Atom-light crystallization of Bose-Einstein condensates in multimode cavities: Nonequilibrium classical and quantum phase transitions, emergent lattices, supersolidity, and frustration

The self-organization of a Bose-Einstein condensate (BEC) in a transversely pumped optical cavity is a process akin to crystallization: when pumped by a laser of sufficient intensity, the coupled matter and light fields evolve, spontaneously, into a spatially modulated pattern, or crystal, whose lattice structure is dictated by the geometry of the cavity. In cavities having multiple degenerate modes, the quasicontinuum of possible lattice arrangements, and the continuous symmetry breaking associated with the adoption of a particular lattice arrangement, give rise to phenomena such as phonons, defects, and frustration, which have hitherto been unexplored in ultracold atomic settings involving neutral atoms. The present work develops a nonequilibrium field-theoretic approach to explore the self-organization of a BEC in a pumped, lossy optical cavity. We find that the transition is well described, in the regime of primary interest, by an effective equilibrium theory. At nonzero temperatures, the self-organization occurs via a fluctuation-driven first-order phase transition of the Brazovskii class; this transition persists to zero temperature and crosses over into a quantum phase transition. We make further use of our field-theoretic description to investigate the role of nonequilibrium fluctuations in the self-organization transition, as well as to explore the nucleation of ordered-phase droplets, the nature and energetics of topological defects, supersolidity in the ordered phase, and the possibility of frustration controlled by the cavity geometry. In addition, we discuss the range of experimental parameters for which we expect the phenomena described here to be observable, along with possible schemes for detecting ordering and fluctuations via either atomic correlations or the correlations of the light emitted from the cavity.

Figure 1

(a) The transversely pumped, quasi-two-dimensional geometry primarily discussed in this paper. (b) Ring cavity geometry. The pump laser beam is perpendicular to the plane defined by the three mirrors, as indicated in the figure. (c) Schematic representation of a concentric cavity, showing the partial rotational symmetry that such a cavity inherits from the sphere of which both cavity mirrors are arcs. (d) Three-dimensional view of a representative mode function for the concentric cavity. This mode function is labeled by $(l,m,n)=(2,1,5)$, or alternatively by ${\mathrm{TEM}}_{21}$. The three numbers enumerate the nodes (one fewer than the number of lobes) in the pump ($z$), angular, and radial directions, respectively. The axial mode index $n$ is fixed by the requirement that $l+m+n$ be constant for a family of degenerate modes, and can therefore be suppressed. (e) The intensity profile of the representative mode TEM${}_{21}$ at one of the cavity’s end mirrors. (f) The intensity profile of the mode TEM${}_{21}$ in the equatorial (i.e., $z=0$) plane of the cavity.

Figure 2

Dispersion relation for low-energy atomic excitations, i.e., those that approximately satisfy $2\pi (m+n)=K_{0}R$; as discussed in the text, the troughlike form of this dispersion enhances fluctuation effects. The inset shows a top view of the dispersion: the black line represents modes at the minimum of the trough, which exactly satisfy $2\pi (m+n)=K_{0}R$; self-organization results in the macroscopic occupation of one of these modes.

Figure 3

(a) Dyson equation for the self-energy at one loop order (i.e., the leading fluctuation correction to $r$). (b) A geometric series of corrections to the vertex (i.e., to $u$), which constitute the primary fluctuation corrections for $(m^{\prime },n^{\prime },{\omega }^{\prime })\ne (m,n,\omega )$. (For the classical case, ${\omega }^{\prime }=\omega =0$.) (c) A geometric series of corrections to $u$ that contribute only when $(m^{\prime },n^{\prime },{\omega }^{\prime })\approx (m,n,\omega )$. It is these contributions that change the sign of $u$, thus causing a first-order transition. (d) Higher-order vertices that emerge under coarse-graining.

Figure 4

Schematic form of the free energy as a function of the order parameter, both above and below threshold, indicating how fluctuations change the character of the phase transition.

Figure 5

(a)–(c) Dependence of coarse-grained, fluctuation-corrected parameters on the bare control parameter $\mathcal{R}$, which is related to the laser strength, for a fixed value of the bare parameter $\mathcal{U}$. (The bars over the parameters signify that they have been rescaled as described in Appendix .) These results are obtained by integrating the renormalization-group equations derived in Appendix . (a) shows the flow of the effective “control parameter” $\overline{r}$ (which remains positive). (b) shows the flow of the effective interaction parameter $\overline{u}$, which changes sign as discussed in the text. (c) shows the flow of the emergent six-point coupling $\overline{w}$. Finally, (d) plots the free energy as a function of the order parameter $A$ for three values of $\overline{\mathcal{R}}$, viz., $-5.3$ (thin solid line), $-5.35$ (dashed line), and $-5.4$ (thick line). The first-order phase transition takes place at $\overline{\mathcal{R}}\approx -5.36$. These results are interpreted in terms of microscopic parameters in Sec. 11.

Figure 6

Contributions to the nonequilibrium vertex having external indices $\mathit{ccqq}$. For an introduction to the Keldysh diagrammatic notation see Ref. [31].

Figure 7

(a) Wulff droplets, corresponding to the TEM${}_{00}$ mode (i), and to a higher-order mode (ii), respectively. The droplets should become less anisotropic (i.e., less “needlelike”) for higher-order modes; it is, however, possible that the optimal droplets in these cases have more complicated shapes. (b) Defected droplets, which are favored for $r\approx r_c$, as discussed in the text. For these, the energetic cost of introducing defects inside the droplet is outweighed by the increase in the fraction of the interface that is transverse.

Figure 8

Elementary excitations of the self-organized state in the concentric cavity. (a) Domains that have self-organized into distinct modes can be separated by analogs of grain boundaries (left half of panel) or by continuous textures (right half of panel). (b) Excitations that are analogous to the splay mode in smectic-$A$ liquid crystals (see Sec. 9b). Lines indicate nodes of the cavity electromagnetic field. The curved wave fronts along the radial direction have been drawn as flat lines to emphasize that the sketched feature is small scale, relative to the size of the cavity.

Figure 9

Case of the large-solid-angle concentric cavity, in which the atom-light system possesses a continuous symmetry associated with the relative phase between the $+m$ and $-m$ components of each mode function. This symmetry, when broken by the self-organized atomic cloud, leads to the existence of both phonon excitations (shown in the left panel of the figure) and true edge dislocations (shown in the right panel of the figure).

Figure 10

Proposed scheme for detecting supersolid order. (a) Profiles of two cavity modes: mode 1 (into which the atoms self-organize) and mode 2 (which can be used to detect phase coherence, as discussed in Sec. 10). The two modes are degenerate; mode 2 possesses more nodes along the $z$ direction (i.e., perpendicular to the plane of the figure). The $\pm$ signs describe the phases of the electromagnetic fields in the two modes relative to some reference (e.g., the pump laser) in various regions of the cavity. (b) Atomic configuration in which the atoms emit constructively into mode 1 and destructively into mode 2. In the insulating phase, this is the typical configuration, as the number of atoms per site is fixed; hence, there is suppressed emission into mode 2. (c) Atomic configuration in which the atoms emit constructively into both mode 1 and mode 2. Such configurations, which involve multiple occupancy, occur in the superfluid phase but are suppressed in the insulating phase; hence, the amount of light emitted into mode 2 is a measure of superfluidity.

Figure 11

Schematic zero-temperature (i.e., quantum) phase diagram for a BEC in a concentric cavity, with the control parameters being the atomic scattering length $a$ and the inverse effective atom-cavity coupling ${\zeta }^{-1}$ (or equivalently the inverse laser intensity ${\Omega }^{-2}$). For weak, repulsive interactions, the superfluid first undergoes self-organization via the Brazovskii transition, thus forming a supersolid. If the laser intensity is increased further, the supersolid undergoes a transition into a normal solid (i.e., a Mott insulator). However, for strong, repulsive interactions, the uniform BEC can lose phase coherence concurrently with the first-order self-organization transition. This situation is to be contrasted with that for the case of a single-mode cavity (inset), in which there should always be a supersolid (SS) region separating the uniform fluid (SF) and normal solid (S) regions. First- and second-order transitions are marked (1) and (2), respectively.

Figure 12

Schematic illustration of the implications of frustration. Atoms are loaded into sheets (i) and (ii), shown as thick lines in (a), which are an integer number of pump-laser wavelengths apart. The dashed and dashed-dotted curves are, respectively, antinodal regions of the modes ${\mathrm{TEM}}_{1m}$, which have low intensity near the centers of sheets (i) and (ii), and ${\mathrm{TEM}}_{2m}$, which have low intensity away from the centers of sheets (i) and (ii). Near the center of each sheet, atoms crystallize into the ${\mathrm{TEM}}_{2m}$ modes; away from the center, they crystallize into the ${\mathrm{TEM}}_{1m^{\prime }}$ modes. Within a sheet, regions may be separated by faults in the ordering, as illustrated in (b). For example, on the left side the fault has the form of a discommensuration (see Sec. 12). By contrast, the fault on the right side is a grain boundary. Between layers, the opposing parity of adjacent modes leads to frustration, which precludes ordering, as in the regions indicated by a $△$ and a $$. Grain boundaries (denoted by $$) are more localized faults, and are therefore less costly, energetically, than discommensurations ($△$).

Figure 13

Feynman diagrams involving the six-point vertices associated with the couplings $w_i$ ($i=1,2,3$), to one-loop order. The six-point vertices are denoted as gray squares. The diagram shown on the left renormalizes the four-point vertices; that on the right renormalizes the six-point vertices.