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Quantum Fluctuations, Temperature, and Detuning Effects in Solid-Light Systems

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Quantum Fluctuations, Temperature, and Detuning Effects in Solid-Light Systems PHYSICAL REVIEW LETTERS week ending PRL 100, 216401 (2008) 30 MAY 2008 Quantum Fluctuations, Temperature, and Detuning Effects in Solid-Light Systems Markus Aichhorn,1 Martin Hohenadler,2 Charles Tahan,2,3 and Peter B. Littlewood2 1Institute for Theoretical Physics and Astrophysics, University of Wu¨rzburg, Germany 2Theory of Condensed Matter, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom 3Booz Allen Hamilton Inc., 3811 N. Fairfax Dr., Arlington, Virginia 22203, USA (Received 14 February 2008; revised manuscript received 27 March 2008; published 29 May 2008) The superfluid to Mott insulator transition in cavity polariton arrays is analyzed using the variational cluster approach, taking into account quantum fluctuations exactly on finite length scales. Phase diagrams in one and two dimensions exhibit important non-mean-field features. Single-particle excitation spectra in the Mott phase are dominated by particle and hole bands separated by a Mott gap. In contrast to Bose- Hubbard models, detuning allows for changing the nature of the bosonic particles from quasilocalized excitons to polaritons to weakly interacting photons. The Mott state with density one exists up to tem- peratures T=g * 0:03, implying experimentally accessible temperatures for realistic cavity couplings g. DOI: 10.1103/PhysRevLett.100.216401 PACS numbers: 71.36.+c, 42.50.Ct, 73.43.Nq, 78.20.Bh X X The prospect of realizing a tunable, strongly correlated ^ y ^ JC ^ H ÿt ai aj Hi ÿ Np; system of photons is exciting, both as a testbed for quan- hi;ji i (1) tum many-body dynamics and for the potential of quantum ^ JC y y simulators and other advanced quantum devices. Three Hi j"iih"i j!0ai ai gj"iih#i jai j#iih"i jai : proposals based on cavity-QED arrays have recently Here !0 is the cavity photon energy, and !0 ÿ shown how this might be accomplished [1–3], followed defines the detuning. Each cavity is described by the by further work [4–8]. Engineered strong photon-photon well-known Jaynes-Cummings (JC) Hamiltonian H^ JC. interactions and hopping between cavities allow photons y (as a component of cavity polaritons) to behave much like The atom-photon coupling g (ai , ai are photon creation electrons or atoms in a many-body context. It is clear that a and annihilation operators) gives rise to formation of polar- particular signature of quantum many-body physics, the itons (combinedP atom-photon excitations) whose number ^ y superfluid (SF) to Mott insulator (MI) transition, should be Np i ai ai j"iih"i j is conserved and couples to the reproducible in such systems and be similar to the widely chemical potential [7]. We consider nearest-neighbor studied Bose-Hubbard model (BHM). Yet, the BH analogy photon hopping with amplitude t, define the polariton is not complete. The mixed matter-light nature of the density n hNpi=L, use g as the unit of energy and set system brings new physics yet to be fully explored. !0=g [13], kB and the lattice constant to one. ‘‘Solid-light’’ systems—so-named for the intriguing MI Hamiltonian (1) represents a generic model of strongly state of photons they exhibit—are reminiscent of cold correlated photons amenable to numerical methods. atom optical lattices (CAOL) [9], but have some advan- Existing theoretical work has focused on mean-field cal- tages concerning direct addressing of individual sites and culations [1], exact diagonalization of few-cavity systems device integration, and the potential for asymmetry con- [2,3,7], and the 1D case [8]. Here we employ a quantum struction by individual tuning, local variation, and far from many-body method for the thermodynamic limit to explore equilibrium devices [4]. Photons as part of the system serve the physics of the model. In particular, quantum fluctua- as excellent experimental probes, and have excellent ‘‘fly- tions on a finite length scale are included. We discuss Mott ing’’ potential so that they can be transported over long lobes (also at experimentally relevant finite temperatures), distances. Temporal and spatial correlation functions are the effect of detuning and for the first time in such systems accessible, and nonequilibrium quantum dynamics may be calculate single-particle spectra, a necessary connection to studied using coherent laser pumping to create initial experiment and also a key metric in early proof of concept states. The possible implementations are many [1]. In calculations of CAOL systems. particular, microcavities linked by optical fibers [2,10], The variational cluster approach (VCA)—introduced small arrays of stripline superconducting Cooper-pair first for strongly correlated electrons [14]—has previously boxes or ‘‘transmon’’ cavities [1,11], condensate arrays been applied to the BHM [15]. The main idea is to ap- [12] and color center or quantum dot periodic band-gap proximate the self-energy of the infinite system by that (PBG) materials [1,5] seem most promising. of a finite reference system. The matrix notation includes Here we focus on the simplest solid-light model [1], orbitals and bosonic Matsubara frequencies, describing L optical microcavities each containing a single i!n. The optimal choice for follows from a gen- two-level atom with states j#i, j"i separated by energy . eral variational principle 0, being the grand The Hamiltonian reads (@ 1) potential. Trial self-energies from isolated clusters (with 0031-9007=08=100(21)=216401(4) 216401-1 © 2008 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 100, 216401 (2008) 30 MAY 2008 p Lc sites) are parametrized by the one-particle parameters ary [17]. The spacing of the points n !0 g n ÿ c c c c c p ft ; ;!0; g of the reference-system Hamiltonian, n 1 where adjacent lobes touch at t=g 0 decreases i.e., c. For bosons [15], quickly with increasing n, in contrast to the BHM where n Un. Convergence with cluster size Lc is surprisingly c ÿ1 ÿ1 c fast. The 1D data agree well with exact results, although Tr ln G0 ÿ ÿ Tr ln G : (2) the VCA slightly underestimates t (the value of t at the lobe tip, t=g 0:2 in [8]). Figure 1(b) represents the first Here, c, Gc, and are the grand potential, Green’s accurate (non-mean-field) phase diagram in two dimen- function, and self-energy of an isolated cluster, and G is 0 sions, arguably the most important case for experimental the noninteracting (g 0) Green’s function. G , Gc, and 0 realizations. are evaluated at bosonic Matsubara frequencies i! , and Q n For t=g 0, the MI states are j i , where the traces include frequency summation. The stationary solu- i n i n H^ JC j;ni tion is given by @ =@c 0. Traces can be evaluated ex- -polariton eigenstate of (the branch in [1]) is j#;ni n actly using only the poles of the Green’s function but not a superposition of photonic ( , with photons) and ÿ1 excitonic states (j";nÿ 1i), their weights [14,15]. The poles !m of G0 ÿ are ob- tained from a bosonic formulation of the Q-matrix method j i ;nj#;ni ;nj";nÿ 1i: (3) [16]. At temperature T>0, the required matrix diagonal- n ization restricts L and T . For simplicity, we restrict c max Lobes with n>1 are much smaller due to the effective ourselves to a single variational parameter c !c. 0 polariton repulsion decreasing with n, and we focus on the The full quantum dynamics are taken into account ex- n 1 case for which quantum effects are strongest, and actly on the length scale of the cluster L , and even for c which can be easily initialized experimentally. L 1 VCA results are beyond the mean-field solution c Figures 1(a) and 1(b) show significant deviations from [1,15]. The present formulation cannot describe the prop- the parabolic lobes predicted by mean-field theory in both erties of the SF phase, as the required symmetry-breaking P one and two dimensions. However, quantum fluctuation ^ y term H i ai ai cannot be cast into a single- effects diminish quickly with increasing coordination num- particle operator. However, this does not affect the accu- ber z [Fig. 1(c)]. In particular, the reentrant behavior with racy of the phase boundary of the MI. The numerical effort increasing t=g [18] and the cusplike tip indicative of the is very moderate as compared to, e.g., the density matrix Berezinskii-Kosterlitz-Thouless transition [19] exist only renormalization group [8], and the VCA provides T 0 for z 2. and T>0 static and dynamic properties in one and two Spectral properties play a key role in understanding dimensions. condensed matter systems, but are notoriously difficult to The T 0 phase diagram in one (chain) and two di- calculate accurately. In cavities, the occupation and spectra mensions (square lattice) is shown in Fig. 1. There exists a can be directly monitored through luminescence spectra, series of Mott lobes with integer polariton density nint with angular emission translating into momentum k, much 0; 1; ... and compressibility @n=@ 0 [1]. In con- y more straightforward than in CAOL. trast to recently proposed photonic MI phases [4], ha ai The VCA yields the single-particle spectral function fluctuates even for constant, integer n. Inside the lobes, A k; !ÿÿ1ImG k; !, shown in Fig. 2 for the pa- where the VCA yields a solution, the system has an energy ~ ~ gap Eg;p (Eg;h) for adding a particle (hole) equal to the 1.00 (a) 1D,I (b) 1D, II (c) 1D, III vertical distance of from the upper (lower) phase bound- 0.80 0.60 ) / g 0.40 L = 1 x 1 µ ∆ = 0 ∆ = 0 c MI (n = 2) (a) 1D, T = 0, (b) 2D, T = 0, - 0.20 0.6 0.5 Lc = 2 x 1 ω L = 2 L = 2 x 2 ( 0.5 c SF c 0.00 0.4 L = 8 Lc = 4 c 0.4 L = 6 -0.20 g g SF c L = 8 0.3 III µ / µ / c 0.3 III II -π 0 π -π 0 π -π 0 π MI (n = 1) MI (n = 1) 0.2 I 0.80 0.2 II (d) 2D,I (e) 2D, II (f) 2D, III I 0.1 0.1 0.60 SF SF 0 0 0.40 0 0.04 0.08 0.12 0.16 0.2 0 0.01 0.02 0.03 0.04 0.05 0.06 ) / g µ 0.20 ∆ = 0 (c) T = 0, - 0.5 1D, Lc = 8 (z = 2) ω 2D, L = 6 (z = 3) ( 0.00 0.4 c 2D, L = 8 (z = 4) g c 0.3 -0.20 µ / SF 0.2 MI (n = 1) (0,0) (π,π) (π,0) (0,0)(0,0) (π,π) (π,0) (0,0) (0,0) (π,π) (π,0) (0,0) 0.1 SF k k k 0 0 0.04 0.08 0.12 0.16 0.2 t / g FIG.
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