A Supplementary Material

Photon solid phases in driven arrays of nonlinearly coupled cavities


We introduce and study the properties of an array of QED cavities coupled by nonlinear elements, in the presence of photon leakage and driven by a coherent source. The nonlinear couplings lead to photon hopping and to nearest-neighbor Kerr terms. By tuning the system parameters, the steady state of the array can exhibit a photon crystal associated to a periodic modulation of the photon blockade. In some cases the crystalline ordering may coexist with phase synchronisation. The class of cavity arrays we consider can be built with superconducting circuits of existing technology.

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Introduction.— Since its beginning, the study of light-matter interaction in cavity and circuit QED has been providing a very fertile playground to test fundamental questions at the heart of quantum mechanics, together with the realisation of very promising implementations of quantum processors (1); (2). The coupling of separate cavities through photon hopping introduces an additional degree of freedom that is receiving increasing interest both theoretically and experimentally.

Cavity arrays, periodic arrangements of neighbouring QED cavities, have been introduced (3); (4); (5) as prototype systems to study many-body states of light. Their very rich phenomenology arises from the interplay between strong local nonlinearities and photon hopping. In the photon blockade regime the array enters a Mott insulating phase, where photon number fluctuations are suppressed. In the opposite regime, where the hopping dominates, photons are delocalised through the whole array with long-range superfluid correlations. The phase diagram has been thoroughly studied by a variety of methods and the location of the different phases, together with the critical properties of the associated phase transitions, have been determined (see, e.g., the reviews (6); (7); (8)).

The properties of cavity arrays resemble in several aspects those of the Bose-Hubbard model (9), as long as particle losses can be ignored. Cavity arrays, however, will naturally operate under nonequilibrium conditions, i.e. subject to unavoidable leakage of photons which are pumped back into the system by an external drive. In this case the situation may change drastically, and, to a large extent, it is an unexplored territory. Only very recently the many-body nonequilibrium dynamics of cavity arrays started to be addressed (10); (11); (12); (13), thus entering the exciting field of quantum phases and phase transitions in driven quantum open systems (14); (15); (16); (17); (18); (19).

Since the very beginning, all the works devoted to cavity arrays studied the case in which adjacent cavities are coupled by photon hopping. In this Letter we introduce a new class of arrays in which the coupling between cavities is mediated by a nonlinear element/medium. Thanks to the flexibility in the design of the nonlinear coupling elements, these finite-range couplings can appear in the form of cross-Kerr nonlinearities and/or as a correlated photon hopping, leading to a steady-state phase diagram that is a lot richer than the cases which have been considered so far. Here we discuss in particular the appearance of a new phase in cavity arrays, a photon crystal, which emerges in the steady-state regime when the array is driven by a coherent homogeneous pump.

A technology that is very well suited for realizing cavity arrays with such features is provided by circuit QED (20), where exceptional light-matter coupling has been demonstrated (21), first experiments with arrays of up to five cavities have been done (22), and great progress towards experiments with lattices of cavities has already been achieved (8).

In the following we first introduce the model for the cavities with their nonlinear couplings, the external drive and the unavoidable leakage of photons. We present a possible implementation in an array of circuit-QED cavities that are coupled via a nonlinear element. We then study the steady-state regime by means of a mean-field approach and Matrix Product Operator (MPO) simulations (23). The scenario that emerges is rather complex, with the appearance of a number of phases and phase instabilities. We focus in particular on the possibility of spatial photon patterns that can emerge. For bipartite lattices the photon blockade is modulated on two different sublattices, furthermore on increasing the photon hopping it may also coexist with a global coherent state.

The model.— The cavity array is sketched in Fig. 1a. The coupling between the cavities is mediated by a nonlinear element. In the specific implementation in circuit-QED this element is a Josephson nano-circuit. When the coupling between the cavities is realized through the circuit described in Fig. 1b, linear tunneling of photons between adjacent cavities can be tuned and even fully suppressed by adjusting the nonlinear coupling circuits to a suitable operating point, see Supplementary Material (SM) (24) for details. In this regime the cavities are coupled via a strong cross-Kerr term and further correlated-hopping terms, which can lead to considerable modifications in the phase diagram (25). Yet there are also more involved approaches involving multiple transmon qubits to realize cross-Kerr interactions in the absence of correlated hoppings (26), see also (27); (28).

Figure 1: (color online). a) An array of QED-cavities described by oscillator modes (red circles) that are coupled via nonlinear elements (crossed boxes). b) and c) Implementation of its building blocks in circuit-QED for one- and two-dimensional lattices. The circuit cavities are represented by a LC-circuit with capacitance and inductance and mutually coupled through a Josephson nano-circuit, with capacitance and Josephson energy , that generates the on-site and cross-Kerr terms in Eq. (1). Details of this implementation can be found in Ref. (24). An alternative approach to cross-Kerr interactions is discussed in Ref. (26).

However, in the regime of parameters we are interested in and where a photon crystal emerges, correlated hopping leads to small quantitative corrections at the expense of complicating considerably the analysis. In the SM we quantify these differences in more details (24).

Here for simplicity, we concentrate on the salient features of the array in Fig. 1 which are captured by the effective Hamiltonian (in the rotating frame),


where the number operator counts the photons in the -th cavity ( being the corresponding creation/annihilation operators). The first three terms describe respectively the detuning of the cavity mode with respect to the frequency of the pump, the coherent pump with amplitude and the hopping of photons between neighboring cavities at rate . The last two terms take into account the nonlinearities through the onsite- and cross-Kerr terms with the associated energy scales and respectively. In the specific case of circuit-QED arrays, the two types of nonlinearities can be realized through the setup of Fig. 1. In order to keep our results as general as possible, we consider the effective model (1) without specifying further the underlying matter-light interaction term.

The dynamics of the array is governed by the Master equation


where is the photon lifetime in each cavity. The model in Eq. (1) together with Eq. (2) encompasses, in some limiting cases, regimes that were already addressed in the literature. The regime of and was considered in Ref. (18), where an antiferromagnetic phase was first predicted in Rydberg atoms. The case of on-site Kerr nonlinearity, i.e. , is the only one studied so far in cavity arrays (12). The model considered here offers a much richer phase diagram. A unique characteristics of the cavity arrays with nonlinear couplings is that the cross-Kerr nonlinearity can even exceed . By coupling an additional qubit locally to each resonator (29), different ranges of the ratio can be explored. Moreover, in devices where on-chip control lines can be used to locally thread magnetic fields through the loops of the coupling circuits and the additional transmons, the ratios respectively and can be tuned on-chip. For this reason we will, in the following, consider , and as independent.

We first discuss the steady-state phase diagram in the mean-field approximation, which becomes accurate in the limit of arrays with large coordination number . The decoupling in Eq. (1) is performed on the hopping and cross-Kerr terms, and , where we assumed a bipartite lattice, and being the two sublattices. The mean-field analysis simplifies the dynamics dictated by Eq. (2) to two coupled equations for the two different sublattices. As a function of all the parameters characterizing the system and its dynamics, one gets a very rich behavior in the asymptotic regime which includes steady-state/oscillating phases, as well as uniform/staggered configurations. Here we highlight what we think are its most intriguing features. All the couplings will be expressed in units of the photon lifetime, .

Figure 2: (color online). Order parameter for the photon crystal in the plane at zero hopping. If the cross-Kerr term exceeds a critical threshold , the steady state is characterized by a staggered order in which . Here we fixed and , for which at , while in the hard-core limit (). In the inset we show as a function of and at a fixed value of . Here and in the next figure the color code signals the intensity of the order parameter, while dashed green lines are guides to the eye to locate the phase boundaries.

Mean-field steady-state diagram.— Fig. 2, where for the moment we set the hopping to zero, shows that, on increasing the cross-Kerr term, the array can reach a steady state in which the photon number is modulated as in a photon crystal, the order parameter being . Here the area above the green line denotes the crystalline phase. In the limit, the transition between the uniform and the crystal phase is located at with , the previous expression holding for small detunings, and coincides with the transition to an antiferromagnetic phase described in Ref. (18). Note that a lower value of favors the crystal phase. In the opposite limiting case of , the transition is found at with . We deliberately considered a regime in the parameter space where since, as already mentioned, it is a peculiar feature of the cavity arrays proposed here. The transition to the crystal phase is reentrant as a function of the drive (inset to Fig. 2). At very small pumping the density is too low to lead to a photon crystal. Vice-versa it also disappears on increasing , since pumping favors an homogeneous photon arrangement. A similar feature has been observed in the limit  (18).

If the hopping between photons is switched on, delocalization will suppress the solid phase and at a critical value of (which depends on , and ) there is a transition to a normal phase. This is shown in Fig. 3, as a function of the cross-Kerr nonlinearity (Fig. 3a) and of the detuning (Fig. 3b). In Fig. 3a we display the case , while at smaller values of the phase diagram shows a reentrance. Although interesting, further analysis is needed to see if this feature is only present at mean-field level. Yet it does not seem improbable that an increased hopping could facilitate the redistribution of particles into a crystalline order imposed by the interactions. We conclude this discussion by pointing out that, as discussed in Ref. (24), under nonequilibrium conditions it is even possible, although much harder, to realize a crystalline phase at . As a matter of fact in that case, for some values of the coupling constants, the steady state can be either uniform or crystalline, depending on the initial conditions.

Figure 3: (color online). Order parameter for the photon crystal at finite hopping. is plotted as a function of and , for and (panel a), and as a function of and , for and (panel b). Here we fixed . At finite values of the detuning, in addition to the normal (white) and crystalline (colored) phases, an intermediate region (shaded green), characterized by an oscillatory behavior in the asymptotic state, appears. As discussed in the main text, we suggest this last regime may be seen as a nonequilibrium analog of a supersolid.

The mean-field phase diagram in the plane and for is depicted in Fig. 3b. As highlighted in the region between the dashed green lines, for , on switching on the photon hopping, a new intermediate phase appears. In this region, even in the long-time limit the state never becomes completely stationary and there is a residual time-dependence of with , i.e. there is an additional time dependence of on top of the trivial oscillation with the frequency of the coherent drive that is hidden in our choice of the rotating frame. At the same time the system shows . In Fig. 4a we show the time evolution of the real and imaginary parts of for the two sublattices. A closer inspection of the properties of the oscillating phase reveals that the reduced density matrix of a single site (in either of the two sublattices) is a coherent state which evolves periodically in time as shown in Fig. 4b. There we plotted the Wigner function of one sublattice at a given time, with , , and being an eigenstate of the position operator . Following the analysis performed in Ref. (19), we are lead to conclude that in this region the dynamical evolution of the whole array is synchronised separately in the two different sublattices. The contemporary presence of checkerboard ordering and global dynamical phase coherence suggests us to view this intermediate phase as a nonequilibrium supersolid phase (30). The intermediate region extends also at finite- values, although the coherent state of Fig. 4b will be progressively deformed on increasing the on-site repulsion.

Figure 4: (color online). a) Time dependent traces of the real and imaginary parts of for the two sublattices as a function of time in the steady-state regime of the intermediate oscillating phase of Fig. 3b. b) The Wigner transform of the reduced single-site density matrix for sublattice A is plotted at a given time in the intermediate oscillating phase of Fig. 3b. Here we used the same parameters as in Fig. 3b and fixed , .

MPO simulations.— Most of the features we discussed can already be seen in small arrays. To show some examples and to further support the mean-field analysis given above, we here present results that were obtained for linear chains of cavities, , with MPO simulations (12) of the Master equation (2), which provide a (numerically) exact description of its nonequilibrium many-body dynamics. Fig. 5a shows the density-density correlation function for a chain of 20 cavities with , , , and various values of . One clearly sees that a staggered dependence of the distance , indicating strong density-density correlations, appears for nonzero , whereas for photons in distinct cavities are uncorrelated (). Fig. 5b shows for a chain of 21 cavities with , , , and various values of . The spatial range of density-density correlations shrinks with increasing tunneling rate , indicating a crossover to an uncorrelated state. A more quantitative analysis of the decay of correlations with increasing distance is not conclusive for the chain length considered here. A true ordering in the steady state can probably only be stabilized in two-dimensions.

Figure 5: (color online). MPO results for linear chains of cavities. a) for , , , and as in the legend. b) for , , , and as in the legend.

Conclusions.— In this Letter we introduced cavity arrays with coupling mediated by nonlinear elements. This opens the way to study a variety of new possibilities, including correlated photon hopping and finite-range photon blockade. We concentrated on this last point studying the effect of a cross-Kerr nonlinearity on the steady state and found a very rich phase diagram. A photon solid characterized by a checkerboard ordering of the average photon number appears for a substantial range of the coupling constants. In addition we see that, for some choice of the parameters, a finite hopping stabilizes a phase where the crystalline ordering coexists with a globally synchronized dynamics of the cavities, suggesting an analogy to a nonequilibrium supersolid. Most of the results presented in this work were obtained in a mean-field approximation. We corroborated the existence of a steady-state solid phase by studying a one-dimensional array by means of a Matrix Product Operator approach. This last analysis confirms that a crystalline ordering of photons can be observed with existing experimental technology.

Acknowledgments.— We acknowledge fruitful discussions with A. Tomadin. This paper was supported by EU - through Grant Agreement No. 234970-NANOCTM, and No. 248629-SOLID, by DFG - through the Emmy Noether project HA 5593/1-1 and the CRC 631 and by National Natural Science Foundation of China under Grant No. 11175033.

Appendix A Supplementary Material

a.1 Cross-Kerr nonlinearities in arrays of circuit cavities

Here we describe a circuit quantum electrodynamics (cicuit-QED) setup that represents an array of cavities coupled by nonlinear elements and can be effectively described by a lattice of harmonic oscillators coupled via cross-Kerr nonlinearities. The setup thus provides an experimentally feasible way for implementing the model described by equations (1) and (2) of the main text. Cicuit-QED is particularly well suited for this task because of the great design flexibility, the tunable nonlinearity provided by Josephson junctions and the exceptionally high coupling between subsystems that can be reached.

For our derivation, we consider lumped element resonators, representing the cavities in our array, which are coupled conductively through capacitively shunted Josephson junctions, see Fig. 6 for a sketch of the circuit representing a linear chain of cavities. We focus on lumped element resonators, in order to keep the derivation simple and transparent. Coplanar waveguide resonators work equally well. For the considered combination of capacitive and inductive coupling between the resonators, the linear parts of the couplings can cancel each other for suitable choices of the parameters, leaving a residual coupling via the nonlinearity of the Josephson junctions.

Figure 6: (color online). Electrical circuit sketch of the setup we envision to realize a system with cross Kerr nonlinearities. Here for the example of a one-dimensional chain of cavities. Lumped element LC-circuits are coupled via capacitively shunted Josephson junctions. The Josephson energies could be tuned in situ by replacing the Josephson junctions with superconducting interference devices.

In terms of the node fluxes , the Lagrangian of the whole setup reads,


with and the inductance and capacitance of the lumped element resonators and and the capacitance and Josephson energy of the Josephson junctions. is the reduced flux quantum. Assuming we invert the capacitance matrix to first order in for performing the Legendre transformation to obtain the Hamiltonian (31),


where , , and .

The charges on the islands ( denotes all connections to site ), defined by the coupling capacitances of the Josephson junctions and the lumped element resonators, and the fluxes associated to the phase drop at the inductance of the lumped element resonators are our canonically conjugate variables. We quantize the Hamiltonian by introducing bosonic lowering and raising operators and according to,


The quantized coupling Hamilton operator thus reads,

where the oscillator frequency is and . We choose such that and the linear tunneling between neighboring oscillator sites vanishes. Furthermore we neglect the terms proportional to in a rotating wave approximation which is justified as long as holds. In this way we arrive at a chain of harmonic oscillators that are decoupled in linear order.

The nonlinear parts of the Hamilton operator however provide us with coupling terms between the neighboring oscillators and on-site nonlinearities. We restrict ourselves to fourth order nonlinearities, perform a rotating wave approximation and arrive at the many-body Hamiltonian,


where is a small correction coming from the normal ordering process of the nonlinearity and is the charging energy of the individual lattice sites of our model. The second term in Eq. (7) is an on-site Kerr term whereas the third term describes cross-Kerr interactions. This specific implementation thus realizes the Hamiltonian of equation (1) in the main text for and , thus leading to .

The term describes correlated hopping of photons between adjacent sites and . For the purposes of the present work, i.e. a description of solid and supersolid phases, this term can be treated at a mean-field level. A mean-field approximation would replace and the first two terms will thus only give rise to a (possibly sublattice dependent) renormalization of the single-photon hopping. The term in turn is a factor of 4 weaker than the cross-Kerr term. Moreover, in regimes of large Kerr interactions, double occupations of lattice sites are small and this term becomes ineffective. For these reasons, in most of our analysis we disregard the correlated hopping at present. A signature of the effects induced by is evident from the data shown in Fig. 7, where we plotted the phase boundary between normal and crystalline phase in the plane at zero hopping. The relatively small discrepancies in the curves with and without such terms points towards the fact that only quantitative modifications are induced in the parameter space we are considering, while all the qualitative features of our results should be unaffected. It should however be noted that a more accurate treatment may lead to new phases in the diagram such as a pair superfluid state, when the corresponding hopping term becomes comparable with the cross-Kerr nonlinearity. This is however beyond the scope of the present analysis and it will be treated separately.

Figure 7: (color online). Emergence of the photon crystal in the diagram at . We highlight the effects induced by the terms in by comparing mean-field results without (green line) and with (blue line) such contributions. Here the strength of the correlated-hopping terms is set equal to , while we set and , as in Fig. 2 of the main text.

Our approximations require that . Nonetheless the cross-Kerr interaction can be much larger than photon losses, i.e. , since, e.g., transmon qubits have GHz and (32).

Finally let us point out that the Josephson junctions linking two neighboring oscillators can be built tunable by replacing them with a dc-SQUID. In this way the and thus the can be modulated by applying an external flux to the dc-SQUIDs and the Hamiltonian (7) can be tuned in real time. Hence, by choosing the external flux such that a linear tunneling of photons between the resonators can be introduced. Moreover larger on-site nonlinearities can be introduced by coupling each resonator locally to a superconducting qubit, e.g. a transmon.

a.2 Photon crystal at

We conclude the discussion of the solid phase by showing that the realization of a steady-state photon solid does not necessarily requires a cross-Kerr nonlinearity. Surely the presence of stabilizes the solid in a wide range of the parameter space, thus making it easier to be observed experimentally. Nevertheless, while in equilibrium is a necessary requirement, under nonequilibrium condition this ceases to be the case. The only requirement is a finite photon hopping and an initial unbalance in the occupation of the two sublattices. In Fig. 8 the regions in which the steady-state solid phase is reached are plotted as a function of the initial unbalance. Note that the value of the crystalline order parameter does not depend on the choice of the initial conditions.

Figure 8: (color online). Phase diagram at in the space of initial conditions, with , , and in the hard-core limit . White areas denote starting occupation values where a steady-state uniform phase is reached, while red areas denote initial conditions leading to an antiferromagnetic phase with .



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