Transverse collisional instabilities of a Bose-Einstein condensate in a driven one-dimensional lattice

Motivated by recent experiments, we analyze the stability of a three-dimensional Bose-Einstein condensate loaded in a periodically driven one-dimensional optical lattice. Such periodically driven systems do not have a thermodynamic ground state but may have a long-lived steady state which is an eigenstate of a “Floquet Hamiltonian.” We explore collisional instabilities of the Floquet ground state which transfer energy into the transverse modes. We calculate decay rates, finding that the lifetime scales as the inverse square of the scattering length and inverse of the peak three-dimensional density. These rates can be controlled by adding additional transverse potentials.

Figure 1

The two protocols of lattice driving: (a) an amplitude-modulated tilted lattice and (b) a shaken lattice.

Figure 2

Schematic showing first (top) and second (bottom) Floquet quasienergy bands of an optical lattice: $\epsilon$ is the single-particle energy, $k$ is the quasimomentum, and $a$ is the lattice spacing. Since Floquet energies are only defined modulo the shaking quanta $\hslash \omega$, the energy of the second band has been shifted down by $\hslash \omega$. Alternatively, this shift can be interpreted as working in a dressed basis, where the energy includes a contribution from the phonons. The mixing between the bands depends on the shaking amplitude. Dashed curves correspond to weak shaking, where the first band has its minimum at $k=0$. Solid curves correspond to strong shaking, where there are two minima at $k=\pm k_0\ne 0$.

Figure 3

Dimensionless decay rate $\Gamma$ as a function of amplitude of shaking $F$ for $\omega =5.5E_R/\hslash$ and $V_0=7.0E_R$. The dotted line shows $\Gamma$ calculated using Eq. (46), while the solid line shows the function ${\left(\frac{\chi F}{{\Delta }_0}\right)}^2$ corresponding to the rate in Eq. (38). The kink shows the paramagnetic-ferromagnetic phase transition.