Bose-Hubbard models coupled to cavity light fields

Recent experiments on strongly coupled cavity quantum electrodynamics present new directions in “matter-light” systems. Following on from our previous work [Phys. Rev. Lett. 102, 135301 (2009)] we investigate Bose-Hubbard models coupled to a cavity light field. We discuss the emergence of photoexcitations or “polaritons” within the Mott phase, and obtain the complete variational phase diagram. Exploiting connections to the super-radiance transition in the Dicke model we discuss the nature of polariton condensation within this novel state. Incorporating the effects of carrier superfluidity, we identify a first-order transition between the super-radiant Mott phase and the single component atomic superfluid. The overall predictions of mean field theory are in excellent agreement with exact diagonalization and we provide details of superfluid fractions, density fluctuations, and finite size effects. We highlight connections to recent work on coupled cavity arrays.

Figure 1

Heuristic diagram showing photoexcitations above the filled bosonic Mott state. In analogy with fermionic band insulators, the coherent superposition of a particle–hole pair and a photon will be referred to as a “polariton”. Within an equilibrium framework these polaritons may Bose condense.

Figure 2

Zero hopping phase diagram in the $U_{\mathit{aa}}\to \infty$ limit with $U_{\mathit{bb}}=U_{\mathit{ab}}=0$. We set ${\epsilon }_b=-{\epsilon }_a=\mathrm{ḡ}=1$ and $\omega =1$ corresponding to $\omega <{\omega }_0$, where ${\omega }_0\equiv {\epsilon }_b-{\epsilon }_a$. The vertical line, $\mathrm{ḡ}={\mathrm{ḡ}}_c$, is the super-radiance transition in the Dicke model, and separates a Mott insulator (MI) with $n_a+n_b=1$ and $\langle {\psi }^{†}\psi \rangle =0$, from a super-radiant Mott insulator (SRMI) with $\langle {\psi }^{†}\psi \rangle \ne 0$. Outside of these regions are the vacuum state, and the unstable region corresponding to macroscopic population of the $b$ states. Whilst the total density is fixed within both Mott phases, $n_a+n_b=1$, the individual $a$ and $b$ populations vary in the super-radiant phase as shown in Fig. 3. Inset: ${\epsilon }_b=-{\epsilon }_a=\mathrm{ḡ}=1$ and $\omega =3$. For $\omega >{\omega }_0$ the upper and lower boundaries may cross and terminate the lobe, since for ${\mu }_2>{\omega }_0$ the lowest stable state is the vacuum; see Appendix D for a derivation of this.

Figure 3

Onset of magnetization $\mathcal{M}=(n_b-n_a)/2$ at the super-radiance transition, traversing a line through Fig. 2 with ${\mu }_1=0$. Whilst the total density $n_a+n_b=1$ remains pinned throughout the entire Mott region, photoexcitation promotes atoms between the bands in the super-radiant phase.

Figure 4

Variational zero hopping diagram for hardcore $a$ and $b$ atoms. The symmetry about ${\mu }_1=0$ reflects the combined particle-hole and species interchange symmetry of the Hamiltonian (6) in the hardcore limit; see text.

Figure 5

Variational phase diagram with $J_a=J_b=J$ and ${\epsilon }_a=-1$, ${\epsilon }_b=1$, $\omega =\mathrm{ḡ}=1$, $U_{\mathit{ab}}=0$. The phases are (i) a Mott insulator (MI, dark blue), (ii) a super-radiant Mott state supporting a condensate of photoexcitations (SR-MI, light blue), (iii) a super-radiant superfluid (SR-SF, light red), and (iv) an $a$-type superfluid (a-SF, dark red). As shown by the solid line, the tetracritical line ultimately bifurcates into two bicritical points connected by a first order transition. For ${\mu }_1>0$, the a-type superfluid is replaced by a $b$-type owing to the particle-hole and species interchange symmetry of the hardcore Hamiltonian (5) when $J_a=J_b$; see text.

Figure 6

Cross section of the phase diagram (5) for ${\mu }_1=-0.6$. The first order transition from the super-radiant Mott state to the $a$-type superfluid is indicated by a dashed line. The locus of this transition is determined analytically from Eq. (C8). The remaining transitions are continuous.

Figure 7

Evolution of the mean field order parameters $\langle a\rangle$, $\langle b\rangle$, $\mathrm{\gamma ̄}=\langle \psi \rangle /\sqrt{N}$ across the transitions depicted in Fig. 6 for ${\mu }_2=0.45$. (I) First order transition from the super-radiant Mott state to the $a$-type superfluid, (II) continuous transition from the $a$-type superfluid to the super-radiant superfluid. Inset: Evolution of the variational energy (relative to the zero hopping Mott phase) across the transitions in Fig. 6 for fixed values of ${\mu }_2$. The solid line at ${\mu }_2=0.45$ has a discontinuity in the first derivative and indicates a first order transition in passing from the super-radiant Mott insulator to the $a$-type superfluid. The remaining transitions are continuous.

Figure 8

Variational phase diagram showing a slice through Fig. 4 with ${\mu }_1=-0.25$ and extended in to the $(J_a,J_b)$ hopping plane. We use the same key as in Fig. 5 and denote the $b$-type superfluid (b-SF) in orange. Single component superfluids are observed for large hopping asymmetry.

Figure 9

Exact diagonalization results for the phase diagram at small hopping, $\mathit{zJ}=0.1$, for hardcore $a$,$b$ atoms and ${\epsilon }_a=-1$, ${\epsilon }_b=1$, $\omega =\mathrm{ḡ}=1$, $U_{\mathit{ab}}=0$. The dashed (solid) lines corresponds to $N=8$ and $M_{\psi }=16$ ($M_{\psi }=64$) photons, and have been obtained from spline fits to the contours at $n=0.99$ and $n=1.01$. The vertical line indicates the location of the super-radiance transition as determined from $\langle {\psi }^{†}\psi \rangle /N⩾0.01$. The results are in good qualitative agreement with the variational zero hopping results in Fig. 4, showing the continuity of this behavior into the Mott phase.

Figure 10

Exact diagonalization results with $N=1,8$ sites, $J_a=J_b=J$, and $\mathit{zJ}=0.1$. We set ${\epsilon }_a=-1$, ${\epsilon }_b=1$, $\omega =\mathrm{ḡ}=1$, ${\mu }_1=U_{\mathit{ab}}=0$. The profiles show the onset of super-radiance within the Mott phase, and evolve with increasing system size toward those for the thermodynamic limit of the Dicke model. Evolution of (a) the photon density $\langle {\psi }^{†}\psi \rangle /N$, (b) the polariton density, $N_2/N$. For $N=1$ the Dicke model reduces to the Jaynes-Cummings model and exhibits quantized integer steps in the polariton density [9, 23]. More generally these are quantized in units of $1/N$. The value of $M_{\psi }$ is of minor importance in the range of ${\mu }_2$ considered as it sets the upper limit on the number of excitations available.

Figure 11

Exact diagonalization results for $N=8$ sites and $M_{\psi }=16$ photons. We set ${\mu }_1=0$, $J_a=J_b=J$, and ${\epsilon }_a=-1$, ${\epsilon }_b=1$, $\omega =\mathrm{ḡ}=U_{\mathit{ab}}=1$. The panels show density plots of (a) total atom density, (b) photon density, (c) $a$-atom density, (d) $b$-atom density, (e) fluctuations of the total atom density, (f) fluctuations of the a-atom density, (g) fluctuations of the $b$-atom density. In particular (a) reveals the Mott-superfluid transition and (b) the super-radiance transition.

Figure 12

Exact diagonalization results for $N=8$ sites and $M_{\psi }=16$ photons. We set ${\mu }_1=0$, $J_a=J_b=J$, and ${\epsilon }_a=-1$, ${\epsilon }_b=1$, $\omega =\mathrm{ḡ}=U_{\mathit{ab}}=1$. The panels show density plots of (a) total atomic superfluid fraction, (b) $a$-atom superfluid fraction, (c) $b$-atom superfluid fraction. The Mott-superfluid transition is evident from the onset in (a), and the single component $a$-type superfluid in (b) and (c).

Figure 13

Exact diagonalization results with $J_a=J_b=J$ and ${\epsilon }_a=-1$, ${\epsilon }_b=1$, $\omega =\mathrm{ḡ}=U_{\mathit{ab}}=1$. We set ${\mu }_1=0$, ${\mu }_2=0.4$, $M_{\psi }=2N$, and show the system size dependence of (a) superfluid fraction, (b) density fluctuations, (c) zero momentum occupation. Within the Mott phase for $J<J_c$, the superfluid fraction decreases with increasing system size. This indicates a Mott-superfluid transition at approximately $z{J}_c\approx 1.1$. This is compatible with the fluctuation onset criterion, $\sigma =0.3$, that we use to determine the overall phase diagram in Fig. 14, and the divergence of $n(k=0)$ with system size in (c).

Figure 14

Overall phase diagram obtained by exact diagonalization with $N=8$ sites and $M_{\psi }=16$ photons. We set $J_a=J_b=J$, ${\epsilon }_a=-1$, ${\epsilon }_b=1$, $\omega =\mathrm{ḡ}=1$, and $U_{\mathit{ab}}=1$, and use the same key as in Fig. 5. We extract the Mott-superfluid boundaries from the onset of density fluctuations, $\sigma ⩾0.3$, and the super-radiance transition from the onset of photons, $\langle {\psi }^{†}\psi \rangle /N⩾0.01$; see text. The simulations show the distinct phases and the bifurcation of the tetracritical line suggested by mean field theory. The bifurcation is indicated by the regions where the two boundaries overlap along a line rather than a single tetracritical point.

Figure 15

The classical light limit of Eq. (1) (where $\psi$ is a replaced by a $c$-number), may be simulated in optical superlattices [92] where the coupling $g$ represents tunneling between different wells, $a$ and $b$.

Figure 16

Evolution of the polariton density, $N_2/N$, in the Dicke model for $N=1,2$ sites and comparison to the $N\to \infty$ thermodynamic limit. For $N=1$ the Dicke model reduces to the Jaynes-Cummings model and the quantized polariton steps correspond to the Mott lobes discussed in Ref. [9]. In the Dicke model these steps are quantized in units of $1/N$ and as $N\to \infty$ we approach the variational results. Inset: Evolution of the photon density. The magnetization (or population imbalance) may be obtained by subtraction. All figures show the onset of super-radiance at ${\mu }_2=(3-\sqrt{5})/2\approx 0.382$, for $\omega \to \mathrm{\omega ̃}=1-{\mu }_2$, ${\omega }_0\to {\mathrm{\omega ̃}}_0=2-{\mu }_2$, and $\mathrm{ḡ}=1$.

Figure 17

Zero hopping phase diagram for the Hamiltonian (6) in the absence of matter-light interaction, with ${\epsilon }_a=-1$, ${\epsilon }_b=1$, $\omega =1$, $g=0$. The loci ${\mathrm{\epsilon ̃}}_a=0$, ${\mathrm{\epsilon ̃}}_b=0$, $\mathrm{\omega ̃}=0$, indicated by the lower, upper, and vertical lines, respectively, delineate the regions of population onset for $a,b,\psi$ in the grand canonical ensemble. When $g$ is switched on the horizontal boundaries bend downward and evolve into those shown in Fig. 2.