Analogue Stochastic Gravity in Strongly-Interacting Bose-Einstein Condensates
Collective modes propagating in a moving superfluid are known to satisfy wave equations in a curved space time, with a metric determined by the underlying superflow. We use the Keldysh technique in a curved space-time to develop a quantum geometric theory of fluctuations in superfluid hydrodynamics. This theory relies on a “quantized” generalization of the two-fluid description of Landau and Khalatnikov, where the superfluid component is viewed as a quasi-classical field coupled to a normal component – the collective modes/phonons representing a quantum bath. This relates the problem in the hydrodynamic limit to the “quantum friction” problem of Caldeira-Leggett type. By integrating out the phonons, we derive stochastic Langevin equations describing a coupling between the superfluid component and phonons. These equations have the form of Euler equations with additional source terms expressed through a fluctuating stress-energy tensor of phonons. Conceptually, this result is similar to stochastic Einstein equations that arise in the theory of stochastic gravity. We formulate the fluctuation-dissipation theorem in this geometric language and discuss possible physical consequences of this theory.
The idea that a curved space-time is an emergent structure has a long history Sakharov (2000); Visser (2002) and has been discussed in various physical contexts BarcelÃ³ et al. (2011) from classical fluid mechanics Unruh (1981); Stone (2000) and crystals with defects Kleinert (1987) to quantum entanglement. Van Raamsdonk (2010) While in the context of fundamental gravity, the emergent scenario remains speculative at this stage, there has been a number of concrete realizations of various aspects of general relativity in “analogue gravity” models, where a non-trivial curved space-time metric arises naturally in the description of collective modes relative to a background solution of the field equations. BarcelÃ³ et al. (2011, 2004) A prominent example of such analogue theory is a strongly-correlated superfluid, Volovik (2009) where the phonon modes propagating relative to a (generally inhomogeneous and non-stationary) superflow satisfy a wave-equation in an effective curved space-time
where is the phonon field – a small deviation from a “mean-field” configuration, is the determinant and is the matrix inverse of the metric
which is determined by the underlying superflow ( and are the superfluid velocity and the speed of sound, is the density of the fluid including the excitations). Many exciting general-relativistic effects immediately follow from this observation, including the formation of sonic horizons and black hole-type physics,BarcelÃ³ et al. (2004) analogue Hawking radiation, Garay et al. (2000) proposed by Unruh Unruh (1981) and recently reported by Steinhauer to have been observed in cold-atom Bose-Einstein condensates, Steinhauer (2016) and a unifying principle for cosmology and high energy physics discussed extensively by Volovik. Volovik (2009, 2001)
This geometric theory of excitations in a superfluid, that this paper focuses on and develops further, is an alternative formulation of the phenomenological Landau-Khalatnikov two-fluid theory of superfluidity, Landau and Lifshitz (1987); Khalatnikov (2000) which has been originally developed as a macroscopic description of superfluid Helium. As the name suggests, this theory separates the fluid flow into two components – one being the zero entropy, zero viscosity superflow and the other being the entropy-carrying, dissipative normal flow. The two-fluid theory relies on the conservation laws for mass, energy, and momentum in a Galilean-invariant continuum made up of these two components. In addition to being an accurate macroscopic description of superfluid helium, the two-fluid theory can be viewed as the first historical example of a long wavelength hydrodynamic limit of a strongly interacting quantum field theory. The low energy effective field theory paradigm offers a number of powerful techniques to analyze strongly interacting field theories and the hydrodynamic limit of high energy theories (e.g., the AdS/CFT and string theory). Kadanoff and Martin (1963); Herzog (2002); Kovtun et al. (2005); Policastro et al. (2002)
The main idea of this work relies on a conceptual analogy between the quantum generalization of the Landau-Khalatnikov two-fluid description and the Caldeira-Leggett-type theories of “quantum friction”, where a closed system is separated into two components – a quantum “particle” and a bath to which it is coupled. Weiss (2008); Altland and Simons (2006) Integrating out the bath leads to classical equations of motion for the particle, which necessarily feature a friction force and a stochastic Langevin force, connected to each other via a fluctuation-dissipation theorem. For a strongly-correlated BEC, this analogy associates the superfluid order parameter field with the Caldeira-Leggett “particle” and the Bogoliubov excitations with the bath. The question we ask here is what is the corresponding Langevin equations of motion that arise? We develop and use a combination of the aforementioned geometric theory of excitations and Keldysh field-theoretical methods in a curved space time to answer this question. The main result is the following stochastic equations of motion \cref@addtoresetequationparentequation
where is the density of the fluid, is the superfluid flow potential, is the irrotational flow fluid of the superfluid component, is the chemical potential of the fluid, is the matrix inverse of the metric tensor (37), is the space-time volume measure, is the stress-energy tensor operator of the phonons (here is a short-hand for the -space-time variable and the Einstein summation convention is in use) and is a stochastic tensor field, describing its fluctuations around the average . That is, the statistics of the Gaussian noise with zero average is determined by the correlator
where and is the anti-commutator. The averages here are calculated relative to a deterministic background. What these equations actually describe are fluctuations in the superfluid, e.g. they yield statistics of density and velocity fluctuations, which in turn determine a stochastic metric. In this sense, there is a strong similarity to the stochastic Einstein equations discussed n the context of stochastic gravity. Hu and Verdaguer (2003); Martin and Verdaguer (1999a)
While these stochastic Einstein equations are interesting in and by themselves, their derivation presents a technical challenge (a non-trivial generalization of the non-equilibrium Keldysh techniques for a curved space-time is required) Calzetta and Hu (1987) and gives rise to a number of additional interesting results along the way, as discussed below. Our paper is structured as follows:
In Sec. II, we discuss the applicability of the metric description of a superfluid by analyzing the relevant length and energy scales. In analogy with cosmology, the geometric description breaks down at an effective “Planck energy,” where both the linear dispersion of phonons and the hydrodynamic description break down.
In Sec. III, we use the background field formalism to write down the Keldysh quantum field theory of quasiparticles. We emphasize that the Keldysh description is necessary for taking the dynamical fluctuations of the phonon field into account.
In Sec. IV, we derive the analogue Einstein equation that governs the background and the excitations – “matter field.” We establish equivalence of the analogue Einstein equation and the covariant conservation law for the phonons to the two-fluid conservation laws of Landau and Khalatnikov. We prove the equivalence of the two descriptions by reducing the covariant conservation law down to the Noether current of the two-fluid system by using the equations of motion. In Appendix A, we provide the technical details of this derivation.
In Sec. V, we take the analogy between the superfluid system and general relativity further to the domain of stochastic fluctuations. We write the response and dissipation kernels in the covariant language, and give the details of this in Appendix B. In global thermal equilibrium, we discuss the notion of temperature on curved space-time. We prove the fluctuation dissipation relation for a metric with globally time-like Killing vectors, that is for a flow that can brought to a stationary form after a Galilean transformation.
Finally in Sec. VI, we linearize the stochastic analogue Einstein equation and obtain a Langevin-type equation for the stochastic corrections to the background. We show that the symmetries of the flow determine structure of the Langevin equation, by considering the Minkowski case.
Throughout the paper, we will use the Einstein summation convention for the indices, unless otherwise stated. The space-time indices are in small case Greek letters while the space indices are in small case Latin letters. We use the sign convention . In addition, the fluid dynamics equations are written in terms of Cartesian tensors, where the distinction between covariant and contravariant tensors is not important.
Ii The model and energy scales
In this section, we review the energy scales involved in the analysis of an interacting system of bosons and its excitations. The analogue“general relativistic” description is an approximation to the exact theory and its applicability is controlled by our ability, or lack thereof, to neglect a quantum pressure term discussed below. The main conclusion of this section is that the stronger the repulsive interactions between bosons composing the superfluid, the less important the quantum pressure term and correspondingly the wider the domain of applicability of the general-relativistic approximation (in the sense of a range of energies and length-scales where the description applies). We discuss these “Planck” energy and length-scales below.
Our starting point is just the standard Lagrangian of interacting bosons
where is the boson field, is the mass of a boson, and the energy describes an external potential and density-density repulsion between the bosons. Though, at this stage we do not specify the external potential and interaction potential between bosons, for , the saddle point of this Lagrangian, satisfies the Gross-Pitaevskii or non-linear Schrödinger equation:
The first step in deriving the hydrodynamic theory is the Madelung transformation of the boson field, Madelung (1927) which is a change of variables to polar coordinates in each point of space-time:
The new variables are the density, , and the phase , which in effect is a “flow potential” for the superfluid, that gives rise to the irrotational flow velocity field
In terms of these variables, the Lagrangian (4) takes the form
Classically, the long-wavelength description of this system in local thermodynamic equilibrium is fluid dynamics. This description can be extended to the quantum regime, where a macroscopic order exists as in the case of a condensate. This macroscopic order is a non-trivial vacuum that solves the mean field fluid equation Eq. (8), which is just the Gross-Pitaevskii equation Eq. (5) in the Madelung parametrization: \cref@addtoresetequationparentequation
These are the Euler equations for an ideal, zero entropy fluid, with an additional energy per unit mass appears in the right-hand side of Eq. (8b) due to the quantum potential term in Eq. (7). Together with the continuity equation Eq. (8a), the gradient of (8b), when multiplied by the density , produces a momentum balance equation. In this equation, the quantum potential leads to a pressure gradient, hence this potential is called “quantum pressure” in Eq. (7).
The excitations over this non-trivial vacuum state are the Bogoluibov quasiparticles. These quasiparticles can be thought of a quantum field propagating on top of the ground state manifold. When treated semiclassically, the quasiparticles constitute sources to the hydrodynamic equations (8), in the spirit of the two-fluid hydrodynamics. Moreover, the entropy-carrying quasiparticle quantum field acts as a bath on the background system. In addition to momentum and energy flux and stresses, the quantum bath creates a noise to the evolution of the background, leading to a stochastic Langevin component.
We now show, by analyzing the appropriate scales, that when the repulsive interactions between the bosons are strong, the quantum pressure term is suppressed. Such a fluid is the basis of the gravitational analogy, as it simulates the space-time on which the matter field – that is, the phonon field – propagates. Volovik (2009, 2001)
If we momentarily ignore the quantum pressure proportional to , we get the Lagrangian density of a vortex free perfect fluid with flow velocity . The excitations of this system are obtained by linearizing the equations of motion and are the sound waves with speed at equilibrium density. When quantized, the excitations have the linear spectrum . Therefore, the dimensional parameters characterizing the vortex free quantum hydrodynamics are the Planck constant, , the equilibrium density , and the equilibrium speed of sound, . The relevant energy scale of this theory is . Since the energy density is , the characteristic length scale is . Therefore, the mass scale is and the time-scale is .
With the addition of the quantum pressure term, the spectrum of Bogoliubov phonons receives a correction as . Therefore the linear spectrum of phonons breaks down at a length scale of , this is the coherence length of the condensate. At this scale, the phonon energy is of order , which can be dubbed the Lorentz violation energy (i.e., where the phonon spectrum deviates from the linear dispersion). Note that, Lorentz violations do not necessarily occur at exactly the inter atomic length scale and thus is generally distinct from the “Planck” energy scale – that is the energy required to resolve the individual atoms separated by a distance (at such length-scales the hydrodynamic description becomes meaningless). Indeed, the ratio of the coherence length to the inter atomic distance is determined by the strength of the atomic interactions. Defining , and setting , we get . In summary the relationships between different scales can be written in terms of the normalized strength of interactions as:
This ratio determines the relative importance of the quantum pressure term compared to the interaction energy. The corresponding ratio is (below, is a length-scale on which the density changes):
In the weak interaction limit, as in a dilute Bose gas, the coherence length and the quantum mass scale are large compared to microscopic counterparts and . This signals that the system behaves like a macroscopic quantum object, hence the condensate fraction is closed to unity. In this regime, Lorentz violation occurs much before the interatomic scales are reached and the excitations are Bogoluibov quasiparticles with the non-linear spectrum. In the strong interaction regime, as in Helium-II or a strongly-interacting BEC, the condensate fraction is small, and the Lagrangian (7) without the quantum pressure term describes the superfluid, as it produces correct equations for a zero entropy dissipationless superfluid. In this regime, the quantum pressure is negligible down to the scale. This means that the collective excitations are sound waves all the way up to the effective Planck energy. Therefore, geometric theory of analogue gravity for the covariant sound waves, that we will summarize in the next section, applies as long as we stay in the hydrodynamic regime, that is to say at length scales larger than the inter atomic distance.
Iii Background field formalism on the Keldysh contour for Bogoliubov quasiparticles
Here, we outline a procedure to extract a field theory of the excitations starting from Eq. (7). We show that an effective curvature and covariance emerge when the amplitude modes of the excitations are integrated out. Finally, we obtain an effective action for the superfluid system and the phonon bath. Unless noted otherwise, we use the units, where
We employ the closed-time path integral or the Keldysh functional integral Kadanoff et al. (1994); Keldysh (1965); Konstantinov and Perel’ (1961); Schwinger (1961) and the background field formalism Peskin and Schroeder (1995); Abbott (1982) to separate the superfluid background and the quantum field theory of excitations. Note that the procedure, outlined below in Eqs. (11) – (27) is completely general and does not rely on a particular form of the initial action, but we will apply it specifically to the superfluid Lagrangian (7).
Generally, given an initial density matrix and the evolution operator that takes the system from to , the density matrix at any point in time is
Suppose is an observable. In the Schrödinger picture, the expectation value of this operator at time is
In the Keldysh formalism, the time contour is made up of two branches that start at an early time and meet at as shown in Fig 1. Each degree of freedom in the system is defined twice, one on the forward and one on the backward branch of the time contour labeled by respectively. The values of all variables are matched at the final point where the branches meet. The observable can be coupled to the system via the source currents that are added to the Hamiltonian . The forward/backward evolution operators associated with the modified Hamiltonian are denoted as . Then the forward/backward expectation values of the observable can be obtained from the following generating function
by differentiating it with respect to the forward/backward source currents
Note that, if we set from the outset, the forward/backward evolution operators cancel due to unitarity and
This means taking the logarithm of the generating function as in ordinary field theory, is redundant in Keldysh theory.
where , the sum goes over the upper and lower Keldysh contours, the factor is for the upper/lower contour.
The simplest observables are the mean fields, that is the expectation values of the fields. Writing
Now, we can separate the system into a classical background and quantum excitations around it as follows. Suppose that the mean fields are given. Then one can express the sources in terms of the mean fields by constructing the effective action, that is the Legendre transform
Now, the sources can be expressed as \cref@addtoresetequationparentequation
Exponentiating the effective action and using (19) one can eliminate the sources in (16) in favor of the mean fields. If we define the deviations from the mean fields: \cref@addtoresetequationparentequation
we can write the following integral equation for the effective action
The effective action, , can be solved for iteratively and be expressed as a series (we restore the Planck constant below to emphasize the semiclassical nature of the expansion)
the classical action
being the zeroth term:
In this paper we consider only the first order or one-loop correction to the classical action. This correction encapsulates the quantum field of Bogoluibov quasiparticles over the background, that become phonons at long wavelengths (hence we use the subscript , a shorthand for “phonon” corresponding to the first loop correction).
We substitute the lowest-order expression( 25) into ’s on the right hand side of Eq. (22). On the left-hand side, we substitute , where is an unknown. Expanding around and , and matching the terms in the equation, we find the phonon effective action i.e. the leading-order correction in and to . Defining
where the , depends on the path index not only through its arguments but explicitly as
in multi-index notation. So far, the results are completely general. Now we use the explicit Lagrangian (7) and obtain the phonon effective Lagrangian, . After suppressing the time-path index , it is
where we defined the “quantum pressure” operator,
iii.1 Covariant phonon action
The one-loop correction can be computed exactly in the strong interaction limit, as the path integral reduces to a Gaussian integral. As noted in Sec. II, at energy scales below , the operator in (28), that results from quantum pressure, can be neglected compared with the term . This yields the following Lagrangian
where we defined the material derivative
Now, the density fluctuations, , can be integrated out. To do the path integral over , we can think of space-time as divided into cubes with volume and discretize the integral. Note that the integrand is a diagonal matrix over space-time and the path integral reduces to a product of Gaussian integrals. At this point, we shorten the notations for the mean-field parameters , and , writing them as simply , and for the sake of brevity. If we define the density matrix as
integrating out the field produces:
with the measure of the path integral being, again suppressing the signs, \cref@addtoresetequationparentequation
and the following covariant action for phonons
This action can also be obtained through a classical treatment of the Lagrangian Eq. (7). Stone (2000) The material derivative is the measure of the time rate of change of in a frame comoving with the fluid. This means the action in Eq. (34) describes non-dispersive waves with speed in the fluid comoving frame. Galilean invariance of the fluid system requires that the sound wave velocity in the lab frame satisfy . This means the sound rays with velocity with are null rays on the manifold with line element
This is the line element for any analogue gravity system with background Galilean symmetry up to a conformal factor, for which the choice of allows us to write the phonon action of Eq (34) in the following suggestive form Unruh (1981); Stone (2000)
Here, the metric can be read off from the line element Eq. (35) by writing as
here convention is used.
The volume measure factor turns out to be . At static equilibrium the line element Eq. (35) is that of Minkowski space.
The measure in Eq. (33) is written as
We note that this measure is manifestly non-covariant due to the coordinate dependent factor . For a field in curved space-time the covariant measure ought to be . Toms (1987); Hawking (1977). This means, although the phonon action Eq. (34) is covariant, the path integral is not, leading to a quantum anomaly. We will come back to this issue in Sec. IV.3 where we derive the conservation law of the stress tensor.
Before going into obtaining equations of motion from the effective action of the Keldysh field theory, it is worthwhile to list a number of properties that Keldysh theory obeys. We draw Fig. 2 to show the forward/backward fields and sources of the phonon field. We identified Eq. (32) with the closed-time-path partition function of phonons . The first order correction to the classical action is then given as:
As a consequence of unitarity, similar to Eq.(15),
i.e. when the background is the same on the forward and backward directions, the product of backward and forward evolution operators is unitary. Also, from Eq. (32),
These properties are handy while computing the Keldysh correlation functions.
Iv analogue Einstein equations and two-fluid hydrodynamics
In general relativity, the relationship between matter and space-time curvature is governed by the Einstein’s equation. Being covariant under coordinate transformations, the matter field obeys the covariant conservation law. However, this is not a total conservation law, as there is energy momentum exchange between fields and the space-time curvature. There are pseudo-tensor constructions like Einstein pseudotensor or the Landau-Lifshitz pseudotensor that quantify the stress energy of the gravitational field. Horský and Novotný (1969); Landau and Lifshitz (1975) These pseudo-tensors, when added to the stress tensor of matter, become a totally conserved quantity.
In this section, in analogy with general relativity, we will start with the first loop effective action
where the metric is a functional of and according to Eq. (37). The effective action Eq. (43) is in agreement with the one postulated in Ref. [dccclxxx(36)] to compute backreaction corrections to acoustic black holes where the quantum effects, i.e. Hawking radiation, is important. We will first write the stress-energy tensor for the covariant phonons starting from the effective action Eq. (19). Then we will write down an analogue Einstein equation that describes the evolution of metric tensor (37) due to the stress-energy of phonons, which play the part of matter. To complete the analogy, we will derive a total conservation law by using the covariant conservation law and the analogue Einstein’s equation. Moreover, we will show that the conserved quantity is a canonical Noether current and therefore describes the total conservation of momentum and energy in the lab frame. This means two fluid hydrodynamics directly follows from the analogue gravity formalism.
iv.1 Hilbert Stress-Energy operator of the covariant phonon field
The expectation value of the stress-energy operator can be defined by using Schwinger’s variational principle Birrell and Davies (1984) as
Then, the stress-energy operator is defined (symmetrized for convenience) as
This expression is problematic because it contains a product of field operators at the same space-time point and generally leads to divergences. These can be cured through a variety of regularization and renormalization schemes. Birrell and Davies (1984) One of these methods is called point splitting where the field operators are taken on different space time points, and the limit of coincidence is taken after performing derivatives and averages. In semiclassical gravity diverging quantities are renormalized into coupling constants in Einstein’s equation. Birrell and Davies (1984); Hu and Verdaguer (2003)
Physically, the zero point quantum fluctuations that add up to an infinite vacuum energy. In the strongly interacting analoguesystem, the divergent quantities are believed to be already accounted for in the background as a part of the internal energy of the fluid. Volovik (2009) This ensures the stability of the liquid droplets, by renormalizing the equilibrium pressure to zero. Recently, the role of zero point energy in the formation of stable macroscopic droplets in strongly interacting BEC’s was investigated both theoretically Petrov (2015) and experimentally. Chomaz et al. (2016) Therefore, assuming that the vacuum energy is already renormalized into the background energy, we will formally discard the divergent piece of the stress-energy expectation value. Note that, in field theory on flat space-time, the divergence is tacitly discarded through the normal ordering of operators.
Let represent the divergent piece in the expectation value for the stress-energy operator. Throughout the paper we will refer to the renormalized stress-energy as:
where is the time ordering operator. The time ordered correlation function is equivalent to the forward-forward correlation function of the Keldysh theory Kamenev (2011)
iv.2 Semiclassical analogue Einstein equations and the “phonon matter”
Having defined the stress tensor of the matter field (phonons), we now write down the equations of motion for the superfluid and the phonons and argue that it is analogous to the semi-classical Einstein’s equation.
After dropping the quantum pressure, and using Eq. (44), the Euler-Lagrange equations that follow from the effective action Eq. (43) in the limit , , are the fluid equations of motion Eq. (8) with semiclassical source terms \cref@addtoresetequationparentequation
Here is the local chemical potential for the superfluid.
Given the initial density operator of phonons and initial values and boundary conditions for the background fields and , one can compute the metric everywhere by solving Eq. (48) and plugging the solutions into the definition Eq. (37). The source terms on the right hand side must be self-consistent with this solution, as the metric appears both explicitly in (48) and also inside the definition of the stress tensor in Eq. (46). This is because the metric tensor determines how the sound field propagates, classically expressed as Eq. (1). In this respect, the Eqs. (48) resembles Einstein’s equations where the Eulerian left hand sides provide dynamics to the metric tensor and are analogous to the Einstein tensor and the stress-tensor corresponds to the matter whose motion is dictated by the curvature.
iv.3 Canonical versus Covariant Conserved Currents
Here, we show that the covariant conservation law and the analogue Einstein equations Eq. (48) leads to a canonical conservation law for energy-momentum.
Classically, the stress tensor in Eq. (44), obeys the covariant conservation law
owing to the fact that it is derived from the effective action,Birrell and Davies (1984) where denotes the covariant derivative. Writing the definitions of Christoffel symbols Eq. (136) in terms of the metric and using Eq. (137) and Eq. (138), we get:
The partial derivative is streamlined with comma notation whenever convenient, i.e. for any quantity , . The Lagrangian density for phonons can be extracted from Eq. (36) as
Let denote the Lagrangian density of the background after the quantum pressure term is dropped
If we define the total Lagrangian of the background-phonon composite system
and noticing that and both vanish, in Eq. (55) is is precisely the conserved Noether current
due to the space time translation invariance of the overall system represented by the effective action Eq. (43).
We define , the stress energy tensor of the analoguegravitational field as
This expression generates the familiar energ and momentum density in the background. For example the energy density of the background in the laboratory frame follows from Eq. (56) as
Similarly, the laboratory frame momentum density of the background follows from Eq. (56) as
This means the mixed canonical stress tensor of the excitations correspond to energy-momentum corrections due to excitations in the laboratory frame.
However, because of the covariance anomaly of the quantum phonon field that manifests itself in the path integral measure Eq. (38), the covaraint derivative of the expectation value of the quantum stress operator must be equal to an anomalous current, i.e. . Furthermore this current should be Galilean covariant due to the overall Galilean invariance of the system. This is the second type of anomaly in the analogue gravity system, the first being the trace anomaly due to Hawking radiation, which occurs when there is a sonic horizon in the system. Garay et al. (2000) In this paper we assume that no sonic horizon exists and that the quantum pressure term is weak everywhere. We defer the rigorous analysis and computation of the anomalous current to another publication. Since anomalies are necessarily quantum effects, in the regimes we work, they should be washed out by thermal contributions. Then for the expectation
the conservation law
iv.4 Mass-energy-momentum balance and the covariant conservation of stress-energy operator
In this section, we complete the connection of the analogue Einstein equation Eq. (48) and the covariant conservation law Eq. (50) and two-fluid hydrodynamics by identifying the normal and superfluid components in terms of the background fields and the stress tensor.