Soliton Diffusion as a Signature of Hawking Radiation in Bose-Einstein Condensates

We propose a scheme to detect analog Hawking radiation (HR) in an atomic Bose-Einstein condensate (BEC) through measuring the diffusion of a dark soliton. The HR is generated by changing the transverse trapping potential of the BEC to obtain a background flow, which is subsonic in downstream and supersonic in upstream, satisfying the condition of black hole horizon. When the system is in thermal equilibrium at Hawking temperature, a dark soliton is created in the upstream. Due to the influence of the HR, the motion of the dark soliton is similar to a Brownian particle and hence exhibits an apparent diffusion, which can be measured and be taken as a signal of the HR. Since the dark soliton is much"heavier"than Hawking quanta, its diffusion is much easier detectable than the Hawking quanta themselves.

Introduction.-In1974, Hawking showed that black holes (BHs) are not entirely "black" objects, but rather emit thermal radiation [i.e.Hawking radiation (HR)] at a temperature inversely proportional to mass [1,2].This is a genuine quantum effect in a classical gravitational background, and in that respect is similar to the Schwinger pair-creation in a strong electric field.HR is also crucial for a connection between BH and thermodynamics [3], and led to a continuing effort to reconcile fundamental principles of quantum mechanics and gravity (for a recent clear and inspiring review, see [4]).
The intensity of HR, however, is extremely weak.The Hawking temperature, T H ≈ (10 −9 /r H )K, where r H is gravitational radius of BH measured in kilometer, is usually much lower than thermal noise, making the detection of HR hopeless in any astrophysical setting.It is hence desirable to look for different physical setups in which similar radiation might be expected and detected.
In 1981, Unruh demonstrated an equivalence between equations that Hawking used to predict his radiation and those governing the propagation of sound waves in inhomogeneous and moving media, such as fluids [5].Unruh argued that if the HR was real, so had to be the quantum radiation of sound waves in an analog hydrodynamic setup.Moreover, the analog setup, because it is so tangible, could be used to resolve some of the theoretical issues with Hawking's original calculation.Following Unruh's observation, many systems were proposed for experimental detection of analog HR [6], e.g., liquid Helium [7], atomic Bose-Einstein condensates (BECs) [8][9][10][11][12][13][14], polariton BECs [15], degenerate Fermi gases [16], slow-light and nonlinear optical media [17,18], surface water waves, etc. [19] (for a review, see [20]).Recent works [21,22] reported the experimental observation of the analog HR in systems of atomic BEC.
In distinction from previous works, here we propose a scheme for detecting HR by using the inherent nonlinearity in an atomic BEC.The essence of the scheme is as follows.Assume the system is first cooled below 10 −10 K, so that thermal fluctuations in the BEC are very small.Then an analog HR is generated by increasing the transverse trapping potential in the downstream of the BEC, with a background flow (BF), allowing the occurrence of an analog BH horizon.After the system equilibrates at Hawking temperature (∼ 10 −9 K), a dark soliton (DS) is generated in the upstream.By the influence of HR, the DS displays a Brownian motion and hence exhibits a diffusive movement.In particular, the DS's width (blackness) asymptotically changes with the law t 3/2 T a H (t −3/2 /T a H ), where t and T a H are respectively time and analog Hawking temperature.This diffusion effect is measured and is regarded as a signal of the HR.Since the DS is a heavy particle comparing with Hawking quanta, its diffusion is much easier to be detected than to detect the Hawking quanta directly.
Linear excitations on BF.-We consider a cigar-shaped BEC formed by a cold alkali (e.g. 87Rb) atomic gas, trapped by the potential , where M is atomic mass and ω ⊥ (ω z ) is transverse (longitudinal) harmonic oscillator frequency, with ω ⊥ ≫ ω z .The dynamics of such quasi one-dimensional (1D) BEC at zero temperature can be described by the wavefunction ψ satisfying the Gross-Pitaevskii (GP) equation [23] , where a s is s-wave scattering length [24].
We assume that there is a BF in the BEC along the z direction, which can be generated by a broad, penetrable  Step 1: the analog HR is produced by increasing the transverse trapping potential in the downstream to satisfy the condition of the BH horizon; Step 2: after the system equilibrates at Hawking temperature, a DS is created in the upstream; Step 3: due to the impact of the HR the DS behaves like a Brownian particle and exhibits an apparent diffusion, which can be measured and regarded as a signal of the analog HR.
barrier moving through the BEC.The moving barrier may be created by, e.g., a laser beam swept through the BEC, resulting in a dipole potential [25].Taking ψ = ρe iσ , the GP equation converts into two equations for ρ and σ, which admit the solution ρ = ρ 0 (constant) and Linear excitations (phonons) on the BF can be found by assuming ρ = ρ 0 + h(z, t) and σ = −(µ/ + ω 0 )t + k 0 z + ς(z, t).Linearizing the equations for h(z, t) and ς(z, t) and taking (h, ς) = ( h, ς)e i(kz−ωt) , one obtains the Bogoliubov spectrum of the excitations , where "+" ("−") corresponds to the excitation propagating to +z (−z) direction relative to the BF velocity v 0 .The sound speed of the Bogoliubov excitations on the BF reads (ω/k) k=0 = v 0 + c s , where c s = ± G 2M ρ 0 is the sound speed for v 0 = 0.
Analog HR in the BEC.-A near-horizon geometry of an analog BH can be mimicked by a transition from a subsonic to a supersonic BF in the BEC.If the central position of the horizon is at z = 0, the upstream (downstream) of the BEC corresponds to the region z < 0 (z > 0), where the BF is subsonic (supersonic).HR is a thermal-equilibrium radiation of phonons emanated from the horizon by Landau instability, where phonons move to the upstream and the downstream, separately [Fig.1(a)].Due to the supersonic feature of the BF in the downstream, where phonons cannot escape from, this region can be taken to mimic the interior region of a BH.
To realize the BH horizon, we assume there is a steplike spatial modulation of the trapping potential, i.e. ω ⊥ is changed into ω ⊥,u for z < 0 and into ω ⊥,d for z > 0, with ω ⊥,u < ω ⊥,d (subscripts "u" and "d" stand for upstream and downstream, respectively).Accordingly, the transverse harmonic-oscillator length 1/2 in the downstream.Mass-current conservation and identical chemical potential in both the upstream and the downstream yield the relations . Thus if ρ 0u and v 0u are given, ρ 0d and v 0d can be obtained by By this the analog Hawking temperature is given by T a H ≈ u/(πk B l 0 ), where l 0 = /(M |c s |) is healing length and k B is the Boltzmann constant.
As a realistic example, a set of system parameters can be chosen from a 87 Rb BEC as N ≈ 200, a s = 94.8 a 0 (a 0 is Bohr radius), ω ⊥,u = 2π × 100 Hz, ω ⊥,d = 2π × 150 Hz, and , which is about 16 nK in our system.Hence the Hawking temperature T a H is much lower than the BEC critical temperature in the system.We see that long wavelength modes [27] in the upstream (subsonic region) are able to propagate in both directions, i.e., in-going and out-going directions with respect to the horizon.However, long wavelength modes in the downstream (supersonic region) are dragged away by the BF and are unable to propagate back to the BH horizon.Thus, long-wavelength modes in the downstream can have only one propagation direction, i.e., the outgoing direction with respect to the horizon.
DS and its diffusion by the HR.-We assume that, after the phonons (emitted from the horizon) has equilibrated at the Hawking temperature T a H , a DS is generated in the upstream near the horizon, which can be realized by employing a phase-imprinting laser field, allowing to set the position and blackness of the DS [28].When T a H is higher than the temperature of other reservoir that may exist [29], due to the influence of the HR, the motion of the DS is similar to a Brown particle and will display an apparent diffusion effect.
The dynamics of the BEC at finite temperature can be described by a stochastic GP equation (SGPE) [30][31][32][33][34], which because of the strong transverse confinement can be reduced into the 1D SGPE [23] is the dissipation rate resulting from the coupling with the thermal reservoir (here the HR), and K (z, t) is Keldysh self-energy due to the collisions between condensed and noncondensed atoms [35].Since the BEC is highly elongated and the DS is far from the edge of the BEC with zero initial velocity (see below), an approximate dissipation rate independent of spatial and temporal coordinates can be used as γ ≈ γ(0) = 3MkB T a H π as 2 [34].The random fluctuation η(z, t) obeys the Gaussian correlation η(z, t)η * (z ′ , t ′ ) = 2 k B T a H γ(z, t)δ(z − z ′ )δ(t − t ′ ), satisfying the fluctuation-dissipation relation.
For investigating the diffusion of the DS, we convert the SGPE into the dimensionless form Since the drift induced by the diffusion from the HR is small, the DS with zero initial velocity will not be affected by the edge of the BEC and the horizon if the DS has some distance from them, and hence one can easily detect the diffusion behavior of the DS due to the influence of the HR.Fig. 2(a) shows the evolution of a typical DS in the upstream by numerically solving the 1D SGPE.The initial position and phase of the DS are chosen as z 0 /l ⊥,u = −10 and φ = 0.68, and hence the initial velocity of the DS is zero.We obtain γ ≈ 0.22 × 10 −4 , corresponding to T a H ≈ 5 nK.The result obtained is averaged from 100 runs for independent noise realizations.
A convenient quantity relevant to experimental measurement is the visibility of the soliton, defined by V = (n max − n min )/(n max + n min ), with n max (n min ) the maximum (minimum) density of the BEC.Shown in Fig. 2(b) is V as a function of ω ⊥,u t obtained by the simulation of Eq. (1).We find that when t < 500/ω ⊥,u ≈ 0.8 s the DS's visibility decreases when t increases.However, when t > 0.8 s, the DS's visibility ceases to decrease.In fact it becomes meaningless as the signal of the DS is lost within the stochastic background.The inset panel shows the dependence of the DS's visibility on T a H for t ≈ 0.8 s.We see that V = 0 at T a H = 0 and it is increased when T a H increases.The vertical lines indicate the standard deviation from the mean value for 100 runs.
We are interested in the DS with a smaller amplitude.The reasons are: (1) a small-amplitude DS is "lighter" than a large-amplitude one, and hence more sensitive to the influence of the HR; (2) for the small-amplitude DS, analytical results on the soliton diffusion can be obtained, which not only can give transparent physical conclusions but also can be compared with numerical results.To this aim, we derived a stochastic KdV equation from the SGPE using the method described in Ref. [36], where u = |F |−u 0 , with u 0 = ρ 0 / √ n 0 , ζ = ζ −( k0 +c s )τ , and cs = ± √ gu 0 .The quantity Π(u, γ) in Eq. ( 2) comes from higher-order contributions (damping, higher-order dispersion, and higher-order nonlinearity, etc.), which are very small and thus can be neglected in the analytical calculation but will be considered in the numerical simulation below [23].
Equation ( 2) for η = 0 admits the DS solu- , where A 0 is a positive constant characterizing the blackness of the DS.Thus, exact to first-order approximation, the DS solution of Eq. ( 1) reads The influence of the HR [i.e. the existence of nonzero fluctuation (η = 0)] will result in a diffusion of the DS.By employing the method described in Refs.[37,38], for large τ we obtain the analytical expression of the soliton diffusion [23] where Λ 0 = 2k B T a H γ/( ω 2 ⊥ ) ∝ T a 2 H , u denotes the ensemble average on u.We see that, under the action of the HR, the DS's width obeys the power law t 3/2 T a H and its blackness (or depth) obeys the power law t −3/2 /T a H , which means that the width (blackness) of the DS is increased (decreased) when the evolution time and the Hawking temperature increase.Fig. 2(c) shows the width (red squares) and the blackness (black circles) of the DS as functions of ω ⊥,u t for T a H ≈ 5 nK, obtained by solving Eq. ( 1) numerically.The result predicted by Eq. ( 3) is also illustrated, which are denoted by the red dashed lines and by the black dashed-dotted lines, respectively.We see that the numerical result from Eq. ( 1) agrees with the analytical result from Eq. ( 2) for large ω ⊥,u t.However, a large disagreement between the numerical and analytical results is observed for small ω ⊥,u t because the stochastic KdV description (2) is valid only for far-field behavior (i.e. for t ≫ ω −1 ⊥,u ).Fig. 2  Based on the Eq. ( 2), the variance of the DS's center position, ζ DS , defined by D(ζ DS ) ≡ (ζ DS − ζ DS ) 2 , is also derived and reads [23] We see that for smaller τ , D(ζ DS ) ∝ τ , in accordance with the Einstein relation for Brownian motion [39]; for larger τ , D(ζ DS ) ∝ τ 3 , i.e. the DS, a specific wavepacket formed by the balance between the nonlinearity and dispersion in the BEC, displays a nonlinear-in-time variance in its position due to the influence of the HR.Furthermore, in contrast to the Einstein relation where the diffusion coefficient ∝ 1/γ, here D(ζ DS ) ∝ γ because the DS has a negative mass [40].Fig. 3(a) shows D(ζ DS ) as a function of ω ⊥,u t obtained by a numerical simulation (red triangles) and from the analytical prediction (4) (black solid line).We see that they are matched quite well.Since the DS suffers no significant distortion during its propagation, it is much easier for extracting the information of the HR than using other kinds of wavepackets.To demonstrate this, we repeat the simulation of the SGPE (1) with initial wavepacket as a Gaussian one, with the result shown in Fig. 3(b).We see that the Gaussian wavepacket is broadened and decays significantly in a short distance and suffers a pairing instability, i.e., it splits into two pulses propagating into opposite directions.
Conclusion.-We have proposed a scheme for detecting the analog HR in a quasi-1D atomic BEC by measuring the DS diffusion, which is completely different from the proposal in Refs.[11][12][13]22] by measuring the correlation between the Hawking quanta on the both sides of BH horizon.Since the amplitude of the DS generated in the BEC is much larger than that of the Hawking quanta, the diffusion of the DS due to Brownian motion is much easier to be detected than to detect the Hawking quanta directly.Because the lowest temperature of

3 FIG. 1 :
FIG. 1: (color online) (a): The setup scheme for detecting the HR, with notations defined in the text.(b) and (c): Bogoliubov spectrum of the excitations excited on the BF in the upstream (a) and downstream (b) of the BEC, with the BF satisfying v0u/|cs,u| ≈ 0.8 and v 0d /|c s,d | ≈ 1.2, respectively.White and shaded domains are for the long wavelength (|k| < 1/l0) modes and shorter wavelength (|k| > 1/l0) modes, respectively.Red dashed-dotted lines denote the slope (sound speed) of long-wavelength phonon excitation near k = 0. "in" and "out" denote in-going and out-going directions with respect to the BH horizon, respectively.(d): Essential steps of the detection of the analog HR in the BEC.Step 1: the analog HR is produced by increasing the transverse trapping potential in the downstream to satisfy the condition of the BH horizon; Step 2: after the system equilibrates at Hawking temperature, a DS is created in the upstream; Step 3: due to the impact of the HR the DS behaves like a Brownian particle and exhibits an apparent diffusion, which can be measured and regarded as a signal of the analog HR.
respectively.The creation of an analog BH requires v 0u < c s,u and v 0d > c s,d , where the sound speeds in the upstream and the downstream are c s,u = ± G 2M ρ 0u and c s,d = ± G 2M ρ 0d , respectively.We consider a simple case with v 0u = |c s,u |−u and v 0d = |c s,d |+u (0 < u ≪ |c s,u |, |c s,d |).
with ζ 0 the initial position of the DS center and φ (|φ| < π/2) the DS's phase angle, characterizing the blackness of the DS (i.e. the depth of the DS relative to the background intensity) [(μ−1)/g] sin φ 2 and the DS's velocity k0 − √ μ − 1 cos φ.Particularly, one can generate a DS having zero initial velocity due to the existence of the BF by setting cos φ = v 0u /c s,u (c s,u < 0).

FIG. 2 :
FIG.2:(color online) (a): |ψ| 2 /n0 as a function of z/l ⊥,u and ω ⊥,u t, obtained for γ ≈ 0.22 × 10 −4 (T a H ≈ 5 nK) by numerically solving the 1D SGPE.Inset: Intensity of the DS at t = 0 (black dashed line) and t ≈ 0.8 s (red solid line).(b): DS's visibility obtained from the simulation of Eq. (1).Shaded region denotes t > 500/ω ⊥,u ≈ 0.8 s, where the visibility becomes meaningless.Inset: Dependence of DS's visibility on T a H for t ≈ 0.8 s.Vertical lines indicate the standard deviation from the mean value for 100 runs.(c) and (d): DS's width (red squares) and blackness (black circles) as functions of ω ⊥,u t for T a H ≈ 5 nK (c) and as functions of T a H for t ≈ 0.8 s (d).The red dashed (black dashed-dotted) lines stand for the asymptotic behavior of the DS's width (blackness) predicted by Eq. (3).

FIG. 3 :
FIG. 3: (color online) (a): The variance of DS's center position ζDS, D(ζDS), as a function of ω ⊥,u t obtained from the numerical simulation (red triangles) and from the analytical prediction (4) (black solid line).(b): Paring instability of Gaussian pulse.Inset: Intensity of the Gaussian pulse at t = 0 (black dashed line) and t ≈ 0.8 s (red solid line).
(d)shows the same quantities as functions of T a H for t ≈ 0.8 s.Since ω ⊥,u t ≫ 1, the numerical result agrees with the analytical one for all T a H .