Pump-probe nonlinear magneto-optical rotation with frequency-modulated light

Specific types of atomic coherences between Zeeman sublevels can be generated and detected using a method based on nonlinear magneto-optical rotation with frequency-modulated light. Linearly polarized, frequency-modulated light is employed to selectively generate ground-state coherences between Zeeman sublevels for which $\mathrm{\Delta }m=2$ and $\mathrm{\Delta }m=4$ in ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ atoms, and additionally $\mathrm{\Delta }m=6$ in ${}^{85}\mathrm{Rb}$. The atomic coherences are detected with a separate, unmodulated probe light beam. Separation of the pump and probe beams enables independent investigation of the processes of creation and detection of the atomic coherences. With the present technique the transfer of the Zeeman coherences, including high-order coherences, from excited to ground state by spontaneous emission has been observed.

Figure 1

Visualization of the averaging of the PMs. The surface distance from the origin in a given direction is equal to the probability of finding the highest projection of the angular momentum $(M=F)$ in that direction. The quadrupole and hexadecapole moments created in the medium due to interaction with light polarized along the $y$ axis are shown in (a) and (d). After creation the atomic polarization rotates around the magnetic field applied along the $z$ axis. If the pumping is modulated at four times the Larmor frequency newly created multipoles (a) and (d) are rotated by $90°$ with respect to the previously created quadrupole (b) and hexadecapole (e) moments. The resultant polarization of the medium after many pumping cycles related to quadrupole and hexadecapole is shown in (c) and (f). Since the averaged quadrupole is symmetric with respect to rotation around the $z$ axis the time-dependent changes of the optical anisotropy of the medium affecting light propagating along the $z$ axis are related only to the hexadecapole moment. The plots are drawn for an $F=2$ state.

Figure 2

The averaged PMs: twofold symmetric quadrupole (a), fourfold symmetric hexadecapole (b), and sixfold symmetric hexacontatetrapole (c). Each of the averaged PMs is associated with the Zeeman coherences in the ground state: averaged quadrupole with $\mathrm{\Delta }m=2$, averaged hexadecapole with $\mathrm{\Delta }m=4$, and averaged hexacontatetrapole with $\mathrm{\Delta }m=6$. The plots are drawn for an $F=3$ state.

Figure 3

Experimental setup. PD1 and PD2 are photodiodes.

Figure 4

The two-beam FM NMOR in-phase (a) and quadrature (b) signals in the modulation-frequency domain with fixed magnetic field $({\Omega }_L\approx 2\pi \times 150\mathrm{Hz})$. Signals at $\sim 300\mathrm{Hz}$, $\sim 600\mathrm{Hz}$, and $\sim 900\mathrm{Hz}$ are related to the twofold symmetric multipoles (mostly quadrupole), fourfold symmetric multipoles (mostly hexadecapole) and sixfold symmetric hexacontatetrapole moment, respectively. The insets show magnification of the hexadecapole and hexacontaterapole resonances. The signals were recorded in ${}^{85}\mathrm{Rb}$ with $I_{pump}=17\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$ and $I_{probe}=20\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$. Pump laser was tuned to the center of the $F=3\to F^{\prime }$ transition group and probe laser to the high-frequency wing of the $F=3\to F^{″}$ transition group. The units of the signal correspond to the rotation of the polarization plane of the probe beam multiplied by the probe-beam intensity. In order to calculate the angle of the rotation the recorded signal should be divided by the probe-light intensity.

Figure 5

Single-beam (a) and two-beam (b) FM NMOR signals measured in ${}^{87}\mathrm{Rb}$. The upper inset shows a magnification of the resonance $({\Omega }_{mod}\approx 2\pi \times 97.5\mathrm{Hz})$ appearing due to the modulation of the probe light in single-beam FM NMOR. The lower inset shows the absence of that signal in the two-beam arrangement. The pump-beam central frequency and probe-beam frequency were tuned to the low-frequency wing of the $F=2\to F^{\prime }=1$ transition of the $D1$ line and the center of the $F=2\to F^{″}$ transition group of the $D2$ line, respectively, in the two-beam experiment. In the single-beam FM NMOR experiment light was tuned to the low-frequency wing of the $D1$ line. The intensities of the pump and probe in the two-beam experiment were the same $(I_{pump}=I_{probe}=20\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2)$, half of the intensity of the incident light in the single-beam experiment $(I=40\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2)$.

Figure 6

The amplitude of the quadrupole and hexadecapole signals vs pump-beam intensity for ${}^{87}\mathrm{Rb}$. The quadrupole signal grows linearly with pump-light intensity at low intensities, and then levels off due to saturation. Under these experimental conditions the hexadecapole signal was not large enough to be observed at intensities much below saturation. The two dashed lines show linear and quadratic slopes in pump intensity. The solid portions show the regions in which the experimental points obey linear and quadratic dependences, respectively. The intensity of the probe, tuned to the center of the $F=2\to F^{″}$ transition group, was $1.6\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$ and the pump-beam central frequency was tuned to the low-frequency wing of the $F=2\to F^{\prime }=1$ transition group.

Figure 7

The widths of quadrupole and hexadecapole resonances as a function of pump-beam intensity. The intensity of the probe is $1.6\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$. Both beams were tuned the same transitions as for the amplitude pump-intensity dependences (Fig. 6).

Figure 8

The ${}^{87}\mathrm{Rb}$ FM NMOR spectra as a function of the pump-light central frequency for the quadrupole (a) and hexadecapole (b) resonances along with the transmission spectrum (c). Note the different spectral dependences of the multipoles. The signal components observed at the $F=1\to F^{\prime }$ transition result from the coherence transfer from the excited state $F^{\prime }$ to the $F=2$ ground state via spontaneous emission. (The inset shows the magnification of the signal related to the transfer of the $\mathrm{\Delta }m=4$ coherences.) The vertical lines in plot (c) show the relative transition strengths of the $F\to F^{\prime }$ hyperfine transitions, where the number below each line indicates the value of $F^{\prime }$. The probe laser was tuned to the center of the $F=2\to F^{″}$ transition group and its intensity was set to be $3.2\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$. The pump intensity was $59\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$. The absorption spectrum (c) was recorded at low light intensity $(2.6\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2)$.

Figure 9

The amplitudes of the quadrupole, and hexadecapole signals vs probe-light intensity at high pump-light intensity $(70\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2)$. The solid lines indicate the region in which the quadrupole has approximately linear and the hexadecapole approximately quadratic dependence on probe-beam intensity. The signals were recorded with the central frequency of the pump laser tuned to the low-frequency wing of the $F=2\to F^{\prime }=1$ transition and the probe beam tuned to the center of the $F=2\to F^{″}$ transition group of ${}^{87}\mathrm{Rb}$.

Figure 10

The widths of the FM NMOR signals vs probe-light intensity. The pump beam was tuned to the low-frequency wing of the $F=2\to F^{\prime }=1$ transition and the probe light was tuned near the center of the $F=2\to F^{″}$ transition group of ${}^{87}\mathrm{Rb}$. The pump intensity was set to $8\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$.

Figure 11

Quadrupole (a) and hexadecapole (b) signals as a function of probe-beam frequency; (c) reference absorption spectrum of ${}^{87}\mathrm{Rb}$. The inset shows the magnification of the quadrupole signal related to the transfer of the coherences $\mathrm{\Delta }m=2$ from the excited $F^{\prime }=1$ or $F^{\prime }=2$ state to the ground state $F=2$. In order to avoid saturation effects due to the pump light, the pump-beam intensity was relatively low $(3.2\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2)$. This entails the use of a high probe-beam intensity $(57\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2)$. The pump-beam central frequency was tuned to the low-frequency wing of the $F=2\to F^{\prime }=1$ transition.

Figure 12

The amplitudes of the FM NMOR resonances related to the quadrupole, hexadecapole, and hexacontatetrapole moments in ${}^{85}\mathrm{Rb}$ vs pump-beam intensity. The dashed lines show linear, quadratic, and cubic slopes while their solid parts indicate the region in which recorded dependences obey theoretical predictions. Under these experimental conditions $(I_{probe}=36\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2)$ the measured amplitude dependence of the hexacontatetrapole signal is weaker than predicted cubic relation which is a result of saturation. The pump-beam central frequency was tuned to the center of the $F=3\to F^{\prime }$ transition group while the probe beam was tuned to the high-frequency wing of the $F=3\to F^{″}$ transition group of ${}^{85}\mathrm{Rb}$.

Figure 13

The amplitudes of the FM NMOR signals related to the quadrupole, hexadecapole, and hexacontatetrapole moments in ${}^{85}\mathrm{Rb}$ as a function of probe-light intensity. The dashed and solid lines have the same meaning as in Fig. 12. The tuning of the pump beam was the same as in the previous case and its intensity was $52\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$. The probe beam tuning was slightly different than in previous case and it was locked to the center of the $F=3\to F^{″}$ transition group of ${}^{85}\mathrm{Rb}$. This difference in tuning of the probe beam is a reason why the amplitudes of the FM NMOR signals presented in this plot are smaller than those in Fig. 12.

Figure 14

The widths of the ${}^{85}\mathrm{Rb}$ quadrupole, hexadecapole, and hexacontatetrapole (inset) FM NMOR resonances vs pump-light intensity. The central frequency of the pump beam was tuned near the center of the $F=3\to F^{\prime }$ transition group. The probe beam was tuned to the high-frequency wing of the $F=3\to F^{″}$ transition group and its intensity was $36\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$.

Figure 15

The widths of the FM NMOR signals related to the quadrupole, hexadecapole, and hexacontatetrapole moments vs probe-light intensity. The pump beam central frequency was tuned to the center of the $F=3\to F^{\prime }$ transition group and its intensity was $52\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$. The probe beam was tuned to the center of the $F=3\to F^{″}$ transition group.

Figure 16

Hexacontatetrapole signal (a) and the reference absorption spectrum taken with low light intensity $(2.5\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2)$ (b) vs pump-beam central frequency recorded with ${}^{85}\mathrm{Rb}$. The strong hexacontatetrapole signal is measured at the $F=3\to F^{\prime }$ transition group which supports generation of this multipole in the ground state. The hexacontatetrapole signal is also observed for the pump light tuned to the $F=2\to F^{\prime }$ transition group which does not support creation of the quadrupole in the ground state. This signal is due to the transfer of the coherences from the excited state $F^{\prime }=3$ to the ground state $F=3$ via spontaneous emission. The probe laser was tuned to the center of the $F=3\to F^{″}$ transition group, while pump-light central frequency was scanned over all hyperfine components of ${}^{85}\mathrm{Rb}$ $D1$ transition. For the measurements of the signal related to the hexacontatetrapole moment the pump-beam intensity was $64\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$ and the probe-beam intensity was $65\mathrm{\mu }\mathrm{W}∕{\mathrm{mm}}^2$.

Figure 17

The signal related to the hexacontatetrapole moment (a) and the reference spectrum taken for low light intensity (b) vs probe-light frequency. The central frequency of the pump beam was tuned to the center of the $F=3\to F^{\prime }$ transition group while the probe-beam frequency was scanned over all hyperfine components of the ${}^{85}\mathrm{Rb}$ $D2$ line. The pump- and probe-beam intensities were chosen to be the same as in Fig. 16.