A polaritonic two-component Bose-Hubbard model

We show that polaritons in an array of interacting micro-cavities with strong atom-photon coupling can form a two-component Bose-Hubbard model. Both polariton species are thereby protected against spontaneous emission as their atomic part is stored in two ground states of the atoms. The parameters of the effective model can be tuned via the driving strength of external lasers. We also describe a method to measure the number statistics in one cavity for each polariton species independently.

Introduction In recent years, significant progress in the theoretical and experimental study of quantum many-body phenomena has been made by employing artificial structures that permit unprecedented experimental control and measurement access. Early activity in this field took place in arrays of Josephson junctions [1] and was followed by several important developments with ultracold atoms in optical lattices [2]. Despite their success, Josephson junction arrays and optical lattices face limitations as it is challenging to access and control individual lattice sites, due to their small separation.
A possibility to overcome these hurdles has very recently been suggested in arrays of coupled micro-cavities, where a scheme for simulating the Bose-Hubbard Hamiltonian [3] and models of interacting Jaynes-Cummings Hamiltonians [4] have been proposed. The phase diagrams of these models were studied in [5], where the existence of a glassy phase has been predicted. These setups, where atoms interact with the resonant modes of the cavities, offer further possibilities to generate effective many-body systems as they can be manipulated with external driving lasers. One of these possibilities, effective spin Hamiltonians, has been studied recently [6].
Here, we show that coupled high-Q cavities can host an effective two-component Bose-Hubbard model, where b † R (c † R ) create polaritons of the type b(c) in the cavity at site R, n and µ c are the polariton energies, U b , U c and U b,c their on-site interactions and J b,b , J c,c and J b,c their tunneling rates.
Bose-Hubbard models of two components [7] can display several interesting phenomena which are partly also known for a Luttinger liquid of low energy excitations in fermionic systems. Among these are spin density separa-tion [8,9], spin order in the Mott regime [10] and phase separation [11].
We consider an array of cavities and study the dynamics of polaritons, combined atom photon excitations, in this arrangement. Since the distance between adjacent cavities is considerably larger than the optical wavelength of the resonant mode, individual cavities can be addressed. Photon hopping occurs between neighboring cavities while the force between two polaritons occupying the same site is generated by a large Kerr nonlinearity [18]. This force can be repulsive and attractive. Each cavity is interacting with an ensemble of these atoms, which are driven by an external laser. By varying the intensity of the driving laser, the parameters of the effective model can be tuned. An experimental realization would require cavities that operate in the strong coupling regime [12,13,14,15,16,17].
The atoms To generate a force between polaritons that are located in the same cavity, we fill the cavity with 4 level atoms, see figure 1: The transitions between levels 2 and 3 are coupled to the laser field and the transitions between levels 2-4 and 1-3 couple via dipole moments to the cavity resonance mode. It has been shown by Imamoglu and co-workers, that this atom cavity system can show a very large nonlinearity [18].
In a rotating frame with respect to , the Hamiltonian of the atoms in the cavity reads, (Ω σ j 23 + g 13 σ j 13 a † + g 24 σ j 24 a † + h.c.) , where σ j kl = |k j l j | transfers level l of atom j to level k of the same atom, ω C is the frequency of the cavity mode, δ, ∆ and ε are detuning parameters (see figure 1), Ω is the Rabi frequency of the driving by the laser and g 13 and g 24 are the parameters of the dipole coupling of the cavity mode to the respective atomic transitions which are all assumed to be real. All atoms interact in the same way with the cavity mode and hence the only relevant states are Dicke type dressed states [22]. Polaritons In the case where g 24 = 0 and ε = 0 , level 4 of the atoms decouples from the dressed state excitation manifolds [18]. If we furthermore assume that the number of atoms is large, N ≫ 1, the Hamiltonian H I can be written in terms of three polariton species. Their creation (and annihilation) operators read, The operators p † 0 , p † + and p † − describe polaritons, quasi particles formed by combinations of atom and photon excitations.
In the relevant Hilbert space spanned by symmetric Dicke type dressed states and for N ≫ 1, they satisfy bosonic commutation relations, [p j , p l ] = 0 and [p j , p † l ] = δ jl for j, l = 0, +, −, where the neglected terms are of order "number of polaritons"/N . p † 0 , p † + and p † − thus describe independent bosonic particles. In terms of these polaritons, the Hamiltonian H I for g 24 = ε = 0 reads, where the frequencies are given by µ 0 = 0 and µ ± = (δ ± A)/2.
We will now consider the case δ ≫ Ω, g. Here, the polaritons and their frequencies read, up to first order in δ −1 . There is no spontaneous emission from the atomic level 2 and hence to leading order, the polaritons p † 0 and p † − do not experience spontaneous emission loss. We therefore define the two polariton species In the rotating frame, the polaritons b † have an energy µ b = 0 and the polaritons c † have an energy µ c = −(B/δ)B. A possible disorder in the resonance frequency of the cavities and hence in δ would thus affect µ b and µ c differently which can have interesting consequences for the phase transitions of the model [19]. The dynamics of these two species is governed by the two component Bose-Hubbard Hamiltonian (1) as we shall see.
Perturbations To write the full Hamiltonian H I , in the polariton basis, we express the operators , and we made use of the rotating wave approximation: Since δ ≫ Ω, g, couplings to the polaritons p † + are negligible, provided that For max(|g 24 gΩ/B 2 |, |g 24 (g 2 − Ω 2 )/B 2 |) ≪ |∆|, the couplings to level 4 can be treated in a perturbative way. If furthermore |g 24 gΩ/B 2 | ≪ |B 2 /δ|, this results in energy shifts of where n b and n c are the numbers of b † respectively c † polaritons. The on-site interactions for the polaritons b † and c † can thus be written as [23] U bc > 0 if ∆ + B 2 /δ < 0 and vice versa. There can thus be repulsive and attractive interactions at the same time, e.g. for ∆ < 0 and |∆| < B 2 /δ we have U b > 0, U c < 0 and U bc < 0. In a similar way, the two photon detuning ε leads to and additional on-site term ε where the transitions b † c + c † b are suppressed if |εgΩ/B 2 | ≪ |B 2 /δ|. Polariton tunneling If the cavities are either coupled by optical fiber tapers or directly via an overlap of evanescent fields, photons can tunnel between neighboring cavities. This process is described by the Hamiltonian α (a † R a R ′ + h.c.), where α is the tunneling rate of the photons. We translate this term into the polariton picture and assume that the tunneling rate is much smaller than δ. In this regime, the tunneling does not induce transitions between the polaritons b † or c † and p † + . Hence the p † + decouple from the polaritons b † and c † whose tunneling terms read, where J bb = αg 2 /B 2 , J cc = αΩ 2 /B 2 and J bc = αgΩ/B 2 . If |J bc | ≪ |B 2 /δ|, transitions between b † and c † are suppressed. This is the case for any Ω as long as g 2 ≫ αδ/2. Parameter range Here we give one example how the parameters of the effective Hamiltonian (1) vary as a function of the intensity of the driving laser Ω. We choose the parameters of the atom cavity system to be g 24 = g 13 , N = 1000, ∆ = −g 13 /20, δ = 2000 √ Ng 13 and α = g 13 /10. Figure 2 shows the interactions U b , U c and U bc , the tunneling rates J bb , J cc and J bc and |µ c − µ b | as a function of Ω/g 13 . For g ≈ Ω we have |U bc | ≪ |U b |, |U c | and J bb ≈ J cc ≈ J bc . Whenever |µ c − µ b | < |J bc |, b † polaritons get converted into c † polaritons and vice versa via the tunneling J bc . With the present choice of α and δ, this happens for 0.16g Ω 1.6g. To avoid such processes, one either needs to choose α smaller or δ larger, where both choices would require higher Q of the cavities to ensure sufficient lifetime. The interactions U b , U c and U bc can furthermore be adjusted by varying the detuning ∆. This can be done by generating a Stark shift to the atomic level 4 with an additional laser that drives the transition between level 4 and a further atomic level in a dispersive (detuned) way.
Numerical results To confirm the validity of the approximations involved in the above derivation, we present a numerical simulation of the full dynamics of polaritons b † and c † in three cavities that each couple to N = 1000 atoms and compare it to the dynamics of the corresponding effective model (1). We consider initial conditions with exactly one polariton b † in cavity 1 and in cavity 2 and exactly one polariton c † in cavity 3. Figure 3a shows the numbers N b = n b and N c = n c of polaritons b † and c † and their number fluctuations and F c = n 2 c − N 2 c for the first cavity. Figure 3b in turn shows differences between the full description and the effective model (1) The effective model describes the dynamics very well.
Spontaneous emission and cavity decay Level 2 of the atoms is metastable and hence its decay rate negligible on the relevant time scales. The decay mechanisms for the polaritons b † and c † thus originate in the cavity decay of the photons and the very small but non-negligible  (2). The resulting effective decay rates, Γ b for the polaritons b † and Γ c for the po- where Θ is the Heaveside step function, κ the cavity decay rate and γ 3 (γ 4 ) the spontaneous emission rates from levels 3 (4). For successfully observing the dynamics and phases of the effective Hamiltonian (1), the interactions U b , U c and U bc need to be much larger than Γ b and Γ c .
The experimentally least demanding case is the onecomponent model for the polaritons b † , for which δ ∼ g. Assuming g 24 = g 13 the maximal achievable ratio of U b /Γ b is here 1 2 g 13 / κ Θ(n b − 2)γ 4 . In particular the Mott state for the polaritons b † , where n b = 2, can even be realized in bad cavities without the strong coupling regime. However, to observe the transition to the superfluid phase, the strong coupling regime with g 13 ≫ √ κγ 4 is required for the single component model, too.
To obtain an estimate for a model with both components, b † and c † , we consider three cases, g ≈ Ω, Ω ≈ 10g and Ω ≈ g/10. Note that g ≪ δ and hence spontaneous emission via level 3 is strongly suppressed. Denoting ζ = g 13 / √ κγ 4 , the achievable ratios of interaction versus decay rates for g ≈ Ω are U b /Γ b ≈ U c /Γ c ≈ ζ/(2 √ 2), while the cross interaction vanishes, U bc ≈ 0. For Ω = 10g (Ω = g/10) the achievable ratios are Realizing these parameters requires cavities that operate in the strong coupling regime with large cooperativity factors, ζ ≫ 1. This regime is currently being achieved in several devices, in photonic band gap cavities [12] (ζ ≈ 3), Fabry-Perot cavities [13] (ζ ≈ 13), toroidal micro-cavities [14] (ζ ≈ 7), fiber cavities [15] (ζ ≈ 17) and micro-cavities on a gold coated silicon chip [16] (ζ ≈ 6) among others. Our scheme should thus be experimentally feasible with current or soon to be available technology. Values of ζ that are predicted to be achievable are as high as 200 for photonic band gap cavities and 3000 for toroidal micro-cavities [17]. Besides the strong coupling, a realization of our scheme also requires trapping the atoms in the location of strong coupling for sufficient time.
Measurements The number statistics of both polariton species b † and c † in one cavity can be measured using state selective resonance fluorescence in a way proposed in [21]. In the one-component BH model [3], the polaritons can therefore be mapped by a STIRAP passage [20] onto the atomic levels. In the two-component case the STIRAP can however not be applied as in [3] because the energies µ b and µ c are similar and the passage would thus need to be extremely slow to still be adiabatic.
For two components, one can do the measurements as follows. First the external driving laser Ω is switched off. Then the roles of atomic levels 1 and 2 are interchanged in each atom via a Raman transition by applying a π/2-pulse. To this end the transitions 1 ↔ 3 and 2 ↔ 3 are driven with two lasers (both have the same Rabi frequency Λ) in two-photon resonance for a time T = πδ Λ /|Λ| 2 (δ Λ is the detuning from atomic level 3). The configuration is shown in figure 4a. This pulse results in the mapping |1 j ↔ |2 j for all atoms j.
Next another laser, Θ, that drives the transition 1 ↔ 4 is switched on, see figure 4b. Together with the coupling g 24 , this configuration can be described in terms of three polaritons, q † 0 , q † + and q † − , in an analogous way to p † 0 , p † + and p † − , where now the roles of the atomic levels 1 and 2 and the levels 3 and 4 are interchanged. Hence, if we choose Θ = Ω the π/2-pulse maps the b † onto the dark state polaritons of the new configuration, q † 0 , whereas if we choose Θ = −Ω it maps the c † onto q † 0 . The driving laser is then adiabatically switched off, Θ → 0, and the corresponding STIRAP process maps the q † 0 completely onto atomic excitations of level 1. This process can now be fast since the detuning ∆ is significantly smaller than δ and hence the energies of all polariton species q † 0 , q † + and q † − well separated. Another π/2-pulse finally maps the excitations of level 1 onto excitations of level 2, which can be measured by state selective resonance fluorescence in the same way as discussed in [3,21].
The whole sequence of π/2-pulse, STIRAP process and another π/2-pulse can be done much faster than the timescale set by the dynamics of the Hamiltonian (1) [3] and b † or c † can be mapped onto atomic excitations in a time in which they are not able to move between sites. The procedure thus allows to measure the instantaneous local particle statistics of each species separately.
Summary We have shown that a two-component Bose-Hubbard model of polaritins can be created in cou- pled arrays of high-Q cavities. As a new feature, the model can display transitions between the two particle species. An experimental realization is feasible with cavities that have cooperativity factors much greater than unity and interact with the atoms for sufficient time. The local particle number statistics of both species can be measured independently with high accuracy. This work is part of the QIP-IRC supported by EP-SRC (GR/S82176/0), the Integrated Project Qubit Applications (QAP) supported by the IST directorate as Contract Number 015848' and was supported by the EP-SRC grant EP/E058256, the Alexander von Humboldt Foundation, the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and the Royal Society.