Quantitative study of optical pumping in the presence of spin-exchange relaxation

We have performed quantitative measurements of the variation of the on-resonance absorption coefficients ${\kappa }_0$ of the four hyperfine components of the Cs $D_1$ transition as a function of laser power $P$, for pumping with linearly and with circularly polarized light. Sublevel populations derived from rate equations assuming isotropic population relaxation (at a rate ${\gamma }_1$) yield algebraic ${\kappa }_0\left(P\right)$ dependences that do not reproduce the experimental findings from Cs vapor in a paraffin-coated cell. However, numerical results that consider spin-exchange relaxation (at a rate ${\gamma }_{\mathrm{se}}$) and isotropic relaxation fit the experimental data perfectly well. The fit parameters, viz., the absolute value of ${\kappa }_0$, the optical pumping saturation power $P_{\mathrm{sat}}$, and the ratio ${\gamma }_{\mathrm{se}}/{\gamma }_1$, are well described by the experimental conditions and yield absolute values for ${\gamma }_1$ and ${\gamma }_{\mathrm{se}}$. The latter is consistent with the previously published Cs-Cs spin-exchange relaxation cross section.

Figure 1

Schematic of the experimental setup: ECDL, external-cavity diode laser; SAFL, saturated absorption frequency lock; SMF, single-mode fiber; EOM, electro-optic modulator; PI, proportional-integral amplifier; PD1,2, photodiode; LP, linear polarizer; $\lambda /4$, quarter waveplate; $I$/$V$, current-voltage converter; DSO, digital storage oscilloscope; and Cs, Cs vapor cell. The magnetic shield and the coils are not shown.

Figure 2

Transmission spectrum showing the hyperfine structure of the Cs $D_1$ transition in a paraffin-coated vapor cell. The colored dots mark the frequencies at which the laser was set for the power scans.

Figure 3

Experimental measurements of the dependence of the optical thickness ${\kappa }_0^{F_g\stackrel{q}{\to }{F}_e}L$ on the power $P_{\mathrm{in}}$ for all four hyperfine transitions of the Cs $D_1$ line. Data were recorded with both (a) linearly polarized and (b) circularly polarized light.

Figure 4

Hyperfine and magnetic sublevel structure used to discuss the optical pumping rate equations.

Figure 5

Anticipated power dependence of the sublevel populations $p_{4,m}$ in the $F_g=4$ state when pumped on the $4\to 3$ and $4\to 4$ transitions with linearly ($q=0$) and circularly ($q=1$) polarized light, assuming isotropic relaxation. The numbers on the right in each graph represent the $m_g$ values, in the same top-to-bottom order as the curves. We note that for $q=0$, the (dashed) lines with a given $m_g$ overlap the (dotted) lines with $-m_g$.

Figure 6

Anticipated power dependence of the sublevel populations $p_{3,m}$ in the $F_g=3$ state when pumped on the $3\to 3$ and $3\to 4$ transitions with linearly ($q=0$) and circularly ($q=1$) polarized light, assuming isotropic relaxation. The numbers on the right in each graph represent the $m_g$ values, in the same top-to-bottom order as the curves. As in Fig. 5, the (dashed) lines with a given $m_g$ overlap the (dotted) lines with $-m_g$ for $q=0$.

Figure 7

Anticipated dependence of the (normalized) peak absorption coefficients on the dimensionless parameter $x={\mathrm{\Gamma }}_{\mathrm{abs}}/{\gamma }_1=P_{\mathrm{in}}/P_{\mathrm{sat}}$ given by the analytical expressions [Eq. (22)] derived from rate equations, assuming isotropic population relaxation for pumping with (a) linearly polarized light and (b) circularly polarized light.

Figure 8

Theoretical predictions of the power dependence of the normalized peak absorption coefficients ${\kappa }_0^{F_g\stackrel{q=0}{\to }{F}_e}\left(x\right)/{\kappa }_{\mathrm{tot}}^{\mathrm{unpol}}$ for pumping with linearly polarized light. Shown from left to right are increasing values of ${\varepsilon }_{\mathrm{se}}$, i.e., the ratio of the spin-exchange rate to the isotropic relaxation rate.

Figure 9

Same as in Fig. 8 for pumping with circularly polarized light.

Figure 10

Power dependence of the optical thickness ${\kappa }_0^{F_g\stackrel{q}{\to }{F}_e}L$ for the four hyperfine transitions measured with (a) linearly polarized and (b) circularly polarized light. The solid lines are fits using the solutions of the rate equation model including both isotropic and spin-exchange relaxation. The dots are the experimental values, where we dropped every second value (compared to Fig. 3) in order to make the fitted curves more visible.

Figure 11

Dependence of ${\gamma }_1^{\mathrm{eff}}/{\gamma }_1$ from Eq. (31) on ${\varepsilon }_{\mathrm{se}}={\gamma }_{\mathrm{se}}/{\gamma }_1$. The green dotted lines show how ${\gamma }_1^{\mathrm{eff}}/{\gamma }_1$ is inferred from the fitted ${\overline{\varepsilon }}_{\mathrm{se}}$ value. The dashed line marks the asymptote for ${\varepsilon }_{\mathrm{se}}\to \infty$.