We consider the quantum dynamics of Bose-Einstein condensates at absolute zero, and demonstrate that an analogue gravity model going beyond the standard linearized analogue gravity paradigm à la Unruh must take into account the backreaction of quasiparticle excitations onto the condensate background. This requires that one expands to second order in perturbation amplitude and thus takes the intrinsic nonlinearity of the theory into account. It is shown that, as a result, significant modifications of the standard paradigm occur. In particular, to obtain a fully Lorentz-covariant equation in curved spacetime for second-order perturbations, we demonstrate that it is necessary to introduce, to leading order in powers of the formal mean-field expansion parameter $N^{-1/2}$ (where $N$ is total particle number), a quantum-fluctuation-renormalized spacetime metric which substantially differs from the Unruh acoustic spacetime metric and, to subleading order $1/N$, two emergent vector fields and a mass term. Both the renormalized metric as well as the vector fields and the mass then keep track of the backreaction of the quasiparticles onto the condensate up to the order in powers of $N^{-1/2}$ considered. Finally, we apply our formalism to an analogue-cosmological Friedmann-Lemaître-Robertson-Walker metric and establish its renormalized form due to the quantum many-body backreaction exerted by the excitation cloud.
We formulate particle creation phenomena in Bose--Einstein condensates in terms of conserving gapless mean field theory for weakly interacting Bose gases. The particle creation spectrum is calculated by rediagonalizing the Bogoliubov--de Gennes (BdG) Hamiltonian in mean field theory. The conservation implies that quasiparticle creation is accompanied by quantum backreaction to the condensates. Particle creation in this mean field theory is found to be equivalent to that in quantum field theory (QFT) in curved spacetime. An expression is obtained for an effective metric affected by quantum back…
Sound Wave in the Backreaction Affected Spacetime in Analogue Gravity Based on Number-Conserving Approach
Natural Science Research Institute, Korea University（韩国大学自然科学研究所）；Division of Semiconductor Physics, Korea University（韩国大学半导体物理系）；Department of Applied Physics, Graduate School, Korea University（韩国大学研究生院应用物理系）；Center for Extreme Quantum Matter and Functionality, Sungkyunkwan University（成均馆大学极端量子物质与功能中心）
We study the quantum fluctuations in a one dimensional Bose-Einstein condensate realizing an analogous acoustic black hole. The taking into account of evanescent channels and of zero modes makes it possible to accurately reproduce recent experimental measurements of the density correlation function. We discuss the determination of Hawking temperature and show that in our model the analogous radiation presents some significant departure from thermality.
Analogue gravity is based on the simple observation that perturbations propagating in several physical systems can be described by a quantum field theory in a curved spacetime. While phenomena like Hawking radiation are hardly detectable in astrophysical black holes, these effects may be experimentally tested in analogue systems. In this Thesis, focusing on Bose-Einstein condensates, we present our recent results about analogue models of gravity from three main perspectives: as laboratory tests of quantum field theory in curved spacetime, for the techniques that they provide to address various…
We have created an analogue of a black hole in a Bose-Einstein condensate. In this sonic black hole, sound waves, rather than light waves, cannot escape the event horizon. A step-like potential accelerates the flow of the condensate to velocities which cross and exceed the speed of sound by an order of magnitude. The Landau critical velocity is therefore surpassed. The point where the flow velocity equals the speed of sound is the sonic event horizon. The effective gravity is determined from the profiles of the velocity and speed of sound. A simulation finds negative energy excitations, by mea…
We theoretically propose a finite-size quasi-one-dimensional Bose-Einstein condensate with coherent source and drain placed at its two ends, which can in principle sustain a stationary sonic black hole with a single event horizon. Our analysis is focused on the condensate persistence against quantum fluctuations. We show that similar to white hole-black hole pairs, dynamical instabilities occur. Investigating in detail the instabilities' dependence on the system parameters, we also identify windows of formally infinite black hole lifetimes. By using quantum depletion of the condensate as a dia…
Inspired by a recent experiment by Steinhauer and co-workers, we present a simple model which describes the formation of an acoustic black hole in a Bose-Einstein condensate, allowing an analytical computation of the evolution in time of the corresponding density-density correlator. We show the emergence of analog Hawking radiation out of a "quantum atmosphere" region significantly displaced from the horizon. This is quantitatively studied both at $T=0$ and even in the presence of an initial temperature T, as is always the case experimentally.
We investigate the structure of quantum correlations in an expanding Bose Einstein Condensate (BEC) through the analogue gravity framework. We consider both a 3+1 isotropically expanding BEC as well as the experimentally relevant case of an elongated, effectively 1+1 dimensional, expanding condensate. In this case we include the effects of inhomogeneities in the condensate, a feature rarely included in the analogue gravity literature. In both cases we link the BEC expansion to a simple model for an expanding spacetime and then study the correlation structure numerically and analytically (in su…
We determine the feasibility of detecting analogue Hawking radiation in a Bose-Einstein condensate in the presence of atom loss induced heating. We find that phonons created by three-body losses overshadow those due to analogue Hawking radiation. To overcome this problem, three-body losses may have to be suppressed, for example as proposed by Search et al. [Phys. Rev. Lett. 92 140401 (2004)]. The reduction of losses to a few percent of their normal rate is typically sufficient to suppress the creation of loss phonons on the time scale of a fast analogue Hawking phonon detection.
In this paper we propose to apply the analogy between gravity and condensed matter physics to relativistic Bose-Einstein condensates (RBECs), i.e. condensates composed by relativistic constituents. While such systems are not yet a subject of experimental realization, they do provide us with a very rich analogue model of gravity, characterized by several novel features with respect to their non-relativistic counterpart. Relativistic condensates exhibit two (rather than one) quasi-particle excitations, a massless and a massive one, the latter disappearing in the non-relativistic limit. We show t…
Using the coherent state functional integral expression of the partition function, we show that the sine-Gordon model on an analogue curved spacetime arises as the effective quantum field theory for phase fluctuations of a weakly imperfect Bose gas on an incompressible background superfluid flow when these fluctuations are restricted to a subspace of the single-particle Hilbert space. We consider bipartitions of the single-particle Hilbert space relevant to experiments on ultracold bosonic atomic or molecular gases, including, e.g., restriction to high- or low-energy sectors of the dynamics an…
The sonic analogue of Hawking radiation can now be experimentally recreated in Bose-Einstein Condensates that contain an acoustic black hole. In these experiments the signal strength and analogue Hawking temperature increase for denser condensates, which however also suffer increased atom losses from inelastic collisions. To determine how these affect analogue Hawking radiation, we numerically simulate creation of the latter in a Bose-Einstein Condensate in the presence of atomic losses. In particular we explore modifications of density-density correlations through which the radiation has been…
We construct quantum field theory in an analogue curved spacetime in Bose-Einstein condensates based on the Bogoliubov-de Gennes equations, by exactly relating quantum particles in curved spacetime with Bogoliubov quasiparticle excitations in Bose-Einstein condensates. Here, we derive a simple formula relating the two, which can be used to calculate the particle creation spectrum by solving the time-dependent Bogoliubov-de Gennes equations. Using our formulation, we numerically investigate particle creation in an analogue expanding Universe which can be expressed as Bogoliubov quasiparticles i…
The Unruh effect predicts a thermal response for an accelerated detector moving through the vacuum. Here we propose an interferometric scheme to observe an analogue of the circular Unruh effect using a localized laser coupled to a Bose-Einstein condensate (BEC). Quantum fluctuations in the condensate are governed by an effective relativistic field theory, and as demonstrated, the coupled laser field acts as an effective Unruh-DeWitt detector thereof. The effective speed of light is lowered by 12 orders of magnitude to the sound velocity in the BEC. For detectors traveling close to the sound sp…
Analogue spacetimes are powerful models for probing the fundamental physical aspects of geometry - while one is most typically interested in ultimately reproducing the pseudo-Riemannian geometries of interest in general relativity and cosmology, analogue models can also provide useful physical probes of more general geometries such as pseudo-Finsler spacetimes. In this chapter we shall see how a 2-component Bose-Einstein condensate can be used to model a specific class of pseudo-Finsler geometries, and after suitable tuning of parameters, both bi-metric pseudo-Riemannian geometries and standar…
Analogue models of gravity have been motivated by the possibility of investigating phenomena not readily accessible in their cosmological counterparts. In this paper, we investigate the analogue of cosmological particle creation in a Friedmann-Robertson-Walker universe by numerically simulating a Bose-Einstein condensate with a time-dependent scattering length. In particular, we focus on a two-dimensional homogeneous condensate using the classical field method via the truncated Wigner approximation. We show that for various forms of the scaling function the particle production is consistent wi…
We investigate quantum entanglement in an analogue black hole realized in the flow of a Bose-Einstein condensate. The system is described by a three-mode Gaussian state and we construct the corresponding covariance matrix at zero and finite temperature. We study associated bipartite and tripartite entanglement measures and discuss their experimental observation. We identify a simple optical setup equivalent to the analogue Bose-Einstein black hole which suggests a new way of determining the Hawking temperature and grey-body factor of the system.
The current article explores the problem of a Bose-Einstein condensate (BEC) trapped in an anisotropic three-dimensional harmonic-oscillator potential, with positive effective interatomic interactions, and the analogue gravity acoustic metric emerging from it. We find that the latter gives rise to a family of conformally equivalent metrics and the space-time interval corresponds to that of an accelerated observer subject to a position-dependent acceleration. Furthermore, such acceleration can be deduced differentiating the potential of the trap, which leads us to infer that, for a Bose-Einstei…
Many-body gravitating quantum systems with Bose-Einstein condensates and dipolar analogue
Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (INO-CNR)（意大利国家光学研究所，意大利国家研究委员会）；Physics Department, University of Naples Federico II（那不勒斯费德里科二世大学物理系）；European Laboratory for Nonlinear Spectroscopy (LENS)（欧洲非线性光谱学实验室）；Physics Department, University of Florence（佛罗伦萨大学物理系）
The recent detections of gravitational waves (GWs) by the LIGO and Virgo collaborations have opened the field of GW astronomy, intensifying interest in GWs and other possible detectors sensitive in different frequency ranges. Although strong GW producing events are rare and currently unpredictable, GWs can in principle be simulated in analogue systems at will in the lab. Simulation of GWs in a manifestly quantum system would allow for the study of the interaction of quantum phenomena with GWs. Such predicted interaction is exploited in a recently proposed Bose-Einstein condensate (BEC) based G…
We show here a general approach to include the quantum potential term in the emergent gravity model of Bose-Einstein condensate by using multiple scales. Our main result shows the emergence of a massive scalar modulating field at larger length scales as a result of Lorentz symmetry breaking at the length scales comparable to the healing length. We also propose that, the nonlocal interactions induced tuning of healing length can be exploited experimentally to observe the systematics of small and large scale coupling as emerges in our present analysis.
We present an analogue spacetime model that reproduces the salient features of the most common ansatz for quantum gravity phenomenology. We do this by investigating a system of two coupled Bose-Einstein condensates. This system can be tuned to have two "phonon" modes (one massive, one massless) which share the same limiting speed in the hydrodynamic approximation [Phys. Rev. D72 (2005) 044020, gr-qc/0506029; cond-mat/0409639]. The system nevertheless possesses (possibly non-universal) Lorentz violating terms at very high energies where "quantum pressure" becomes important. We investigate the p…
For almost a century, the cosmological constant has been a mysterious object, in relation to both its origin and its very small value. By using a Bose-Einstein condensate analogue model for gravitational dynamics, we address here the cosmological constant issue from an analogue gravity standpoint. Starting from the fundamental equations describing a system of condensed bosons, we highlight the presence of a vacuum source term for the analogue gravitational field, playing the role of a cosmological constant. In this simple system it is possible to compute from scratch the value of this constant…
We study the properties of a $2+1$ dimensional Sonic black hole (SBH) that can be realised, in a quasi-two-dimensional two-component spin-orbit coupled Bose-Einstein condensate (BEC). The corresponding equation for phase fluctuations in the total density mode that describes phonon field in the hydrodynamic approximation is described by a scalar field equation in $2+1$ dimension whose space-time metric is significantly different from that of the SBH realised from a single component BEC that was studied experimentally, and, theoretically meticulously in literature. Given the breakdown of the irr…
We discuss the backreaction force exerted by quantum fluctuations in dilute Bose-Einstein condensates onto the motion of the classical background, derived by an ab initio approach from microscopic physics. It is shown that the effective-action method, widely employed in semiclassical quantum gravity, fails to give the full backreaction force. The failure of the effective-action method is traced back, inter alia, to the problem of the correct choice of the fundamental variables and the related operator ordering issues.
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 h…
We investigate the formation dynamics of sonic horizons in a Bose gas confined in a (quasi) one-dimensional trap. This system is one of the most promising realizations of the analogue gravity paradigm and has already been successfully studied experimentally. Taking advantage of the exact solution of the one-dimensional, hard-core, Bose model (Tonks-Girardeau gas) we show that, by switching on a step potential, either a sonic (black-hole-like) horizon or a black/white hole pair may form, according to the initial velocity of the fluid. Our simulations never suggest the formation of an isolated w…
We investigate the properties of the condensates of cold atoms at zero temperature in the tunable binary Bose-Einstein condensate system with a Rabi transition between atomic hyperfine states. We use this system to examine the effect of quantum fluctuations in a tunable quantum gas on phonon propagation. We show that the system can be represented by a coupled two-field model of a gapless phonon and a gapped mode, which are analogous to the Goldstone and Higgs particles in particle physics. We then further trace out the gapped modes to give an effective purely phononic theory using closed-time-…
Analogue models for gravity intend to provide a framework where matter and gravity, as well as their intertwined dynamics, emerge from degrees of freedom that have a priori nothing to do with what we call gravity or matter. Bose Einstein condensates (BEC) are a natural example of analogue model since one can identify matter propagating on a (pseudo-Riemannian) metric with collective excitations above the condensate of atoms. However, until now, a description of the "analogue gravitational dynamics" for such model was missing. We show here that in a BEC system with massive quasi-particles, the …
Bose-Einstein condensates (BEC) have recently been the subject of considerable study as possible analogue models of general relativity. In particular it was shown that the propagation of phase perturbations in a BEC can, under certain conditions, closely mimic the dynamics of scalar quantum fields in curved spacetimes. In two previous articles [gr-qc/0110036, gr-qc/0305061] we noted that a varying scattering length in the BEC corresponds to a varying speed of light in the ``effective metric''. Recent experiments have indeed achieved a controlled tuning of the scattering length in Rubidium 85. …
We investigate the possibility to generate quantum-correlated quasi-particles utilizing analogue gravity systems. The quantumness of these correlations is a key aspect of analogue gravity effects and their presence allows for a clear separation between classical and quantum analogue gravity effects. However, experiments in analogue systems, such as Bose-Einstein condensates, and shallow water waves, are always conducted at non-ideal conditions, in particular, one is dealing with dispersive media at nonzero temperatures. We analyze the influence of the initial temperature on the entanglement ge…
Arising out of a Non-local non-relativistic BEC, we present an Analogue gravity model upto $\mathcal{O}(ξ^{2})$ accuracy in the presence of the quantum potential term for a canonical acoustic BH in $(3+1)$-d spacetime where the series solution of the free minimally coupled KG equation for the large length scale massive scalar modes is derived. We systematically address the issues of the presence of the quantum potential term being the root cause of a UV-IR coupling between short wavelength `primary' modes which are supposedly Hawking radiated through the sonic horizon and the large wavelength …
The dynamics of sound in a fluid is intrinsically nonlinear. We derive the consequences of this fact for the analogue gravitational field experienced by sound waves, by first describing generally how the nonlinearity of the equation for phase fluctuations back-reacts onto the definition of the background providing the effective space-time metric. Subsequently, we use the the analytical tool of Riemann invariants in one-dimensional motion to derive source terms of the effective gravitational field stemming from nonlinearity. Finally, we show that the consequences of nonlinearity we derive can b…
In this diplom-arbeit I consider a specific class of "analogue models" of curved spacetime that are specifically based on the use of Bose-Einstein condensates. As is usual in "analogue models", we are primarily interested in the kinematics of fields and quanta immersed in a curved-space background. We are not directly concerned with the Einstein equations of general relativity. Over the last few years numerous papers concerning "analogue models" have been published, the key result being that in many dynamical systems the perturbations have equations of motion that are governed by an "effective…
Analogue models of gravity have played a pivotal role in the past years by providing a test bench for many open issues in quantum field theory in curved spacetime such as the robustness of Hawking radiation and cosmological particle production. More recently, the same models have offered a valuable framework within which current ideas about the emergence of spacetime and its dynamics could be discussed via convenient toy models. In this context, we study here an analogue gravity system based on a relativistic Bose--Einstein condensate. We show that in a suitable limit this system provides not …
General types of Bose-Einstein condensates are considered. The formation of black-hole analogues is examined for both short- and long-range interactions for arbitrary spatial dimensions greater than two. The former case includes non-linear derivative terms plus an inevitable external potential, while the latter one consists solely of gravity-like self-interactions for which intrinsic formation of analogue Schwarzschild-type background space-times is possible. The corresponding geometries are studied, and it is shown how they can be made stable. Their Hawking temperature is estimated, and it is…
The quasiparticle content of a quantum field depends on the observer, in particular on its motional state, on the way the observer's detector couples to the quantum field, and on the frequency standard in which the detector carried by the observer measures the quanta to be detected. I review a procedure of making this fundamental property of quantum field theory experimentally manifest using quantum-optical means in Bose-Einstein condensates.
We study the back-reaction associated with Hawking evaporation of an acoustic canonical analogue black hole in a Bose-Einstein condensate. We show that the emission of Hawking radiation induces a local back-reaction on the condensate, perturbing it in the near-horizon region, and a global back-reaction in the density distribution of the atoms. We discuss how these results produce useful insights into the process of black hole evaporation and its compatibility with a unitary evolution.
A Bose-Hubbard model on a dynamical lattice was introduced in previous work as a spin system analogue of emergent geometry and gravity. Graphs with regions of high connectivity in the lattice were identified as candidate analogues of spacetime geometries that contain trapped surfaces. We carry out a detailed study of these systems and show explicitly that the highly connected subgraphs trap matter. We do this by solving the model in the limit of no back-reaction of the matter on the lattice, and for states with certain symmetries that are natural for our problem. We find that in this case the …
Artificial black holes, such as sonic holes in Bose-Einstein condensates, may give insights into the role of the physics at the event horizon beyond the Planck scale. We show that sonic white holes give rise to a discrete spectrum of instabilities that is insensitive to the analogue of trans-Planckian physics for Bose-Einstein condensates.
In 1981 Unruh proposed that fluid mechanical experiments could be used to probe key aspects of the quantum phenomenology of black holes. In particular, he claimed that an analogue to Hawking radiation could be created within a fluid mechanical `dumb hole', with the event horizon replaced by a sonic horizon. Since then an entire sub-field of `analogue gravity' has been created. In 2016 Steinhauer reported the experimental observation of quantum Hawking radiation and its entanglement in a Bose-Einstein condensate analogue black hole. What can we learn from such analogue experiments? In particula…
We observe spontaneous Hawking radiation, stimulated by quantum vacuum fluctuations, emanating from an analogue black hole in an atomic Bose-Einstein condensate. Correlations are observed between the Hawking particles outside the black hole and the partner particles inside. These correlations indicate an approximately thermal distribution of Hawking radiation. We find that the high energy pairs are entangled, while the low energy pairs are not, within the reasonable assumption that excitations with different frequencies are not correlated. The entanglement verifies the quantum nature of the Ha…
Recent developments in simulating fundamental quantum field theoretical effects in the kinematical context of analogue gravity are reviewed. Specifically, it is argued that a curved space-time generalization of the Unruh-Davies effect -- the Gibbons-Hawking effect in the de Sitter space-time of inflationary cosmological models -- can be implemented and verified in an ultracold gas of bosonic atoms.
We show that Schwarzschild black hole solutions in asymptotically Anti-de Sitter (AdS) and de Sitter (dS) spaces may, up to a conformal factor, be reproduced in the framework of analogue gravity. The aforementioned derivation is performed using relativistic and non-relativistic Bose-Einstein condensates. In addition, we demonstrate that the (2+1) planar AdS black hole can be mapped into the non-relativistic acoustic metric. Given that AdS black holes are extensively employed in the gauge/gravity duality, we then comment on the possibility to study the AdS/CFT correspondence and gravity/fluid d…
Generalizing the one-component case, we demonstrate that the propagation of sound waves in two-component Bose-Einstein condensates can also be described in terms of effective sonic geometries under appropriate conditions. In comparison with the one-component case, the two-component setup offers more flexibility and several advantages. In view of these advantages, we propose an experiment in which the evolution of the inflaton field, and thereby the generation of density quantum fluctuations in the very early stages of our universe during inflation, can be simulated, realizing a {\em quantum si…
We investigate gravitational analogue models to describe slowly rotating objects (e.g., dark-matter halos, or boson stars) in terms of Bose-Einstein condensates, trapped in their own gravitational potentials. We begin with a modified Gross-Pitaevskii equation, and show that the resulting background equations of motion are stable, as long as the rotational component is treated as a small perturbation. The dynamics of the fluctuations of the velocity potential are effectively governed by the Klein-Gordon equation of a "Eulerian metric," where we derive the latter by the use of a relativistic Lag…
