Transient dynamics of a nonlinear magneto-optical rotation

We analyze nonlinear magneto-optical rotation (NMOR) in rubidium vapor subjected to a continuously scanned magnetic field. By varying the magnetic-field sweep rate, a transition from traditionally observed dispersivelike NMOR signals (low sweep rate) to oscillating signals (higher sweep rates) is demonstrated. The transient oscillatory behavior is studied versus light and magnetic-field parameters, revealing a strong dependence of the signals on magnetic sweep rate and light intensity. The experimental results are supported with density-matrix calculations, which enable quantitative analysis of the effect. Fitting of the signals simulated versus different parameters with a theoretically motivated curve reveals the presence of oscillatory and static components in the signals. The components depend differently on the system parameters, which suggests their distinct nature. The investigations provide insight into the dynamics of ground-state coherence generation and enable application of NMOR in detection of transient spin couplings.

Figure 1

(a) Schematic diagram of experimental setup used to measure NMOR signals. PBS is the polarizing beam splitter; $\lambda /2$ is the half-wave plate; P is the polarizer; GLP is the Glan-laser crystal polarizer; PD1 and PD2 are the photodetectors; and $\mu \mathrm{DAVLL}$ is the dichroic atomic vapor laser lock system, exploiting a microsize rubidium vapor cell. (b) Two-level atomic system $F_g=2\to F_e=1$ interacts with left (${\sigma }^{-}$) and right (${\sigma }^{+}$) circular polarized components of the linear polarized light in the $\mathrm{\Lambda }$ configuration. Optical coherences between ground-state and excited-state Zeeman sublevels are given by ${\rho }_{-e}$ and ${\rho }_{+e}$, and ${\rho }_{-+}$ represents the ground-state Zeeman coherences with $\mathrm{\Delta }m=2$ (solid curved arrows) and $\mathrm{\Delta }m=4$ (dashed curved arrow).

Figure 2

Experimentally measured (top row) and numerically simulated (lower row) NMOR signals as a function of time and magnetic field at different magnetic-field sweep rates. For small sweep rates, $4.5\mu \mathrm{G}/\mathrm{s}$ in (a) and (d), the signals are quite similar to a conventional NMOR signal (with a small asymmetry between two wings). For larger rates, $700\mu \mathrm{G}/\mathrm{s}$ in (b) and (e), $1800\mu \mathrm{G}/\mathrm{s}$ in (c) and (f), the oscillations of polarization rotation are observed while crossing $B=0$. The period of oscillations varies with time due to the change of magnetic field. The laser beam intensity is set to 2 $\mu \mathrm{W}/{\mathrm{mm}}^2$ at the cell. The parameters used in the simulations are $\gamma /2\pi =2.5\mathrm{Hz}$ and $\mathrm{\Gamma }/2\pi =5.75\mathrm{MHz}$. For better agreement with the experimental results, the Rabi frequency ${\mathrm{\Omega }}_R/2\pi$ is 20 kHz in (d) and (e), and 28 kHz in (f).

Figure 3

Experimentally measured (top row) and numerically simulated (lower row) NMOR signals as a function of time and magnetic field with magnetic-field sweep rate 914 $\mu \mathrm{G}/\mathrm{s}$ at different light intensities: (a) 0.71 $\mu \mathrm{W}/{\mathrm{mm}}^2$, (b) 4.2 $\mu \mathrm{W}/{\mathrm{mm}}^2$, (c) 21.2 $\mu \mathrm{W}/{\mathrm{mm}}^2$, and Rabi frequencies (${\mathrm{\Omega }}_R/2\pi )$ (d) $12\mathrm{kHz}$, (e) $30\mathrm{kHz}$, and (f) $70\mathrm{kHz}$ (other parameters are the same as before). The results show that damping of the oscillations depends on light intensities and prove that the damping is a manifestation of faster relaxation due to power broadening and polarization repumping.

Figure 4

Experimentally measured (a) and numerically simulated (b) NMOR signals as a function of time at a magnetic-field sweep rate of 914 $\mu \mathrm{G}/\mathrm{s}$. The dashed curve (red) and solid curve (black) show the rotation signals for $B_0=457\mu \mathrm{G}$ and ${\mathrm{\Omega }}_n/2\pi =2\mathrm{Hz}$, and for $B_0=914\mu \mathrm{G}$ and ${\mathrm{\Omega }}_n/2\pi =1$ Hz, respectively. The measurements are performed with light intensity (a) 0.71 $\mu \mathrm{W}/{\mathrm{mm}}^2$ and (b) Rabi frequency ${\mathrm{\Omega }}_R/2\pi =12\mathrm{kHz}$.

Figure 5

Amplitude ${\phi }_o$ (a) and decay rate ${\gamma }_o$ (b) of the oscillation component and amplitude ${\phi }_s$ (c) and decay rate ${\gamma }_s$ (d) of the static component extracted using Eq. (10) as functions of scanning field frequencies and amplitudes. The parameters used in simulations are ${\mathrm{\Omega }}_R/2\pi =7\mathrm{kHz}$, $\gamma /2\pi =2.5\mathrm{Hz}$, and $\mathrm{\Gamma }/2\pi =5.75\mathrm{MHz}$.

Figure 6

Amplitude of the oscillatory component ${\phi }_o$ of the observed NMOR signal versus the sweep rate for two Rabi frequencies at scanning frequency ${\mathrm{\Omega }}_n=8$ Hz. The modulation of the amplitude, observed for much weaker Rabi frequency, is the effect of the interference between initially created coherence $t<t_0$ and the coherences generated for $t>t_0$. If both coherences are in phase, the amplitude of the signal is enhanced, whereas if they are out of phase, it is deteriorated.

Figure 7

Amplitude ${\phi }_o$ (a) and decay rate ${\gamma }_0$ (b) of the oscillating component of the signal and amplitude ${\phi }_s$ (c) and decay rate ${\gamma }_s$ (d) of the static component of the observed signals as functions of the Rabi frequency ${\mathrm{\Omega }}_R$ and ground-state relaxation rate $\gamma$ at a given sweep rate (5 mG/s).