Stability of a Floquet Bose-Einstein condensate in a one-dimensional optical lattice

Motivated by recent experimental observations [Parker, Hua, and Chin, Nat. Phys. 9, 769 (2013)], we analyze the stability of a Bose-Einstein condensate (BEC) in a one-dimensional lattice subjected to periodic shaking. In such a system there is no thermodynamic ground state, but there may be a long-lived steady state, described as an eigenstate of a “Floquet Hamiltonian.” We calculate how scattering processes lead to a decay of the Floquet state. We map out the phase diagram of the system and find regions where the BEC is stable and regions where the BEC is unstable against atomic collisions. We show that Parker et al. performed their experiment in the stable region, which accounts for the long lifetime of the condensate $(\sim 1\mathrm{s})$. We also estimate the scattering rate of the bosons in the region where the BEC is unstable.

Figure 1

Floquet spectra of shaken 1D lattices for lattice depths of $V_0/E_R=2.02,7$ and 7. In (a) and (c), the shaking frequency is blue detuned while in (b), the shaking frequency is red-detuned. The parameters in (c) are similar to those in Ref. [31]. Quasimomentum, $k$ and quasi-energy, $\epsilon$ are measured in terms of the lattice spacing $a$ and the period $T=\frac{2\pi }{\omega }$. Solid and dashed lines represent bands and their periodic repetition and circles show the location of band minima. Right panels are magnified views. In (a) and (b) arrows represent scattering processes which cause a condensate at the band minima to decay. In (a) this is an intraband scattering process, where the final state of the scattered particles have the same Bloch index. In (b), it is an interband process where the Bloch indices are different. Case (c) is stable: there are no two-body processes that conserve energy and momentum.

Figure 2

Phase diagram of the floquet BEC for a variety of lattice depths and detunings in the limit of infinitesimal driving.

Figure 3

Phase diagram of the Floquet BEC in a shaken one-dimensional lattice of depth $V_0=7.0E_R$. The zero-momentum band gap ${\Delta }_0$ is $4.96E_R$. The vertical arrow shows the parameters found in Ref. [31]. The BEC is stable in the blue-detuned regime. In the red-detuned regime, the BEC is unstable below a critical driving strength and stable above it. The thick black line shows the critical driving strength.

Figure 4

Plot of dimensionless decay rate $\Gamma$ [defined in Eq. (18) in the text] as a function of amplitude of shaking, $F_{0}a/E_R$ for $\omega =4.95E_R/\hslash$ and $V_0=7.0E_R$. Within the rotating wave approximation, with this $\omega$ the transition from unstable ferromagnetic phase to stable ferromagnetic phase happens at $F_0=0.037E_R/a$.