Understanding superradiant phenomena with synthetic vector potentials in atomic Bose-Einstein condensates

We theoretically investigate superradiance effects in quantum field theories in curved space-times by proposing an analog model based on Bose-Einstein condensates subject to a synthetic vector potential. The breaking of the irrotationality constraint of superfluids allows us to study superradiance in simple planar geometries and obtain intuitive insight into the amplified scattering processes at ergosurfaces. When boundary conditions are modified to allow for reflections, dynamical instabilities are found, similar to the ones of ergoregions in rotating space-times. Their stabilization by horizons in black hole geometries is discussed. All these phenomena are reinterpreted through an exact mapping with the physics of one-dimensional relativistic charged scalar fields in electrostatic potentials. Our study provides a deeper understanding of the basic mechanisms of superradiance: By disentangling the different ingredients at play, it shines light on some misconceptions on the role of dissipation and horizons and on the competition between superradiant scattering and instabilities.

Figure 1

Schematic representation of our setup for the realization of an analog ergosurface in a Cartesian geometry using synthetic vector potential fields.

Figure 2

$\mathbf{k}$-space locus of modes at a given $\omega$ for the Klein-Gordon equation in the slow lower $y<0$ region (solid line) and the fast upper $y>0$ region (dashed lines). The slow region is taken at rest $v_x(y<0)=v_x^s=0$ and the fast one is moving with supersonic speed $v_x(y>0)=v_x^f=3c_s$. The speed of sound $c_s$ is the same on both sides. Positive norm (see Appendix) branches are shown as thin black lines, negative norm branches are shown as thick red lines. For each mode, the arrows indicate the direction of the group velocity ${\mathbf{v}}_g={\mathbf{\nabla }}_{\mathbf{k}}\omega$. The different levels of gray (I–IV) indicate the $k_x$ intervals for which the different scattering processes occur (see text). The vertical white line in the darkest region (I) indicates an amplified superradiant scattering process, with the filled black dot indicating the incident mode, the empty black dot the reflected one, and the red dot the transmitted one.

Figure 3

Schematic representation of the physical mechanism underlying the massive bosonic Klein paradox. The dispersion relations in the asymptotic regions are plotted along with the spatial profile of the electrostatic potential. The condition for the amplification to occur is to have both particle and antiparticle modes available at the same frequency, as indicated by the light gray shading in the plot.

Figure 4

Constant-$k_x$ cuts of the hydrodynamic Klein–Gordon (upper panels) and the Bogoliubov (lower panels) dispersion relations for increasing values of $k_x$ (left to right). Solid lines refer to the slow region at rest with $v_x^s=0$ and dashed lines to the fast region with $v_x^f=2.8c_s$. Black (thinner) lines are the positive-norm branches, red (thicker) lines are the negative-norm ones. In the Klein-Gordon case, if the condition (14) is satisfied, superradiant scattering remains possible at all $k_x$ thanks to the enduring intersection between the positive-norm branch in the slow region and the negative-norm one in the fast region (upper panels). In the Bogoliubov case, this intersection is only present for low or moderate values of $k_x$ (bottom-left and bottom-center panels) and disappears for high enough values of $k_x$ for which superradiant scattering is no longer possible (bottom-right panel).

Figure 5

Cuts of the Bogoliubov dispersion relation at constant $\omega$. Solid (dashed) lines refer to the slow (fast) region at $y<0$ ($y>0$). The speed of sound $c_s$ is the same on both sides. Black (thin) lines are positive-norm modes and red (thicker) ones negative-norm ones. Arrows indicate the direction of the group velocity, that is, outward for the black curves and inward for the red ones. (a) Cuts at different values of $\hslash \omega /M{c}_s^2$ (indicated by the numbers on the curves) in a uniform region with a supersonic speed $v_x^f=2c_s$: Because of the superluminal dispersion, these curves have an oval shape also in the supersonic case, rather than the hyperbolic one of the Klein-Gordon case. (b) Dispersions at $\hslash \omega =M{c}_s^2$ in the two zones for a slow region at rest $v_x^s=0$ and a fast region with a subsonic speed $v_x^f=0.83c_s$. In the darker gray region, one has total reflection while in the lighter gray one negative refraction occurs; the dots and the arrows indicate the modes involved in an example of such process and their group velocities. This process will be addressed in the GPE simulation shown in the lower panels of Fig. 8. (c) Dispersions at $\hslash \omega =1.5M{c}_s^2$ for a slow region at rest $v_x^s=0$ and a fast one with a supersonic speed $v_x^f=\pi c_s$: In the gray region, superradiant scattering is possible. (d) A closeup of the same plot; the dots and the arrows indicate the modes involved in an example of such a process and their group velocities. This specific process will be addressed in the GPE simulations shown in Fig. 6 and in the upper panels of Fig. 8.

Figure 6

Snapshots of the time evolution of a superradiant scattering process as predicted by a numerical simulation of the GPE. The color plot shows the spatial profile of the density modulation with respect to the uniform unperturbed BEC. We impose a perturbation wave packet on top of a uniform BEC in the presence of a vector potential $A_x=-\pi M{c}_s$ in the $y>0$ half plane and a compensating external potential as specified in the text. The interaction constant $g$ is spatially uniform to give a constant speed of sound $c_s$. The initial Gaussian wave packet has $k_x=0.63/\xi$, a central $k_y=0.63/\xi$ [indicated as dot in the dispersion plot of Fig. 5] and is spatially centered around $y=-30\xi$ with a variance ${\sigma }_y=8\xi$. Times are expressed in units of $\mu /\hslash$. The white arrows indicate the group velocity of the wave packets along $y$. The thick black (red) arrows indicate the directions of the wave vector (i.e., of the phase velocity) of the positive (negative) norm wave packets. One can recognize the negative norm wave packet from the opposite directions of the group velocity and of the wave vector along $y$. The simulations have been carried out in an integration box of size $L_x=20\xi$ along $x$ and $L_y=200\xi$ along $y$. Grid spacings $\mathrm{\Delta }x=\mathrm{\Delta }y=0.2\xi$ are taken and a time step $\mathrm{\Delta }t={10}^{-3}\mu /\hslash$. Thanks to the periodic boundary conditions along $x$, visibility of the figure was improved by expanding the $x$ domain by repeating the data multiple times.

Figure 7

Time dependence of the Bogoliubov norm (19) of the wave packets, normalized to the one of the initial ingoing wave packet, for the parameters of the simulation of Fig. 6. One can see that the reflected wave packet is amplified by roughly $70\%$.

Figure 8

Upper panels: Snapshots of the scattering process for the same configuration as in Fig. 6 except for a finite width ${\sigma }_x={\sigma }_y=10\xi$ of the wave packet along $x$. The integration box has the same size $L_y=200\xi$ along $y$ but a wider size $L_x=200\xi$ along $x$. The color plot shows the spatial profile of the density modulation with respect to the uniform unperturbed BEC. The thick black and red arrows indicate the direction of the wave vector (i.e., of the phase velocity) of the positive-norm and negative-norm wave packets, respectively. The dashed lines in the rightmost panels are the trajectories of the center of the wave packets during the scattering and the thin arrows on them indicate the direction of the group velocities. Lower panels: Analogous plots in the case of negative refraction. The vector potential in the upper region is $A_x=-1.26M{c}_s$ and the initial Gaussian wave packet is centered in $k_x=0.3/\xi$ and $k_y=0.58/\xi$ and has variances ${\sigma }_x={\sigma }_y=10\xi$. Notice the different scale used in the bottom-right panel, showing the reduced amplitudes of the reflected and negative-refracted wave packets as compared to the superradiant case shown in the upper-right panel.

Figure 9

Snapshots of the time evolution of a dynamically unstable condensate as predicted by a numerical GPE calculation starting from different initial states. The condensate is confined along $y$ in a box of length $y_{\text{max}}=80\xi$. The interaction constant $g$ is constant, giving a spatially uniform speed of sound $c_s$. In the $0\le y\le L$ region, a transverse vector potential $A_x=-3M{c}_s$ and a compensating external potential $V=-A_x^2/\left(2M\right)$ are applied. Fluctuations are absorbed via an imaginary potential when approaching the lower edge at $y_{\text{max}}$ to mimic an open system geometry along that direction. On the upper row, the initial state features an incident wave packet traveling in the upward direction toward the cavity of length $L=30\xi$; the white arrows indicate the directions along $y$ of the group velocities of the wave packets. On the lower row, the initial state only features a weak white noise and the cavity length is $L=15\xi$.

Figure 10

Spectra of the effective one-dimensional Bogoliubov problem (20) for a condensate confined in a box $0\le y\le y_{\text{max}}$ in the presence of a transverse vector potential of variable intensity $A_x$ restricted to the region $y\in [0,L]$ with $L=2\xi$ and $y_{\text{max}}=30\xi$. The transverse momentum is fixed at $k_x=-0.5/\xi$ and the speed of sound $c_s$ is spatially uniform. Black solid (red dotted) lines indicate the real-valued frequencies of the positive (negative) norm modes; the blue thick lines are the real part of the frequencies of dynamically unstable zero-norm modes. The inset is a zoom of the region where dynamical instability emerges. The lower panel shows the complete calculation, the upper part is the prediction of the hydrodynamic Klein-Gordon approximation as discussed in the Appendix.

Figure 11

$k_x$-dependent spectra of the same effective one-dimensional Bogoliubov problem (20) as studied in Fig. 10. System parameters: $y_{\text{max}}=30\xi$, $L=4\xi$, and $A_x=-3M{c}_s$. The upper panel shows the result of the hydrodynamical Klein-Gordon approximation, the lower panel illustrates the complete Bogoliubov problem. The effects of superluminal dispersion are evident in the different spacing of the modes and in their curvature as a function of $k_x$, which result in the suppression of the instability at high transverse momenta.

Figure 12

Log-scale color plots of the absolute value of the determinant (arbitrary units) of the linear matching problem with open boundary conditions on both sides of the interface for $k_x=0.5/\xi$. The three plots correspond to $A_x^f=0$, $A_x^f=-1.5M{c}_s$ and $A_x^f=-3M{c}_s$. A vanishing value of the determinant at a positive imaginary part of the frequency $\Im \left(\omega \right)$ is a necessary (but not sufficient) condition for instability. Here, zeros of the determinant are present only in the first (trivial) case and only for real values of the frequency, meaning that the single interface configuration is dynamically stable.

Figure 13

Log-scale plots of the absolute value of the determinant (arbitrary units) in the case of a reflecting boundary condition in the faster region at a distance $L$ from the interface. Here $k_x=0.5/\xi$ and $A_x^f=-3M{c}_s$ and the tree plots correspond to $L=5\xi$, $L=10\xi$, and $L=20\xi$. Dynamical instabilities are signalled by the presence of roots in the upper half complex plane $\Im \left(\omega \right)>0$.

Figure 14

(a) Comparison of the cuts at $\hslash \omega /M{c}_1^2=1$ of the Bogoliubov dispersion relations inside (solid lines) and outside (dash-dotted lines) the horizon for $v_x^s=0$, $v_x^f=2.12M{c}_1$, $v_y^s=v_y^f=0.85c_1$, and $c_2=0.3c_1$. Dashed lines are instead the cut of dispersion relation of the corresponding Klein-Gordon problem inside the horizon. (b)–(d) Referring to the configuration of Fig. 15: Comparison of the $\hslash \omega /M{c}_1^2=0.2$ cut of the Bogoliubov dispersion relations in the exterior region (dash-dotted lines) with the one in the ergoregion (b) and with the ones inside a region of supersonic longitudinal flow $v_y^f>c_2$ (c), (d). Parameters: $A_E=-2.12M{c}_1$, $v_y^s=v_y^f=0.85c_1$, $c_2=0.3c_1$. In panel (c), $A_H=0$. In panel (d), $A_H=A_E$. The vertical blue lines in (b)–(d) indicate the value $k_x\xi =0.1$ considered in the following Figs. 15 and 16.

Figure 15

Top row: Scheme of a configuration including both an ergoregion and a horizon. The ergoregion is included by means of a vector potential oriented along $x$. The horizon is created by changing the speed of sound in the third region. Here, the vector potential $A_H$ may or may not be present. Bottom row: Plots of the dispersion relations at fixed transverse momentum $k_x\xi =0.1$ for the same parameters considered in Figs. 14–14, namely, $A_E=2.12M{c}_1$, $v_y=0.85c_1$, $c_2=0.3c_1$, $A_H=A_E$ (dashed) or 0 (solid). The gray region indicates the frequency interval in which superradiant scattering can occur.

Figure 16

Snapshots of the time evolution of the moduli of the fluctuation spinor components given by the reduced one-dimensional Bogoliubov problem at a fixed transverse wave vector $k_x\xi =0.1$ for the configuration sketched in Fig. 15 with $A_H=0$ and ${\ell }_0=5\xi$. The evolution starts from a noisy configuration and absorbing regions are included at the boundary of the integration box to avoid spurious reflections. The black (thinner) and red (thicker) lines show the modulus of the $u,v$ components of the Bogoliubov spinor. Time is measured in units of the external $\mu =m{c}_1^2$.

Figure 17

Space dependence of the minimum frequency of the upper branch of the dispersion relation ${\text{min}}_{+}\left(y\right)$ (black thin lines) and of the maximum of the lower dispersion branch ${\text{max}}_{-}\left(y\right)$ (red thick lines) in two configurations realizing the sketch in Fig. 15. Both panels refer to fixed $k_x\xi =0.1$, $c_2=0.3c_1$, $v_y=0.85c_1$, $A_E=2.12M{c}_1$ and the thickness of the two transition regions are ${\ell }_E={\ell }_H=20\xi$. The upper panel corresponds to the case $A_H=0$, with the gray region indicating the frequency range in which ergoregion instabilities occur. In the inset, the corresponding nondispersive Klein–Gordon case. The lower panel corresponds to $A_H=(2/3)A_E$, for which no ergoregion instabilities occur; the horizontal blue line indicates a generic frequency at which stable superradiant scattering occurs.