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Photon-Blockade-Induced Mott Transitions and XY Spin Models in Coupled Cavity Arrays

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Photon-Blockade-Induced Mott Transitions and XY Spin Models in Coupled Cavity Arrays RAPID COMMUNICATIONS PHYSICAL REVIEW A 76, 031805͑R͒͑2007͒ Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays Dimitris G. Angelakis,1,* Marcelo Franca Santos,2 and Sougato Bose3 1Centre for Quantum Computation, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, United Kingdom 2Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, 30161-970, Minas Gerais, Brazil 3Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, United Kingdom ͑Received 15 December 2006; revised manuscript received 26 June 2007; published 28 September 2007͒ We propose a physical system where photons could exhibit strongly correlated effects. We demonstrate how a Mott-insulator phase of atom-photon excitations ͑polaritons͒ can arise in an array of individually addressable coupled electromagnetic cavities when each of these cavities is coupled resonantly to a single two-level system ͑atom, quantum dot, or Cooper pair͒. This Mott phase is characterized by the same integral number of net polaritonic excitations with photon blockade providing the required repulsion between the excitations in each site. Detuning the atomic and photonic frequencies suppresses this effect and induces a transition to a photonic superfluid. Finally, on resonance the system can straightforwardly simulate the dynamics of many-body spin systems. DOI: 10.1103/PhysRevA.76.031805 PACS number͑s͒: 42.50.Pq, 05.70.Fh, 03.67.Ϫa, 03.67.Lx Strongly coupled many-body systems described by the wave resonators ͓8,13͔. This has already stimulated propos- Bose-Hubbard models ͓1͔ exhibit Mott-insulating phases als to use such systems to implement quantum computation whose realization in optical lattices ͓2͔ has opened varied ͓18͔ using coupled CCWs ͓19͔ and in a two-dimensional possibilities for the simulation of many-body physics ͓3͔. cluster state setup ͓20͔. The production of entangled photons Can there be another engineered quantum many-body system ͓21͔ and the interaction of polaritons ͓22͔ have been studied which displays such phases? This will be especially interest- as well. The Hamiltonian corresponds to a series of quantum ing if the strengths of this system are “complementary” to harmonic oscillators coupled through hopping photons and is that of optical lattices—for example, if it allowed the coex- given by H=͚N ␻ a†a +͚N A͑a†a +H.c.͒, where a† ͑a ͒ istence of accessibility to individual constituents of a many- k=1 d k k k=1 k k+1 k k are the localized eigenmodes ͑Wannier functions͒. The pho- body system and a strong interaction between them, or if it ton frequency and hopping rate are ␻d and A, respectively, allowed the simulation of arbitrary networks rather than and no nonlinearity is present yet. Assume now that the cavi- those derivable from superposing lattices. It will be particu- ties are doped with two-level systems ͑atoms, quantum dots, larly arresting to find such phases by modifying a system of or superconducting qubits͒, and ͉g͘k and ͉e͘k are their ground photons which, by being noninteracting, are unlikely candi- and excited states at site k. The Hamiltonian describing the dates for the studies of many-body phenomena. Here we pro- system is the sum of three terms, Hfree, the Hamiltonian for pose such a system consisting of coupled electromagnetic the free light and dopant parts, Hint, the Hamiltonian describ- cavities ͓4–7͔, doped with single two-level systems ͓8–15͔. ing the internal coupling of the photon and dopant in a spe- Using the nonlinearity generated from the corresponding cific cavity, and Hhop, for the light hopping between cavities: photon blockade effect ͓16͔, we show the possibility of ob- serving an insulator phase of total ͑atomic+photonic or sim- N free † ply polaritonic͒ excitations and its transition to a superfluid H = ␻d͚ akak + ␻0͚ ͉e͘k͗e͉k, ͑1͒ ͑SF͒ of photons. Compared to the optical lattice case, the k=1 k different Hubbard-like model that describes the system in- volves neither purely bosonic nor purely fermionic entities; N int † the transition from insulator to SF is also accompanied by a H = g͚ ͑ak͉g͘k͗e͉k + H.c.͒, ͑2͒ transition of the excitations from polaritonic to photonic; and k=1 excitations, rather than physical particles such as atoms, are involved. In addition, the possibility of simulating the dy- N hop † namics of an XY spin chain with individual spin manipula- H = A͚ ͑akak+1 + H.c.͒, ͑3͒ tion, using a mechanism different from that used in optical k=1 lattices ͓3͔, has been suggested. where g is the light-atom coupling strength. The Hfree+Hint Assume a chain of N coupled cavities. A realization of part of the Hamiltonian can be diagonalized in a basis of this has been studied in structures known as coupled resona- mixed photonic and atomic excitations, called polaritons. tor optical waveguides or couple-cavity waveguides ͑CCWs͒ These polaritons, also known as dressed states, involve a in photonic crystals ͓6,17͔, in tapered fiber-coupled toroidal mixture of photonic and atomic excitations and are microcavities ͓10͔, and in coupled superconducting micro- ͑±,n͒ defined by the operators Pk =͉g,0͘k͗n±͉k, where ͉n͘k denotes the n-photon Fock state in the kth cavity. The polari- tons of the kth atom-cavity system denoted by ͉n±͘k are *[email protected] given by ͉n+͘k =͑sin ␪n ͉g,n͘k +cos ␪n ͉e,n−1͘k͒/ͱ2 and 1050-2947/2007/76͑3͒/031805͑4͒ 031805-1 ©2007 The American Physical Society RAPID COMMUNICATIONS ANGELAKIS, SANTOS, AND BOSE PHYSICAL REVIEW A 76, 031805͑R͒͑2007͒ “on-site” photonic repulsion, and leads to the Mott state of the net excitations per site being the ground state for com- mensurate filling. Reducing the strength of the effective non- linearity, i.e., the blockade effect, through detuning for ex- ample, should drive the system to the SF regime. This could be done by Stark-shifting the atomic transitions from the cavity by an external field. The new detuned polaritons are not as well separated as before, and their energies are merely shifts of the bare atomic and photonic ones by ±g2n/⌬, re- spectively. In this case it costs no extra energy to add exci- tations ͑excite transitions to higher polaritons͒ in a single site, and the system moves to the SF regime. Note here that the mixed nature of the polaritons could in principle allow for mostly photonic excitations and a photonic Mott state. The required values of g and ⌬ for the corresponding non- linearity, though, seem to be unrealistic within current tech- nology ͓8–15͔. FIG. 1. ͑Color online͒ A series of cavities coupled through light To quantify the transition of the system from a Mott phase and the polaritonic energy levels for two neighboring cavities. to a SF phase as the detuning ⌬=␻0 −␻d is increased, we have performed a numerical simulation using 3–7 sites with the Hamiltonian of Eqs. ͑1͒–͑3͒͑numerical diagonalization ͉n−͘k =͑cos ␪n ͉g,n͘k −sin ␪n ͉e,n−1͘k͒/ͱ2 with energies 1 E± =n␻ ±gͱn+⌬2 /g2, tan 2␪ =−gͱn/⌬, and atom-light of the complete Hamiltonian without any approximations͒. n d n In the Mott phase the particle number per site is fixed and its detuning ⌬=␻0 −␻d. They are also eigenstates of the sum of the photonic and atomic excitation operator variance is zero ͑every site is in a Fock state͒. In such a † phase, the expectation value of the destruction operator for =a a +͉e͗͘e͉ with eigenvalue n ͑Fig. 1͒. k k k k the relevant particles, the order parameter, is zero. In the N We will now justify the statement that the lowest-energy traditional mean-field and thus necessarily approximate state of the system consistent with a given number of net ͑ ͒ picture, this expectation value becomes finite on transition to excitations per site ͑or filling factor͒ becomes a Mott state of a SF, as a coherent superposition of different particle num- the net ͑polaritonic͒ excitations for integer values of the fill- bers is allowed to exist per site. However, our entire system ing factor. To understand this, we rewrite the Hamiltonian is a “closed” system and there is no particle exchange with ͑for ⌬=0͒ in terms of the polaritonic operators as the outside. SF states are characterized by a fixed “total” N ϱ ϱ number of particles in the finite-site system and the expecta- ͑−,n͒† ͑−,n͒ ͑+,n͒† ͑+,n͒ tion of a destruction operator at any given site is zero even in H = ͚ ͚ n͑␻d − g͒Pk Pk + ͚ n͑␻d + g͒Pk Pk k=1 ͩn=1 n=1 the SF phase. Thus this expectation value cannot be used as ϱ ϱ an order parameter for a quantum phase transition. Instead, ͱ ͑−,n͒† ͑−,n͒ ͱ ͑+,n͒† ͑+,n͒ we use the variance of the total number of excitations per + ͚ g͑n − n͒Pk Pk + ͚ g͑ n − n͒Pk Pk n=1 n=1 ͪ site, the operator k, at a given site ͑we choose the middle N N cavity, but any of the other cavities would do͒ to characterize † the Mott-to-SF phase transition. This variance, var͑ k͒, has + A͚ ͑akak+1 + H.c.͒. ͑4͒ N k=1 been plotted in Fig. 2 as a function of log10 ⌬ for a filling factor of one net excitation per site. For this plot, we have 2 1 The above implies ͑assuming the regime An gͱn ␻d͒ taken the parameter ratio g/A=10 ͑g/A=10 gives very that the lowest-energy state for a given number,≪ say ␩,≪ of net similar results͒, with ⌬ varying from ϳ10−3g to ϳg and 4 excitations at the kth site would be the state ͉␩−͘k ͑this is ␻d ,␻0 ϳ10 g.
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