Energy conservation and quantum backreaction in Bose-Einstein condensates

The problem of determining how backreaction occurs in semiclassical gravity is one of the most intricate and rewarding ones in quantum field theory. In simple terms, the problem consists in determining how quantum fluctuations of test fields on top of a classical curved background influence the spacetime. Examples of quantum backreaction effects include the spontaneous Hawking radiation by black holes, which is believed to diminish the black hole mass [1], and certain chronological protection mechanisms in cosmic string spacetimes [2, 3]. Other interesting example is known as the gravity-induced vacuum dominance [4], where the quantum fluctuations become sufficiently strong and force the spacetime into a different configuration via decoherence [5, 6]. Yet, despite the plethora of interesting examples, some backreaction problems of fundamental importance remain not fully solved [6]. We cite, for instance, the problem of determining how quantum fluctuations affect gravitational collapse in general. Only partial results exist [7, 8], and they suggest that quantum fluctuations can play a prominent role on black hole formation.

Another difficulty in studying backreaction in semiclassical gravity is related to the available experimental data, which, for some weak effects like the Hawking radiation, might be virtually impossible to obtain. One possible route for studying such small effects is offered by analogue gravity in Bose-Einstein condensates (BEC). This program, that gained momentum with [9], is well-developed [10], and counts with examples ranging from black hole to cosmological analogues [11, 12, 13, 14, 15]. Also, the analogue gravity community has achieved an impressive milestone: The measurement of an analogue Hawking radiation [16, 17]. BECs are also systems that can be used to study some quantum backreaction effects [18, 19, 20]. Within the analogue gravity program, we cite the possibility of simulating the evolution of quantum fluctuations on top of analogue black hole spacetimes [21, 22], one of the necessary ingredients to find the corresponding corrections to the background analogue spacetime.

Different methods can be implemented in order to study quantum backreaction in BECs. In an ideal scenario, one seeks a solution for the full non-linear dynamics of a gas of bosons trapped in some region and subjected to initial conditions. This goal, however, cannot be fulfilled in general due to the elevated complexity of the field equations. Yet, in some cases, when it is possible to identify small quantum fluctuations, approximate solutions can be found by means of the Bogoliubov expansion or other equivalent methods [23, 24]. This idea was explored in [25], where the authors developed an approach to the quantum backreaction in BECs based on the particle number conservation. This method was recently applied for a concrete condensate configuration [26] and a solution to the quantum backreaction was found.

In this work we go a step further in the analysis initiated in [26] by discussing another crucial feature of the backreaction analysis, which is the energy conservation. By starting with a non-relativistic gas of bosons allowed to move in one spatial dimension, we implement an expansion of the field operator in powers of the number of particles in the system, thus identifying a notion of background condensate, quantum fluctuations, and condensate corrections. We then adopt the canonical stress tensor, $T^{\mu}_{\ \nu}$, as a measure of the system energy and momentum densities, and identify the different contributions to $T^{\mu}_{\ \nu}$ coming from the quantum fluctuations and the condensate corrections, as well as the associated conservation laws. It is shown that the backreaction scheme of [25, 26] can be restated in terms of one of the conservation laws linked to $T^{\mu}_{\ \nu}$, namely, the conservation of momentum. The latter is capable of revealing the role played by the various contributions of the Euler equation of [26], pinpointing the sources of stresses in the system and forces due to external agents.

The backreaction equations are solved for a homogeneous condensate trapped in a ring configuration which undergoes a quantum quench from an initially non-interacting regime to an interacting one. This is important in order to avoid the vacuum state indeterminacy coming from the phenomenon of phase diffusion [27]. In order to keep the model experimentally appealing, we employ a continuous transition between the different regimes which contains the sudden transition as a particular case and such that quantization can be performed exactly. The solution to the backreaction problem is then used to fully characterize the energy of the system.

This work is organized as follows. In Sec. II the general description of the non-relativistic gas of bosons is reviewed, and the backreaction equations are obtained. In Sec. III the energy-momentum tensor is presented and its various contributions are identified, as well as the corresponding conservation laws. Section IV discusses in details the construction of a quantum field theory for the quantum fluctuations in the system. In particular, the connection between the system vacuum state and different instantaneous quasiparticle vacua is presented. In Sec. V the condensate depletion and the system energy are thoroughly studied, and we finish with some final remarks in Sec. VI.

We consider a non-relativistic gas of bosons allowed to move in only one dimension, hereafter parameterized by $x$. We let $\phi=\phi(t,x)$ be the complex field describing the gas according to the Lagrangian (in units of $\hbar=1$)

where in general $V=V(t,x),g=g(t,x)$, $m,g>0$, with $m$ being a constant. Note that the above Lagrangian is invariant under the global $U(1)$ transformation $\phi\rightarrow\exp(i\alpha)\phi$, for any constant real parameter $\alpha$. This in turn implies by means of Noether’s Theorem the local conservation law [28]

where

Therefore $\rho$ can be interpreted as the particle density, with

being the total number of gas particles in the system. Finally, by treating $\phi$ and $\phi^{*}$ as independent variables, the Euler-Lagrange equation for $\phi^{*}$ reads

As in any quantum field theory, given the system parameters $m,g,V$, the quantum features of the system can be studied by looking for an operator-valued distribution, $\phi$, solution of Eq. (6), such that

holds true, where

is the momentum canonically conjugate to $\phi.$

There exists a plethora of possible routes to quantize the above theory, and here we adopt the Bogoliubov expansion [25]. Specifically, we are interested in the regime $N\gg 1$, while $gN\sim 1$. When this occurs it is possible to separate the field $\phi$ into a “classical” part plus “small” quantum fluctuations by expanding $\phi$ in powers of $\sqrt{N}$ as

where formally $\phi_{0}$ is proportional to $\sqrt{N}$, $\chi$ is proportional to $N^{0}$, and $\zeta$, to $N^{-1/2}$. Thus it follows from this ansatz and the Euler-Lagrange equation that $\phi_{0}$ is solution of the Gross-Pitaevskii equation [26],

with $\rho_{0}=|\phi_{0}|^{2}$, $\chi$ is ruled by the the Bogoliubov-de Gennes (BdG) equation [26]

and finally a BdG-like equation with $\chi$-dependent source terms for $\zeta$ [26],

appears at order $N^{-1/2}$. We stress that it is crucial to keep the field $\zeta$ in the expansion in order to ensure that quantum fluctuations are correctly accounted for in the conservation laws, as we discuss in the next section.

Within the Bogoliubov expansion, which holds true as long as the hierarchy in separation (9) is preserved, the quantum properties of theory are assumed to be encapsulated in the field $\chi$ subjected, by means of Eq. (7), to the canonical commutation relation

Physically, the above theory describes condensates composed of a sizable background described by $\phi_{0}$, and on top of which small quantum fluctuations, described by $\chi$, exist. The field $\zeta$ then models eventual corrections to the background condensate. This is the idea behind the quantum backreaction discussed in [25, 26].

We now turn our attention to the conservation laws associated to the system particle number, energy, and momentum within the quantum backreaction scheme presented in the previous section. The particle number conservation was briefly discussed in Eq. (2) in order to identify number of particles, $N$, of the gas. For the system energy and momentum we adopt the canonical energy-momentum (stress) tensor as their measure. At the classical level, by letting $\phi=\phi_{1}$, $\phi^{*}=\phi_{2}$, and $x^{\mu}=(t,x)$, the canonical energy-momentum tensor, $T^{\mu}_{\ \nu}$, for a given Lagrangian $\mathcal{L}$ is defined by [28]

Therefore, it follows from Eq. (1) and the Euler-Lagrange equation that

where

We thus immediately recognize $T^{0}_{\ 0}$ to be the system Hamiltonian density obtained from the Lagrangian (1), which in turn implies that $T^{1}_{\ 0}$ is the system energy flux, whereas $-\mbox{Re}(T^{0}_{\ 1})=mJ$ is the system momentum density, and $\mbox{Re}(T^{1}_{\ 1})$ is the corresponding system stress [29].

We note also that if $\partial_{\nu}g=0$ and $\partial_{\nu}V=0$, then Eq. (15) becomes a continuity equation: $\partial_{t}T^{0}_{\ \nu}+\partial_{x}T^{1}_{\ \nu}=0$ and the corresponding “charge” $\int\mathrm{d}xT^{0}_{\ \nu}$ is conserved.

In order to establish the conservation laws at the quantum level it is necessary to adopt some ordering prescription when building operators from the classical observable quantities, e.g, $\rho=|\phi|^{2}$. Indeed, as $\chi$ and $\chi^{\dagger}$ are noncommuting variables, in general $\langle\chi^{\dagger}\chi\rangle\neq\langle\chi\chi^{\dagger}\rangle$. This affects, for instance, the evolution of $\zeta$, which depends on products of $\chi$ and $\chi^{\dagger}$. In this work we adopt the normal ordering prescription, which corresponds to define quantum operators by putting all $\chi^{\dagger}$ to the left of all $\chi$ in all monomials. Also, we assume that $\chi$ is prepared in a given vacuum state, for which $\langle\chi\rangle=0$.

With the normal ordering prescription, the c-number $\zeta$ is taken to satisfy the vacuum expectation of Eq. (12)

thus fixing how the condensate is corrected by the quantum fluctuations.

The first conservation law we discuss is the one given by Eq. (2). Upon quantization, we redefine the gas density to be $\rho=\langle\phi^{\dagger}\phi\rangle$ and the particle flux as $J=(1/m)\mbox{Im}(\langle\phi^{\dagger}\partial_{x}\phi\rangle)$. Thus the expansion (9) implies [26]

where $J_{0}=(1/m)\mbox{Im}(\phi^{*}_{0}\partial_{x}\phi_{0})$ is the background particle flux,

are the density of depleted particles and the phonon flux, and

are the corrections to the condensate density and particle flux, respectively. The importance of keeping the field $\zeta$ in the expansion resides in the fact that $\partial_{t}\rho_{0}+\partial_{x}J_{0}=0$ always, whereas [26]

which is the mathematical expression for the breakdown of particle number conservation by the Bogoliubov expansion. Although the number of depleted particles alone is not conserved, $\partial_{t}(\rho_{\chi}+\rho_{\zeta})+\partial_{x}(J_{\chi}+J_{\zeta})=0$. A detailed account of these various contributions was recently considered in [26].

Following the normal ordering procedure, we define the system energy density as $\mathcal{E}=\langle:T^{0}_{\ 0}:\rangle$, the system energy flux by $S=\langle:T^{1}_{\ 0}:\rangle$, the system momentum density $\mathcal{P}=-\mbox{Re}(\langle:T^{0}_{\ 1}:\rangle)$, and $\Theta=\mbox{Re}(\langle:T^{1}_{\ 1}:\rangle)$ is the system stress. Thus, the expansion (9) implies

and similarly for $S,\mathcal{P}$, and $\Theta$. The dominant (order $N$) contributions, $\mathcal{E}_{0}$, $S_{0}$, $\mathcal{P}_{0}$, $\Theta_{0}$ are obtained from Eqs. (16), (17), (18), and (19) by making the substitution $\phi\rightarrow\phi_{0}$, whereas for the remaining terms, we find

which are identified as the energy density of the depleted cloud and the correction to the background condensate energy density, respectively. Here, $G^{(2)}=\langle:(\rho-\langle\rho\rangle)^{2}:\rangle$ is the local density variance. For the energy flux, it follows from Eq. (17) that

Thus, the general equation (15) implies, for $\nu=0$, that

The above equation, which rules the energy exchange between condensate and depleted cloud, holds at order $N^{0}$, and it shows, in particular, that for $\partial_{t}V=\rho_{0}\partial_{t}g=0$, the energy $\int\mathrm{d}x(\mathcal{E}_{\chi}+\mathcal{E}_{\zeta})$ is conserved for confined condensates.

The last conservation law is the one linked to the system momentum, $\mathcal{P}$. We find that $\mathcal{P}=mJ=m(J_{0}+J_{\chi}+J_{\zeta})$, and

Therefore, Eq. (15) for $\nu=1$ implies

The right hand side of the above equation is the force density applied on the system by external laboratory agents. Naturally, for $\partial_{x}V=\rho_{0}\partial_{x}g=0$, the total momentum $\int\mathrm{d}x(\mathcal{P}_{\chi}+\mathcal{P}_{\zeta})$ is conserved in confined condensates.

We conclude this section with some remarks regarding the connection of the conservation laws (33) and (36) with previous works. Equation (36) is essentially the Euler-like equation upon which the backreaction analysis of [26] is based. The novelty of Eq. (36) is that it identifies the sources of the various contributions to the system momentum and stress. Furthermore, although in principle it is possible to determine the evolution of $J_{\zeta}$ and $\rho_{\zeta}$ once the quantum fluctuations on the system are known, it is not clear, from the results in [26], how to determine the system energy from these data, in a similar fashion to what occurs to the determination of the electromagnetic stress tensor from Maxwell’s equations [29]. We recall that at a fundamental level the true stress tensor of a given field theory is the one that sources gravity through Einstein’s equations. Nevertheless, although the canonical stress tensor might not coincide with the true stress tensor, for our non-relativistic case the system energy and momentum are indeed correct, and Eqs. (29) and (30) give a clear separation between the energy density of the depleted cloud and of the corresponding correction to the condensate, whereas Eq. (33) expresses how the latter evolve in the condensate.

In this section we present the quantization of $\chi$ for a family of condensates of great experimental appeal: Homogeneous condensates, at rest with respect to the laboratory frame, trapped in ring configurations. It is important to notice that some works treat condensates in the thermodynamic limit, which are idealized infinitely extended systems. For instance, in [25] it was shown how the condensate density is corrected for a 3D gas in the thermodynamic limit. However, in actual experiments it is not completely clear how to decide whether or not the thermodynamic limit is a good approximation, and for finite condensate configurations phase diffusion might play a prominent role.

We consider a trapping potential for which $\phi$ appearing in Eq. (1) is defined for $-\ell/2<x\leq\ell/2$, and the following boundary conditions

are naturally imposed by the field equation. Accordingly, because of their independence at each order in the expansion in powers of $N^{1/2}$, the fields $\phi_{0}$, $\chi$, and $\zeta$ are also subjected to the same conditions.

Another prominent aspect in any quantum backreaction analysis in condensates is the difficulty in choosing a quantum vacuum state that can describe a given experimental realization. This difficulty emerges from the fact that the field $\chi$ can be dynamically unstable, as occurs for instance in the phenomenon of gravity-induced instability [4], or it might describe a condensate undergoing phase diffusion [27]. In both cases, even when the background condensate is stationary, the quantum fluctuations on top of it can grow, leading to the spontaneous breakdown of the time-translation symmetry and the consequent difficulty in knowing the condensate state at future instants of time. In such scenarios, it is necessary to provide extra assumptions in order to select a meaningful vacuum state.

A homogeneous condensate is thus obtained by assuming a constant potential $V$ and a background order parameter $\phi_{0}=\sqrt{\rho_{0}}\exp(-i\mu t)$, for constant $\rho_{0}$ and $\mu$, such that the GP equation (10) reduces to an equation for the chemical potential $\mu$:

An interesting strategy to avoid the indeterminacy of the system vacuum is to assume that the condensate under consideration starts from a non-interacting regime ($g=0$), say, for $t<0$, and then reaches an interacting regime after some switching time $\tau_{s}>0$, for which $g=g_{0}>0$ becomes constant. Thus, $\mu=V\equiv\mu_{0}$ for $t<0$ and $\mu=V+g_{0}\rho_{0}\equiv V+\Delta\mu$ for $t>\tau_{s}$, where $\Delta\mu=g_{0}\rho_{0}$. During the switching time, we assume that the interactions are linearly turned on, according to

Henceforth we call the periods $t<0$, $0\leq t\leq\tau_{s}$, $\tau_{s}<t$ the non-interacting regime, the switching regime, and the interacting regime, respectively. This profile has the advantage of allowing for the construction of exact solutions.

We call attention to the spatial translation invariance of this condensate realization, which implies that there are no particle fluxes during the condensate evolution, i.e., $J_{0}=J_{\chi}=J_{\zeta}=0$, and the continuity equation reduces to $\partial_{t}(\rho_{\chi}+\rho_{\zeta})=0$. Accordingly, $\rho_{\zeta}=-\rho_{\chi}$, and the condensate density is determined by the quantum depletion alone. We stress that this property does not occur for more complex condensate configurations. We cite [26] for the study of a case where the particle fluxes are relevant.