Photon Devil’s staircase: photon long-range repulsive interaction in lattices of coupled resonators with Rydberg atoms

The realization of strong coherent interactions between individual photons is a long-standing goal in science and engineering. In this report, based on recent experimental setups, we derive a strong photon long-range repulsive interaction, by controlling the van der Waals repulsive force between Cesium Rydberg atoms located inside different cavities in extended Jaynes-Cummings-Hubbard lattices. We also find novel quantum phases induced by this photon long-range repulsive interaction. For example, without photon hopping, a photon Devil’s staircase, induced by the breaking of long-range translation symmetry, can emerge. If photon hopping occurs, we predict a photon-floating solid phase, due to the motion of particle- and hole-like defects. More importantly, for a large chemical potential in the resonant case, the photon hopping can be frozen even if the hopping term exists. We call this new phase the photon-frozen solid phase. In experiments, these predicted phases could be detected by measuring the number of polaritons via resonance fluorescence.

rational states are separated by many states. If photon hopping exists, we predict a photon-floating solid phase, due to the motion of particle-and hole-like defects. More importantly, for a large chemical potential in the resonant case, photon hopping can be frozen even if the hopping term exists. We denote this new phase the photon-frozen solid phase. In experiments, these predicted phases could be detected by measuring the number of polaritons via resonance fluorescence 29 .

Results
Extended Jaynes-Cummings-Hubbard model. We first propose a possible way to realize an extended Jaynes-Cummings-Hubbard model with long-range atom-atom interactions in different cavities, based on recent experimental setups [30][31][32][33][34] . As shown in Fig. 1, a series of SiO 2 nanofibers are arranged in the same direction of a specific plane, and an ensemble of Cesium (Cs) Rydberg atoms are trapped close to each nanofiber. Each nanofiber, with radius b = 0.25 μm, acts as a 1D photonic crystal cavity, due to its fabricated fiber Bragg-grating (FBG) structure 31,32 [see Fig. 2(a)]. A guided field, whose evanescent field acts as the quantum cavity mode, propagates along the cavity y axis. The cavity decay rate is characterized by the parameter κ, which induces the photon hopping in the cavity array 35 , and the distance between nearest-neighbor cavities is about x i+1 − x i ≈ 2.4 μm. Since the evanescent field strength is sufficiently weak at the radial distance of about b-4b away from the surface of the nanofiber 36,37 , each adjacent nanofiber pairs located at such a distance will not lead to an efficient overlap of different cavity modes, which guarantees that the ith ensemble of Cs Rydberg atoms can interact only with the ith cavity 33,36 .
By using the red-and blue-detuned evanescent light fields around the optical nanofiber, a two-color optical dipole trap can be formed. This optical dipole trap should allow an ensemble of Cs Rydberg atoms to be prepared at a few hundred nanometers from the nanofiber surface 30,38 . For Cs Rydberg atoms, we can choose the fine-structure states | , = 〉 / S F 6 4 1 2 and | , ′ = 〉 / P F 6 5 3 2 as the ground state g and the intermediate state p , respectively, while the Rydberg state is assumed as 70S 1/2 . As shown in Fig. 2(b), the photon induced by the evanescent field, with wavelength 852 nm, governs the transition between the ground state g and the intermediate state p , whereas the other transition between the intermediate state p and the Rydberg state r is controlled by a classical driving laser, with wavelength 510 nm, as shown in Fig. 1.
Formally, the total Hamiltonian of the system considered in Fig. 1 is In the Hamiltonian (1), H JC describes the interaction between the photons and the ensemble of Cs Rydberg atoms for all nanofiber photonic crystal cavities. We first consider the interaction between the photon and a single three-level Cs Rydberg atom at one cavity. In the current experimental setups 30-34 , the interaction between photons and the single Cs Rydberg atom is of the order of MHz (the detailed estimation will be shown in the next subsection). Therefore, in the framework of the rotating-wave approximation, the corresponding Hamiltonian is where ω = ω c −ω l is the effective photon frequency, ε = E r − E g − ω l + Ω 2 /Δ p is the effective transition frequency of the two-level Rydberg atom, g 1 = g 0 Ω /Δ p is the effective interaction strength, and λ = /Δ g p 0 2 . For large detuning, λ is very small and we thus can omit the interaction term † a a g g . In addition, for large detuning, g 1 is also weak. In order to enhance the effective atom-photon interaction strength, here we consider an ensemble of Cs Rydberg atoms in the center of each cavity. For simplicity, we also assume that the number of Cs Rydberg atoms in each cavity is a constant N R . The strong van der Waals repulsive interaction between Cs Rydberg atoms in the same cavity generates a Rydberg-blocked effect, which excites only one Cs Rydberg atom 39 . In such case, we should introduce the collective ground state = | , …, and the collective excitation state Thus, the first term of the Hamiltonian (1) becomes The second term in the Hamiltonian (1) governs the photon hopping between two adjacent cavities, and is where κ = / π t F 2 is the photon hopping rate and F is the cavity finesse. The third term in the Hamiltonian (1) governs the long-range van der Waals interaction between Cs Rydberg atoms in different cavities, and is 6 with C 6 being the van der Waals coefficient, and x i being the position of the ith cavity 40 . The long-range van der Waals interaction can induce a strong correlation between Cs Rydberg atoms in different cavities. Hereafter, we use the nearest-neighbor interaction to represent the entire van der Waals interaction, i.e., ≡ V V 1 , because V 2 = V 1 /2 6 , and V 3 = V 1 /3 6 , . In the last term of the Hamiltonian (1), the chemical potential μ is the Lagrange multiplier, and the total number of polaritons is It should be noted that a dielectric medium placed near dipoles will alter the spatial distribution of the electromagnetic field. However, for the parameters of the nanofiber and Cs Rydberg atoms considered here, this alteration can be regarded as a higher-order small quantity, compared with the direct atom-atom interaction [41][42][43] . This allows us to safely treat the interaction between Cs Rydberg atoms in different cavities as the standard long-range van der Waals force. Typical parameters. Before proceeding, we estimate the relevant parameters of the Hamiltonian (1) in terms of the above proposal.
• The effective photon frequency ω = ω c − ω l and the effective atom transition frequency ε = E r − E g − ω l + Ω 2 /Δ p . These two parameters can be well controlled by the driving frequency ω l of the classical laser. Thus, these can have suitable values as required experimentally.
. In our considered nanofiber photonic crystal cavity, , where η c is the channeling efficiency, c is the light velocity, L is the cavity length 44,45 . It should be noted that since the Cs Rydberg atoms considered here are tightly trapped, the decay γ of the Rydberg superatom is enhanced 46 by γ = N R Γ , where Γ is the decay of an isolated Cs Rydberg atom in the state 70S 1/2 , due to the supperradiant effect 47 . The Rabi frequency and the detuning are chosen here as Ω /2π ~ 100 MHz and Δ p /2π ~ 1 GHz, respectively, which fulfill the adiabatic elimination condition, Δ , . In addition, for the two-color optical dipole trap, with wavelengths 33 1064 nm and 780 nm, respectively, the number of Cs Rydberg atoms of each ensemble can be of the order of 10 4 . Therefore, the collective atom-photon interaction strength reaches π / .  g 2 203 GHz, when η c /2π = 0.01 (see Ref. 33), γ = 27.5 MHz (Γ /2π = 0.55 kHz), L = 10 mm, and N R = 5 × 10 4 . If the atomic number density is increased, this collective atom-photon interaction strength g can increase rapidly, because it is proportional to N R .
• The van der Waals interaction strength V(i − j) = C 6 /(x i − x j ) 6 . Based on the aforementioned energy level structures 48,49 , the van der Waals coefficient is C 6 ≈ 610 GHz·μm 6 . For the distance x i+1 − x i ≈ 2.4 μm, the interaction strength between the nearest-neighbor sites is V 1 /2π ≈ 500 MHz, i.e., V/2π = V 1 /2π ≈ 500 MHz. This interaction strength can be modified by changing the distance of the nearest-neighbor cavities. • The cavity decay rate κ and the photon hopping rate t. In the nanofiber photonic crystal cavity considered in Fig. 2(a) 44,45 , κ = πc/FL. In current experimental setups 34 , F ≈ 500. Thus, κ/2π = 30 MHz and t/2π = 628 MHz, when L = 10 mm. Both the cavity decay rate and the photon hopping rate can be controlled by changing the cavity length.
The above parameters show two basic features: } implies that we may safely neglect the influence of the decay of both cavity and atom, because these only change slightly the phase boundaries 50,51 . In addition, using the above parameters, we also estimate that the atomic number density of each cavity is of the order of 10 12 cm −3 . For such a typical density, the dephasing time of the collective states G i and R i which are induced by the atomic collision, can, at least, reach the order of microseconds. This is much larger than the time scales of κ −1 and g −1 , and can thus be neglected 48,52 . This guarantees the validity of our effective two-level model in Eq. (4) 39,52 .
Photon long-range repulsive interaction. We now construct a strong PLRRI in terms of the Hamiltonian H V . We begin to address the simplest case, κ = V = 0, in which the Hamiltonian (2) reduces to The eigenstates of the Hamiltonian H S are given by Since here we investigate the lower-energy behavior, only the lower polariton branch − n is considered 17 . Thus, the Hamiltonian H S is rewritten as The second term of the Hamiltonian H S leads to an even distribution of polaritons, which provides an effective on-site repulsive interaction between photons 17 . When  t g the rotating-wave approximation is reasonable, and thus the hopping term becomes In addition, since the upper polariton branch + n has the higher probability of Rydberg excitation (stronger repulsive interaction), we also only consider the projection of the van der Waals interaction into the lower polariton branch − n . Thus, the corresponding Hamiltonian becomes is the effective interaction strength. Since V(i − j) > 0, and moreover, V = V 1 ~ g, Eq. (13) demonstrates explicitly that the van der Waals interaction generates a strong PLRRI. As will be shown below, this strong PLRRI leads to non-trivial quantum phases exhibiting photon solid states.
Quantum phases. We investigate quantum phases and phase diagrams by perturbation theory and a mapping into an effective Hamiltonian. For instance, when the chemical potential μ is weak, the high-occupancy-photon states (n > 1) of the Hamiltonian (2) are not considered. In such case, we rewrite the Hamiltonian (2) in a reduced Hilbert space, with n = 0, 1, as and is the single-particle energy of the 1 state. This effective photon hopping rate ⊥ J can be easily tuned by the detuning δ, since θ 1 = arctan(2g/δ)/2. In addition, for the low-energy effective Hamiltonian (15), it is convenient to introduce a renormalized nearest-neighbor van der Waals interaction θ = ∼ V V sin 4 1 to simplify the discussions about phase diagrams, as shown below.
We first consider the case without photon hopping ( ⊥ J = 0). At the initial time, we assume that every cavity is in its vacuum state, as shown in Fig. 3(a). When increasing the chemical potential μ, photons in some cavities can be excited, due to the existence of the PLRRI (without the PLRRI, all cavities are excited identically 17 ), and some 1 states emerges, as shown in Fig. 3(b). The corresponding critical point is with p and q being both integers. In order to quantitatively determine the filling factor ρ, we introduce X i 0 and X i l , where X i 0 is the position of the ith 1 state and X i l is the distance to the lth next 1 state, . When the ground-state energy is minimized for all sites, we have where r l < l/ρ < r l + 1, and satisfy the relation 53,54 In Eq. (19), N 0 is the total number of cavities. For a given filling state, the repulsive interaction energy of the 1 states can be estimated by applying the relations in Eqs. (18)- (19) to the Hamiltonian (15). Moreover, the corresponding phases are stable if it costs energy to add or remove a particle and rearrange the structure.
Photon solid phase. We define the photon solid phase, with the filling factor ρ, as c q . If we add one 1 state, c q becomes p q and the 1 states are crowded. To minimize the repulsive energy, the summation of distances between the 1 states must be a minimum. Thus, the most likely rearrangement structure is that some pairs of the adjacent 1 states are shortened by one site 53,55 . By considering the periodic boundary condition and relations in Eqs.  The expression for Δ μρ shows that the stability interval is only dependent on q, and moreover, decreases rapidly when increasing q. This means that the photon solid phases with p = 1, i.e., ρ = 1/q = 1/2, 1/3, 1/4,…, are more likely to be observed. Below, we mainly address these phases.
Scientific RepoRts | 5:11510 | DOi: 10.1038/srep11510 staircase could be detected experimentally by measuring the mean-photon number / † a a L, since ρ / = / † a a L 2, and thus here called the photon Devil's staircase. However, when increasing ∼ V, ρ varies jumpily from high to low because the PLRRI prevents the photon excitation.
Recently, the photon nearest-neighbor interaction was studied and a photon solid state was predicted 24 . In that case, the Z 2 symmetry, translated by one site, has been broken. Here the PLRRL generates a long-range translation symmetry, whose breaking induces the photon Devil's staircase. Moreover, it leads to other non-trivial phases when the photon hopping exists.
Notice that between the adjacent photon solid phases, with ρ = 1/q and ρ = /( )  q 1 1, respectively, there are many transition states which have different numbers of defects. Here we define the pairs of the 1 states with shorter (longer) distance as a particle-(hole-) like defect structure. Since these states have very small stability intervals, they should be hard to observe when ⊥ J = 0, and thus not plotted in Fig. 3(b). However, when ⊥ J ≠ 0, they play an important role for the ground-state properties, because of the motion of the defects, as shown in Fig. 3(c). Especially, when the hopping energy is negative, the states with defects may be more stable than the adjacent photon solid states. Thus, the photon solid phases melt and a photon-floating solid phase 57 can emerge. In general, it is difficult to fully characterize this process. However, in the region close to the phase-transition point, the repulsive interaction between the defects only allow one defect. Thus, the phase boundary can be estimated by comparing the energy of the photon solid state c q with that of the state with one defect. Using a perturbative method, we obtain the following phase boundaries (see Methods section): Equation (23) shows that the hopping energies of the defects reduce the regions where the photon solid phases exist, because μ up down , and thus the energy bands of the particle-and hole-like defect states cross and the photon solid phases cannot exist. This is the reason why only the photon solid phases, with ρ = 1/2 and ρ = 1/3, can emerge in Fig. 4(b). From Fig. 4(b), we also see that the regions where the photon solid phases exist are very small, and are melted for a smaller ⊥ J ( ⊥ J /g = 0.001). This implies that the hopping term can be treated as a perturbation. So the results from the phase boundaries in Eq. (23) are reasonable. Strictly speaking, in the photon-floating solid phase, the total number of the 1 states is sensitive to the fluctuation of the parameters, and also ρ and / † a a L are hard to calculate in that phase. Recently, the quantum Monte Carlo method has been used to solve this problem 58 . When = ∼ V 0, the photon-floating solid phase disappears [see the blue line in Fig. 4(b)].
Photon-frozen solid phase. Finally, we address the case of a strong chemical potential μ, in which the higher-photon-occupancy states in some cavities can occur, and moreover, the single-particle energy of the  2 state, μ − E 2 , is close to that of the 1 state, μ − E 1 , (here we omit the case n > 2). In this case, there are three kinds of repulsive interactions: between the 1 and 1 states, between the  2 and  2 states, and between the 1 and  2 states. Moreover, the photon hopping has two channels, from the 0 to 1 states and from the 1 to  2 states. These two channels are very complex. However, in the resonant case (δ = 0), sin 2 θ n = 1/2, and  (15), is thus composed of the 0 and 1 states. By increasing μ, ρ increases from 0 and reaches 1. Further increasing μ, all cavities can be excited with uniform photon numbers, which is similar to that of the standard Jaynes-Cummings-Hubbard model, as shown in Fig. 5(a).
However, there is a non-trivial case for a strong PLRRI, as shown in Fig. 5(b). In such case, the pho ton solid phases can exist in the strong-μ region. But we cannot ensure that the lattice is fully filled by the 1 states, due to inversion of μ − E 1 and μ − E 2 . This process can be determined by comparing μ c1 ≈ ω − g + 1. . Since the photon hopping is always frozen even if t exists. We denote the corresponding phase as the photon-frozen solid phase. In this phase, the fractional filling structure of the  2 states is robust, i.e., it is not easily destroyed by the photon hopping. In terms of the Hamiltonian (24), when further increasing μ; to satisfy μ μ ω , the lattice can be fully filled by the  2 states, as shown in Fig. 5

Discussion
In summary, we have achieved a strong PLRRI by controlling the van der Waals interaction of Rydberg atoms located in different cavities in extended Jaynes-Cummings-Hubbard lattices, and then predicted novel quantum phases. Since the atom-cavity polariton can be easily controlled experimentally 59,60 , our proposal offers a new way to control the interaction between individual photons. In addition, our proposal might help to explore rich many-body phenomena of light and quantum nonlinear optics, as well as potential applications to quantum information and computing.

Derivation of Eqs. (20) and (21).
We have described the low-energy behavior of the Hamiltonian (1) by an effective Hamiltonian (15). Moreover, we have also pointed out that when ⊥ J = 0, there is a succession of photon crystal states with different filling factors, denoted as a photon Devil's staircase structure, and the energy gap of the photon crystal states can be calculated in terms of Eqs. (18) and (19), i.e., = X r i l l or r l + 1, and ∑ = X lN i i l 0 . For example, we define the crystalline ground state, with the filling factor ρ = p/q, as c q . By adding one 1 state, the crystalline ground state c q becomes p q . After rearranging the 1 states, the distance r l between the 1 states is changed. Using Eqs. (18) and (19)  is calculated as Figure 5. Schematics of the ground-state phase diagrams as functions of the chemical potential μ and the photon hopping rate t, when δ = 0. In (a) the PLRRI is weak and all cavities are excited to the 1 states before the higher-photon-occupancy states emerge. This can be determined by considering μ c1 < μ c2 . In (b), the PLRRI is strong and the photon-frozen solid phase occurs. This can be determined by considering μ c1 > μ c2 . When μ > μ c1 and μ > μ c3 , all cavities in (a) and (b) are excited identically, respectively. Here, SF, PS, PF, and FS denote the following phases: superfluid, photon solid, photon-floating solid, and photonfrozen solid, respectively. JCH stands for Jaynes-Cummings-Hubbard. This figure is not to scale.