid line) and probe beam ${\sigma }_{-}(\nu ,x)$ (the dashed line). Plotted as solid lines along the vertical axis at the center figure are the first-order population shifts $\left(x\right|{\chi }_{\mu \mu }^{\left\{1\right\}})$ of (148) as a function of velocity $x$ for the ground-state sublevel $|\mu \rangle$ with low-field quantum numbers $f=2$ and $m=2$, produced by a pump beam with a frequency detuning of 0 GHz. Plotted as a dashed line on the same scale is the velocity dependence of the factor $\mathrm{Re}\left(\right(s_0|A_{\mathrm{dp}}\left(x\right)\left|\mu \mu \right)$ of (148) for the probe beam. Collisions in a 10 mTorr buffer gas at $T=50{}^{\circ }$C have transferred atoms in the sublevel $|\mu \rangle$ from their four resonant excitation velocities to other velocities, some of which are in resonance with the probe beam. For these curves, $a=0.1$ cm, $b=0.5$ cm, and $D_0=0.1$ cm${}^2$ s${}^{-1}$. The dot-dashed line is a Maxwellian distribution of velocities, $\left(x\right|v_0)$ of (48). Strong VSOP resonances may be observed at laser frequencies that make pump and probe resonant velocities equal, within the envelope of the Maxwellian velocity distribution.

Figure 5

A MCD signal modeled with (148) for measurements like those shown in Fig. 4 of Aminoff et al. [18]. The modeling parameters were temperature $T=150{}^{\circ }$C, optical-coherence linewidth ${\gamma }_{\mathrm{oc}}/2\pi =10$ MHz, longitudinal field $B=1$ G, buffer-gas pressure $p=30$ mTorr, spatial diffusion coefficient $D_0=0.6$ cm${}^2$ s${}^{-1}$, laser beam diameter $a=3.5$ mm, and a cell diameter $b=10$ mm. The cusp-kernel parameters of Fig. 2 were used with no adjustment.

Figure 6

Magnetic depolarization signal predicted with (147) for a transverse field of $B=0.1$ G, but with all other modeling parameters the same as those of Fig. 5. The “collisional background” of the saturated absorption signals is greatly modified and even reversed in sign. Such experiments could be used to determine the optimal cups kernels for describing velocity-changing collisions.

Figure 7

Schematic of a VSOP experiment, described in more detail in the text. For the somewhat similar experiment of Aminoff et al. [18], modeled in Fig. 5, and for the proposed experiment of Fig. 6, the intensity chopper is replaced by a device that modulates the circular polarization of the pumping light, and a weak probe beam has fixed circular polarization. Alternatively, one could modulate the polarization of the probe beam and use fixed circular polarization for the pump beam, which would eliminate any concerns about transient buildup and decay of the optically pumped spin polarization.

Figure 8

The left panels are VSOP signals from potassium vapor, measured with the apparatus of Fig. 7. The right panels are signals modeled with (148) for monoisotopic ${}^{39}$K vapor. The buffer-gas pressures were (a) no intentionally added buffer gas, (d) 1 mTorr of N${}_2$, (b) or (e) 100 mTorr of Kr, and (c) or (f) 1500 mTorr of N${}_2$. The modeling parameters were a diffusion coefficient $D_0=0.10$ cm${}^2$ s${}^{-1}$ for both Kr and N${}_2$ at 1 atm, laser beam diameter $a=1$ mm, and cell diameter $b=5$ mm. The cusp-kernel parameters of Fig. 2 were used with no adjustment.