A multimode cavity QED Ising spin glass

Abstract

We realize a driven-dissipative Ising spin glass using cavity QED in a novel ``4/7" multimode geometry. Gases of ultracold atoms trapped within the cavity by optical tweezers serve as effective spins. They are coupled via randomly signed, all-to-all Ising cavity-mediated interactions. Networks of up to n = 25 spins are holographically imaged via cavity emission. The system is driven through a frustrated transverse-field Ising transition, and we show that the entropy of the spin glass states depends on the rate at which the transition is crossed. Despite being intrinsically nonequilibrium, the system exhibits phenomena associated with Parisi's theory of equilibrium spin glasses, namely replica symmetry breaking (RSB) and ultrametric structure. For system sizes up to n = 16, we measure the Parisi function q(x), Edwards-Anderson overlap q_EA, and ultrametricity K-correlator; all indicate a deeply ordered spin glass under RSB. The system can serve as an associative memory and enable aging and rejuvenation studies in driven-dissipative spin glasses at the microscopic level.

Figures (13)

• Figure 1: (a) Cartoon of the $4/7$ cavity QED apparatus. Ultracold atomic gases (green) trapped in the cavity midplane serve as pseudospins. Light scattered by the atoms from a transverse pump (red) and into the cavity mediates spin interactions. A camera images cavity emission for spin state detection. A processed experimental image shows a $5{\times} 5$ array of spins; color indicates spin state. (b) The transverse pump strength is exponentially ramped to $4\times$ the critical power ${\propto}\Omega_c^2$ over a time $t_R$ before quenching to a higher power for imaging. (c) 1D cartoon of a rugged, spin glass energy landscape. The actual space is high-dimensional. (d) Example dendrogram showing ultrametric structure arising from overlap distances $d_{\alpha\beta} = 1 - |q_{\alpha\beta}|$. (e) Example overlap matrix $q_{\alpha\beta}$ with fractal ultrametric structure seen as block-diagonal correlations.
• Figure 2: Study of spin configurations arising from the disorder realization $J_1$ of an $n= 16$ network. (a) $J_1$ connectivity diagram normalized to the absolute amplitude of the largest element. (b,c,d) Images of spin-state configurations of three replicas, normalized to the maximum spin amplitude in each. Scale bars are equal to the length of the Gaussian mode waist $w_0=35$$\mu$m. (e) Spin amplitude distribution over all spins in 200 replicas, normalized by the total RMS spin amplitude; Ramp time $t_R=5$ ms. (f) Base-2 Shannon entropy of the binarized spin-state distribution versus $t_R$. Each point is derived from an ensemble of 200 replicas, and $t_R=\{0.1, 1, 5, 10, 15, 20\}$ ms from I to VI. (g) Hierarchical clustering of replicas by the overlap distance $d$. Columns I to VI correspond to ramp times in panel (f). (h) Overlap matrix $q_{\alpha\beta}$ and (i) overlap distribution. Error bars are standard error both here and in figures below.
• Figure 3: (a) Gallery of overlap distributions for seven disorder realizations of the coupling matrix, $J_1$--$J_7$, for $n= 16$. (b) Overlap distributions for these same disorder realizations after spin binarization. (c) The Parisi distribution of all 14 disorder realizations. (d) Parisi distributions after spin binarization. (e) Parisi functions for the continuously valued spins (blue) and spins after binarization (red). Dashed lines show least-squares fits to linear-constant piecewise functional forms.
• Figure 4: (a) Measured Parisi function $q(x)$ for system sizes $n=8$, 12, and 16 versus the cumulative overlap probability $x$. (b) Ultrametricity $K$-correlator distributions, normalized to the peak probability density of each system size. A comparison to that of the paramagnetic phase is provided.
• Figure S1: Visualization of the time-dependent trap depth for the preparation of an array of $n_x=4$ by $n_y=4$ sites. Trap depth is shown in units of the recoil energy $E_r$ for $^{87}$Rb at 1064 nm. (a) The normalized optical potential generated in the $x$-direction and (b) in the $y$-direction. The 2D potential is the sum of the two trap depths in each direction. (c) The initial 2D flat-bottom trap potential at $t=0$ ms (green dashed lines) corresponds to the end of the optical evaporation sequence. (d) By $t=175$ ms (black dashed lines), the 2D flat-bottom trap has grown in size and started to develop separated wells, thereby splitting the atomic cloud into localized sites. (e) The final trap shape at $t=300$ ms (red dashed lines), resulting in cold atom clouds arranged in a rectilinear array at the intersections of the XODT beams.