Observation of Quantum Phase Synchronization in Spin-1 Atoms

With growing interest in quantum technologies, possibilities of synchronizing quantum systems have garnered significant recent attention. In experiments with dilute ensemble of laser cooled spin-1 ${}^{87}\mathrm{Rb}$ atoms, we observe phase difference of spin coherences to synchronize with phases of external classical fields. An initial limit-cycle state of a spin-1 atom localizes in phase space due to dark-state polaritons generated by classical two-photon tone fields. In particular, when the two couplings fields are out of phase, the limit-cycle state synchronizes only with two artificially engineered, anisotropic decay rates. Furthermore, we observe a blockade of synchronization due to quantum interference and emergence of Arnold-tongue-like features. Such anisotropic decay induced synchronization of spin-1 systems with no classical analog can provide insights in open quantum systems and find applications in synchronized quantum networks.

Figure 1

(a) Energy level diagram of a spin-1 atom with coherent couplings ${\eta }_{-1,0}$ and ${\eta }_{0,1}$ along with incoherent decay rates ${\gamma }_g$ and ${\gamma }_d$. (a.i) Coherent couplings are engineered using control (${\mathrm{\Omega }}_c^{\pm }$) and probe fields (${\mathrm{\Omega }}_p^{\pi }$). Here ${\mathrm{\Delta }}_B$ is the ground state energy shift due to a magnetic field along the quantization axis. (a.ii) Decay rates ${\gamma }_g$ and ${\gamma }_d$ are engineered with two fields coupling states $|1,-1⟩$ and $|1,1⟩$ to the excited state $|0,0⟩$. (b) Experimental setup, showing propagation directions of storing [C(S), green] and retrieving [C(R), red] control fields, probe field (P, blue), and decay beams (D1 and D2, in violet, at small angles to control fields). Here BS: beam splitter. (c) Experimental timing sequence with intervals for state preparation (SP), decay beams (DB), and control-probe (DSP) fields. Here MOT: magneto optical trap; OP: optical pumping; D: decay beams. (d) Typical probe field time traces with varying storage times and with ${\varphi }_c={\varphi }_d=0$. Reconstructed Husimi-$Q$ function from numerical simulations (corresponding to red time trace) are plotted at different times (with the horizontal and vertical axes corresponding to $\varphi$ and $\theta$, respectively, and $0\le \theta \le \pi$ and $-\pi \le \varphi \le \pi$). (e) In presence of magnetic field (with dynamic phase ${\varphi }_d/\pi =3.22$, and tone phase ${\varphi }_c=0$), the retrieved intensity oscillates with changing storage time. Corresponding Husimi-$Q$ function gets localized and entrained with the dynamic phase, precessing in equatorial plane. Here $I_{\mathrm{in}}$ and $I_t$ are input and transmitted probe intensities, respectively.

Figure 2

(a), (b) Plots of retrieved intensity with dynamic (${\varphi }_d$) and tone (${\varphi }_c$) phases and with decay rate ratios: ${\gamma }_d/{\gamma }_g=1.00$ (a) and 11.90 (b). ${\varphi }_d$ is varied with magnetic field while keeping the storage time ($\tau =600\mathrm{ns}$) fixed. (c), (d) Numerically simulated retrieved intensity for decay rate ratios: ${\gamma }_d/{\gamma }_g=1.00$ (c) and 11.00 (d). (e) Husimi-$Q$ plots for the regions A, B, C, and D of (c) and (d). (f)–(i) Maximum of synchronization function [${\tilde{S}}_{\max }(\varphi )$] calculated from experimental and simulated plots corresponding to (a),(b),(c), and (d). Here experimental parameters ${\mathrm{\Omega }}_p^{\pi }$, ${\mathrm{\Omega }}_{c(S)}^{\mathrm{lin}}$, ${\mathrm{\Omega }}_{c(R)}^{\mathrm{lin}}$, and ${\gamma }_g$ are set to $0.64\gamma$, $1.02\gamma$, $1.44\gamma$, and 107 kHz, respectively, and the simulation parameters are as tabulated in the Supplemental Material [33]. Color bars for experimental plots are in units of $\mu \mathrm{W}/{\mathrm{cm}}^2$ and for theory plots, in units of $2|{\mathrm{\Omega }}_R{|}^2/{\gamma }^2$, where $|{\mathrm{\Omega }}_R|$ and $\gamma$ correspond to retrieved field and excited state decay, respectively.

Figure 3

Retrieved intensity at a region of destructive interference is plotted as a function of increasing ${\gamma }_g$, for a fixed ${\mathrm{\Delta }}_B$. Blue and yellow are for ${\gamma }_d/{\gamma }_g=1.00$ and 11.90, respectively. Inset shows ${\tilde{S}}_{\max }(\varphi )$, as extracted from experimental data. Blue and red correspond to Figs. 2 and 2, respectively. Here the experimental parameters ${\mathrm{\Omega }}_p^{\pi }$, ${\mathrm{\Omega }}_{c(S)}^{\mathrm{lin}}$, and ${\mathrm{\Omega }}_{c(R)}^{\mathrm{lin}}$ are set as in Fig. 2 with ${\varphi }_c=140°$ and ${\mathrm{\Delta }}_B=67\mathrm{kHz}$.

Figure 4

(a),(b) Retrieved intensity, plotted with increasing probe field strength (${\mathrm{\Omega }}_p^{\pi }$) and dynamic phase (${\varphi }_d$) (storage time $\tau =600\mathrm{ns}$), with ${\gamma }_d/{\gamma }_g=1.00$ and 11.90 for (a) and (b), respectively. (c),(d) Simulated plots for ${\gamma }_d/{\gamma }_g=1.00$ (c), and 11.00 (d). Insets of (c) and (d) are the corresponding ${\tilde{S}}_{\max }(\varphi )$. (e) Husimi-$Q$ plots for regions A, B, C and D, E, F of (c) and (d), respectively. Here experimental parameters ${\mathrm{\Omega }}_{c(S)}^{\mathrm{lin}}$, ${\mathrm{\Omega }}_{c(R)}^{\mathrm{lin}}$, and ${\gamma }_g$ are set as in Fig. 2 and ${\varphi }_c$ is set to 140°. Simulation parameters are as tabulated in the Supplemental Material [33]. Color bars for all the plots are as in Fig. 2.