Optically induced spin echoes in rubidium atoms: On- and off-resonant manipulations of spins

We studied the optical manipulation of spin coherence in rubidium atoms by using polarization spectroscopy with the pump-probe technique. Two types of optically induced spin echoes, namely on- and off-resonant manipulations, whose creation mechanism are the optical pumping effect and the light shift effect, respectively, are studied theoretically and experimentally. A theory of optically induced spin echoes that can be applied to both types of spin echoes is considered by using the density matrix, and a general expression for the refocusing efficiency of the control pulse is presented. The effect of the off-resonant manipulation was examined by using light pulses of two frequencies provided by two lasers. Vector models for the two types of spin echoes are presented, and the observed dependences of the frequency and the duration of the control pulse on the spin-echo intensity are suitably explained by the theoretically derived refocusing efficiency. The off-resonant manipulation gives the possibility of arbitrary-angle spin rotation around an arbitrary axis.

Figure 1

Diagram of the four-level system. The solid arrow indicates the transition by the ${\sigma }^{+}$ circular pumping, and the broken arrows indicate the transitions by the spontaneous emission.

Figure 2

Pump and probe of the optically induced magnetization for the (a) on- and (b) off-resonant manipulations.

Figure 3

Typical experimental result of the observed spin transients.

Figure 4

(a) Magnetic-field dependence of the observed free induction decay signals in the ground state of ${}^{87}\text{R}\text{b}$. (b) Fourier transform of the free induction decay signals in (a).

Figure 5

Frequency dependence of the signal intensity of the observed FID. The probe pulse is tuned to the ${}^{87}\text{R}\text{b}$ $\left(F=2\to F^{\prime }=1\right)$ line, and the frequency of the pump pulse is changed. The solid circles show the observed signal intensity. The solid curve serves as a visual guide. The transmitted intensity of the pump pulse for a reference cell is also shown. A characteristic error bar is shown on one of the observed data, which is determined by the stability of the laser frequency and the electrical noise of the detection system.

Figure 6

Frequency dependence of the signal intensity of the optically induced spin echoes observed for the on-resonant manipulation. The generation, control, and probe pulses are provided by a single laser, and their frequency is swept simultaneously. The solid circles show the signal intensity of the observed spin echoes. The black solid line shows the calculated curve obtained from the sum of the contributions from the four transition lines. The transmitted intensity for a reference cell is also shown. A characteristic error bar is shown on one of the observed data, which is determined by the stability of the laser frequency and the electrical noise of the detection system.

Figure 7

Frequency dependence of the signal intensity of the optically induced spin echoes observed for the off-resonant manipulation. The generation and probe pulses are provided by a laser diode, which is tuned to the ${}^{87}\text{R}\text{b}$ $\left(F=2\to F^{\prime }=1\right)$ line. The control pulse is provided by a Ti:sapphire laser, and its frequency is changed. The solid circles show the signal intensity of the observed spin echoes. The solid curve shows the theoretical frequency dependence of the spin-echo intensity obtained from Eq. (8). The transmitted intensity for a reference cell is also shown. A characteristic error bar is shown on one of the observed data, which is determined by the stability of the laser frequency and the electrical noise of the detection system.

Figure 8

Dependence of the spin-echo intensity on the control pulse duration for the on-resonant manipulation. The solid circles show the observed spin-echo intensity. The solid curve shows the theoretical pulse-duration dependence of the spin-echo intensity obtained from Eq. (8). A characteristic error bar is shown on one of the observed data, which is determined by the stability of the laser frequency and the electrical noise of the detection system.

Figure 9

Dependence of the spin-echo intensity on the control pulse duration for the off-resonant manipulation. The solid circles show the observed spin-echo intensity. The solid curve shows the theoretical pulse-duration dependence of the spin-echo intensity obtained from Eq. (8). A characteristic error bar is shown on one of the observed data, which is determined by the stability of the laser frequency and the electrical noise of the detection system.

Figure 10

Schematic diagram for the light shift. (a) Energy shift in a two-level system interacting with an off-resonant light field. (b) Energy splitting in a $\mathbf{J}=1/2$ ${\mathbf{J}}^{\prime }=1/2$ system interacting with a circularly polarized light field, for example ${\sigma }^{+}$ light.

Figure 11

Vector model for the optically induced spin echoes for the on-resonant manipulation.

Figure 12

Vector model for the optically induced spin echoes for the off-resonant manipulation.