Nucleation of superfluid-light domains in a quenched dynamics

Strong correlation effects emerge from light-matter interactions in coupled resonator arrays, such as the Mott-insulator to superfluid phase transition of atom-photon excitations. We demonstrate that the quenched dynamics of a finite-sized complex array of coupled resonators induces a first-order like phase transition. The latter is accompanied by domain nucleation that can be used to manipulate the photonic transport properties of the simulated superfluid phase; this in turn leads to an empirical scaling law. This universal behavior emerges from the light-matter interaction and the topology of the array. The validity of our results over a wide range of complex architectures might lead to a promising device for use in scaled quantum simulations.


Integration time
In the definition of the order parameter, namely, using the variance of the polariton number or the bipartite fluctuations of the polariton number, we integrate fluctuations up to a time T = 1/J, which is a characteristic time scale where the hopping strength dominates the dynamics, thus producing a delocalized wave function over different lattice points.
However, for longer simulation times, there are no significant differences in the behavior of the studied order parameter. We have carried out numerical simulations for the dimer, and for a linear array of three coupled resonators; the results are shown in Fig. (1). We see that the order parameter for different integration times exhibits the same qualitative behavior. Also, for times t < 1/J, not shown here, the studied order parameter almost vanishes and we do not see the crossover of the Mott-insulator to superfluid.

Describing Simulated First Order Phase Transition using Bipartite Fluctuations
The polariton number is conserved in our system since it does not exchange particles with the outside. Hence for both detuning ∆ = 0 and ∆ g, where g stands for the light-matter coupling strength, the corresponding (simulated) Mott and superfluid states are described by Fock states. This way, if one considers the standard order parameter studied in mean-field approximation, that is, the average value of the annihilation operator per site, it will always be zero and will not capture any crossover from Mott insulator to superfluid. This is why we choose to study the simulated phase transition using the onsite variance of the polariton number. However, we demonstrate that the first-order like-behavior is universal, in the sense that it does not depend on the choice of the order parameter. Indeed, if we use the bipartite fluctuations of subsystems we find that this approach also provides a correct description of the Mott to superfluid phase transition. Certainly, the dispersion of the polariton number on a given partition M − th can be assessed by the variance of this subsystem, which can be obtained from the two-point correlation function C i,j = n i n j − n i n j , using the parameter i,j∈M C ij , where n i denotes the polariton occupation number at the i−th site 1, 2 .
We have performed numerical calculations for bipartite fluctuations in our finite arrays. Thus we can demonstrate that a simulated first order phase transition appears regardless of the choice of the order parameter; see left-panel of Fig. (2). As shown in the right-panel of the Fig. (2), using the bipartite fluctuations approach, the averaged standard deviation also displays a linear dependence on the connectivity of the partition. Where the connectivity of the partition has been defined as the ratio between the external connectivity (total number of links between one node in the partition and the external nodes) of each node and the total number of nodes. The connectivity of the partition are summarized in Table 1. These results demonstrated the linear dependence of the order between the connectivity and the order parameter, in both case, when the order parameter is identified as the standard deviation or as the variance of the bipartite fluctuations.
Partition External connectivity Number of Nodes Connectivity of the partition Table 1. Summary of the connectivity of each partition corresponding to the array of the inset of Fig. (2-left). The variance of the superfluid phase depends linearly on the connectivity of the partition, as shown in Fig. (2-right)

Bose-Hubbard model
When we investigate the order parameter in the Bose-Hubbard model, in the framework of the grand canonical ensemble, it is apparent that it exhibits a similar behavior to the Jaynes-Cummings model, as shown in Figs. 3 The above results have been obtained for the Bose-Hubbard model in the grand canonical ensemble and for a set of CRAs from two to eight interconnecting resonators, as shown in b) are considered. In order to reproduce an analogous quench dynamics, as compared with the Jaynes-Cummings-Hubbard model, we have adopted the parameters µ = 10 −6 , J = 10 2 µ, and U in the interval [10 2 J, 10 −2 J]. As the chemical potential is not zero, and U > µ, the lowest energy state per site in the Mott regime U/J 1 is the Fock state |1 . Thus, for each simulation our initial state is |ψ 0 = L i=1 |1 i .