Atom-light crystallization of BECs in multimode cavities: Nonequilibrium classical and quantum phase transitions, emergent lattices, supersolidity, and frustration
The self-organization of a Bose-Einstein condensate 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 quasi-continuum 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 of a new universality class. We make further use of our field-theoretic description to investigate the role of nonequilibrium fluctuations on 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.
Over the course of the past fifteen years, many phenomena of long-standing interest in condensed-matter physics have been realized in ultracold atomic settings bloch:review (). Such realizations are considerably different from condensed-matter systems: in particular, ultracold atomic systems are highly controllable—i.e., they are isolated from the environment and are governed by thoroughly understood microscopic Hamiltonians—and tunable—i.e., the interaction strength, lattice depth, etc. are governed by quantities such as laser intensities, which are easy to alter. The high degree of control and tunability has made it possible both to explore emergent phenomena in a simpler setting than is typical in condensed matter and to address hitherto experimentally inaccessible questions, such as the dynamics of ordering in systems that are quenched past a quantum critical point greiner:quench (). To date, most ultracold atomic realizations have focused on simulating the physics of electrons propagating through static lattices (via, e.g., realizations of the Hubbard model moritz ()) or on constructing novel quantum fluids (e.g., Tonks-Girardeau gases bloch:tonks () or unitary Fermi gases ufg ()). Areas of condensed matter such as soft matter, supersolidity supersolids (), and glassiness—which involve emergent, compliant lattices capable of exhibiting dynamics, defects, melting etc.—have proven inaccessible to ultracold atomic physics because the lasers that create the lattice potentials in typical experiments are essentially insensitive to the atomic motion in those potentials. Aspects of such condensed matter phenomena remain unsettled in their traditional settings (e.g., the dynamics of glassy media and of supersolids), and therefore ultracold atomic realizations of them are especially desirable.
A possible approach to realizing phenomena dependent on the emergent, compliant character of the lattice is to have the atoms interact with a potential created by dynamical, responsive quantum light, instead of static lasers. Exploring precisely such interactions has been the central theme of cavity QED walls (). Traditionally, cavity QED has aimed to realize systems involving a single atom coupled to a single mode of the electromagnetic field; however, the physics of many atoms coupled to one (or more) electromagnetic modes, i.e., many-body cavity QED, has also been studied extensively in recent work ritschprl (); ritschpra (); vuletic:prl (); ringcav (); domokos08 (); morigi08 (); morigi10 (); esslinger06 (); baumann (). In particular, it was predicted in in Refs. ritschprl (); ritschpra () that a cloud of atoms confined in an optical cavity would exhibit collective effects such as self-organization; these effects were subsequently observed vuletic:prl (). The atoms considered in these works were essentially classical, thermal particles: however, the dynamics of many quantum atoms (e.g., a Bose-Einstein condensate or a Mott insulator) confined in a cavity has since been explored both theoretically domokos08 (); morigi08 (); morigi10 () and experimentally esslinger06 (); baumann (). Much of the work in this area, to date, has explored the novel implications of the atom-cavity coupling for standard ultracold-atomic phenomena such as the superfluid-Mott insulator transition morigi08 (); morigi10 () or the collective excitations of a Bose-Einstein condensate domokos08 (); esslinger06 (). The objective of the present work is to suggest that a quite different class of condensed-matter problems, involving the emergence and dynamics of spatially ordered states, can be realized and explored using ultracold atoms confined in optical cavities. Elements of this work were reported in Ref. us (); related issues, involving the simulation of phonons in optical lattices, were previously raised in Ref. lewenstein:multimode ().
A central idea in the extant literature as well as the present work is the idea of cavity-induced self-organization ritschprl (); ritschpra (), which may be explained as follows: Consider two-level atoms in a single-mode optical cavity, interacting with the cavity mode and a pump laser oriented transverse to the cavity axis (see Fig. 1). The atoms coherently scatter light between the pump and cavity modes. Atoms arranged at every other antinode of the cavity field (i.e., one cavity-mode wavelength apart) emit in phase; therefore, -period fluctuations of the atomic density increase the number of photons in the cavity, thus drawing the atoms into -spaced wells at either the even or odd antinodes. This leads to greater constructive interference in the emitted light, stronger atomic trapping, and so on. The system reaches a spatially modulated steady state when the energetic gain from the atom-light interaction is balanced by the cost, in kinetic energy or repulsive interactions, of confining the atoms to either the even or the odd sites of the emergent lattice.
Although self-organization in a single-mode cavity results in the spontaneous breaking of a discrete symmetry, the locations of the antinodes are not themselves emergent, but are fixed by the cavity geometry. In other words, the non-crystalline state does not possess continuous translational invariance; thus self-organization resembles, e.g., a phase transition between crystal structures, rather than true crystallization. With multimode cavities, by contrast, self-organization results in the breaking of continuous symmetries, both in the collective choice of which mode(s) to populate with photons and in the choice of relative phases between the modes. For example, in the case of the ring cavity (which consists of two counter-propagating traveling-wave modes ringcav ()), the atoms must collectively choose the (continuous) relative phase between the two counter-propagating modes, thus setting the location of the antinodes of the cavity field. As with real solid-state crystallization, the breaking of this continuous symmetry induces rigidity with respect to lattice deformations. With larger families of modes, one can envision realizing such characteristically crystalline notions as dislocations and geometrical frustration.
In a previous article us (), we developed a field-theoretic description of the interacting many-atom, many-mode system, which we applied to the case of a transversely pumped concentric cavity. We observed: (i) that a quasi-two-dimensional atomic cloud undergoes a weakly first-order transition into a spatially ordered state, and that this transition becomes a quantum phase transition at ; and (ii) that for the case of a strongly layered three-dimensional cloud, inter-layer frustration precludes global ordering, and the system instead breaks up into inhomogeneous, static domains. In addition, we suggested that both fluctuation phenomena and signatures of supersolidity may be observed via the spatial and temporal correlations of the light emitted from the cavity. The formalism developed in Ref. us () assumed, however, that because the flux of energy through the atom-cavity system is negligible in the regime of interest, one could adopt a quasi-equilibrium description of this phase transition. One of our objectives in the present work is to justify this assumption within a general nonequilibrium formalism; furthermore, we compute the leading corrections to the effective theory of Ref. us () that arise because of nonequilibrium effects.
In the present work, our approach towards meeting these objectives is as follows: (i) we develop a fully nonequilibrium, field-theoretic description of the atom-cavity system, using the Schwinger-Keldysh functional-integral formalism schwinger (); keldysh (); (ii) we show that the nonequilibrium description can be reduced, in the regime in which the cavity’s photon decay time is longer than the timescales for atomic motion, to an effective equilibrium description; (iii) we analyze this effective equilibrium theory using diagrammatic and renormalization-group techniques to establish the nature of the self-organization transition, in the specific case of a concentric cavity; and (iv) we reintroduce the nonequilibrium effects, due to the leakage of photons out of the cavity, using perturbation theory, and account for their effects on critical behavior near the self-organization transition. Our analysis of the equilibrium theory extends that used in our previous work us () via an adaptation of Shankar’s renormalization-group treatment of the Fermi liquid shankar (), as well as an analogy with the -invariant vector model of magnetism, to establish that for an interacting Bose-Einstein condensate the self-organization transition is always discontinuous in the concentric cavity. Furthermore, we show that the chief consequence of nonequilibrium effects is to decohere quantum correlations on a timescale related to the linewidth of the cavity; thus, the self-organization transition is always classical on the longest timescales. (For sufficiently high-finesse cavities, there should, however, be an extended crossover regime of timescales for which the phase transition appears quantum.)
After presenting our analysis of the phase transition itself, we turn to the properties of the ordered (i.e., self-organized) state. We show that the ordered state has low-lying excitations, associated with the continuous symmetry-breaking, that resemble the excitations of smectic liquid crystals. We also expand on our previous observation us () that, for a Bose-condensed atomic cloud, the ordered state would be a “supersolid” (or, more accurately, a “super-smectic”) in that it simultaneously possesses emergent (liquid-)crystalline and superfluid order. The properties and even the existence of supersolidity in He are much-discussed topics in the condensed-matter literature supersolids (); toner08 (); shevchenko (); west09 (); davis:superglass (); the appeal of ultracold-atomic realizations is that one can explore the characteristic phenomenology of supersolids in contexts where it is less challenging to establish that they do in fact possess supersolidity. In order to explore the relevant phenomenology, it is necessary that the solidity be associated with a broken continuous spatial symmetry; for this purpose a continuum supersolid such as the one explored in the present work should serve as a more suitable setting than would, e.g., the lattice supersolids proposed in Refs. lew02 (); sun07 (). The present scheme has the added advantage of being more readily realizable. The self-organization of a BEC in a cavity was, in fact, recently demonstrated experimentally by Baumann et al. baumann () for the case of a single-mode cavity; generalizing this experiment to the multimode case, in which one has continuous symmetry-breaking, should be technically straightforward. Continuing with the theme of supersolidity, we propose a scheme for detecting such as state, and develop a schematic phase diagram for the system (Fig. 11), which exhibits three phases: the supersolid, the normal solid, and the uniform superfluid. The phase diagram for a multimode cavity differs from that for a single-mode cavity in that the multimode case features a direct transition from the uniform superfluid to the normal solid, whereas in the single-mode case there is always a supersolid regime separating the uniform superfluid and the normal solid.
This paper is organized as follows. In Sec. II we describe the microscopic model of the atom-cavity system that is used in the rest of the paper, and in Sec. III we discuss the qualitative behavior one might expect from this model. In the next three sections we construct and analyze a nonequilibrium field-theoretic formulation of this model: Sec. IV introduces the relevant field-theoretic formalism, Sec. V applies this formalism to derive an atoms-only action, and Sec. VI describes the quasi-equilibrium limit of the atoms-only action. In Sec. VII we derive an effective Ginzburg-Landau free energy, valid near the phase transition, which realizes a version of Brazovskii’s model brazovskii () of ordering at a finite wavelength, and analyze the effects of fluctuations on the self-organization transition in both the classical () and quantum () cases. In Sec. VIII we turn to the effects of departures from equilibrium, both on the fluctuations near the transition and (following the work of Hohenberg and Swift swift:bubbles ()) on the nucleation of ordered states. In the next two sections we focus the properties of the ordered state: Sec. IX reviews the properties and elementary excitations of the ordered state, and Sec. X discusses the supersolid aspects of the ordered state. In Sec. XI we discuss the experimental feasibility of the phenomena that we are investigating, showing that most of them should be readily detectable in the laboratory. In Sec. XII we briefly consider the case of strongly layered three-dimensional systems, which have the feature that geometrical factors tend to frustrate the development of globally coherent long-range order. Finally, in Sec. XIII we summarize the results of this work, and discuss its relationship with other problems involving phase transitions and related collective effects in atom-light systems. Various supplementary issues are addressed in four appendices.
The system analyzed in this paper consists of two-level atoms confined in a multimode optical cavity, together with the electromagnetic modes of the cavity. The system is pumped by an external laser, which is oriented transverse to the cavity axis, as shown in Fig. 1. In addition, the system is coupled to a set of extracavity electromagnetic modes, which constitute a bath for the system. The complete Hamiltonian governing the system and bath comprises three elements—viz., , the atoms-only Hamiltonian; , the light-only Hamiltonian; and , the atom-light interaction Hamiltonian—which we discuss in detail in the rest of this section.
ii.0.1 Two-level atoms
The atoms are described by the Hamiltonian
where is the number of atoms (indexed by ), each of which has mass ; the position and momentum of atom are, respectively, and . The operator denotes the Pauli operator, which acts on the internal state of the atoms, which, for the sake of simplicity, we take to be two-state atoms, with being the energy splitting between the ground (g) and excited (e) states. (Our conclusions do not hinge in any essential way on this restriction to two states.) We model the interaction between atoms via a repulsive contact potential, which is parametrized in terms of the (positive) parameter . To ensure the correct handling of the Bose-Einstein quantum statistics of the atoms, we employ the framework of second quantization. Thus, becomes
in which the pair of bosonic field operators, and , respectively represent the ground- and excited-state atoms. In the last line of Eq. (II.0.1), all the coordinate indices are , as the interaction is taken to be local in space.
Next, we assume that the density of excited-state atoms is sufficiently low that collisions between such atoms can be neglected. This assumption necessarily holds in the low-temperature regime, which is the regime of primary interest to us, because the rate of spontaneous decay (which is proportional to the number of excited atoms) must be kept low in order to avoid the heating associated with it. With this regime in mind, we approximate the interaction term as
In what follows we shall replace by the frequency , in order that its magnitude be expressed in the same units as those conventionally used to describe the atom-cavity coupling.
ii.0.2 Electromagnetic modes
Optical cavities typically consist of two or more mirrors that support localized modes of the electromagnetic field between them siegman (). The modes might be standing waves, as in the single-mode cavity, or the traveling waves that one can have in the ring cavity. In general, we shall be concerned with transverse electromagnetic (TEM) modes. The vector character of the electromagnetic field can be absorbed into the effective atom-mode coupling walls (); hence, such modes can effectively be described in terms of harmonic solutions to the scalar wave equation siegman (). Furthermore, as the mirrors are not perfectly reflective, these modes are in fact weakly coupled to the extracavity modes, and thus cavity-mode photons tend to “leak” out of the cavity at some nonzero rate (which sets the intrinsic linewidth of the cavity). These considerations lead us to model the electromagnetic field—both intracavity and extracavity—in terms of the Hamiltonian
where and respectively represent the intracavity and extracavity photons, and are the intracavity and extracavity mode frequencies, and describes the coupling between the intracavity mode and the extracavity mode . We assume that the intracavity-extracavity coupling is weak enough (i.e., the cavity is of sufficiently high finesse) that it is meaningful to separate the modes into intracavity and extracavity ones.
The modes of a generic standing-wave cavity are not frequency-degenerate: the typical frequency spacing, or longitudinal “free spectral range,” is of order , where is the linear dimension of the cavity: e.g., for a cm cavity, the free spectral range is about GHz. For such a cavity, the frequency spacing between higher-order transverse modes is an appreciable fraction of this number; thus there are no degenerate modes. However, certain specific cavity geometries do support degenerate modes. The simplest of these is the ring cavity, which is a three-mirror arrangement that supports two counterpropagating traveling-wave modes (see Fig. 1a). Even larger degeneracies are possible, e.g., in confocal or concentric cavities. Let us label the cavity-mode structure by the integers , where is the number of nodes along the cavity’s axial direction, and are the numbers of nodes along the transverse directions; see Fig. 1. (The corresponding mode functions are approximately sinusoidal in the axial direction, and Hermite-Gaussian (or Laguerre-Gaussian) in the transverse directions, although this approximation breaks down in the limit of a concentric cavity.) In the confocal cavity, the condition for frequency degeneracy is that (see, e.g., Ref. siegman ()) for some fixed integer ; in the concentric cavity, the condition becomes . In principle, these conditions imply that there are of order degenerate modes, where is roughly the number of optical wavelengths across the cavity, commonly or more. In practice, higher-order modes are increasingly lossy, because their profiles (i.e., spot sizes) are larger, and therefore more of their light leaks out of the sides of the cavity mirrors, which typically occupy only a modest amount of solid angle. We approximately account for this effect by assuming that of the modes have a common loss rate , and that the other modes are perfectly lossy (i.e., for them ).
ii.0.3 Atom-light interactions
In the dipole and rotating-wave approximations walls (), the atom-light coupling has the generic form h.c., where the operators raise or lower the atomic internal state; is either an intracavity or extracavity mode; and , in which is the coupling between the atom and the mode, and is an appropriately normalized mode function. Note that is proportional to , where is the system volume walls (); hence stays finite in the thermodynamic limit. In the special case of the pump laser mode, we shall treat the corresponding as a classical variable, so that the atom-laser coupling takes the form , where , with being the local Rabi frequency (which is proportional to the local amplitude) of the laser. In our analysis, we shall largely neglect the term that governs spontaneous atomic emission into extracavity modes (i.e., spontaneous decay); this term is proportional to the linewidth of the transition, which is denoted as . The reason we can neglect spontaneous decay processes is not that such processes are always weak. Rather, it is that their effects amount to the heating up of the sample via the random impulses they give to the atoms, and therefore the timescale for spontaneous decay [which is given by ] acts as an upper limit on the duration of an experiment: quantum dynamics on timescales slower than is likely to be washed out as a result of spontaneous decay. Neglecting, then, spontaneous decay processes, the Hamiltonian governing the atom-light interactions is given by
Iii Physical expectations
iii.1 Single-mode case: semiclassical picture
As discussed in the Introduction, the atom-cavity system is unstable towards crystallization when transversely pumped at sufficient intensity. A more quantitative, albeit semiclassical, picture of the crystallization is as follows. Assume that the atoms are pumped by a pair of counterpropagating lasers perpendicular to the equatorial plane of the cavity (see Fig. 1a), so that the electric field due to the lasers is given by , and that the field in the cavity mode, which is a standing wave, is given by . The polarizations of the two modes, being parallel, can be neglected for the present purposes. Atoms in the plane are subject to an effective electric field of intensity . The first term is spatially constant; of the spatially varying terms, for small the second is the dominant one. The potential energy of the (high-field-seeking) atoms in this field can therefore be taken to be
Furthermore, the magnitude of is set by the atomic distribution, i.e.,
where is the pump laser-cavity phase difference. From Eqs. (6) and (7) it follows that . When exceeds the kinetic-energy cost of self-organizing, the atoms undergo self-organization. As depends only on , it is invariant under , i.e., under translating the distribution of atoms through half a mode wavelength (e.g., from even to odd antinodes) along with changing the sign of the cavity field. This is the even-odd symmetry (which is an inversion symmetry, provided the cavity has an even number of total wavelengths) that is broken when the atom-light system crystallizes. This crystallization transition can be thought of in one of two equivalent ways: either as a crystallization of the atoms or as a locking of the phases of the laser and the cavity modes. This phase-locking is associated with the condensation of photons into a cavity mode; analogously, conventional crystallization can be thought of as a condensation of phonons.
Note that, for a single-mode cavity, the presence of the term in that goes as does not qualitatively affect the picture just outlined, unless is of the same order of magnitude as . If , the energetics would still favor symmetry breaking; however, the dynamics of ordering would then be complicated by the fact that there are local electric-field minima at the “minority” antinodes—i.e., the ones at which the atomic density is supposed to be low—and these local minima can trap atoms, as discussed in Ref. ritschpra (). We shall not discuss this regime further in the present work.
iii.2 Multimode case
The question that arises when one attempts to extend the idea of atom-light self-organization to the setting of multimode cavities is this: Which mode(s) do the atoms crystallize into? In this section, we offer some heuristic general considerations aimed at addressing this issue. We shall return to this question in the specific context of the concentric cavity, once we have developed the relevant field-theoretic techniques.
iii.2.1 Traveling waves
The argument given in Sec. III.1 on self-organization for a single-mode cavity proceeds similarly for the case of multimode cavities, except that the individual mode-function must be replaced by the (as yet unspecified) set of cavity mode functions . There are, however, two crucial differences. The first is pertinent whenever the cavity supports traveling-wave modes (e.g., the ring cavity). In this case, the potential energy is given by
and the dynamics of each mode is coupled to that of its partner under time-reversal. This has an important consequence, which is easiest to illustrate in the case of the ring cavity. Here, , and is invariant under the transformation , which involves shifting the atomic distribution along the cavity axis (and adjusting the antinodes of the cavity mode accordingly). Therefore, crystallization in multimode cavities having traveling-wave modes necessarily involves the spontaneous breaking of a continuous translational symmetry.
iii.2.2 Mode selection
If two cavity mode functions and are associated with a pair of frequency-degenerate harmonic solutions of the wave equation for the same (homogeneous) boundary conditions, any linear combination is also a legitimate cavity mode. Therefore, it might seem that, in an -fold degenerate cavity, any normalized mode of the form would be an “equally good” arrangement for crystallization, i.e., there is an -dimensional degenerate subspace. This is not generally true, as a result of terms in the energy, omitted so far in the present section, that lift this degeneracy, such as the interatomic contact repulsion. Consider the extreme simplification involving two modes having the respective mode functions and (as would arise, e.g., from two cavities, perpendicular to one another and to the laser): any function of the form , with , is a legitimate mode function. If the atoms are self-organized in the state , the expectation value of the atomic field is given by , and the interaction energy, Eq. (5), goes as . This can readily be checked to be smallest when either or , i.e., for a stripe-like arrangement along either the axis or the axis.
A similar effect arises from the mode-mode scattering term (i.e., the term),
In the two-mode example discussed in the previous paragraph, in which the modes are at right angles to one another, this term is essentially diagonal in the mode indices for either of the stripe-like states. Suppose, however, that the two cavities lie at a small angle rather than at right angles to one another, so that the modes are and with . In this case, atomic density fluctuations of wave-vector —which, for small , could be excited either thermally or quantum-mechanically—would suffice to mix the cavity modes. The effect of such mixing would be to lock the relative phases of the two modes.
Iv Field-theoretic formulation
Our objective in this and the subsequent two sections, Sec. V and Sec. VI, is to construct a useful field-theoretic formulation that will enable us to explore the quantum statistical mechanics of correlated many-atom, many-photon systems in multimode cavities. Having done that, in Sec. VII we employ this formulation to address issues such as the emergence and nature of the spatial structure and spatio-temporal atomic and photonic correlation properties of such systems, focusing on the vicinity of the transition to the self-organized state. The atom-cavity systems of interest here are neither closed nor in thermal equilibrium, because they are driven by an external pump laser (which adds energy) and leak photons into the continuum of modes that lie outside the cavity (hence losing energy); thus, even in its steady states there is a flux of energy through the system. For these reasons, the structure and correlation properties must be computed within a nonequilibrium formalism. The one we employ is the closed-time-path formalism, due to Schwinger schwinger () and Keldysh keldysh (), which enables the use of diagrammatic methods as well as renormalization-group techniques to analyze fluctuations. (For a discussion of the differences between the equilibrium and nonequilibrium formalisms, see Sec. VI and Ref. kamenev ().)
Although, as we have just discussed, a full analysis of the problem demands a nonequilibrium approach, we find that, for systems that are near the threshold for self-organization and in the dispersive regime (i.e., the pump laser is far-detuned from the atomic resonance), an effective equilibrium description is valid, to a reasonable approximation. As we shall see, e.g. in Sec. VIII, the description of this regime can be improved, and other regimes (such as the strongly organized regime) can be analyzed, using the full machinery of the Schwinger-Keldysh nonequilibrium approach.
iv.1 Schwinger-Keldysh functional integral
The quantities of interest in quantum many-body dynamics are the expectation values of observables and their response and correlation functions. Formally, the task of computing these may be stated as follows: suppose that we know the state of the system in the infinite past, as described by its density matrix , when it is taken to be isolated and noninteracting. The system is then coupled to an environment (or environments), which generically force the composite of system and environment(s) to be out of equilibrium, and the intra-system interactions are adiabatically switched on. The question then becomes: What are the expectation values and response and correlation functions of the various system observables, once the system and environment have relaxed to a steady state?
Let us first consider the case of a single harmonic oscillator with Hamiltonian , i.e., a free bosonic degree of freedom having the characteristic frequency . Suppose, as a simple example, that we are interested in the expectation value of some observable at time , given that the system was at some time in a thermal state at temperature , i.e., governed by the density matrix . Thus, we wish to compute the quantity
One can expand the trace in terms of bosonic coherent states altland () at the times and , thus arriving at the expression
Note that the primary motivation for inserting the complete set of states is to make the above expression more symmetric between initial and final times. The relevant matrix element of the initial density matrix is given by , the overlap is given by , and the two other expressions, which are transition amplitudes, can be rewritten as coherent-state path integrals:
in which the signs indicate whether the final state is the ket () or the bra (), and the action is given by
One can thus rewrite Eq. (10) as follows:
If we omit the functional derivative from the right hand side of Eq. (14), the remaining formula is the quantity commonly denoted as (by analogy with the partition function). is a generating functional for the correlation functions of the oscillator: it can be differentiated repeatedly with respect to either of the two source functions in order to generate all requisite correlation functions of . By differentiating with respect to the and sources appropriately, one can compute expectation values that involve various orderings of the operators, e.g., the time-ordered, retarded, and advanced Green functions. By contrast, in the zero-temperature and Matsubara nonzero temperature equilibrium formalisms, the only correlation functions that can be computed directly are the time-ordered ones; the (physically relevant) retarded Green functions are then to be inferred using identities that hold in equilibrium or at zero temperature.
Because, in Eq. (14), the initial and final values of the paths are integrated over (with an exponential measure), the path integral in Eq. (14) is in effect an unconstrained path integral over the two sets of paths . Moreover, these paths are uncoupled from one another except at the two endpoints of the path integrals. For the oscillator in question one can write the action in the form
in which integration is implied over time. We shall sometimes denote as the matrix comprising the four block matrices . The diagonal blocks and are given by Eq. (13); the off-diagonal blocks and are zero, except at the time-endpoints; they cannot, however, be neglected, because the presence of such off-diagonal terms changes the inverse of , denoted , which contains the two-point correlation functions and has the following structure:
where is the Bose-Einstein distribution function. Evidently, and are the time-ordered and anti-time-ordered Green functions. It is convenient for our purposes to rotate into its “classical” and “quantum” components, defined as follows: , , in Eq. (15). This has the advantage of reducing the number of independent Green functions by one:
and are the retarded and advanced Green functions—which describe the response of the system to an external perturbation—whereas , the “Keldysh Green function,” depends on the system’s initial density matrix and its correlations. For a system in equilibrium with a bath, the correlations and response are related by the fluctuation-dissipation theorem; by contrast, for an isolated, noninteracting (and hence nonequilibrating) system such as the harmonic oscillator, the two properties—response and correlation—are independent. In interacting systems that are away from equilibrium, there is in general a complicated interplay between the correlations and the response; therefore, all the Green functions contain information about both correlations and response. The nature of this relation, however, varies from system to system, and hence the information contained in the three Green functions is not redundant.
iv.2 Application to atom-photon system
The prescription for proceeding from the second-quantized Hamiltonian for a generic set of bosonic degrees of freedom to the appropriate coherent-state path integral (in real time) is as follows:
In this expression, is the chemical potential for field —in the present case, the chemical potential for the photons is zero, whereas that for the atoms determines the atomic density—and the symbol indicates a formally identical set of terms in which fields are replaced by the corresponding fields. This prescription can be carried out for each of the terms in the Hamiltonian introduced in Sec. II, and generates all the and components of the action. The off-diagonal ( and ) blocks in the bare (i.e., microscopic) theory depend on the appropriate Bose-Einstein distributions of the free photonic modes and the free atomic modes. Because these blocks are infinitesimal (as they arise from the end-point couplings discussed in the previous section), it is more useful to write down the relevant blocks in the inverse action, viz., the bare Green functions and . For the cavity photon modes these have precisely the forms in Eqs. (16), i.e.,
whereas for bosonic atoms (in their internal ground state) and in a single-particle eigenstate of the kinetic energy having eigenvalue , these have the form
Other degrees of freedom can be treated similarly.
V Constructing the atoms-only action
In this section we derive an effective action, involving the ground-state atoms, which we shall use to determine expectation values and correlators involving atoms and/or intracavity photons. This is accomplished by integrating out all other degrees of freedom—a task that is straightforward, owing to the fact that they appear quadratically in the complete action.
v.1 Eliminating the atomic excited state
The functional integral over the quadratically occurring in Eq. (V.1) can be performed exactly. In the regime of interest, the atom-laser detuning is much greater than the energy scale associated with atomic motion; therefore, one can simplify matters by dropping the gradient term for . (Put heuristically, excited-state atoms, being short-lived and massive, “decay” before they have time to move, so that the interactions they mediate are local in space and time.) Thus, one can integrate out the excited state, determining the necessary kernel via the standard technique of solving the classical equations of motion for and to obtain
where, for convenience, we have made the change of photon variables to enable us later to exploit the approximate degeneracy of the laser and cavity modes. By inserting the classical solutions, Eq. (25) into the action, Eq. (V.1), we arrive at the following contributions to the action:
In what follows we shall approximate by , using the fact that the interatomic interaction is typically many orders of magnitude weaker than .
v.2 Eliminating the photon states
Next, we integrate out the photon states, doing so in two steps: (i) by integrating out the environment modes to arrive at an effective action for the cavity photons, and (ii) by integrating out the cavity photon modes to arrive at an effective action for ground-state atoms alone. To achieve step (i) we use the result of Caldeira and Castro-Neto [i.e., Eqs. (36) of Ref. castroneto () in the limit ] for the path integral over extracavity modes, and thus we identify the following contributions to the action that involve the cavity modes:
where is the local atomic density. Note that we have dropped the atomic internal-state index (i.e., we have made the relabeling ).
To achieve step (ii), we observe that the action is quadratic in the cavity photon modes, so they too can be integrated out, to produce the desired atom-only action which, for convenience, we express in terms of the classical-quantum (i.e., ) basis for the fields rather than the basis (see Sec. IV):
in which is the matrix
and the quantities are defined as follows:
where the Keldysh components of the atomic density are given by
How one should proceed from here depends on the relative magnitudes of , , and . Our objective in this paper is to analyze the self-organization transition in a multimode cavity. Physically, this transition is associated with the laser-cavity interference term, as discussed in Sec. III, and is most straightforward to analyze when (i.e., when ); moreover, as discussed in Ref. ritschprl () (and as we shall show), the steady-state temperature of the system is proportional to . Therefore, in order to explore self-organization at low-temperatures, it is natural to take Note1 (). We may therefore expand and in powers of and , thus arriving at a simplified atom-only effective action:
where the five terms—which are labeled by their corresponding coupling constants—respectively account for: , the kinetic energy of the atoms; , the -periodic interaction caused by the scattering of photons between the laser and cavity modes; , the interaction due to the scattering of photons between cavity modes; the contact repulsion between the atoms; and , dissipative processes due to the leakage of photons through the cavity mirrors. The terms have the following explicit forms:
To streamline the notation we have introduced the coupling constants and . For the regime in which is large compared with the other frequencies in the effective action (e.g., the typical atomic kinetic energy), a further simplification is possible: the integrals over and can be expanded in a gradient expansion in terms of . In what follows, we shall keep only the zeroth order term in this expansion, thus arriving at the following “instantaneous” forms of and , in which we have expanded and in terms of the atomic field operators:
Note that if we neglect the effects due to , the nonequilibrium character of the theory would apparently disappear: the nonequilibrium laser- and cavity-mediated interactions and have precisely the same form in terms of Keldysh and time indices as the contact repulsion ; thus, an effective equilibrium description of these terms should be possible. In the rotating-wave approximation (see Sec. VI below and Ref. walls ()), the laser, although a nonequilibrium element, only influences the density matrix by raising the system’s “apparent” energy eigenvalues by . As explained in the following section, we can use this fact to develop an effective equilibrium description of the self-organization transition, to which the effects of a nonzero can later be added as a perturbative correction.
Vi Effective equilibrium theory
In this section we offer some considerations on self-organization in pumped cavities in general and, in particular, on the quasi-equilibrium character of this transition in the high-finesse limit. The fundamental difference between the zero-temperature and nonequilibrium formalisms is the assumption—valid in the former case—that the initial state (in this case, the ground state) of the system without interactions evolves adiabatically into a pure energy eigenstate (in this case, the ground state) of the interacting system, provided the interactions are switched on sufficiently slowly. This assumption does not, in general, hold away from equilibrium, but there are certain nonequilibrium systems—e.g., an atom pumped by a far-off-resonant laser—for which it does hold: in this example, Fermi’s golden rule implies that the laser does not stimulate real transitions between the atomic levels unless the laser is resonant with the atomic transition. (For a broadened atomic level, Fermi’s golden rule implies that real transitions are negligible as long as the laser-atom detuning, , exceeds the linewidth, , of the atomic transition.)
In the present case, the excitation gap between the trivial ground state (i.e., the photon-free cavity and a uniform distribution of atoms) and the lowest excited state, which has photons and a -periodic atomic density modulation (a “phonon”), consists of two parts: (i) the energy cost of adding a cavity photon and (ii) the energy gain due to the photon-phonon coupling, which in second-order perturbation theory would have the form , where . Thus the total gap must have the form:
For the system at hand, the assumption of adiabaticity holds as long as the cavity is of sufficiently high finesse, i.e., . As we shall see in Sec. XI, the self-organization transition is weakly discontinuous, and the range of values of the control parameter over which the ordered and disordered phases coexist can be made larger than . Under these conditions, the self-organization transition should be well-described by the effective equilibrium theory sketched in Ref. us (). Only for large values of (i.e., bad cavities) would nonequilibrium effects have a chance of playing a leading role. For an alternative construction of such an equilibrium theory, which illustrates the relation between its quantum phase transitions and conventional equilibrium quantum phase transitions, see App. A.
The assumption of adiabatic switching implies that if interactions are turned on adiabatically in the distant past and turned off adiabatically in the distant future, an initial energy eigenstate would evolve into itself, up to a (physically unimportant) phase factor agd (). This consequence in turn implies that a single path integral can capture all the dynamical information; the second path integral in Eq. (13), which was demanded by the necessity of summing over all final states, would be superfluous because the final state would be known. In terms of Green functions, this implies that the components and become redundant; indeed, it can be proved directly, in terms of Keldysh diagrammatics biroli (), that the expansion for (and ) is closed, containing all information about correlations and response.
Formally, the case of nonzero temperatures is less straightforward because the adiabatic switching assumption is not available for mixed states, and therefore the derivation of the imaginary-time Matsubara formalism from the real-time Keldysh formalism is, in general, nontrivial. In this case, we are guided by the following consideration: if the fluctuation-dissipation theorem holds for the exact Green functions, the Matsubara formalism is valid; in the present case, we can use the Keldysh diagrammatic technique (see Sec. VIII) to show that violations of the fluctuation-dissipation theorem are small, and hence that the Matsubara technique is approximately valid. This is what one expects on physical grounds: the chief effect of the laser photons is to mediate an effective atom-atom interaction, rather than to cause an energy flux.
Our strategy in the next section, Sec. VII, will be to analyze the equilibrium critical behavior of the system at both and , using established results from statistical mechanics and quantum field theory, and neglecting dissipative processes. After that, we shall return to the Keldysh formalism, in Sec. VIII, to reinstate the dissipative processes and explore their consequences.
Vii Quasi-equilibrium Landau-Wilson description
In this section we derive coarse-grained, Landau-Wilson forms of the atom-only action, Eq. (33), both for zero and nonzero temperatures, valid in the quasi-equilibrium regime. These Landau-Wilson actions are closely related to the one first introduced by Brazovskii brazovskii (); we exploit this relationship in order to describe the impact of collective fluctuations on self-organization in multimode cavities. Finally, we discuss how the correlations of these fluctuations can be detected via the light emitted from the cavity.
In an effective equilibrium theory, the prescription for going from the Keldysh to the Matsubara or zero-temperature formalisms is to trace, in reverse, the steps one would have taken to go from the equilbrium to the Keldysh formalism: i.e., keep the component of the action and drop the Keldysh indices kamenev (). It is, furthermore, convenient to reformulate the action in terms of an order parameter, i.e., a quantity that is zero in the uniform phase and nonzero in the self-organized phase. The considerations of Sec. III suggest that the appropriate order parameter for detecting crystallization into cavity mode should be given by
The details of this procedure are dependent on the cavity geometry; we shall focus in this work on the case of the concentric cavity, as it is the most straightforward case.
vii.1 Mode structure of the concentric cavity
A concentric cavity can be thought of as consisting of two mirrors that cover antipodal regions of the same sphere. Its mode structure is derived from the solutions of the Helmholtz equation inside the sphere. The effects of the edges of the cavity mirrors can be hard to compute accurately; the standard technique is to solve the Helmholtz problem approximately, by requiring the electric field to vanish at the the mirrors edges. (As with the subsequent approximations that we shall make, this one works best for large cavities.) As the atomic distribution is quasi-two-dimensional, being confined near the equatorial plane, it breaks the spherical symmetry of the cavity, and it is therefore convenient to employ cylindrical coordinates . Thus, the planar dependence of the cavity modes takes the form , where is a Bessel function, is quantized by the requirement that (see Fig. 1), and by the requirement that the field should vanish at the mirrors. The quantization of solutions along the direction (i.e., the pump laser axis; see Fig. 1) yields a third mode index . For large cavities, modes having a fixed value of are frequency-degenerate. Because, as discussed in Sec. II, the free spectral range of the cavity is larger than the energy scales relevant to self-organization, we can restrict ourselves to cavity modes having a certain fixed value of .
The atomic density, which is quasi-two-dimensional, can be expanded over a similar set of mode functions, indexed by , provided one retains all such modes and not just those satisfying . In addition, the boundary conditions on these mode functions are not in general the same as those on the cavity modes, as the atoms are confined by an external, confining laser field rather than by the cavity mirrors. For sufficiently large traps, however, this distinction is not expected to have important effects, and we shall neglect it.
The cavity modes for which are expected to be favored for crystallization, as they have the highest amplitude in the equatorial plane of the cavity, to which the atoms are confined. However, as we shall see, modes having are also of importance in setting the range of the effective atom-atom interaction, and thus in determining, e.g., the extent of the fluctuation-dominated regime, as well as the size and stability properties of droplets of the ordered phase Note2 ().
In the basis for the atomic density, the order parameter, Eq. (37), is given by . Note, also, that provided that the atomic density is spread out over a large number of optical wavelengths, the following asymptotic result holds:
In this expression, we have introduced the sign-insensitive Kronecker delta