Quantum simulation of Cooper pairing with photons

We propose a scheme to observe the crossover from weakly to strongly bound pairs with stationary polaritons. We first show how stationary light-matter excitations (polaritons) can realize an optically tunable two-component Bose-Hubbard model with repulsive intraspecies interactions and attractive interspecies interactions. We then discuss the feasibility of generating an effective Fermi-Hubbard model of polaritons exhibiting the crossover behavior. The predicted behavior of the system characterized by a crossover from short- to long-range spatial correlations as interactions are tuned can be efficiently observed by measuring correlation functions on the light exiting the waveguide.

Figure 1

(a) A schematic diagram of the system under study. In a fiber setup (a hollow core version is shown here [43] but a tapered fiber approach [38] could also be used), cold atoms are interacting with a pair of quantum fields ${\widehat{E}}_{1,2}$, and a pair of classical fields ${\Omega }_{a,b}$. The resulting stationary light-matter excitations in the waveguide can be steered to a strongly interacting regime mimicking an effective FH model with highly tunable attractive interactions. (b) The atomic level structure for a type-$a$ atom. Type-$b$ atoms have similar structure but primarily coupled to ${\widehat{E}}_2$ (c) Schematic illustration of interesting phases discussed in the text. Coherently mapping the stationary excitations to propagating photon pulses allows for the efficient probing of the BCS-BEC crossover by measuring the temporal correlations of the photon pulses leaving the fiber.

Figure 2

Achievable polaritonic (a) interspecies interaction $V/U$ and (b) hopping strength $t/U$ as functions of the quantum pulse frequency difference ${\delta }_1^b$, single photon detuning ${\Delta }_4$, and the atomic distribution modulation $n_1/n$. Here the parameters are set as ${\Gamma }_{1\mathrm{D}}=0.2\Gamma$, $n/n^{\mathrm{ph}}={10}^4$, ${\Delta }_2=-5\Gamma$, $\Omega =\Gamma$, and ${\Delta }_3=-0.01\Gamma$.

Figure 3

Correlation functions for polaritons: In (a) and (b) the second-order cross-species correlations $g_{\uparrow \downarrow }^{\left(2\right)}\left(l\right)$ are plotted as functions of the interspecies interaction for the same site ($l=0$) and neighboring sites ($l=1$), for two values of hopping strengths $t=0.01$ and $t=0.1$. The colored gradient background, proportionate to the on-site cross-species correlation, portrays the BCS-BEC-BB crossover. (c,d) show the correlation between the difference in populations for the two species. Note the sensitivity to the BCS-like phase. The coherent transfer of the polaritonic correlations to propagating photonic ones allows for witnessing the different phases of the system using photon coherence measurements and energy resolving photodetectors.

Figure 4

Cross-species second-order correlation function for $t/U=0.01$ and different values of $\left|V\right|/U=1.4,0.99,0.5,0.001$ as a function of the distance $l$ in units of the effective photonic lattice spacing.