PHYSICAL REVIEW LETTERS week ending PRL 100, 216401 (2008) 30 MAY 2008

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

Markus Aichhorn,1 Martin Hohenadler,2 Charles Tahan,2,3 and Peter B. Littlewood2 1Institute for Theoretical 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 of the bosonic particles from quasilocalized excitons to polaritons to weakly interacting . 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 t ai aj Hi 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 energy, and !0 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 iai 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 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 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 1 1 c fast. The 1D data agree well with exact results, although Tr lnG0  Tr lnG : (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 1 excitonic states (j";n 1i), their weights [14,15]. The poles !m of G0  are ob- tained from a bosonic formulation of the Q-matrix method

j i;nj#;ni;nj";n 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, Ak; !1ImGk; !, 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. 2. Single-particle spectra at T 0 in one dimension [(a)– FIG. 1 (color online). Ground-state phase diagram: (a) 1D, (c), Lc 6] and 2D [(d)–(f), Lc 2 2]. I–III refer to the (b) 2D, and (c) different coordination numbers z. Lines are marks in Fig. 1, i.e., (a) t=g 0:16, (b) 0.08, (c) 0.01, (d) 0.05, guides to the eye. (e) 0.02, (f) 0.005. 216401-2 PHYSICAL REVIEW LETTERS week ending PRL 100, 216401 (2008) 30 MAY 2008 rameters marked in Fig. 1. Important for detection in 0.5 (a) 1D, T = 0, ∆ / g = 2 experiment, the MI state is characterized by cosinelike 0 Lc = 2 particle and hole bands, separated by the Mott gap Eg, g L = 4 -0.5 c µ / MI (n = 1) L = 6 minimal at k 0, which decreases with increasing t=g, c and eventually closes in the SF phase. The phase bounda- -1 ries are related to the particle or hole dispersions -1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 "p=hk 0;t via t"p0;t and t"h0;t, 0.68 and the Mott gap is given by E t" 0;t" 0;t g p h (b) 1D, T = 0, ∆ / g = -2 [20]. 0.66 0.64 For t t , particles are much lighter than holes (e.g., g

µ / MI (n = 1) tmp 0:33, tmh 0:81 in Fig. 2(c); obtained from para- 0.62 bolic fits). The particle or hole bandwidth scales almost 0.6 linearly with t, confirming the weakly interacting Bose gas 0.58 0 0.005 0.01 0.015 0.02 0.025 picture for the superfluid fraction of particles/holes doped t / g into the MI [19]. The hole bandwidth in both one and two 4 (c) ∆= 2g (d) ∆ = -2g dimensions is about zt. For the BHM model, the ratio of the 3 0.1 bandwidths is Wp=Wh 2 for the MI with n 1 due to g 2 ) / µ the effective particle (hole) hopping tp n 1t (th - 1 0 ω ( nt)[15,21]. For the JCM, the matrix elements for hop- 0 ping of one particle or hole in the MI have to be evaluated -0.1 j i -π 0 π -π 0 π pusing the dressed states n , and we find Wp=Wh 3=2 k k 2 2:91 in one dimension, in good agreement with Figs. 2(a)–2(c). The bandwidth ratio is fairly independent FIG. 3 (color online). Phase diagram for detuning (a) =g 2, (b) =g 2, and spectra for (c) =g 2, t=g 0:7, and of t=U, respectively, t=g because the interaction energy of (d) =g 2, t=g 0:0:018 in one dimension at T 0. a single particle or hole is the same at every site. In the BHM, emergent particle-hole symmetry leads to mp mh ! 0 for t ! t [22]. This behavior is also seen in the Solid state quantum devices will operate at finite tem- JCM (mp=mh 1:3 for t 0:16 in 1D), but the VCA does perature, much higher than in CAOL (estimated at nK). not permit a detailed analysis near t . This leads to the important question of the stability of the In contrast to CAOL, the parameters and !0 and hence MI state at T>0, which will ultimately determine their the detuning !0 can easily be changed experi- technological usefulness. Strictly speaking, there is no true mentally. This permits to tune the character of the bosonic MI at T>0 due to thermal fluctuations. However, there particles. The states j ni are excitonlike for !0 , exist regions where fluctuations are small enough for the photonlike for !0 , and polaritonlike for !0 . system to behave like a MI for experimental purposes [23]. The 1D phase diagrams for =g 2 are shown in We determine the region of existence of the MI using the 4 Figs. 3(a) and 3(b). The excitonic system (=g 2) with stringent criterion n jn nintj10 , corresponding small photon-mediated hopping exhibits a large n 1 to a ‘‘worst case scenario’’, since experimentally the MI Mott lobe, whereas the latter is very small in the photonic will survive as long as the Mott gap g is large compared system with small exciton-mediated interaction. Reentrant to thermal fluctuations. behavior due to quantum fluctuations is seen for the ex- Results for 0 are shown for one and two dimensions citonic case, but is absent in the photonic case for large in Fig. 4. Finite-size effects diminish quickly with increas- enough Lc. ing temperature [see Fig. 4(a)]. The size of the n 1 Mott Single-particle spectra in one dimension near the lobe lobe is significantly reduced with increasing temperature, tips are shown in Fig. 3(c) and 3(d). The excitonic system and n exceeds 104 at T=g 0:03 in both one and two shows a very large ratio of particle and hole bandwidths dimensions. This value is consistent with the onset of (Wp=Wh 7), whereas Wp=Wh 2:1 (very close to the deviations of n from 1 in the exact atomic-limit results, BHM, since j 1ij#; 1i) for the photonic case. These and agrees with the few-cavity results of [7]. Lobes with values result from the different admixture of the states n>1 disappear at much lower temperatures (not shown). j"; 0i and j#; 1i to j 1i depending on . In particular, the In contrast to T 0, Mott lobes with n and n 1 do not approximate relation Wh zt found for 0 does not touch at t=g 0, but are surrounded by the normal fluid hold.Q The excitonic MI with n Q1 is approximately given (NF) phase for sufficiently small t=g, and a transition NF- by ij"; 0ii, whereas we have ij#; 1ii for the photonic SF occurs at larger t=g. However, while the VCA can case. Finally, the incoherent features in Fig. 3(c) may distinguish between the MI and the NF, the phase boundary originate from the finite cluster size Lc, and should be to the SF at T>0 cannot be determined accurately. The addressed using other exact methods. gap in Ak;! increases with T [23]. Particle or hole bands 216401-3 PHYSICAL REVIEW LETTERS week ending PRL 100, 216401 (2008) 30 MAY 2008

features of Bose-Hubbard models and possible experi- (a) 1D, ∆ = 0 T/g = 0.00 (L = 6) 0.5 c T/g = 0.01 (Lc = 2) ments, and the novel physics emerging from detuning has T/g = 0.01 (L = 4) c been explored. 0.4 T/g = 0.02 (L = 2) c ¨ g T/g = 0.03 (L = 2) This work was supported by the FWF Schrodinger Grant 0.3 c µ / No. J2583, the DFG research unit FOR538, and by NSF 0.2 Grant No. DMR-0502047. C. T. would like to acknowledge useful conversations with A. Greentree et al. 0.1

0 0 0.04 0.08 0.12 0.16 0.2

T/g = 0 (b) 2D, ∆ = 0 [1] A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. 0.5 T/g = 0.005 (L = 1 x 1) T/g = 0.01 Hollenberg, Nature Phys. 2, 856 (2006). c T/g = 0.02 0.4 T/g = 0.03 [2] M. J. Hartmann, F. G. S. L. Branda˜o, and M. B. Plenio,

g Nature Phys. 2, 849 (2006). 0.3 µ / [3] D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev. A 0.2 76, 031805(R) (2007). [4] M. J. Hartmann and M. B. Plenio, Phys. Rev. Lett. 99, 0.1 103601 (2007); Phys. Rev. Lett. 100, 070602 (2008). 0 [5] N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, Phys. 0 0.01 0.02 0.03 0.04 0.05 0.06 t / g Rev. A 77, 031803(R) (2008). [6] S.-C. Lei and R.-K. Lee, Phys. Rev. A 77, 033827 (2008). FIG. 4 (color online). Mott-like regions for different T. [7] M. I. Makin, J. H. Cole, C. Tahan, L. C. L. Hollenberg, and A. D. Greentree, arXiv:0710.5748 [Phys. Rev. A (to be published)]. are still well defined at T>0, and spectra (not shown) look [8] D. Rossini and R. Fazio, Phys.Rev. Lett. 99, 186401 (2007). very similar to T 0. Excitation spectra of the NF and SF [9] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. deserve detailed future studies. Sen, and U. Sen, Adv. Phys. 56, 243 (2007). The quasi-MI lobes in Fig. 4 are dominated by the point [10] D. E. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, arXiv:0712.1817 (unpublished). (t 0, 0:3) where E~g is maximal and by the T 0 lobe tip (, t). The MI is destroyed by thermal excitation [11] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schu- ster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and of particles or holes, and therefore survives longest near R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007). 0:3 where the energy cost is largest. Besides, T is [12] C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H. determined by the size of the lobe at T 0, t 0. Hence, Deng, M. D. Fraser, T. Byrnes, P. Recher, N. Kumada, T. the system at T>0 is dominated by atomic-limit physics Fujisawa, and Y. Yamamoto, Nature (London) 450, 529 (yielding the same T in 1D and 2D) and the T 0 fixed (2007). point (quantum critical point) of the MI-SF transition [19]. [13] As in [1]; !0 is not an absolute frequency, so that this For the 1D BHM [19], the VCA yields T=U 0:059, choice does not affect results. lower than T=U 0:2 found in [24] using a different [14] M. Potthoff, M. Aichhorn, and C. Dahnken, Phys. Rev. criterion [25]. The ratio of the critical temperatures for Lett. 91, 206402 (2003); M. Potthoff, Eur. Phys. J. B 36, the 1D JCM and 1D BHM is 0.56, close to the ratio of the 335 (2003). t 0, T 0 Mott gaps (0.59). [15] W. Koller and N. Dupuis, J. Phys. Condens. Matter 18, 9525 (2006). Solid state cavity-QED systems offer the possibility of 10 [16] M. Aichhorn, E. Arrigoni, M. Potthoff, and W. Hanke, large g. Taking a conservative estimate g 10 Hz [1] Phys. Rev. B 74, 235117 (2006). gives for the effective temperature of the quantum system [17] For t=g 0 and density n, a particle or hole corresponds T * 14 mK. Even taken as the actual temperature, this is to one site having occupation n 1 (n 1). well accessible experimentally. The hopping rate 1=t has to [18] T. D. Ku¨hner and H. Monien, Phys. Rev. B 58, R14741 be fast enough to permit equilibration before photon loss (1998). and dephasing set in. Indeed, taking t=g 0:01 (inside the [19] M. P.A. Fisher, P.B. Weichman, G. Grinstein, and D. S. Mott lobe in Fig. 4(b)], we obtain a realistic t1 * 0:14 ps. Fisher, Phys. Rev. B 40, 546 (1989). Finally, T for the n 1 MI is enhanced by detuning > [20] N. Elstner and H. Monien, Phys. Rev. B 59, 12 184 (1999). 0 [Fig. 3(a)], which additionally increases t and jj, [21] D. van Oosten, P. van der Straten, and H. T. C. Stoof, Phys. estimated as ;t0:08; 0:02 meV in two dimen- Rev. A 63, 053601 (2001). [22] B. Capogrosso-Sansone, N. V. Prokof’ev, and B. V. sions for T 0. Svistunov, Phys. Rev. B 75, 134302 (2007). In summary, motivated by theoretical predictions and [23] D. van Oosten, P. van der Straten, and H. T. C. Stoof, Phys. ongoing experimental advances, we have studied polariton Rev. A 67, 033606 (2003). Mott phases by means of a versatile quantum many-body [24] F. Gerbier, Phys. Rev. Lett. 99, 120405 (2007). approach. Phase diagrams, single-particle spectra and fi- [25] There are no well-defined plateaus in the density even nite temperature effects have been related to the known below T=U 0:2 in [24]. 216401-4