Stability diagram and growth rate of parametric resonances in Bose-Einstein condensates in one-dimensional optical lattices

A Bose-Einstein condensate in an optical lattice exhibits parametric resonances when the intensity of the lattice is periodically modulated in time. These resonances correspond to an exponential growth of the population of counter-propagating Bogoliubov excitations. A suitable linearization of the Gross-Pitaevskii (GP) equation is used to calculate the stability diagram and the growth rates of the unstable modes. The results agree with the ones extracted from time-dependent GP simulations, supporting our previous claim [M. Krämer et al., Phys. Rev. A 71, 061602(R) (2005)] concerning the key role of parametric resonances in the response observed by Stöferle et al. [Phys. Rev. Lett. 92, 130403 (2004)] in the superfluid regime. The role of the seed excitations required to trigger the parametric amplification is discussed. The possible amplification of the quantum fluctuations present in the quasiparticle vacuum, beyond GP theory, is also addressed, finding interesting analogies with similar processes in nonlinear quantum optics and with the dynamic Casimir effect. Our results can be used in exploiting parametric instabilities for the purpose of spectroscopy, selective amplification of a particular excitation mode and for establishing a new type of thermometry.

Figure 1

Lowest bands in the Bogoliubov spectrum of a cylindrical condensate in a periodic lattice, obeying Eq. (2) with $s=s_0=4$ and $gn=0.72E_R$. The range of $k$ corresponds to half of the first Brillouin zone. We use the shorthand notation ${\omega }_0$, ${\omega }_1$ and ${\omega }_2$ to indicate the Bogoliubov bands whose dispersion relations are ${\omega }_{0k}$, ${\omega }_{1k}$ and ${\omega }_{2k}$.

Figure 2

Position of parametric resonances for $s_0=4$, $gn=0.72E_R$ and for a small modulation amplitude, $A\to 0$. See Fig. 1 for the Bogoliubov spectrum ${\omega }_{jk}$ with the same parameters.

Figure 3

Growth rates of parametric amplification for $s_0=4$, $gn=0.72E_R$ in the small $A$ limit. The resonances are the same as in Fig. 2. Lines: results of the stability analysis of Eq. (14). Different values of $A$ in the range $A<0.01$ give the same curves for $\gamma ∕A$. Circles: Eqs. (27, 34) of the two-mode approximation, with ${\Gamma }_{+-}^{j{j}^{\prime }}$ given in (19) and calculated by solving the Bogoliubov equations (6) at $s_0=4$.

Figure 4

Stability diagram in the $(k,\Omega )$ plane obtained with the procedure of Sec. 2C for $A=0.1$, $s_0=4$ and $gn=0.72E_R$. The lightest level of grey in the grey-scale represents all modes having ${10}^{-10}<\gamma <{10}^{-3}$. The main branch corresponds to the $2{\omega }_0$ resonance, whose maximal growth rate ${\gamma }_{\max }$ grows with $k$ as shown in Fig. 5. The other weak branches are higher order parametric resonances.

Figure 5

Growth rate $\gamma ∕A$ for the $2{\omega }_0$ resonance $(\Omega =2{\omega }_{0k})$ obtained with the stability analysis of Eq. (14) with $s_0=4$, $gn=0.72E_R$ and $A<0.01$ (solid line) and $A=0.05$ (dashed) and 0.1 (dot-dashed).

Figure 6

Growth rate $\gamma$ as a function of the modulation frequency $\Omega$. The lines are the results of the stability analysis of Sec. 2 for $A=0.2$, $s_0=4$ and $gn=0.72E_R$. In particular, the two solid lines are the resonances of order $A$ at $2{\omega }_0$ and ${\omega }_0+{\omega }_1$, while the dashed and dot-dashed lines are the weaker resonances of order $A^2$ at $({\omega }_0+{\omega }_1)∕2$ and $({\omega }_0+{\omega }_2)∕2$, respectively. Empty circles are extracted from the time evolution of the incoherent fraction $\delta N∕N$ in GP simulations on a cylindrical condensate, unbound along $z$ and harmonically confined in the transverse direction. Solid circles come from GP simulations as well, but for a condensate which is harmonically confined also along $z$, similar to the trapped condensates of the experiments of Refs. 15, 16, 17.

Figure 7

Coupling ${\Gamma }_{+-}^{00}$ between counter-propagating modes of the first Bogoliubov band as a function of quasimomentum $k$ at $gn=0.1E_R$ (top panel) and $gn=0.72E_R$ (bottom panel) for different values of the lattice depth $s_0=4$, 10 and 16. Lines: definition (19) with the quasiparticle amplitudes taken from the numerical solution of the Bogoliubov equations (6). Points: Tight binding result (42).

Figure 8

Momentum distribution of a condensate which is harmonically confined in the radial direction and periodic along $z$, obtained by performing GP simulations. The interaction parameter is $gn=0.72E_R$, where $n$ is the average linear density. The optical lattice has $s_0=4$ and is modulated with $A=0.2$ and $\Omega ={\omega }_R$. The modulation starts at $t=0$ and the momentum distribution is plotted at $t=t_A$ (top panel) and $t=t_B$ (bottom panel), where $A$ and $B$ are the points indicated in Fig. 9. The arrows show the position of the parametrically unstable modes.

Figure 9

Incoherent fraction $\delta N∕N$ as a function of modulation time obtained by performing time-dependent GP simulations. Solid lines correspond to $s_0=4$, $A=0.2$ and two different modulation frequencies: $\Omega ={\omega }_R$ and $\Omega =2{\omega }_R$. The growth rate $\gamma$ is extracted from a linear fit (dashed lines). The momentum distribution of the condensate at the instants $A$ and $B$ is shown in Fig. 8.