Spontaneous quantum superradiant emission in atomic Bose-Einstein condensates subject to a synthetic vector potential

We theoretically investigate the spontaneous quantum emission of phonon pairs by superradiant processes in an atomic Bose-Einstein condensate subject to a synthetic vector potential. Within the analog gravity perspective, this effect corresponds to the spontaneous emission of radiation from the ergosurface of rotating black holes. A general input-output formalism is built and used to characterize the spectral and correlation properties of the emission. Experimentally accessible signatures of the emission are pointed out in the correlation functions of the atomic gas.

Figure 1

Sketch of the physical configuration under examination. The condensate velocity is induced by the synthetic vector potential only in the upper $y>0$ part of the system, while the lower $y<0$ part is at rest. The wiggly lines show the propagation direction of the modes indicated by dots on the dispersion curves of Fig. 2 and involved in the superradiant processes.

Figure 2

Dispersion relations of the Bogoliubov modes at a given $k_x\xi =0.5$ in the two regions. The left and right panels correspond to the slow and fast regions, respectively with $v_x^s=0$ and $v_x^f=2.5c_s$. Black thin and red thick lines correspond to positive and negative norm modes. The light gray regions indicate the frequency ranges where ordinary scattering occurs, while the dark grey ones indicate the superradiant scattering range. The white line indicates an example of frequency in the superradiant range; the dots on the dispersion curves indicate the modes involved in the superradiant scattering process.

Figure 3

Reflection $|S_{ss}{|}^2$ (black thin line) and transmission $|S_{fs}{|}^2$ (red thick line) coefficients obtained with the matching of the modes at a steplike interface with $v_x^f=2.5c_s$ for an ingoing wave from the lower region with for $k_x=0.5\xi$. The dark and light grey regions correspond to the ones of Fig. 2: In particular, in the dark grey superradiant range, one can see that the amplified reflection coefficient goes well above 1.

Figure 4

Main plot: Incident-energy-dependent transmission coefficient for an ingoing wave from the lower region and a fixed $v_x^f=2.5c_s$. The different (thick red dashed, thin black dashed, thick red solid, think black solid) curves correspond to different values of the transverse incident wave vector $k_x\xi =0.3,0.5,0.75,1$. Inset: Log-log plot of the $k_x$ dependence of the transmission maximum. The solid red line shows the numerical data, the superimposed black dashed line is a fit of the $\frac{\alpha }{k_x^2}-\beta$ form, which reproduces the data very accurately.

Figure 5

Top and central panels: Snapshots at different times $t\mu /\hslash =0,150,200,350$ of the time evolution of a superradiant scattering process on an steplike ergosurface separating regions with $v_x^s=0$ and $v_x^f=2.5c_s$. The incident wave packet is Gaussian with a transverse momentum $k_x\xi =0.5$ and a longitudinal momentum centered around $k_y^{\text{in}}\xi =0.32$, that is $\hslash \omega =0.63M{c}_s^2$. It is initially located in space around $y_0=-100\xi$ with $\sigma =20\xi$. The black thin lines are the modulus of the first component $\left|U\right|$ of the Bogoliubov spinor, the red thick ones are the modulus of the second one, $\left|V\right|$. The amplification can be measured by computing the total Bogoliubov norm (18) in the two regions, whose time dependence is shown in the lower panel, with the black thin line being the modulus of the norm $|N^s|$ in the slow region and the red thick line the one, $|N^f|$, in the fast region. The amplification coefficient is compatible with the maximum value found in Fig. 3 and indicated here by the horizontal dotted line.

Figure 6

Main plot: Frequency-dependent spectrum of the spontaneous quantum superradiant emission after integration over transverse momentum $k_x$. Inset: Momentum-dependent spectrum after integration over frequencies $\omega$. The different curves correspond to different values of the synthetic vector potential in the fast region, such that $v_x^f=2.5c_s$ (lower solid red curves) and $v_x^f=3c_s$ (upper solid blue curves). In all curves, the synthetic vector potential (and hence the velocity) vanishes, $A_x^s=0$, in the slow region. The black dashed lines are fits of the numerical curves with a shifted exponential law of the form $f\left(\omega \right)=\alpha exp(-\beta \omega )-\gamma$ (and analogous expression with $k_x$ instead of $\omega$) with $\alpha ,\beta ,\gamma$ positive parameters. Such fits are found to describe very accurately the numerical data.

Figure 7

Color plots of the position space correlation function $G^{\left(2\right)}$ of density fluctuations for a pair of points $(x,y)$ and $(x^{\prime },y^{\prime })$ located respectively in the slower ($y<0$) and faster ($y^{\prime }>0$) regions. Because of the translational invariance along $x$, the correlation function only depends on $|x-x^{\prime }|$. The upper plot shows the $y,y^{\prime }$ dependence of $G^{\left(2\right)}$ for $x-x^{\prime }=0$, while the lower one shows its dependence on $x-x^{\prime }$ and $y$ for symmetrically located $y^{\prime }=-y$. The black dashed line in the lower plot has a slope ${[{(v_x^f/c_s)}^2-4]}^{-1/2}$ and well reproduces the boundary of the main contributions to the correlation function. The value of the synthetic vector potential in the faster region is taken so that $v_x^f=2.5c_s$, while in the slower region $v_x^s=0$. The numerical calculation was performed using the scattering matrix approach for a system initially in the ground state.

Figure 8

Plot of the normalized two body correlation function in momentum space for $v_x^f=3c_s$. For each value of $k_x$, nontrivial correlations are localized on a line indicating the momenta $k_y,k_y^{\prime }$ of the modes involved in the superradiant emission. The union of all these lines describes the surface that is visible in the plot. The coloring of the surface indicates the strength of the normalized correlation function (39).