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Impact of trans-Planckian excitations on black-hole radiation in dipolar condensates

We consider a quasi-one-dimensional dipolar condensate in an analogue black hole setup. It is shown that the existence of a roton minimum in the condensate dispersion relation leaves deep imprints onto the Hawking radiation spectrum. In particular, the emitted radiation can be either more intense or suppressed, depending on the depth of the roton minimum in the excitation spectrum. In addition, we find that spontaneous particle creation occurs even when the horizon is removed. Our results establish that dipolar condensates offer a richer and more versatile environment for the simulation of particle production from the quantum vacuum in the presence of horizon-interfaces than their contact-interaction counterparts.

Black holes are a versatile laboratory to probe particle production from the quantum vacuum in the presence of horizon-interfaces separating distinct regions of spacetime [1][2][3]. A suitably clean and controllable arena to produce analogue black holes are Bose-Einstein condensates [4], in which the first unambiguous detection of spontaneous quantum Hawking radiation has been achieved [5,6]. It has been argued in the years since its inception that Hawking radiation is a rather universal phenomenon, as the thermality of the Hawking spectrum is generally robust against trans-Planckian deformations of the spectrum breaking Lorentz invariance [7][8][9][10][11], even though the emitted quanta at infinity, when traced back to the horizon, experience an infinite blueshift [12]. Black holes as well as their analogues are however conventionally set up in the field-theoretical context of contact interactions. In the following, we demonstrate that admitting nonlocal field theories offer a much richer arena to harness the impact of trans-Planckian excitations on black-hole radiation. Specifically, we show that due to the increased complexity of the scattering problem at the horizon, novel features emerge which clearly distinguish dipolar black holes from their contact counterparts. To this end, we take into account that interactions can be long-range and in particular anisotropic, for which dipole-dipole interactions between atoms or molecules with magnetic or electric dipole moments are the archetype. Due to the roton minimum in their elementary excitation spectrum [13][14][15][16], black holes in dipolar condensates sensitively probe the robustness of Hawking radiation thermality to high frequency dispersion, thereby also thoroughly addressing one of the major original motivations of the whole analogue black hole program [17].
We thus provide below the first simulation of black hole (BH) analogues from dipolar Bose-Einstein condensates. Our system is assumed to be an elongated radially harmonic trapped flowing quasi-one-dimensional (quasi-1D) dipolar condensate, systems routinely realized in ex-periment [18]. We assume that the system is stationary and sufficiently strongly elongated such that the details of how the flow is sustained can be neglected to a first approximation cf., e.g., Refs. [19][20][21][22][23][24][25][26][27].
We recall that three ingredients are necessary to trigger spontaneous particle production of quantum origin in stationary condensates: The existence of negative energy excitations, a mechanism of mode conversion [20,23,28], also cf. Ref. [29], and that the fundamental commutation relations for the field in question are fulfilled [30]. For contact BH analogues, negative energy superluminal excitations reach the event horizon from within the BH, which converts a part of these modes into outgoing spontaneous radiation, thus giving rise to the system's vacuum decay [20,23]. We show in what follows that spontaneous particle production also occurs in dipolar condensates when these three criteria are met.
For quasi-1D dipolar condensates, after integrating out the radial directions transverse to the long x axis, the ratio β = ⊥ /ξ u , where ⊥ = 1/ √ ω ⊥ is the transverse harmonic oscillator length and ξ u is the subsonic healing length outside the analogue BH (setting = m = 1), measures how deep into the quasi-1D regime the system is [15,31]. There is a critical β c ∼ 0.776 for which the system becomes unstable against a proliferation of rotonic excitations in the crossover to three spatial dimensions. In the limit β → 0, indicating a quasi-1D condensate with effective contact interactions, our model system coincides with the one studied in [22], see for further details below. At finite β, for which the anisotropy of the dipolar interaction becomes manifest, we find important differences between contact and dipolar BHs that can be summarized as follows: (i) β > 0 increases the radiation power when the black hole exists; (ii) The rotonic/maxonic dispersion relation typical of dipolar gases leads to a strong nonthermality of the Hawking radiation spectrum in comparison to local black hole analogues; (iii) We find, due to the rotons, radiating systems even when a horizon is absent. These findings demonstrate that dipolar condensates offer a much richer environment to simulate the Hawking phenomenon in comparison to contact condensates. Within the mean-field approximation, the evolution of the condensate order parameter φ is described by the nonlocal Gross-Pitaevskii equation (GPE) [32] where U = U (x) is the trap potential, and g dd > 0 is the quasi-1D dipolar interaction strength. We assume that the dipoles have a common direction relative to the long x axis, which fixes g dd [31,33]. Note that we have separated off the contact part of the dipolar interaction (third term in square brackets) and we assume other contact interaction contributions coming from swave scattering to be negligible. This regime can be achieved by using Feshbach resonances [18]. For computational convenience, we discretize the quasi-1D dipolar kernel G in Eq. (1) [32,33]. Denoting the convolution by where the limit N → ∞, ∆q → 0 is understood, leading to the continuum expression G(x) − δ(x) = −(1/2 ⊥ )f (|x|/ ⊥ ), f (y) = −y + (1 + y 2 ) exp(y 2 /2)Erfc(y/ √ 2) π/2, Erfc being the complementary error function [31]. This form of writing the quasi-1D dipolar kernel enables the construction of essentially analytical solutions: We solve for finite N and ∆q and take the limits afterward. Also, by keeping N and ∆q finite we can produce approximate solutions to the problem to the desired accuracy. For instance, as shown in [33], for N = 10 and ∆q = 1/3.4 the error is below 1%, maintained throughout our simulations.
To convey the essential physics, we construct our stationary dipolar BH analogue as the A subscript on local quantities "u" here and in what follows denotes the upstream region (x < 0), and "d" the downstream region (x > 0) ( Fig. 1 upper panel). This model was recently studied for contact condensates in [22]. We assume zero temperature throughout. Equation (1) then implies continuity fixes the external potential U (cf. Fig. 1 upper panel). We scale lengths in units of ξ u , e.g. x = x[ξ u ], so that the energy and (inverse time unit) becomes 1/ξ 2 u . A dipolar BH is then specified by the set of parameters {m u , m d , β = β u = ⊥ }, where the local Mach number is m = v/c and c = √ g dd ρ is the local sound speed. Then m u < 1 < m d defines an analogue BH. Our goal is to determine how small fluctuations over this BH background lead to spontaneous radiation. The Bogoliubov expansion of the wave function readsΨ = ( √ ρ +ψ) exp(−iµt + ivx), where the bosonic operatorψ models the small quantum fluctuations. In order to study the Hawking radiation in our system, we expand the quantum fieldψ in the unambiguous basis of quasiparticle modes whose vacuum state represents a zero flux of phonons sent towards the event horizon in the laboratory frame [20]. Hence for this vacuum choice any spontaneous radiation is linked to Hawkinglike processes. Our representation for the interaction kernel (2) can be used to find such a basis as follows [32]. We first define the Nambu spinor in particle-hole space, is a complete set of positive norm quasiparticle solutions with respect to the Bogoliubov scalar product: dxρΦ † n σ 3 Φ n = δ n,n . The functions Φ n are solutions of the Bogoliubov-de Gennes equation We find that σ 1 Φ * n is a negative norm solution, and σ i , i = 1, 2, 3 are Pauli matrices, with σ 4 = 1 + σ 1 .
Solutions to Eq. (3) can be found with the time dependence exp(−iωt) for ω > 0. Furthermore, because Eq. (2) is suppressed for |x| 1, ⊥ , all solutions of Eq. (3) far from the horizon are written in terms of the local homogeneous condensate perturbations, i.e., we have superposition of plane waves in the form Φ(t, x) = exp(−iωt + ikx)Φ k for constant Φ k . This gives rise to the dispersion relation Equation (4) enables us to identify a quasiparticle basis whose vacuum state is by definition characterized by no quasiparticles propagating towards the event horizon. This state is, therefore, suitable for studying the spontaneous Hawking radiation in BH analogues. A positive (negative) sign of the dimensionless group velocity V (k) = dω/dk (sign of the slopes in Fig. 1 lower panel) determines whether the plane wave is propagating to the right (left), and each plane wave propagating towards the event horizon gives rise to a distinct quasiparticle mode found by solving the scattering problem within Bogoliubov theory [20,23,33,34].
Let us recollect the salient features of the dispersion relation in the contact BH regime (dotted curves of Fig. 1  bottom). When β = 0 there is only one plane wave going towards the horizon from the upstream region (  . Left: Dispersion for the upstream region. Notice the strong deviation from a contact-interaction-dominated regime β = 0 (red dotted curve), caused by the dipolar interactions. Each plane wave propagating to the right gives rise to a distinct quasiparticle mode, indexed by kin1, kin2, kin3, and kr. Note that the "rotonic" branch kr (dashed curve) has negative energy (negative norm), and that for contact-only BHs, only kin1 exists. Right: Downstream dispersion relation. Each leftwards propagating wave gives rise to a quasiparticle, indexed by pin and the negative energy pH (Hawking branch). Dipolar interactions, in particular, increase the cut-off frequency of the local maximum on the right. bottom left) for each ω, whereas in the downstream region ( Fig. 1 bottom right) for frequencies below the local maximum (Hawking cut-off frequency) there are always two plane waves propagating towards the horizon, with the negative energy quasiparticles indexed by p H from the "Hawking branch" (dashed curve in Fig. 1 bottom right) leading to the spontaneous radiation process [23]. When β > 0, the observed effect inside the black hole is the increase of the Hawking frequency, suggesting a stronger radiation power [23]. Furthermore, novel phenomena are expected to occur outside the black hole. As β continuously increases from zero, initially no qualitative distinction from the contact case occurs (dotted curves in Fig. 1 bottom). However, when the roton minimum emerges (continuous curves in Fig. 1 bottom), the new plane wave branch k in2 approaches the horizon from the upstream region. This was recently shown to have a great impact on the scattering process at interfaces [33], and thus also affects the Hawking radiation as we show in the below. Finally, if β is increased even further (Fig. 1 bottom left, dashed curve), beyond β = β r ( 0.628 for m u = 1/2), a novel negative energy quasiparticle branch indexed by k r emerges from the upstream region, which we call the "rotonic branch" for simplicity. The occurrence of the latter is indicative of a strong departure from a contact-dominated BH. Indeed, whereas the maximal number of quasiparticles at given frequency is 3 in the contact case, for the dipolar BH we find up to 6 quasiparticles, depending on β (cf. Fig. 1 bottom, showing that different mechanisms of mode conversion take place for this dipolar BH. Therefore, the scattering problem in our dipolar case is significantly more intricate than for the contact BH analogues.
We show in the supplement [32] that the normalized quasiparticles for ω > 0 can be written as where we denote the downstream wave vectors by p.
determines whether an incoming mode propagating towards the horizon has positive or negative energy for real k [32]. Furthermore, the quasiparticles in Γ (+) have positive norm, whereas the ones in Γ (−) have negative norm. Accordingly, the full field operator expansion readŝ We adopt the convention that for each index α, botĥ a similar to the contact BH [20]. The Hamilton operator above demonstrates that exciting a quasiparticle mode with index in Γ (−) diminishes the system energy. We can calculate the corresponding radiation spectrum deep into the upstream region (x → −∞) via the following evolution equation for the Hamiltonian densityĤ [32], whereŜ = −(∂ tψ † )(∂ x − iv)ψ/2 + H.c. andĤ is the Hamiltonian density. Notice that only when β = 0 and thus G = 0 (contact-only case) energy is locally conserved in the system, whereas for any finite β there is no local energy conservation. Nevertheless, we find generally that energy is globally conserved [32]: ∂ t H = −S ∞ + S −∞ = 0, H = dx Ĥ is the system energy, and the outgoing flux becomes the following ω-integral The sum in k is performed over all real upstream propagating wave vectors only. The quantity S −∞ is then identified as the power radiated at x → −∞, and F ω is the corresponding power spectrum, containing the negative energy modes from Eq. (7). We depict F ω in Fig. 2 top left panel for several values of β and m d = 1.2, m u = 0.5. For the contact BH analogue β = 0 the spectrum is approximately thermal in the low-frequency regime [23]. We note that as β increases from zero (β = 0.2, 0.3, 0.4 in Fig. 2 top left), the effect of the dipolar interactions is to increase the spectrum cut-off frequency and the radiated power ( Fig. 2 top right). Dipolar black holes therefore represent a more promising scenario to probe Hawking radiation from the point of view of the radiated energy than local condensates. For larger values of β, the emergence of new quasiparticle branches with excitations of low group velocity then leads to a decrease in the radiated power, see Fig. 2 top right and bottom panels. As β approaches the deep roton regime and the rotonic quasiparticle branch (k r ) appears beyond β r , a marked distinction of the dipolar BH when compared to contact BHs is observed, as two (negative energy) mode conversion mechanisms originating in the quasiparticles k r and p H compete in the radiation process. Figure 2 top right and bottom panels show that the rotonic branch can, Top right: Radiation power as function of β. As β increases from zero, S−∞ increases, achieves a local minimum at about β ∼ 0.55 and eventually becomes negative when the system approaches the deep roton minimum (close to βc ∼ 0.776), for which the radiation direction is reversed, indicating the proximity to instability of the quasi-1D dipolar gas, and a strong departure from thermal radiation is observed (bottom left). The local minimum in S−∞ corresponds to the appearance of upstream quasiparticle branches with small group velocities (bottom right). In the bottom part, we also display the correlation between spectral distribution for β = 0.55 and 0.776 (left) and the dispersion relation branch with negative sign in Eq. (4) (right), from which the rotonic kr and the Hawking pH modes originate, cf. Fig. 1, lower panel.
in principle, even suppress the radiation power or revert its direction, which represents an instance of significant impact of trans-Planckian physics. In Fig. 2 top left panel we also depict the radiation spectrum for two non-BH configurations, the dashed and dotted curves corresponding to a flow with m d = 0.9, β = 0.5 and m d = 0.6, β = 0.65, respectively. We recall that spontaneous particle creation can occur if both negative energy excitations exist and a mechanism that converts such negative energy excitations into outgoing radiation is present. In the flowing dipolar condensate, the negative energy rotonic and Hawking-type quasiparticles might be present independently of the value of m d , and can be scattered at the interface at x = 0 owning to outgoing radiation even without a horizon (m d < 1), leading to the (strongly nonthermal) radiation spectrum presented in Fig. 2 top left panel.
In conclusion, we considered here for the first time the possibility of simulating BHs in dipolar Bose-Einstein condensates. Our analysis shows that the presence of dipolar interactions leads to a marked departure from contact condensates, including a novel mechanism of mode conversion of the rotonic branch of quasiparticles at the interface, leading to radiating scenarios even when there is no horizon. The presence of a roton minimum leads to strong nonthermality of the spectrum as in the analogue Unruh effect [35], and to enhanced radiation power, making dipolar condensates especially promising candidates to probe Hawking radiation. Generally, we expect in dipolar gases significant alterations from the predictions of a Lorentz-invariant theory, also and in particular in cosmological scenarios which employ a similar dispersion relation see, e.g., [36][37][38][39]. Finally, our analysis focuses on the existence of quasiparticle radiation mechanisms which characterize a dipolar BH analogue model. Future studies will investigate the effects of "zero modes" on the very dipolar condensate existence [25,40,41], and the radiation and pair-entanglement verification procedure via density-density correlations [27,[42][43][44] We consider a near-quasi-1D dipolar condensate modeled by the order parameter φ solution of the nonlocal GPE where ρ = |φ| 2 is the condensate density. Here, the symbol G * ρ indicates the convolution G * ρ( where the interaction kernel G is defined in terms of its Fourier transform by G( The advantage of writing the interaction kernel as in the equation above comes from the fact that for N ∼ 10 and ∆q ∼ 1/3.4 one already has an excellent approximation to the exact kernel. Also, ⊥ is the characteristic radial size o the condensate. Equation (S1) can also be expressed using the Madelung representation φ = √ ρ exp(iθ) in terms of the system density and phase θ as where v = ∂ x θ. Equation (S3) is just the continuity equation, and Eq. (S4) is a nonlocal, dipolar generalization of the Euler equation. Furthermore, the GPE follows from the Lagrangian Given a solution φ of Eq. (S1), we want to study small perturbations of the form φ → φ + δφ, that correspond to L → L + δL, where δL = dx i 2 δφ * ∂ t δφ − i 2 (∂ t δφ * )δφ − 1 2 |∂ x δφ| 2 − (U + 2g dd ρ − 3g dd G * ρ) |δφ| 2 − g dd 2 (φ 2 δφ * 2 + φ * 2 δφ 2 ) + 3g dd 2 (φ * δφ + φδφ * )G * (φ * δφ + φδφ * ) , in view of the GPE. Thus, by starting from the solution φ = √ ρ exp(−iµt + ivx) to the GPE, and defining our field variable ψ by δφ = exp(−iµt + ivx)ψ, we obtain the Lagrangian L ψ for the field ψ and the Euler-Lagrangian equation known as the Bogoliubov-de Gennes equation. In order to find the solutions of Eq. (S8), it is convenient to work with the Nambu spinor defined by which is seen to satisfy the reflection property σ 1 Ψ * = Ψ, and the BdG equation in the form Here, σ i , i = 1, 2, 3 are the Pauli matrices, and σ 4 = 1 + σ 1 . Solutions of the equation above are such that Ψ, ρ∂ x Ψ are everywhere continuous functions, properties used in our model as boundary conditions at the event horizon. Equation (S10) implies that the quantity is conserved in time, which we use as a scalar product on the space of solutions to (S10).