Amplifying the Hawking Signal in BECs

Abstract

We consider simple models of Bose-Einstein condensates to study analog pair-creation effects, namely, the Hawking effect from acoustic black holes and the dynamical Casimir effect in rapidly time-dependent backgrounds. We also focus on a proposal by Cornell to amplify the Hawking signal in density-density correlators by reducing the atoms’ interactions shortly before measurements are made.

1. Introduction

Analogue models in condensed matter systems are nowadays an active field of investigation, not only on the theoretical side but, more importantly, also on the experimental one. The underlying idea is to reproduce in a condensed matter context peculiar and interesting quantum effects predicted by Quantum Field Theory in curved space, whose experimental verification in the gravitational context appears at the moment by far out of reach.

Many efforts are devoted to find the most famous of these effects, namely, the thermal emission by black holes predicted by Hawking in 1974 [1]. Among the condensed matter systems under examination, Bose-Einstein condensates appear as the most promising setting to achieve this goal [2–13]. The major problem one has to face experimentally is the correct identification of the signal corresponding to the analogue of Hawking radiation, namely, a thermal emission of phonons as a consequence of a sonic horizon formation, since it can be covered by other competing effects, like large thermal fluctuations.

A major breakthrough to overcome this problem came in 2008, when it was predicted that, as a consequence of Hawking radiation being a genuine pair creation process, a characteristic peak in the density correlation function of the condensate should appear for points situated on opposite sides with respect to the horizon [14]. This is the “smoking gun” of the Hawking effect. Soon after this proposal, Eric Cornell at the first meeting on “experimental Hawking radiation” held in Valencia in 2009 suggested that one can amplify this characteristic signal by reducing the interaction coupling among the atoms of the BEC shortly before measuring the density correlations [15].

Here we review in a simple pedagogical way, using toy models, how the analogous of Hawking radiation occurs in a supersonic flowing BEC and how the corresponding characteristic peak in the correlation function can be amplified according to Cornell’s suggestion. It should be stressed that nowadays correlation functions measurements are becoming the basic experimental tool to investigate Hawking-like radiation in condensed matter systems.

2. BECs: The Gravitational Analogy and Hawking Radiation

A Bose gas in the dilute gas approximation is described by a field operator Ψ with equal-time commutator (see, e.g., [16]) Ψ𝑡,𝑥^→,Ψ^†𝑡,𝑥^→′=𝛿𝑥^3→−𝑥^→′ (1) satisfying the time-dependent Schrödinger equation 𝑖ℏ𝜕_𝑡−ℏΨ=²2𝑚^→∇²+𝑉_ext+𝑔Ψ^†ΨΨ, (2) where 𝑚 is the mass of the atoms, 𝑉_ext is the external potential, and 𝑔 is the nonlinear atom-atom interaction coupling constant. At sufficiently low temperatures a large fraction of the atoms condense into a common ground state which is described, in the mean field approach, by a 𝑐-number field Ψ₀(𝑡,𝑥^→).

To consider linear fluctuations around this classical macroscopic condensate, one writes the bosonic field operator Ψ as Ψ∼Ψ₀1+𝜙, (3) where 𝜙 is a small (quantum) perturbation. Ψ₀ and 𝜙 satisfy, respectively, Gross-Pitaevski 𝑖ℏ𝜕_𝑡Ψ₀=−ℏ²2𝑚^→∇²+𝑉_ext+𝑔𝑛₀Ψ₀, (4) where 𝑛₀=|Ψ₀|² is the number density, and Bogoliubov-de Gennes equations 𝑖ℏ𝜕_𝑡ℏ𝜙=−²2𝑚^→∇²+ℏ²𝑚^→∇Ψ₀Ψ₀^→∇𝜙+𝑚𝑐²𝜙+𝜙^†, (5) with 𝑐=𝑔𝑛₀/𝑚 being the speed of sound.

Contact with the gravitational analogy (see, e.g., [13]) is achieved in the (long wavelength) hydrodynamic approximation, more easily realised by considering the density-phase representation for the Bose operator Ψ=√𝑛𝑒^𝑖𝜃 and the splitting 𝑛=𝑛₀+𝑛₁, 𝜃=𝜃₀+𝜃₁ in which 𝑛₁,𝜃₁ represent the linear (quantum) density and phase fluctuations, respectively. In terms of 𝜙 and 𝜙^† we have 𝑛₁=𝑛₀𝜙+𝜙^†,𝜃₁𝑖=−2𝜙−𝜙^†. (6) Provided that the condensate density 𝑛₀ and velocity ^→𝑣₀=ℏ∇𝜃₀/𝑚 vary on length scales much bigger than the healing length 𝜉=ℏ/𝑚𝑐 (the fundamental length scale of the condensate), the BdG equation reduces to the continuity and Euler equations for 𝑛₁ and 𝜃₁ and these can be combined to give a second-order differential equation for 𝜃₁ which is mathematically equivalent to a Klein-Gordon (KG) equation □𝜃₁=0, (7) where □ is the covariant KG operator from the acoustic metric 𝑑𝑠²=𝑛₀−𝑐𝑚𝑐²−^→𝑣²₀𝑑𝑡²−2^→𝑣₀𝑑𝑡𝑑𝑥^→+𝑑𝑥^→2. (8) For a flow which presents a transition from a subsonic (|𝑣₀|<𝑐) flow to a supersonic one (|^→𝑣₀|>𝑐 in some region) the metric (8) describes an acoustic black hole, with horizon located at the surface where |𝑣₀|=𝑐. The same analysis performed by Hawking in the gravitational case can be repeated step by step, leading to the prediction [17] that acoustic black holes will emit a thermal flux of phonons at the temperature 𝑇_𝐻=𝜅,2𝜋 (9) where 𝜅=(1/2𝑐)(𝑑(𝑐²−^→𝑣²)/𝑑𝑛)∣_hor, with 𝑛 being the normal to the horizon, is the horizon’s surface gravity.

3. The Model

To simplify the mathematics involved in the process, we will consider a 1D configuration (in 1D one should more correctly speak of quasi-condensation [18, 19]) in which 𝑛₀ and 𝑣₀ are constant and where the only nontrivial quantity is the speed of sound 𝑐. As explained in [20], this can be achieved by varying the coupling constant 𝑔 (and therefore 𝑐) and the external potential but keeping the sum 𝑔𝑛₀+𝑉_ext constant. In this way, the plane-wave function Ψ₀=√𝑛₀𝑒^{𝑖𝑘0𝑥−𝑖𝑤0𝑡}, where 𝑣₀=ℏ𝑘₀/𝑚 is the condensate velocity and ℏ𝑤₀=ℏ²𝑘²₀/2𝑚+𝑉_ext+𝑔𝑛₀, where ℏ𝑤₀ is the chemical potential of the gas, is a solution of (4) everywhere. Note that such a stationary configuration is difficult to reach experimentally; nevertheless it gives results similar to those obtained by more realistic configurations [21].

The fluctuation operator 𝜙 is expanded in the usual form in terms of positive and negative norm modes as 𝜙(𝑡,𝑥)=_𝑗𝑎_𝑗𝜙_𝑗(𝑡,𝑥)+𝑎^†_𝑗𝜑^∗_𝑗(𝑡,𝑥), (10) where 𝑎_𝑗 and 𝑎^†_𝑗 are quasi-particle’s annihilation and creation operators. From (5) and its Hermitean conjugate, we see that the modes 𝜙_𝑗(𝑡,𝑥) and 𝜑_𝑗(𝑡,𝑥) satisfy the coupled differential equations 𝑖𝜕_𝑡+𝑣₀𝜕_𝑥+𝜉𝑐2𝜕²_𝑥−𝑐𝜉𝜙_𝑗=𝑐𝜉𝜑_𝑗,−𝑖𝜕_𝑡+𝑣₀𝜕_𝑥+𝜉𝑐2𝜕²_𝑥−𝑐𝜉𝜑_𝑗=𝑐𝜉𝜙_𝑗. (11) The normalizations are fixed, via integration of the equal-time commutator obtained from (1), namely, 𝜙(𝑡,𝑥),𝜙^†𝑡,𝑥^′=1𝑛₀𝛿𝑥−𝑥^′, (12) by 𝑑𝑥𝜙_𝑗𝜙^∗_𝑗′−𝜑^∗_𝑗𝜑_𝑗′=𝛿_𝑗𝑗′𝑛₀. (13)

In order to get simple analytical expressions, in the following we will consider simple models with step-like discontinuities in the speed of sound 𝑐 and impose the appropriate boundary conditions for the modes that are solutions to (11). For more general profiles a numerical analysis can be performed; see, for example, [22–25].

3.1. Acoustic Black Holes and the Hawking Effect

A simple analytical model of an acoustic black hole [26] can be obtained by gluing two semi-infinite stationary and homogeneous 1D condensates, one subsonic (𝑥<0) and the other supersonic (𝑥>0), along a spatial discontinuity at 𝑥=0 (see [27], to which we refer for more detailed explanations throughout this paragraph, and the references therein): 𝑐(𝑥)=𝑐_𝑙𝜃(−𝑥)+𝑐_𝑟𝜃(𝑥). We take 𝑣₀<0; that is, the flow is from right to left, and 𝑐_𝑙<|𝑣₀|<𝑐_𝑟. We denote the modes solutions in each homogeneous region and corresponding to the fields 𝜙 and 𝜑 as 𝜙_𝜔=𝐷(𝜔)𝑒^{−𝑖𝑤𝑡+𝑖𝑘(𝜔)𝑥},𝜑_𝜔=𝐸(𝜔)𝑒^{−𝑖𝑤𝑡+𝑖𝑘(𝜔)𝑥}, (14) so that (11) become 𝜔−𝑣₀𝑘−𝜉𝑐𝑘²2−𝑐𝜉𝐷(𝜔)=𝑐𝜉𝐸(𝜔),−𝜔−𝑣₀𝑘−𝜉𝑐𝑘²2−𝑐𝜉𝐸(𝜔)=𝑐𝜉𝐷(𝜔), (15) while the normalization condition (13) gives |𝐷(𝜔)|²−|𝐸(𝜔)|²=12𝜋𝑛₀𝑑𝑘.𝑑𝑤 (16) The combination of the two equations (15) gives the Bogoliubov dispersion relation for a one-dimensional Bose liquid flowing at constant velocity 𝜔−𝑣₀𝑘²=𝑐²𝑘²+𝜉²𝑘⁴4 (17) containing the positive and negative norm branches 𝑤−𝑣₀𝑘=±𝑐𝑘²+(𝜉²𝑘⁴/4)≡±Ω(𝑘) which, for the subsonic and supersonic regions, are given, respectively, in Figures 1 and 2. Moreover, inserting the relation between 𝐷 and 𝐸 from (15) into (16) we find the mode normalizations 𝐷(𝜔)=𝜔−𝑣𝑘+𝑐𝜉𝑘²/24𝜋𝑛₀𝑐𝜉𝑘²(𝜔−𝑣𝑘)(𝑑𝑘/𝑑𝜔)⁻¹,𝐸(𝜔)=−𝜔−𝑣𝑘−𝑐𝜉𝑘²/24𝜋𝑛₀𝑐𝜉𝑘²(𝜔−𝑣𝑘)(𝑑𝑘/𝑑𝜔)⁻¹, (18) where 𝑘=𝑘(𝜔) are the roots of the quartic equation (17) at fixed 𝜔.

Figure 1 

Dispersion relation in the subsonic region (𝜔 is given in units of the chemical potential and 𝑘 in units of the healing length).

Figure 2 

Dispersion relation in the supersonic region (again, 𝜔 is given in units of the chemical potential and 𝑘 in units of the healing length).

In the subsonic case (17) admits two real and two complex solutions. Regarding the real solutions, Figure 1, we call 𝑘_𝑣(∼𝜔/(𝑣₀−𝑐)+𝑂(𝜉²)) and 𝑘_𝑢(∼𝜔/(𝑣₀+𝑐)+𝑂(𝜉²)) the ones corresponding to, respectively, negative and positive group velocity 𝑣_𝑔=𝑑𝜔/𝑑𝑘 (the other two complex conjugated solutions correspond to, respectively, spatially decaying 𝑘_𝑑 and growing 𝑘_𝑔 modes). In the supersonic case (see Figure 2) we see that, for the most interesting regime (𝜔<𝜔_max∼1/𝜉), there are now four real solutions, corresponding to four propagating modes: 𝑘_𝑣,𝑘_𝑢 (present also in the hydrodynamical limit 𝜉=0) and 𝑘₃,𝑘₄(∼1/𝜉), two of which (𝑘_𝑢 and 𝑘₄) belong to the negative norm branch.

To find modes evolution for all 𝑥 one needs to write down the general solutions for 𝜙 (𝜑) in the left supersonic (𝑙) and the right subsonic (𝑟) regions (we restrict to the most interesting case 𝜔<𝜔_max) 𝜙^𝑙_𝜔=𝑒^{−𝑖𝜔𝑡}×𝐷^𝑙_𝑣𝐴^𝑙_𝑣𝑒^{𝑖𝑘𝑙𝑣𝑥}+𝐷^𝑙_𝑢𝐴^𝑙_𝑢𝑒^{𝑖𝑘𝑙𝑢𝑥}+𝐷^𝑙₃𝐴^𝑙₃𝑒^{𝑖𝑘𝑙3𝑥}+𝐷^𝑙₄𝐴^𝑙₄𝑒^{𝑖𝑘𝑙4𝑥},𝜙^𝑟_𝜔=𝑒^{−𝑖𝜔𝑡}×𝐷^𝑟_𝑣𝐴^𝑟_𝑣𝑒^{𝑖𝑘𝑟𝑣𝑥}+𝐷^𝑟_𝑢𝐴^𝑟_𝑢𝑒^{𝑖𝑘𝑟𝑢𝑥}+𝑑_𝜙𝐴^𝑟_𝑑𝑒^{𝑖𝑘3𝑑𝑥}+𝐺_𝜙𝐴^𝑟_𝑔𝑒^{𝑖𝑘𝑟𝑔𝑥} (19) (the expansions for 𝜑 are the same up to the replacement 𝐷→𝐸) and impose, from (11), the matching conditions 𝜙=0,𝜙^′=0,𝜑=0,𝜑^′=0, (20) where [] indicates the variation across the jump at 𝑥=0, allowing us to write down the relations between left and right amplitudes 𝐴 through a scattering matrix 𝑀_scatt in the form 𝐴^𝑙_𝑣𝐴^𝑙_𝑢𝐴^𝑙₃𝐴^𝑙₄=𝑀_scatt𝐴^𝑟_𝑣𝐴^𝑟_𝑢𝐴^𝑟_𝑑𝐴^𝑟_𝑔. (21) This allows us to construct explicitly the decomposition of the field operator 𝜙 in terms of the “in” and “out” bases. The “in” basis is constructed with 𝜙ⁱⁿ modes propagating from the asymptotic regions (𝑥→±∞) towards the discontinuity (𝑥=0), while the “out” basis is constructed with modes 𝜙^out propagating away from the discontinuity to 𝑥=±∞. Looking at Figures 1 and 2, we see that unit amplitude modes defined on the left moving 𝑘^𝑟_𝑣 and right moving 𝑘^𝑙₃,𝑘^𝑙₄(∼1/𝜉) momenta define the ingoing scattering states, while unit amplitude modes defined on the right moving 𝑘^𝑟_𝑢 and left moving 𝑘^𝑙_𝑣,𝑘^𝑙_𝑢 momenta define the outgoing scattering states. One can then write down the “in” decomposition in terms of the “in” scattering states 𝜙=^𝜔max₀𝑑𝜔𝑎^𝑣,in_𝜔𝜙^𝑣,in_𝜔+𝑎^3,in_𝜔𝜙^3,in_𝜔+𝑎^4,in†_𝜔𝜙^4,in_𝜔+ℎ.𝑐. (22) or, equivalently, on the basis of the “out” scattering ones. Note that since 𝑘^𝑙₄ belongs to the negative norm branch the corresponding “in” mode 𝜙ⁱⁿ_4,𝑙 is multiplied by a creation operator 𝑎^4,in†_𝜔 (the same thing happens, in the “out” decomposition, for 𝜙^out_𝑢,𝑙). Using (21) one can construct the 3×3 S-matrix relating 𝜙ⁱⁿ and 𝜙^out modes 𝜙^𝑣,in_𝜔=𝑆_{𝑣𝑙,𝑣𝑟}𝜙^𝑣,out_𝜔+𝑆_{𝑢𝑟,𝑣𝑟}𝜙^𝑢𝑟,out_𝜔+𝑆_{𝑢𝑙,𝑣𝑙}𝜙^𝑢𝑙,out_𝜔,𝜙^3,in_𝜔=𝑆_{𝑣𝑙,3𝑙}𝜙^𝑣,out_𝜔+𝑆_{𝑢𝑟,3𝑙}𝜙^𝑢𝑟,out_𝜔+𝑆_{𝑢𝑙,3𝑙}𝜙^𝑢𝑙,out_𝜔,𝜙^4,in_𝜔=𝑆_{𝑣𝑙,4𝑙}𝜙^𝑣,out_𝜔+𝑆_{𝑢𝑟,4𝑙}𝜙^𝑢𝑟,out_𝜔+𝑆_{𝑢𝑙,4𝑙}𝜙^𝑢𝑙,out_𝜔, (23) which is not trivial since it mixes positive and negative norm modes. As a consequence, the Bogoliubov transformation between “in” and “out” creation and annihilation operators is also not trivial because it mixes creation and annihilation operators. This has the crucial consequence that the “in” and “out” Hilbert spaces are not unitary related; in particular the corresponding vacua are different; that is, |0,in⟩≠|0,out⟩. The physical consequence is that if we prepare the system in the |0,in⟩ vacuum state, so that there are no incoming phonons at 𝑡=−∞, we will have, at late times, outgoing quanta on both sides of the horizon: the vacuum has spontaneously emitted phonons, mainly in the 𝑘^𝑟_𝑢 channel (Hawking quanta) and 𝑘^𝑙_𝑢 (partners). The analytical calculations show that the number of emitted Hawking quanta [26] 0,in𝑎^{𝑢𝑟,out†}_𝜔𝑎^𝑢𝑟,out_𝜔0,in=𝑆_{𝑢𝑟,4𝑙}²∼1𝜔 (24) and partners 0,in𝑎^{𝑢𝑙,out†}_𝜔𝑎^𝑢𝑙,out_𝜔0,in=𝑆_{𝑢𝑙,4𝑙}²∼1𝜔 (25) follow an approximate (low-frequency) thermal (1/𝑤) spectrum [21], the proportionality factor allowing the identification of a Hawking temperature (∼1/𝜉) in this idealised setting. We can understand the mechanism by which Hawking radiation is emitted by looking, using (6), at the equal-time density-density correlator 𝐺⁽²⁾𝑡;𝑥,𝑥^′≡1𝑛²₀lim_{𝑡→𝑡′}0,in𝑛¹(𝑡,𝑥),𝑛¹𝑡^′,𝑥^′0,in, (26) whose main contribution in the 𝑥<0 and 𝑥^′>0 sector comes from the 𝑢𝑙-𝑢𝑟 term 1𝑛²₀⟨𝑛₁𝑛₁⟩_Hawking∼Re^𝑤max₀𝑑𝑤𝑆_{𝑢𝑙,4𝑙}𝑆^∗_{𝑢𝑟,4𝑙}𝜙^𝑤,out_𝑢,𝑙+𝜑^𝑤,out_𝑢,𝑙×𝜙^𝑤,out∗_𝑢,𝑟+𝜑^𝑤,out∗_𝑢,𝑟∼sin𝜔_max𝑥^′/𝑣₀+𝑐_𝑟−𝑥/𝑣₀+𝑐_𝑙𝑥^′/𝑣₀+𝑐_𝑟−𝑥/𝑣₀+𝑐_𝑙. (27) The existence of the peak at 𝑥^′𝑣₀+𝑐_𝑟=𝑥𝑣₀+𝑐_𝑙 (28) was first pointed out in [14] in the hydrodynamical approximation using QFT in curved space techniques. The physical picture that emerges is that Hawking quanta and partners are continuously created in pairs from the horizon at each time 𝑡, propagate on opposite directions at speeds 𝑣₀+𝑐_𝑙<0 and 𝑣₀+𝑐_𝑟>0, and after time Δ𝑡 are located at 𝑥 and 𝑥^′ related as in (28). The existence of the Hawking peak was nicely confirmed by numerical “ab initio” simulations with more realistic configurations performed in [20]. How correlation measurements can reveal the quantum nature of Hawking radiation is discussed in [28–30].

3.2. Analog Dynamical Casimir Effect

A distinct type of pair-creation takes place in time-dependent backgrounds, with one important example being quantum particle creation in cosmology. We will study the analogue of these phenomena in BECs with another simple model in which the speed of sound has a step-like discontinuity in time at 𝑡=0 separating two “initial” and “final” infinite homogeneous condensates: 𝑐(𝑡)=𝑐_in𝜃(−𝑡)+𝑐_fin𝜃(𝑡); see for more details [31, 32]. The modes solutions in the initial (𝑡<0) and final (𝑡>0) regions are now of the type 𝜙_𝑘=𝐷(𝑘)𝑒^{−𝑖𝑤(𝑘)𝑡+𝑖𝑘𝑥},𝜑_𝑘=𝐸(𝑘)𝑒^{−𝑖𝑤(𝑘)𝑡+𝑖𝑘𝑥}, (29) for which (11) become −(𝜔−𝑣𝑘)+𝑐𝜉𝑘²2+𝑐𝜉𝐷(𝑘)𝑐=−𝜉𝐸(𝑘),(𝜔−𝑣𝑘)+𝑐𝜉𝑘²2+𝑐𝜉𝐸(𝑘)𝑐=−𝜉𝐷(𝑘), (30) while the normalization condition (13) yields |𝐷(𝑘)|²−|𝐸(𝑘)|²=12𝜋𝑛₀ (31) giving 𝐷(𝑘)=𝜔−𝑣𝑘+𝑐𝜉𝑘²/24𝜋𝑛₀𝑐𝜉𝑘²,𝐸(𝑘)|(𝜔−𝑣𝑘)|=−𝜔−𝑣𝑘−𝑐𝜉𝑘²/24𝜋𝑛₀𝑐𝜉𝑘².|(𝜔−𝑣𝑘)| (32) Here, 𝜔=𝜔(𝑘) corresponds to the two real solutions to (17), which is quadratic in 𝜔 at fixed 𝑘. These read 𝜔₊(𝑘)=𝑣𝑘+𝑐²𝑘²+𝑐²𝑘⁴𝜉²4(𝑘),𝜔≡𝑣𝑘−Ω₋(𝑘)=𝑣𝑘−𝑐²𝑘²+𝑐²𝑘⁴𝜉²4≡𝑣𝑘−Ω(𝑘), (33) where 𝜔₊(𝑘) corresponds to the positive norm branch and 𝜔₋(𝑘) corresponds to the negative norm one. To find modes evolution for all 𝑡 one first writes down the general solutions in the initial and final regions for 𝜙 (𝜑) 𝜙^fin(in)_𝑘=𝑒^𝑖𝑘𝑥×𝐷⁺_fin(in)(𝑘)𝐴_fin(in)𝑒^{−𝑖𝜔fin(in)+(𝑘)𝑡}+𝐷⁻_fin(in)(𝑘)𝐵_fin(in)𝑒^{−𝑖𝜔fin(in)−(𝑘)𝑡} (34) (for 𝜑 we have the same expansion with 𝐷 replaced by 𝐸) and impose the matching conditions, from (11), 𝜙=0,𝜑=0, (35) where [] now indicates the variation across the discontinuity at 𝑡=0. They allow the final amplitudes 𝐴^fin to be related to the initial ones 𝐴ⁱⁿ through the matrix 𝑀_Bog: 𝐴_fin𝐵_fin=𝑀_bog𝐴_in𝐵_in, (36) where 𝑀_bog=12Ω_inΩ_outΩ_in+Ω_finΩ_fin−Ω_inΩ_fin−Ω_inΩ_in+Ω_fin. (37) One can easily construct “in” and “fin” decompositions for the field 𝜙 by considering initial (final) unit amplitude positive norm modes (𝐴_in(fin)=1,𝐵_in=0) 𝜙^in(fin)_𝑘 (𝜑^in(fin)_𝑘) modes for all 𝑡 using (36): 𝜙(𝑡,𝑥)^in(fin)=^∞_−∞𝑑𝑘𝑎^in(fin)_𝑘𝜙^in(fin)_𝑘+𝑎^in(out_𝑘^)†𝜑^in(fin_𝑘^)∗. (38) The modes are related through a nontrivial Bogoliubov transformation mixing positive and negative norm modes 𝜙ⁱⁿ_𝑘=𝛼_𝑘𝜙^fin+𝛽_𝑘𝜙^out∗_−𝑘, (39) where 𝛼_𝑘=Ω_in+Ω_fin2Ω_inΩ_fin,𝛽_𝑘=Ω_fin−Ω_in2Ω_inΩ_fin, (40) and, consequently, also the relation between “in” and “fin” annihilation and creation operators will mix annihilation and creation operators, implying again that the two decompositions are inequivalent and in particular the two vacuum states |0,in⟩ and |0,fin⟩ are different. The physical consequence is that the step-like discontinuity at 𝑡=0 will induce particle creation, the features of which can be understood by looking at the time-dependent terms of the one-time density-density correlator which in the hydrodynamical limit reads 𝐺⁽²⁾(𝑡,𝑥,𝑥^′)_dyncas∼12𝑐_fin𝑡−𝑥−𝑥^′2+12𝑐_fin𝑡−𝑥^′−𝑥². (41) At 𝑡=0 and everywhere in space correlated pairs of particles with opposite momentum are created out of the vacuum state, with velocities 𝑣−𝑐_fin (left moving) and 𝑣+𝑐_fin (right moving). At time 𝑡 such particles are separated by a distance 𝑥−𝑥^′=2𝑐_fin𝑡, (42) which is indeed the correlation displayed in (41). This effect was recently observed in [33] by considering homogeneous condensates with trapping potential 𝑉_ext rapidly varying in time, where correlation functions in velocity/momentum space were measured.

3.3. Amplification of the Hawking Signal in Density Correlators

It has been argued by Cornell [15] that a way to amplify the Hawking signal in density-density correlators is to reduce the interactions shortly before measuring the density correlations. Since 𝑐=𝑔𝑛₀/𝑚, reducing 𝑔 means that the speed of sound is also reduced. We will model this situation by matching our idealised acoustic black hole configuration of Section 3.1 with a final infinite homogeneous condensate characterised by a small sound velocity 𝑐_fin (<𝑐_𝑙<𝑐_𝑟); see Figure 3. To study this situation, in which a spatial step-like discontinuity in 𝑐 at 𝑥=0 is combined with a temporal step-like discontinuity at some 𝑡=𝑡₀, we will use the tools introduced in the previous two subsections. We will calculate the density-density correlator in the “in” vacuum by expanding the density operator in the “in” decomposition 𝑛¹(𝑡,𝑥)≃𝑛₀^𝜔max₀𝑑𝑤𝑎^𝑣,in_𝜔𝜙ⁱⁿ_𝑣,𝑟+𝜑^𝑣,in_𝑣,𝑟+𝑎^3,in_𝜔𝜙ⁱⁿ_3,𝑙+𝜑ⁱⁿ_3,𝑙+𝑎^4,in†_𝜔𝜙ⁱⁿ_4,𝑙+𝜑ⁱⁿ_4,𝑙+ℎ.𝑐. (43) to get 1𝑛²₀0,in𝑛₁𝑛₁=0,in_𝑤max0𝑑𝑤𝜙ⁱⁿ_𝑣,𝑟+𝜑^𝑣,in_𝑣,𝑟𝜙^in∗_𝑣,𝑟+𝜑^𝑣,in∗_𝑣,𝑟+𝜙ⁱⁿ_3,𝑙+𝜑ⁱⁿ_3,𝑙𝜙^in∗_3,𝑙+𝜑^in∗_3,𝑙+𝜙^in∗_4,𝑙+𝜑^in∗_4,𝑙𝜙ⁱⁿ_4,𝑙+𝜑ⁱⁿ_4,𝑙. (44) By expressing the “in” modes in terms of the “out” modes and in the absence of the temporal step-like discontinuity (say 𝑡₀→+∞) the Hawking signal is given by 1𝑛²₀0,in𝑛₁𝑛₁0,in_Hawking∼Re^𝑤max₀𝑑𝑤𝑆_{𝑢𝑙,4𝑙}𝑆^∗_{𝑢𝑟,4𝑙}𝜙^𝑤,out_𝑢,𝑙+𝜑^𝑤,out_𝑢,𝑙×𝜙^𝑤,out∗_𝑢,𝑟+𝜑^𝑤,out∗_𝑢,𝑟. (45) In the presence of the temporal step-like discontinuity we need to evolve the relevant modes 𝜙^out_𝑢,𝑙 (𝜑^out_𝑢,𝑙) and 𝜙^out_𝑢,𝑟 (𝜑^out_𝑢,𝑟), at the same value of 𝑤, across the discontinuity at 𝑡=𝑡₀. Going to the 𝑘 basis and considering 𝑘,𝑘^′ small (𝑤∼(𝑣+𝑐_𝑙)𝑘∼(𝑣+𝑐_𝑟)𝑘^′) we have 𝜙^𝑘,out_𝑢,𝑙+𝜑^𝑘,out_𝑢,𝑙∼|𝑘|𝑐_𝑙𝑒^{−𝑖(𝑣+𝑐𝑙)𝑘𝑡+𝑖𝑘𝑥}⟶|𝑘|𝑐_fin𝑒^𝑖𝑘𝑥𝛼𝑒^{−𝑖(𝑣+𝑐fin)𝑘𝑡}+𝛽𝑒^{−𝑖(𝑣−𝑐fin)𝑘𝑡}, (46) where 𝛼=((𝑐_𝑙+𝑐_fin)/2√𝑐_𝑙𝑐_fin)𝑒^{𝑖(𝑐fin−𝑐𝑙)𝑘𝑡0}, 𝛽=((𝑐_fin−𝑐_𝑙)/2√𝑐_𝑙𝑐_fin)𝑒^{−𝑖(𝑐fin+𝑐𝑙)𝑘𝑡0}, and 𝜙^𝑘′,out_𝑢,𝑟+𝜑^𝑘′,out_𝑢,𝑟∼𝑘^′𝑐_𝑟𝑒^{−𝑖(𝑣+𝑐𝑟)𝑘′𝑡+𝑖𝑘′𝑥′}⟶𝑘^′𝑐_fin𝑒^{𝑖𝑘′𝑥′}𝛼^′𝑒^{−𝑖(𝑣+𝑐fin)𝑘′𝑡}+𝛽^′𝑒^{−𝑖(𝑣−𝑐fin)𝑘′𝑡}, (47) with 𝛼^′=((𝑐_𝑟+𝑐_fin)/2√𝑐_𝑟𝑐_fin)𝑒^{𝑖(𝑐fin−𝑐𝑟)𝑘′𝑡0}, 𝛽^′=((𝑐_fin−𝑐_𝑟)/2√𝑐_𝑟𝑐_fin)𝑒^{−𝑖(𝑐fin+𝑐𝑟)𝑘′𝑡0}.

Figure 3 

Spacetime diagram sketch of a spatial step-like discontinuity (acoustic black hole-like) followed by a temporal one leading to a final homogeneous configuration.

It is useful to rewrite (46) and (47) in terms of 𝑤 to compare with the standard result without the temporal step-like discontinuity𝜙^𝑤,out_𝑢,𝑙+𝜑^𝑤,out_𝑢,𝑙∼𝑤𝑐_𝑙𝑒^{−𝑖𝑤𝑡+𝑖(𝑤𝑥/(𝑣+𝑐𝑙))}⟶𝑤𝑐_fin𝑒^{𝑖(𝑤𝑥/(𝑣+𝑐𝑙))}×𝛼𝑒^{−𝑖((𝑣+𝑐fin)/(𝑣+𝑐𝑙))𝑤𝑡}+𝛽𝑒^{−𝑖((𝑣−𝑐fin)/(𝑣+𝑐𝑙))𝑤𝑡},𝜙^𝑤,out_𝑢,𝑟+𝜑^𝑤,out_𝑢,𝑟∼𝑤𝑐_𝑟𝑒^{−𝑖𝑤𝑡+𝑖(𝑤𝑥′/(𝑣+𝑐𝑟))}⟶𝑤𝑐_fin𝑒^{𝑖(𝑤𝑥′/(𝑣+𝑐𝑟))}×𝛼^′𝑒^{−𝑖((𝑣+𝑐fin)/(𝑣+𝑐𝑟))𝑤𝑡}+𝛽^′𝑒^{−𝑖((𝑣−𝑐fin)/(𝑣+𝑐𝑟))𝑤𝑡}. (48) The standard result is, from (45), a stationary peak at 𝑥𝑣+𝑐_𝑙−𝑥^′𝑣+𝑐_𝑟=0 (49) weighted by the 𝑤-independent factor (see (24) and (25)) 𝑤√𝑐_𝑙𝑐_𝑟𝑆_{𝑢𝑙,4𝑙}𝑆^∗_{𝑢𝑟,4𝑙}. (50) The effect of the temporal step-like discontinuity at 𝑡=𝑡₀ (see (48)) is to modify this signal into a main signal located at 𝑥𝑣+𝑐_𝑙−𝑥^′𝑣+𝑐_𝑟=𝑣+𝑐_fin𝑣+𝑐_𝑙−𝑣+𝑐_fin𝑣+𝑐_𝑟𝑡−𝑡₀ (51) with strength 𝑤𝛼𝛼^′𝑐_fin𝑆_{𝑢𝑙,4𝑙}𝑆^∗_{𝑢𝑟,4𝑙} (52) and three smaller signals located at 𝑥/(𝑣+𝑐_𝑙)−𝑥^′/(𝑣+𝑐_𝑟) = ((𝑣+𝑐_fin)/(𝑣+𝑐_𝑙) − (𝑣−𝑐_fin)/(𝑣+𝑐_𝑟))(𝑡−𝑡₀), 𝑥/(𝑣+𝑐_𝑙) − 𝑥^′/(𝑣+𝑐_𝑟) = ((𝑣−𝑐_fin)/(𝑣+𝑐_𝑙) − (𝑣+𝑐_fin)/(𝑣+𝑐_𝑟))(𝑡−𝑡₀), and 𝑥/(𝑣+𝑐_𝑙) − 𝑥^′/(𝑣+𝑐_𝑟) = ((𝑣−𝑐_fin)/(𝑣+𝑐_𝑙) − (𝑣−𝑐_fin)/(𝑣+𝑐_𝑟))(𝑡−𝑡₀) with strengths given by (52) in which 𝛼𝛼′ is substituted, respectively, by 𝛼𝛽^′, 𝛼^′𝛽, and 𝛽𝛽^′.

We see immediately that we loose the stationarity of the Hawking signal (49) and that the main signal is multiplied by √𝜂=𝑐_𝑟𝑐_𝑙𝑐_fin𝛼𝛼^′=𝑐_fin+𝑐_𝑙𝑐_fin+𝑐_𝑟4𝑐²_fin (53) with respect to the standard result. This results indeed in an amplification (i.e., the above term is >1) when 𝑐_fin<𝑐_𝑙,𝑐_𝑟. Being the Hawking peak expected to be of order 5×10⁻³ for realistic experimental settings [20] (where 𝑐_𝑟=2𝑐_𝑙 was considered), we obtain an amplification factor 𝜂=15/4 for 𝑐_𝑟=2𝑐_𝑙=4𝑐_fin.

4. Conclusions

In this paper we have briefly reviewed the analysis of the analog Hawking effect and of the analog dynamical Casimir effect by considering simple analytical models of Bose-Einstein condensates in which the speed of sound has step-like discontinuities. We focussed on the study of the density-density correlators which show, in the former case, the existence of a characteristic stationary Hawking quanta-partner peak located at (28) and, in the latter, that of a time-dependent feature (42). Following a suggestion by Cornell, we combined these two analyses to construct a model in which the atoms’ interactions are rapidly lowered (and so the speed of sound) before the correlations are measured. This results in an amplification of the main Hawking peak, now time-dependent and located at (51), by the factor (53) that could be useful in the experimental search.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank I. Carusotto for useful discussions.