# A Quantum Theory for Bose–Einstein Condensation of the Ideal Gas

This is a Perspective on "Analytical framework for non-equilibrium phase transition to Bose–Einstein condensate" by V. Yu. Shishkov, E. S. Andrianov, and Yu. E. Lozovik, published in Quantum 6, 719 (2022).

By Fabrice P. Laussy.

## Historical background: insights of a genius.

The achievements of Einstein in formulating the laws governing nothing short of space and time are so  monumental that they often overshadow his contributions to his other predilection — statistical physics. There, by comparison, his accomplishments appear as sporadic glimpses into the mind of a genius, briefly contemplating a variety of unrelated problems, such as Brownian motion explaining small fluctuations of large objects in a fluid, his $A$ and $B$ coefficients regulating light-matter interactions, or his theory of solids. All these seminal contributions, as ingenious as they look simple, remain at the heart of the fields that later emerged from them. Other breakthroughs were made with hardly any attempt by Einstein at claiming authorship, as is the case for one of the most famous results of statistical physics, the Wiener-Khinchine theorem, which was actually first proposed by Einstein in an obscure private meeting of the Société Suisse de Physique, recording the insight in a memo of less than two pages, 14 years before Wiener himself would develop the theory of generalized harmonic functions to arrive at the same result through 142 pages of mathematical rigor. One of the most far-reaching statistical results of Einstein, the theory of Bose-Einstein condensation (BEC), is a climax in the history of science. This describes the process by which a large fraction of particles in a system coalesce to their ground state. Einstein conceived the mechanism following Bose’s personal communication of results on the statistics of photons that he, as an unknown Bengali scientist, could not publish. Not only did Einstein pay attention to a letter that would nowadays remain in most prominent scientists’ spam folder, and not only did he immediately recognize the value of the reasoning and personally recommended it for publication [1], he actually also translated the text into German himself for the editor of Zeitschrift für Physik!

If one counts how many particles can be in the various available states of the system when they have the statistics that Bose said to Einstein was able to explain Planck’s black-body spectrum, one gets a broken equation: particles can not all fit below a critical temperature. Einstein thus postulated, not that the theory is incorrect, but that the extra particles pile up in the lowest energy state (the ground state) [2]. Bose’s statistics applied to photons, or light, which is known to be able to be superposed with itself (it just gets brighter). Bose, in fact, did not derive any new result, but found an alternative explanation for Planck’s radiation law, let alone that he was counting cells in phase-space rather than particles. Einstein, in contrast, predicted a new phenomenon altogether by extending Bose’s statistics to massive particles, for which it is more unsettling that they could occupy the same state. This has considerable consequences for the central quantity of statistical physics: entropy. The system frees itself from statistics: particles that were previously a chaotic collection of innumerable balls bouncing around, now agglutinate in a strange new phase of matter: a sort of macroscopic one-particle state, that would later be known to be a wave function. With Bose’s way of counting, one could get rid of Boltzmann’s statistics. Einstein was again anticipating quantum physics.

Few have initially taken such a simple and bold idea seriously. One does not just predict a new phase of matter by making up a candid assumption to collect what spills out of an integral. Ehrenfest even qualified such a statistical approach as “disgusting”. His PhD student, Uhlenbeck, raised a more serious criticism based on the fact that discrete sums do not exhibit a divergence and therefore no condensation would take place in a finite-size system. This turned out to be a more general concern for second-order phase transitions rather than for BEC. London was an exception in welcoming Einstein’s idea at a time when even Einstein himself had lost interest, to describe superfluids and superconductors. Also with quantum mechanics emerging (ironically, thanks to Schrödinger’s learning of de Broglie wave-duality from Einstein’s BEC manuscript) and with quantum statistics becoming more clear-cut (ironically, linked to Uhlenbeck’s spin), the issue of the Bose gas became trendy again and soon turned into that of its kinetics of formation, bringing back some of Uhlenbeck’s original rebuttal in the form of infinite times of formation for the nucleation of the condensate. Interactions came handy to make the problem so complicated as to require approximations and ad hoc hypotheses to account for the possibility of the phenomenon, which eventually materialized in the laboratory in a series of works that are now famous [3,4].

## Quantum Boltzmann Master Equations

Einstein’s theory is for the populations of the states, which is the most  immediate and obvious observable, mainly due to the fact that this is also the most important one in classical physics. Einstein’s condensation occurs when the population of the ground state becomes commensurable with the population of the full system. In quantum mechanics, however, populations are just a particular case of the general picture that consists of an infinite set of correlators to describe the system. In a modern theory of BEC, for instance, beyond the population, a so-called “off-diagonal long-range order” (ODLRO), in terms of the correlations between points in space with large separation, is a more accurate and meaningful measure of the amount of condensation. Another central quantity, developed to describe coherence in lasers, is Glauber’s second-order correlation function, which relates to particle fluctuations. A density matrix $\rho$ embodies all this, also including purely statistical information in addition to the quantum correlations that are described by a wave function. The Schrödinger equation for a density matrix of an open system is called a master equation. The “off-diagonal” in ODLRO refers precisely to the corresponding elements in the density matrix. The population is one element only on the diagonal. The variance of the population, that enters in Glauber’s coherence, makes another one, and follows an infinite series of higher-order correlators. While classical statistical physics also features moments of any order, they are typically related to the lower-order ones or play a minor role, while in a quantum system, such correlators can become critical and no longer follow from the populations. Instead, they have a dynamic of their own. In the case of a classical gas, the populations are described by Boltzmann equations. The simplest upgrade to a quantum version merely renormalizes the scattering rates to account for stimulated processes (the larger the population of a state, the larger the probability to scatter there), and is known as the Uehling–Uhlenbeck equations or Quantum Boltzmann equations. They fail to start or nucleate the condensate: populations alone, even with Bose stimulations, have no access to what brings coherence and quantum correlations into the system, so this must be put in by hand. The next simplest upgrade provides a similar equation but now for the density matrix. It has been introduced by Gardiner and Zoller [5] in their quantum kinetic theory of Bose-Einstein condensation. While they focused on interactions and potential traps, as well as transport in space (in which case they deal with a more general version that they call the “quantum kinetic master equation”), the formalism turns out to be nontrivial even without these elements, and provides a largely overlooked quantum theory of Bose–Einstein Condensation for the ideal gas, that is to say, the simplest nontrivial quantum extension of Einstein’s theory. Related and enlightening insights based on master equations were similarly obtained [6,7] although by focusing on the reduced density matrix of the condensate, they obfuscate the mechanism which allows particle-number correlations to entail condensation, especially in cases where the number of particles is not fixed. One can understand this underlying mechanism in terms equally simple as those captured by Einstein, but now going beyond populations only, to also consider correlations between populations. The basic idea can even be illustrated with two oscillators only.

## The less-known (Lindblad) coupling of two quantum oscillators

Coupling two quantum oscillators $a$ and $b$ is a basic textbook problem in quantum mechanics, with well-known Rabi oscillations (the exact solution to Einstein’s perturbative $A$ and $B$ coefficients), anti-crossing as well as superpositions and/or entanglement of the states. This Hamiltonian picture does not include dissipation, typical in a classical treatment of oscillators. For that, one needs a master equation in the Lindblad form $i\hbar\partial_t\rho=\mathcal{L}\rho$, that generalizes the Schrödinger equation to include non-Hermitian (irreversible) processes through super-operators $\mathcal{L}_\Omega$ that act on the density matrix as $\mathcal{L}_\Omega\rho\equiv2\Omega\rho\Omega^{\dagger}-\Omega^{\dagger}\Omega\rho-\rho\Omega^{\dagger}\Omega$ for any operator $\Omega$. Göran Lindblad indeed formulated the most general such type of equation that preserves the properties of the density matrix to provide a valid physical picture [8]. When including dissipation, one arrives at a more advanced textbook level describing weak and strong coupling.  Although of no higher technical complexity, the direct dissipative coupling between two oscillators is much less studied, appearing, when it does, as a sidekick to other Hamiltonian or dissipative processes and little considered on its own. It reads:

\label{eq:Wed10Aug131917CEST2022}
\mathcal{L}={W_{a\to b}\over2}\mathcal{L}_{a{b}^{\dagger}}+
{W_{b\to a}\over2}\mathcal{L}_{{a}^{\dagger}b},

with the first term transferring a particle from $a$ to $b$ at the rate $W_{a\to b}$ (and vice-versa for the second term). This classical-looking coupling of the two oscillators — in particular with possibly different rates from one to the other — is nevertheless able to extend Einstein’s theory of BEC of the ideal gas to the quantum regime all by itself. In particular, it captures the condensation process beyond the populations only (this was done by Einstein), but also for the system’s quantum correlators, in particular describing the increase of coherence by the system’s evolution towards a coherent state. The density matrix’s diagonal elements $p(n,m)$, representing the probability of finding $n$ quanta in oscillator $a$ and $m$ in oscillator $b$ can be expressed exactly [9]:

\label{eq:Wed10Aug133031CEST2022}
p(n,m)={\xi-1\over\xi^{n+m+1}-1}\xi^n\sum_{k+l=n+m}p(k,l),

where the sum on the right-hand side is time-independent and so is given by the initial condition, while $\xi$ has been defined as $W_{b\to a}/W_{a\to b}$. We shall assume that $\xi>1$, which defines $a$ as the ground state (since $W_{b\to a}/W_{a\to b}=e^{-(E_a-E_b)/k_\mathrm{B}T}$, which is how temperature is introduced in the model). For instance, starting with the two oscillators each in a thermal state, with respective effective temperature $\theta$ and $\nu$ (between 0 and 1), one has

\label{eq:Wed10Aug133309CEST2022}
p(n,m)=(1-\theta)(1-\nu)\theta^n\nu^m,

which is a solution of Eq. (2) provided that $\xi=\theta/\nu$, relating the states’ temperature to the system’s temperature. This corresponds to a classical, uncorrelated and uncondensed solution, regardless of the population $\theta/(1-\theta)$ of the ground state which, at equilibrium, is always larger than that of the excited state. As a characteristic feature of thermal states, however, the highest probability of occupation is that of the vacuum and high populations merely mean huge fluctuations from thermal kicks in and out of the oscillators. Bose condensation is an altogether different distribution of the populations which, in Einstein’s picture, manifests itself with a Dirac $\delta$ function. In our quantum case, lowering the temperature (increasing $\xi$), one can see from the exact result (2) how correlations arise between the oscillators due to the Lindblad coupling, with the effect of, not only transferring particles to the ground state, but also developing correlations between the states: the joint probability distribution is not a product state like Eq. (3) anymore. This locking of the ground state’s population to that of the other oscillator results in increasing the probability for its occupied states at the detriment of its vacuum: the system develops coherence in the sense of Glauber. It stabilizes its population. No interactions are needed, only the conservation of particle numbers through the Lindblad coupling, i.e., the correlated transfer: one particle can be added to a mode because it was removed from another. This is in contrast with Boltzmann equations where populations are balanced in the same proportion, but with no correlations. There are many interesting and pretty features already captured by this model. For instance, one needs at least two particles on average for the ground state to possibly have a higher probability to be excited than the probability of its vacuum. In other words, one needs at least two bosons to condense! 1.9 on average won’t suffice, however hard those fluctuating bosons are pressed into the same oscillator.

Joint probability $p(n,m)$ to have $n$ quanta in the ground state $a$ and $m$ in the excited state $b$, starting from two thermal states (no condensation) at $\xi\approx 1.17$ and lowering temperature (increasing $\xi$) close to full condensation. In between these two limits where the states are uncorrelated, the nucleation of the condensate is observed to take place by the development of strong correlations between them. As a result, the reduced probability for the ground state $\sum_{m}p(n,m)$ (red) grows a peaked-distribution, characteristic of quantum coherence.  Condensation is complete when the full system’s distribution has been copied to the ground state. Note that the probability of the vacuum $p(0,0)$ is a constant that the system can never improve regardless of its degree of condensation.

The condensation itself is better seen from a plot of the density matrix than from working out the equations. In the animation below, we put seven particles in the ground state $a$ ($\theta=7/8$) and three in the excited state $b$ ($\nu=3/4$) and raise $\xi$ (i.e., lower the temperature) from the thermal equilibrium situation $\xi\approx 1.17$ to large values, up to $\xi\approx 50$, which brings over $99.8\%$ of the particles into the ground state (and cycle back and forth). One can see how the initial uncorrelated product state of thermal distributions evolve into a configuration where the excited state remains thermal, though with fewer particles, while the ground state grows not only its population but also acquires a peaked structure: the vacuum is no longer the most likely state, which is that with 4 particles instead, with probability $p(4)\approx6.9\%$. This distribution has essentially converged to the limit of full condensation, $\xi\to\infty$, where all particles are in the ground state and its reduced distribution becomes identical to $p(n,0)$. The ground state has detached itself from the rest of the system, which is again in a product state, but now with a condensate. It is easy to see that this distribution is that of the total number of particles: $P(n)\equiv\sum_{k+l=n}p(k,l)$. Not only has the ground state detached itself but, to an even larger extent, as far as statistics are concerned, it became the full system. One could now focus on this quantum state in isolation, and turn to a Schrödinger (or with interactions, Gross-Pitaevskii) equation for the condensate alone. In the intermediate regime, as the condensate nucleates, the two states become strongly correlated, which is why Boltzmann equations fail to capture this intermediate but crucial bridge between the two product states: they cannot describe the intermediate correlations that bring one type of product state to the other. This also explains the fluctuations of the condensate: they are still very large as can be seen from the long tail, but this is the best that two oscillators can do out of thermal states. To improve this situation, one needs either more oscillators, or better statistics of the individual states, as is discussed in the next Section.

## An exactly-solvable quantum theory for the ideal Bose gas

The long tail of the condensate in a two-harmonic oscillators toy model can be lifted by generalizing it to include more oscillators, or by adding a mechanism to relax constraints on the statistics of the whole system, since this is where the statistics of the condensate come from. One such escape is provided by open systems, where the finite lifetime of particles is compensated by continuous pumping, a situation that is realized for instance by exciton polaritons in microcavities [10]. In this case, the two-oscillators picture then becomes able to describe the condensation of so-called polariton lasers [11] very well, which can be supplemented by standard Boltzmann equations to describe the dynamics of relaxation of the excited states and focus on coherence for the ground state only. This is already enough to study photon-number statistics of such condensates [12].

A more fundamental and complete quantum theory of Bose condensation, at or out of equilibrium, in a closed or open system, would nevertheless require one to include all the oscillators into the quantum picture, and to worry about their correlations. These are very small, as are the average populations of each state, so the prospect of keeping track of all this looks like overkill, but their combined effect does allow for a coherent state (with Poisson distribution) to emerge spontaneously out of thermal states everywhere else in the system, without the need of interactions. The mechanism of condensation is indeed the same as with two oscillators: a single state (the ground state) acquires, or copies, the properties of the system as a whole, not only its population (as is the case with Boltzmann) but also its statistics. With many oscillators, however, the central limit theorem states that $P(N)$ is normally distributed, which is the origin for the familiar Bell-shaped distribution of the coherent state in BEC. Such generalizations have been studied under still considerably simplifying assumptions, with a large number — but degenerate in energy — of oscillators [13]. A more ambitious program is to consider the Hamiltonian of the ideal gas: with an infinite number of oscillators corresponding to the dispersion of a continuous system in either 2D or 3D and with different relaxation rates as mandated by the temperature (from, say, phonon relaxations). Despite the apparent complexity, with no interactions (the gas is ideal), the problem can be solved exactly, even when including pumping and decay, which also provides the out-of-equilibrium theory. This has been done by Shishkov, Andrianov and Lozovik in a paper recently published in Quantum [14]. They could find, under the innocuous assumption of fast-enough thermalization and generalizing a previous effort for finite sets [15], an analytical solution for the full system, i.e., for the full density matrix. We reproduce here their result for the partition functions $Z_N$ that normalize the density matrices with a fixed number of bosons and from which, together with the probabilities of occupations corresponding to the type of steady state (thermal equilibrium or the interplay of pumping and decay), one can extract a wealth of statistical quantities for the system in various scenarios. Besides, they provide this result only in an appendix, preferring to highlight a recurrence formula which is numerically more efficient:

\label{eq:Sat13Aug155139CEST2022}
Z_N=\sum_{n=0}^N{1\over n!}B_n\left({1!G_{3D}\over 1^{5/2}},{2!G_{3D}\over2^{5/2}},\cdots,{n!G_{3D}\over n^{5/2}}\right),

where $B_n$ is the $n$-th complete exponential Bell polynomial, which is a sum full of combinatorics, pointing at all the ways the bosons can redistribute themselves among the oscillators. The authors also find $G_{3D}\equiv V(2mT/[4\pi\hbar^2])^{3/2}$ for a quadratic dispersion of 3D particles with mass $m$ in a system with volume $V$ at temperature $T$, that quantifies the number of states above the ground state with energy less than $k_\mathrm{B}T$. This defines a lower bound for the number of particles required for a possible condensation in a realistic system, as opposed to $N=2$ in the two-oscillators model. They also find the partition function in 2D, in which case the result depends on a corresponding $G_{2D}$ and the $5/2$ exponents turn into squares.

This is a beautiful and fundamental result that takes a direct step forward from Einstein’s historical theory, with no reliance on the considerable bulk of works that followed for the interacting gas. This also makes a rare addition to the scarce list of exactly solvable quantum systems. Beyond the solution itself, the authors have also highlighted several notable if not surprising results for quantities that follow straightforwardly from their analytical expression. One can mention for instance their finding that the well-known Shallow-Townes formula for the linewidth of the coherent state breaks down in 2D or that although pumping any state is beneficial for the population increase of the condensate, its coherence requires pumping through excited-states only, in which case a thresholdless randomly phased but perfect coherent state is formed at zero temperature. Much more remains to be explored and calculated even from their sole partition function. The full-fledged theory of the ideal gas à la Einstein (for populations) is in fact far from trivial [16] but has remained textbook material, with known pathologies, such as its infinite compressibility. Its fully-solvable quantum upgrade and extension to finite lifetime and incoherently pumped cases may provide precious insights into basic and fundamental questions of quantum phase transitions, such as the role and importance of U(1) symmetry breaking, since no absolute phase is needed, or the actual role of interactions, since none are needed either. It should also rekindle the interest for the dynamics of formation of condensation, which has remained one of the most challenging problems in the field, and, of course, serve as a new starting point for dealing with interactions, for instance to develop a Bogoliubov version with interactions in the ground state alone and, possibly, also provide an exact and complete analytical theory by following a similar route for the bogolons. It remains to squeeze out the essence of what the quantum theory of the ideal gas can still contribute to the phenomenon. The authors called attention to their solution in the framework of non-equilibrium condensation, with external pumping compensating for decay, but their approach probably captures deeper and more general features, including the relationship between at- and out-of-equilibrium steady states.

## Conclusions

Einstein’s seemingly naive insight into the statistic of the ideal gas gave rise to what remains the most emblematic quantum phase of matter — the Bose-Einstein condensate — which exploded into a thriving field requiring the most sophisticated and complex theoretical modeling: strongly correlated quantum systems. It is remarkable that, without prohibitively complex mathematical tools or numerically involved treatments, one can still, in a spirit very much reminiscent of Einstein’s “as simple as possible but not simpler” reduction of the physics to a toy model, get closer to the essence of the phenomenon and capture hallmarks such as the coherence buildup and quantum correlations of the gas in its various phases. Following the treatment of Shishkov et al. who grace us with an exact solution for the complete density matrix, a rare and precious object, one can now dive deeper into features of the condensation itself, including transport terms (inhomogeneous case) and potential traps, its connection to other phases such as superfluidity [17] or study the line-narrowing of condensation due to the mechanism itself, as opposed to interactions alone, bringing back, of course, such interactions, as well as spin and a wealth of other factors relevant in the laboratory. It might well be the case that this tractable picture captures enough physics to account for most observations, or that one even stumbles upon more refined treatments otherwise lost in the complexity of more complicated models. The analytic solution will also allow us to validate numerical methods, such as Monte Carlo simulations. From this perspective, one can also reflect on how Einstein’s, not annus, but properly vita mirabilis, was even possible. As simple as possible, but not simpler. If many-body perturbation theory, Popov diagrams or machine learning and other sophisticated numerical simulations are eventually unavoidable to achieve the required level of understanding, then one has to learn the trade and practice it. Einstein did, after all, master differential geometry to generalize his picture of space and time. But this should not exclude more straightforward approaches that, more often than not, rely on a couple of harmonic oscillators only and can — in the hands of skilled theorists — lead to exact closed-form results and analytical expressions for nontrivial systems. Shishkov et al.’s exact solution for the quantum condensation of the ideal gas, in or out-of-equilibrium (the underlying mechanism is the same), in two or three dimensions, is doubtlessly a remarkable result of great significance. As is typical of such a breakthrough when it is of deceiving simplicity, as was indeed the case with Einstein’s own theory, it will probably take time before its value and importance are fully appreciated.

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