Harmonic Fine Tuning and Triaxial Spatial Anisotropy of Dressed Atomic Spins

The addition of a weak oscillating field modifying strongly dressed spins enhances and enriches the system quantum dynamics. Through low-order harmonic mixing, the bichromatic driving generates additional rectified static field acting on the spin system. The secondary field allows for a fine tuning of the atomic response and produces effects not accessible with a single dressing field, such as a spatial triaxial anisotropy of the spin coupling constants and acceleration of the spin dynamics. This tuning-dressed configuration introduces an extra handle for the system full engineering in quantum control applications. Tuning amplitude, harmonic content, spatial orientation, and phase relation are control parameters. A theoretical analysis, based on perturbative approach, is experimentally tested by applying a bichromatic radiofrequency field to an optically pumped Cs atomic vapour. The theoretical predictions are precisely confirmed by measurements performed with tuning frequencies up to the third harmonic.

Figure 1

$B_{0x}$ calibration data obtained for the dressed $B_t=0$ case at ${\omega }_{0z}/2\pi =5.979(2)\mathrm{kHz}$ and $\omega /2\pi =30\mathrm{kHz}$. The ratio of the ${\mathrm{\Omega }}_L^0$ measurements at the $\xi =0$ and $\xi =1.833(5)$ values vs $B_{0x}$ is marked by the black dots. The fit to the theoretical predictions determines the $B_{0x}$ scale at the four percent precision level given by the horizontal error bars. The $B_{0x}=0$ data point provides the $\xi$ calibration.

Figure 2

Absolute ${\mathrm{\Omega }}_L$ frequency vs $\xi ={\mathrm{\Omega }}_d/\omega$ scanning ${\mathrm{\Omega }}_d/2\pi$ in the (0,200) kHz range, for ${\omega }_{0z}/2\pi =2.040(1)\mathrm{kHz}$, and ${\omega }_{0x}={\omega }_{0y}=0$. Parameters $p$, $\mathrm{\Phi }$, $\omega /2\pi$ and ${\mathrm{\Omega }}_t/2\pi$, both in kHz: in (a) $[1,\pi /2,9,4.97(25)]$, in (b) $[2,0,10,2.23(10)]$, and in (c) $[3,\pi /2,9,4.25(20)]$. Black dots for data. Error bars on the frequency at two per thousand, and on $\xi$ at the three per thousand level, both smaller than the dot size. Theory in continuous blue lines from perturbative treatment: in (a) black line from the numerical analysis, in (b) dashed red one for a refined ${\mathrm{\Omega }}_t/2\pi$ value, see text.

Figure 3

${\mathrm{\Omega }}_L$ vs the $\mathrm{\Phi }$ phase difference, for ${\omega }_{0x}={\omega }_{0y}=0$ and ${\omega }_{0z}/2\pi =2.040(1)\mathrm{kHz}$. Parameters $p$, $\xi$, $\omega /2\pi$, and ${\mathrm{\Omega }}_t/2\pi$, both in kHz: in (a) [1,1.38,20,1.49(8)], in (b) [2,3.83,10,2.23(12)], and in (c) [3,1.54,9,1.23(6)]. Black dots data. Error bars two per thousand on the frequency, and of one degree for the phase. Theoretical predictions given by the blue continuous; in (b) also by the red dashed line for a refined ${\mathrm{\Omega }}_t/2\pi$ value, both presented in the text.

Figure 4

${\mathrm{\Omega }}_L$ vs ${\omega }_{0x}$ as a test of the spin anisotropy, for an applied $p=1$ tuning field, ${\mathrm{\Omega }}_t/2\pi =0.354(3)\mathrm{kHz}$, and other parameters as in Fig. 1. Blue dots for data and continuous line for the theoretical predictions. Error bars two per thousand on the frequency, and 4% on the ${\omega }_{0x}$ scale.