Particle creation in the spin modes of a dynamically oscillating two-component Bose-Einstein condensate

We investigate the parametric amplification of the zero-point fluctuations in the spin modes of a two-component Bose-Einstein condensate, triggered by the dynamical evolution of the condensate density. We first make use of a Thomas-Fermi approximation to develop a tractable theoretical model of the quantum dynamics of the Bogoliubov excitations in a harmonically trapped condensate with a time-dependent trapping frequency. The predictions of this model are then compared to an ab initio numerical study of the correlation functions of density and spin fluctuations for general spatially inhomogeneous configurations. Results are shown for the two cases of expanding and oscillating condensates: while the quantum excitation of spin modes remains weak and relatively featureless in the case of an expanding condensate, clear and experimentally promising signatures of particle creation are anticipated for the oscillating case under suitable resonance conditions between the density and the spin modes.

Figure 1

Time evolution of the (spin) density (solid black line) and (relative) phase (dashed blue line) components of a spin mode of initial angular frequency ${\omega }_s/{\omega }_0=50$, in a one-dimensional condensate that is expanding after reverting the sign of the harmonic potential at $t=0$. In the inset we show the dynamics of the expansion parameter $\ell (t)$: After the initial transient regime, whose duration is of the order $\sim 1/{\omega }_0$, the expansion is exponential, and characterized by the Hubble parameter $H(t)=\dot{\ell }(t)/\ell (t)={\omega }_0$. The three stages of the evolution of the modes discussed in the main text, driving the system from the under- to the over-damped regime, are clearly visible.

Figure 2

Panels (a–d) (upper and middle row): time evolution of the number of excitations $n_q$, including the initial vacuum fluctuations, $n_q+1/2\equiv W[{\phi }_q,{\varrho }_q^{*}]/(2{\omega }_q(t))$ (solid black lines), of the modulus of the anomalous correlations $c_q\equiv W[{\phi }_q,{\varrho }_q]/(2{\omega }_q(t))$ (dashed blue lines) and of the correlation functions of the density fluctuations $G_2^{(2)}(q,-q)$ (dot-dashed red line), generated in the spin mode of frequency ${\omega }_{s,1}/{\omega }_0=0.5$, of a condensate that is linearly (a,c) and exponentially (b,d) expanding in one (a,c) and three (b,d) dimensions. The inset shows the time evolution of the energy of the mode, which goes to zero because of the freezing effect. Panels (e,f) (bottom row): same quantities for a condensate that oscillates in the breathing density mode. The two panels refer respectively to a resonant case with ${\omega }_{s,2}\approx {\omega }_{d,2}/2\approx \sqrt{3}/2$ (e) and to a nonresonant case with ${\omega }_{s,1}/{\omega }_0=0.5$. In all panels, we have taken $g_d/g_s=4$.

Figure 3

Bogoliubov spectrum of the density (black dots) and spin (red and blue markers) modes in a one-dimensional, harmonically trapped two-component condensate of chemical potential $\mu /{\omega }_0=28.25$. We notice that the spin modes of quantum number $n=3$ (for $g_d/g_s=13.3$, red marker) and $n=5$ (for $g_d/g_s=73.3$, blue marker) are close to resonance with the breathing mode ($n=2$) in the density branch of the excitations. We reported in the inset the density profile of the condensate.

Figure 4

Time evolution of the density-density correlation function for a one-dimensional, harmonically trapped two-component condensate of chemical potential $\mu /{\omega }_0=28.25$. Panel (a) shows the contribution to the correlations due to the density excitations, for the case of a linearly expanding system. In panels (b) and (c) we report the contribution to the correlations due to the spin excitations, respectively for the linearly and exponentially expanding systems.

Figure 5

Time evolution of the density-density correlation function for a one-dimensional, harmonically trapped two-component condensate, of chemical potential $\mu /{\omega }_0=28.25$ whose overall density is oscillating in the breathing mode. In panel (a) we report the contribution to the correlations due to the density excitations. We see that the zero-point fluctuations that populate these modes are not excited and the correlations do not evolve in time. In panels (b,c) is reported the contribution to the correlations due to the spin excitations. We clearly see here the structure of the resonant mode, whose vacuum fluctuations are parametrically amplified because of the oscillations in the density.