Soliton Diffusion as a Signature of Hawking Radiation in Bose-Einstein Condensates
Chao Hang Gregory Gabadadze Guoxiang Huang
quant-ph cond-mat.quant-gas






    ^1State Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences,
                         East China Normal University, Shanghai 200062, China
        ^2Department of Physics, New York University, New York, NY 10003, USA
        ^3NYU-ECNU Institute of Physics at NYU Shanghai, Shanghai 200062, China
        
        
        The analog Hawking radiation (HR) emanating from a sonic black hole horizon formed
        in a cigar-shaped Bose-Einstein condensate (BEC) at finite temperature is investigated.
        In particular, we study the effect of HR on a dark (topological) soliton.  We show that, due to thermal fluctuations,
        the dark soliton in the BEC displays a nonlinear Brownian motion resulting in a specific asymmetric diffusion.
        Based on numerical simulations we argue that the analog HR formed in the BEC can be detected indirectly
        through measurement of the dark soliton diffusion.
        
        
        
        03.75.Kk, 03.75.Lm, 04.70Dy  Keywards: Analog Hawking radiation; Solitons, Bose-Einstein condensationSoliton Diffusion in a Bose-Einstein Condensate:  
         A Signature of the Analogue Hawking Radiation
            Chao Hang^1,3,  Gregory Gabadadze^2,3, and Guoxiang Huang^1,3
            December 30, 2023
        =================================================================================================
        
         Introduction.  The goal of this work is to propose  a
        scheme for an indirect detection  of the analog Hawking radiation in a certain
        quantum system, using a topological soliton that the same system supports.
        
        Hawking radiation (HR) is a quantum effect in a
        classical background of a black hole  <cit.>. It has been a subject of a numerous
        studies aimed at better understanding  of the fundamental  principles of quantum physics
        in the presence of a black hole (BH) geometry  (see, e.g., <cit.> and references therein).
        
        In a commonly discussed astrophysical setting – i.e., for a BH in the center of a galaxy –
        the intensity of HR is negligible as compared to that of the in/out-flows
        triggered by accretion of surrounding  matter and radiation onto the BH.
        Hence, detection of HR in such settings  appears to be unlikely.
        
        It is then desirable to look for different setups in which similar quantum radiation,
        in otherwise classical background, might be expected and detected. Such a program had been
        initiated by Unruh  who noticed the equivalence between the equations used to predict HR
        and those governing wave propagation in certain inhomogeneous and moving media <cit.>.
        The radiation of quantum counterparts of such waves, potentially
        accessible in a lab, is referred as the analog HR.
        
        A numerous interesting platforms  have been proposed for the detection of the analogue HR <cit.>. Among them,
        atomic Bose-Einstein condensates (BECs) were suggested as potentially useful  for the
        explorations. Moreover, a direct observation of the analog HR has  recently been claimed in such systems <cit.>.
        
        
        Given the complexity of such experiments and subtleties of the detection <cit.>,
        it is highly desirable to have complementary methods of observing effects of  HR in BEC's.
        For  instance,  indirect detection by observing  influence of HR on object within BEC
        that can be controlled and manipulated would be welcome. In
        this work we propose one such scheme using peculiarities of
        diffusion of a topological (dark) soliton in BEC caused by a thermal radiation, the analog
        HR in our case.
        
        Solitons are fascinating large-amplitude excitations, stabilized by the interplay between nonlinearity and
        dispersion/diffraction. They appear in nonlinear media or in nonlinear relativistic systems.
        Among diverse types of solitons discovered in many branches of physics, matter wave
        solitons in atomic  BECs have been  broadly studied <cit.>.  Their  diffusion
        has also been explored recently in various settings <cit.>.
        
        In this work we use the accumulated knowledge on the analog HR, on  solitons, and on
        field theory techniques describing both of these phenomena,
        to investigate the HR driven diffusion of dark (topological)
        solitons in a cigar-shaped atomic BEC with a repulsive atom-atom interaction.
        
        
        By developing an analytic approach based on effective field theory methods,
        we show that a dark soliton immersed in BEC displays a
        nonlinear temporal variance of its position under
        the influence of thermal fluctuations,
        corresponding to a nonlinear Brownian motion. Thus,
        it exhibits an apparent, asymmetric diffusion during its propagation.
        In particular, we find that the width and depth of the dark soliton change in time t respectively
        as,   t^3/2 T and t^-3/2/T,
        with T being temperature of thermal radiation.
        Using these results, we propose a scheme for an indirect detection of  HR by measuring the diffusion
        of the soliton caused by  the analog
        HR emanating from a sonic black hole horizon in a moving BEC.
        
        
        The proposed scheme  is entirely based on the inherent nonlinearity of  the BEC, and by that, it
        differs  from previous studies. Since the dark soliton can be regarded as a large classical particle,
        its diffusion  should be easier to detected than it is to detect the Hawking quanta directly.
        
        
         0.05in
        
         The Model.– We consider a cigar-shaped atomic BEC with repulsive atom-atom interaction that is
        trapped by the potential V_ ext(𝐫)=M/2[ω_^2(x^2+y^2)+ω_z^2z^2],
        where M is the atomic mass and ω_ (ω_z) is transverse (axial) harmonic oscillator
        frequency, with ω_⊥≫ω_z. At  low energies and momenta the dynamics of such BEC
        can be described by an effective Lagrangian for
        a space-time dependent order parameter of the condensed state, ψ(z,t), that can be referred as
        a wave-function of the condensate.  In the long-wavelength approximation
        both the condensate  and its fluctuations  are captured  by a classical solution
        for ψ(z,t) and its perturbations.  Below we will directly deal with the equation of motion  that follows
        from the effective Lagrangian,  and also incorporates  the terms due to the external harmonic trap.
        Thus, the wave-function ψ(z,t),  satisfies  the Gross-Pitaevskii (GP) equation:
        
            iħ∂ψ/∂ t=(-ħ^2/2M∂^2/∂ z^2+M/2ω_z^2z^2+ħω_+g_1D|ψ|^2)ψ,
        
        where g_1D≡ 2ħω_⊥a_s, with a_s>0 is the s-wave scattering length determined  by
        the quartic repulsive self-interaction term in the effective Lagrangian <cit.>.
        The condensate solution and spectrum of its long-wavelength perturbations ( the Bogliubov spectrum of
        phonons) are well known;  we just emphasize for further use how  that spectrum changes
        when one looks at a background flow of BEC with velocity v_0 along the z direction
        
            ω=v_0 k±[k^2/2M(ħ^2k^2/2M+2g_1Dρ_0)]^1/2,
        
        where “+” (“-”) corresponds to the phonon propagating in the  +z (-z) direction, while
        ρ_0 is the one-dimensional density. The sound phase velocity at low momentum reads (ω/k)_k=0=v_0+c_s,
        where c_s=±√(g_ 1Dρ_0/M) is the sound speed in the BEC frame.
        
        The above description does not capture the effects of finite temperature.  In particular,
        at any nonzero temperature a finite fraction of the atoms will not be in the condensed state and
        will scatter incoherently off the condensed atoms, leading to dissipation in the condensate. Furthermore,
        thermal fluctuations will give rise to randomness. To account for both of these effects
        in a parametric way  one  modifies the GP equation
        into a stochastic GP equation (SGPE) <cit.>
        iħ∂ϕ/∂
        t   =   (1-iγ)(-ħ^2/2M∂^2/∂ z^2+M/2ω_z^2z^2+ħω_.
              .+g_1D|ϕ|^2)ϕ+η(z,t),
        
        where γ≡ iβħΣ^K(z,t)/4 is the dissipation rate (with β=1/(k_BT) and k_B the Boltzmann constant), and ħΣ^K is the Keldysh self-energy due to the incoherent collisions between condensed and noncondensed atoms <cit.>. The last term η parametrizes  the
        thermal fluctuation, obeying the fluctuation-dissipation relation
        
            ⟨η(z,t)η^∗(z',t')⟩=2ħ k_B T γ(z,t)δ(z-z')δ(t-t'),
        
        with ⟨⋯⟩ denoting the averaging over different noise realizations.
        Since the BEC is highly elongated and the region of interest is far away from the edges,
        an approximate space- and time-independent dissipation rate can be used in the analytical treatment,
        i.e., we  adopt the approximation γ≈γ(0)=3Mk_BTa_s^2/(πħ^2) <cit.>.
        
        
        For convenience of later calculations, we rewrite the SGPE (<ref>) into the dimensionless form
         i∂ F/∂τ   =   (1-iγ)(1-μ̃-ik̃_0∂/∂ζ-1/2∂^2 /∂ζ^2+Ω^2/2ζ^2.
              .+g|F|^2)F +Λ, 
        where τ=ω_t, ζ=l_^-1z, g=2a_sρ_0, F=ϕ e^iμ̃τ-ik̃_0ζ+iω̃_0τ/√(ρ_0), Ω=ω_z/ω_⊥≪1, μ̃=μ/(ħω_), and
        Λ=η e^iμ̃τ-ik̃_0ζ+iω̃_0τ/(ħω_), with l_⊥≡√(ħ/(Mω_⊥)) being the transverse harmonic-oscillator length, k̃_0= k_0l_ being the flow wavenumber, and ω̃_0=ω_0/ω_ [ω_0≡ Mv_0^2/(2ħ)] the flow frequency. The correlator of the dimensionless fluctuation fields  reads ⟨Λ^∗(ζ,τ)Λ(ζ',τ')⟩=2 k_B Tγδ(ζ-ζ')δ(τ-τ')/(ħω_^2).
        
        
        If γ, Ω, Λ vanish, then the dimensionless SGPE (<ref>) admits an
        exact dark soliton solution <cit.>
            F=√((μ̃-1)/g)(cosϕ_0tanhZ̃+isinϕ_0),
         with Z̃=√(μ̃-1)cosϕ_0[ζ-(k̃_0+√(μ̃-1)sinϕ_0)τ-ζ_0].
        Here, ζ_0 denotes the initial position of the soliton center and ϕ_0 (|ϕ_0|<π/2) is a phase
        characterizing the blackness (the difference between the minimum soliton intensity and the
        background intensity)  as [(μ̃-1)/g]cos^2ϕ_0, its width  3.3/(√(μ̃-1)cosϕ_0), and  its
        velocity k̃_0+√(μ̃-1)sinϕ_0.
        
        In the next section we will consider small but nonzero γ, Ω,  and Λ,
        and their effects on the soliton.
        
         0.05in
        
         Diffusion of the soliton.– We start by looking  for analytical results on soliton diffusion,
        to get a clear  physical picture and then to  compare with numerical simulations.  Starting with the
        SGPE (<ref>) in a weak nonlinear regime
        a stochastic Korteweg-de Vries (KdV) equation  can be  derived by using the the multi-scale
        method developed in Ref. <cit.>; the result reads as follows:
        
        ∂ u/∂τ+3c̃_s/u_0u∂ u/∂ζ̃-1/8c̃_s∂^3 u/∂ζ̃^3=-1/4c̃_s∂Λ/∂ζ̃-ℛ(u,γ), 
        where ζ̃=ζ-(k̃_0+c̃_s)τ and u=Fexp(-iφ)-u_0 <cit.>. The first term on the right hand side of Eq. (<ref>) results from the thermal fluctuation; the second term, ℛ, comes from the higher-order effects  – e.g., the
        high-order thermal fluctuations, higher-order dispersion and nonlinearity, the trapping potential along the z direction, etc. –
        which are all small and neglected in the analytical treatment,
        but will be included in our numerical simulations.
        
        
        If Λ is set to zero, then Eq. (<ref>) admits an exact dark soliton solution
        u=-A_0 sech^2[√(2c̃_s^2A_0/u_0)(ζ̃+c̃_s
        A_0τ/u_0-ζ_0)], where A_0 is a positive constant characterizing the amplitude of the soliton.
        In the leading-order approximation the dark soliton solution of Eq. (<ref>) can be expressed as
        
            F(ζ,τ)=u_0[1-Ã_0 sech^2(√(2c̃_s^2Ã_0)X)]
            e^iφ,
        
        with Ã_0=A_0/u_0, X=ζ-[k̃_0+c̃_s(1-Ã_0)]τ-ζ_0, and
        φ=-√(2Ã_0)tanh(√(2c̃_s^2Ã_0)X). We should note that:
         (i) only the small-amplitude dark solitons can be described by the KdV equation;
         (ii) dark solitons can be at rest in the lab frame if k̃_0+c̃_s(1-Ã_0)=0,
         which can only be satisfied if there is a background flow (i.e. k̃_0≠ 0).
        
        
        If the thermal fluctuations are weak enough (Λ≪ 1), the first term on the right hand side of Eq. (<ref>) can be treated as a small perturbation. Thus, its effect on the dark soliton can be studied by perturbing the non-perturbative soliton solution  <cit.>; this leads to the  following  perturbed solution
        
        
        u   =   -(A_0+√(2A_0)W_1) sech^2[(√(2c̃_s^2Ã_0)
        +c̃_s/√(u_0)W_1)X.
              .+W_2],
        with W_1=[1/(2√(u_0))]∫_0^τdτ'∫_-∞^∞ ds Λtanh s  sech^2s, and
        W_2=(c̃_s^2Ã_0/u_0)∫_0^τdτ'
        ∫_0^τ'dτ”∫_-∞^∞ ds Λtanh s  sech^2s-[1/(2u_0√(2Ã_0))]∫_0^τdτ'∫_-∞^∞ ds Λ (ssinh s-cosh s)/cosh^3 s. 
        
        
        
        In what follows we'll be  interested in the long-time behavior of the dark soliton under the influence of the thermal fluctuations. To this end, we compute the ensemble average ⟨ u ⟩, and consider the case of τ≫1. Following the method of Ref. <cit.>, and using the  relations ⟨ W_1^2 ⟩ =u_0^2Λ_0σ/(15c̃_s^4), ⟨ W_1W_2 ⟩=2u_0^2A_0Λ_0σ^2/(15c̃_s^4), and ⟨ W_2^2 ⟩=u_0^2Λ_0σ/(15c̃_s^4)[(30+π^2)/(24A_0)
        +4A_0^2σ^2], we obtain the asymptotic solution for τ≫1
        
        ⟨ u ⟩   =   -u_0^5/4/3√(π)c̃_s√(10/3Λ_0τ^3) exp ( -5√(u_0)/3Ã_0Λ_0τ^3X^2 ) ,  
        where Λ_0≡2 k_B T γ/(ħω_⊥^2)∝ T^2. From Eq. (<ref>) we see that for a long-time diffusion, the width of the dark soliton grows as  t^3/2T while the blackness decreases as 1/ (t^3/2T). Even though  the soliton gets deformated because of the diffusion,  the  area enclosed by its shape-function remains  constant.
        
        
        The obtained soliton diffusion is due to its Brownian motion induced by the thermal fluctuation in the BEC. From the solution (<ref>),
        we can easily extract the time dependence of the position of the center of the soliton, ζ_ DS=ζ_0+[k̃_0+c̃_s(1-Ã_0)]τ
        -√(u_0)W_2/[c̃_s(√(2A_0)+W_1)]. From this, we can calculate the variance of the position, D(ζ_ DS)≡⟨ (ζ_ DS-⟨ζ_ DS⟩)^2 ⟩,  which reads
        
         D(ζ_ DS)≈1/√(u_0)(30+π^2/451/16c̃_s^4 Ã_0^2+2/15Ã_0τ^2)Λ_0τ. 
        We see that for small τ, D(ζ_ DS)∝τ, in accordance with the Einstein relation for Brownian motion <cit.>; however for larger τ, D(ζ_ DS)∝τ^3, that is,  the dark soliton displays a nonlinear time-variance of its position, which can be regarded  as a nonlinear Brownian motion. Furthermore, in contrast with the Einstein relation where the diffusion coefficient ∝1/γ, here D(ζ_ DS)∝γ since  the dark soliton has a negative mass <cit.>.
        
        
        Having the theoretical part clarified we adopt realistic parameters based on a recent experiment reported in Ref. <cit.>,
        where a cigar-shaped almost pure ^87Rb BEC of N≈5×10^4 atoms in the ⟨ 5^2S_1/2,F=1,m_F=-1⟩ state was
        prepared, with (ω_,ω_z)=2π×{133,5.9} Hz and the peak density around 5.8×10^13 cm^-3.
        Furthermore, the velocity of the background flow is taken to be v_0≈0.7 mm s^-1, and the temperature of the BEC is assumed to be
        T≈5 nK (which is much lower than the critical temperature of one-dimensional BEC, T_c=Nħω_z/[ ln(2N)k_B]≈1.2μK <cit.>). With the above parameters at hand, one calculates l_≈1μm, c_s≈±0.8 mm s^-1, and γ≈0.22×10^-4.
        
        
        To test the obtained analytical results, we solve the SGPE  numerically by using the spectral method together with the
        fourth-order Runge-Kutta method for time stepping. The numerical evolution of a typical dark soliton is shown in Fig. <ref>(a), with the initial position and phase respectively taken as ζ_0=0 and ϕ_0≈-0.86 rad (the initial blackness and width are approximately 0.6 and 4.3, respectively).
        
        
        Fig. <ref>(b) shows the width (red squares) and the blackness (black circles) of the dark soliton as functions of τ for T≈5 nK, obtained by averaging over 50 runs of independent noise realizations. The respective analytic curves obtained  are also shown by the red dashed and black dashed-dotted lines, respectively. We see that the numerical results agree with the analytic ones for large τ, however, a disagreement occurs for small τ since  the asymptotic solution (<ref>) is valid only in the long-time regime (i.e. τ≫1). Fig. <ref>(c) shows the same quantities as functions of T for τ=500 (t≈600 ms). Since τ≫1, the numerical results agree well with the analytic predictions in the entire range of T.
        
        
        The dependence of D(ζ_ DS) on τ is illustrated in Fig. <ref>(d), where the red triangles are obtained from the numerical
        calculation and the black solid line is  the analytic  result (<ref>). One can see that the numerical and analytic results are matched quite well.
        
        The left lower inset of the figure shows D(ζ_ DS) (black solid line) and its linear part (blue dashed line) in the range of 0≤τ≤5. One sees that they are matched only for a very short time, 0<τ≲2 (0<t≲2.4 ms). Thus, for a long-time behavior, the dark soliton displays a nonlinear Brownian motion instead of a linear one. The dependence of ζ_ DS on T for τ=500 (t≈600 ms) is also given, as shown in the right-upper inset.
        
         0.05in
        
        
         Detection on Analogue HR via soliton diffusion.– We now turn to demonstrate that the measurement of the dark soliton diffusion might be an effective technique to detect the analogue HR in the BEC. To this end, we assume that the background flow in the BEC is generated along the z direction; this can be realized by, say, adiabatically accelerating the trapping potential in the z direction until the BEC reaches a constant velocity v_0 <cit.>.
        The near-horizon geometry of an analogue sonic black hole can be mimicked by a transition region from a subsonic to supersonic background flow [see Fig. <ref>(a)].
        
        
        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 flow is subsonic (supersonic). The analog HR is a thermal radiation of the phonons emanating from the analog horizon. Since the background flow
        is supersonic  in the downstream region, and the  phonons cannot escape from it, that region can be regarded as the interior of an analog black hole.
        
        In practice, to create a horizon of the analogue black hole, we assume there is a stationary, step-like change of the radial confinement of the trapping potential, i.e. ω_⊥=ω_⊥,u for z<0 and ω_⊥=ω_⊥,d for z>0, with ω_⊥,u<ω_⊥,d (hereafter, subscripts “u” and “d” refer to the upstream and downstream, respectively).
        Accordingly, the transverse harmonic-oscillator length in the upstream (downstream) is l_⊥,u≡√(ħ/(Mω_⊥,u)) (l_⊥,d≡√(ħ/(Mω_⊥,d))). The quantities in the upstream and  downstream regions can be related to  one another by the mass-current conservation and the condition of equal chemical potential in the whole BEC <cit.>: ρ_0u v_0u = ρ_0d v_0d, and M/2v_0u^2+ħω_,u+g_ 1D,uρ_0u=M/2v_0d^2+ħω_,d+g_ 1D,dρ_0d, respectively, with g_ 1D,u=2ħω_⊥,ua_s and g_ 1D,d=2ħω_⊥,da_s.
        
        
        
        The sonic black hole requires that v_0u<|c_s,u| and v_0d>|c_s,d|, where the sound
        speeds in the upstream (downstream) is c_s,u=±√(g_ 1D,uρ_0u/M) (c_s,d=±√(g_ 1D,dρ_0d/M)). For 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|), the analogue Hawking temperature
        is determined by the velocity gradient (see, e.g.,   <cit.>)
        
            T_H ≈ħ/2π k_B|∂/∂ z(v-c_s)|≈ħ u/π k_B l_0,
        
        where l_0=ħ/(M|c_s|) is the healing length. The latter is assumed to be the shortest length scale
        in the hydrodynamical limit where the wavelength of the Bogoliubov excitations are much longer than l_0 <cit.>.
        
        
        For the numerical calculations, we take a_s=94.8 a_0, ω_,u=2π×133 Hz (l_,u≈0.94μm), ω_,d=2π×140 Hz (l_,d≈0.91μm), ρ_0u≈1.0×10^6 cm^-1, v_0u=0.25 mm s^-1, and c_s,u≈±0.78 mm s^-1. With the above parameters we get ρ_0d≈2.67×10^5 cm^-1, v_0d≈0.94 mm s^-1, and c_s,d≈±0.41 mm s^-1. Thus, the Mach number in the upstream (downstream) reads: m_u=v_0u/|c_s,u|≈0.3<1 (m_d=v_0d/|c_s,d|≈2.3>1), u≈0.5 mm s^-1, and the Hawking temperature T_H≈1.4 nK.
        
        
        Fig. <ref>(b) and Fig. <ref>(c) show the Bogoliubov spectrum in the upstream and  downstream regions, respectively. We see that long-wavelength excitations in the upstream are able to propagate in both directions, i.e., they'd be in-going and out-going with respect to the horizon. However, they are dragged away by the background flow and are unable to propagate back to the black hole horizon in the downstream, i.e. they can only propagate along the out-going direction with respect to the horizon. The minimum wavelength of the trapped, long-wavelength excitations in the downstream is approximately 5.5 μm.
        
        
        Since the system is essentially  one-dimensional, we assume that a uniform density region will quickly equilibrated at the Hawking temperature T_H, as long as the latter is higher than the ambient temperature of the BEC. Once this happens, a dark soliton will be injected in the upstream region, somewhat  close to the horizon, but in a way for it not to back-react on the analog horizon significantly; this can by done by employing the known phase-imprinting laser field <cit.>, allowing to precisely set the soliton  position, blackness, and velocity.
        
        If T_H is higher than the temperature of all other reservoirs that may exist, the dark soliton will behaves as a nonlinear Brownian particle due to the influence of the HR, as detailed in the previous section. Note that the coolest  BEC realized up to now can  have temperatures as low as
        ∼10 pK <cit.>. In such setups the role of the BEC thermal fluctuations will  indeed be negligible and only the HR  will remain responsible for the soliton diffusion. Furthermore, since the dark soliton is set  with zero initial velocity [when taking Ã_0=k̃_0u/c̃_s,u+1≈0.24 (c̃_s,u<0)] and a small amplitude, it is “slow", and hence sensitive to the influence of the HR.
        
        
        Last but not least, we perform the numerical simulation of the  entire process of detection of the analogue HR. In the simulation, we first let the BEC (with the length ≈500μm and a background flow velocity v_0=0.25 mm s^-1) to undergo  the change in the radial confinement at z=0, enabling
        the emergence of the analogue black hole horizon  [Fig. <ref>(a)].
        
        
        As soon as  the uniform density region reaches the horizon, a dark soliton with zero initial velocity is injected in the upstream (we set this time as t=0) [Fig. <ref>(b)]. Shown in Fig. <ref>(c) and Fig. <ref>(d) are numerical results on the HR-induced-diffusion of the dark soliton at τ≈300 (t≈359 ms) and τ≈600 (t≈718 ms), respectively. From Fig. <ref>(c), we see that the soliton width (blackness) has increased (decreased) by nearly 1.5 times as compared with the initial values.  These changes   are in good agreement with the results given in Fig. <ref>(b), and  should be possible to measure in a realistic experiment (see the third panel in the middle row of Fig. <ref>). In Fig. <ref>(d), an asymmetric soliton diffusion is observed. Specifically, we measure the distance between the soliton center and its left (right) "edge", denoted by δ_1 (δ_2). We find that δ_1/δ_2≈1.3>1 (the last panel in the lower row of Fig. <ref>). Such an asymmetric soliton diffusion might be caused by the inhomogeneous density of the BEC along the longitudinal direction or by an asymmetry in the pressure acting on the soliton due to the HR. Furthermore, we calculate the velocity of the whole BEC (i.e. the velocity of the center-of-mass of the BEC), v_ BEC, at different times,  and  the influence of the HR is also calculated (see the first panel of the lower row in Fig. <ref>). One can see that v_ BEC increases from the value of the background flow velocity in the upstream to that in the downstream. Our simulation confirms  that the dark soliton propagation in BEC is robust, and  the soliton has  the lifetime longer than 718 ms.
        
        
         Conclusion.–  The Hawking radiation is one of the intriguing long studied quantum phenomenon in
        a  classical background field of a black hole. Its detection in any  realistic astrophysical context is  unlikely. The analog Hawking radiation
        is governed by the mathematical equations that are identical to those used in the Hawking  calculation. Therefore,
        it is interesting to study the analog HR in table-top experimental systems where it can be potentially detected directly or indirectly.
        
        In this work the analog HR emanating from a sonic black hole horizon formed
        in a cigar-shaped Bose-Einstein condensate was investigated.
        In particular, we studied the effect of HR on a dark (topological) soliton.  We showed that under the thermal HR
        the dark soliton in the BEC  would display a nonlinear Brownian motion resulting in a specific asymmetric diffusion.
        We performed numerical simulations of this process  and based on the obtained results   argued that the analog HR
        can be detected indirectly through the measurement of the dark soliton diffusion.
        
        Our scheme is complementary to the   previous proposals  (see works <cit.>) that are based
        on the measurements   of the correlation between a pair of Hawking quanta.
        The method proposed and developed here may be extended to study the stochastic dynamics of other
        nonlinear waves <cit.> and other analogue gravity problems <cit.>.
        
        
         This work was supported by the National Natural Science Foundation of China (NSFC) under Grants Nos. 11474099 and  11475063, National Key Research and Development Program of China under Grant No. 2017YFA0304201, Shanghai Program of Shanghai Academic/Technology Research Leader under Grant No. 17XD1401500, and the US National Science Foundation under Grant No. PHY-1620039.
        
        
        
        
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