Stability of a Bose-Einstein condensate in a driven optical lattice: Crossover between weak and tight transverse confinement

We explore the effect of transverse confinement on the stability of a Bose-Einstein condensate loaded in a shaken one-dimensional or two-dimensional square lattice. We calculate the decay rate from two-particle collisions. We predict that if the transverse confinement exceeds a critical value, then, for appropriate shaking frequencies, the condensate is stable against scattering into transverse directions. We explore the confinement dependence of the loss rate, explaining the rich structure in terms of resonances.

Figure 1

Schematic of shaken optical lattices: (a) 1D lattice with weak transverse confinement, (b) 1D lattice with tight transverse confinement, and (c) 2D lattice with weak transverse confinement. Ellipsoids represent edges of cloud in each well of the optical lattice sites and arrows illustrate motion of trap. A typical spacing between lattice sites is 532 nm (half the laser wavelength, ${\lambda }_L=1064$ nm) and a typical shaking amplitude is 15 nm.

Figure 2

Schematic showing first (top) and second (bottom) Floquet quasienergy bands of an optical lattice: $\epsilon$ is the single-particle energy (arbitrary units used for schematic), $k$ 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 $\hslash \omega$ so that it lies below the first band. 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

Schematic showing the dispersion of the first Floquet band of a shaken square lattice beyond a critical amplitude. Color represents energy in units of the recoil energy, $E_R$ (see scale). We see that the superfluid order parameter develops a $D_4$ symmetry in momentum space.

Figure 4

Adimensional scattering rate $\mathrm{\Gamma }$ as a function of the forcing amplitude $F_0$ in the limit of weak confinement into a 1D lattice [Fig. 1]. Blue dotted line: $\hslash \mathrm{\Omega }/E_R=0.04$; red dashed line: $\hslash \mathrm{\Omega }/E_R=0.08$; black solid line: analytic result from Ref. [55].

Figure 5

Logarithm of the adimensional scattering rate $\mathrm{\Gamma }$ in a 1D lattice [Figs. 1 and 1] as a function of the transverse trapping frequency $\mathrm{\Omega }$ for a fixed value of the forcing amplitude, $F_0=F_c$, where $F_c$ is the amplitude where the dispersion of the ground band is quartic near $k=0$. Red vertical lines denote resonances at $\mathrm{\Omega }={\mathrm{\Omega }}_n^{\left(a\right)},{\mathrm{\Omega }}_n^{\left(b\right)}$ corresponding to the closing of scattering channels (see text). The black dashed line shows the value of ln $\left(\mathrm{\Gamma }\right)$ for different values of the transverse confinement for which the BEC lifetime is greater than 10 s [assuming the parameters quoted after Eq. (13)].

Figure 6

Schematic illustrating conservation of energy and momentum in two-body collisions in a shaken pancake lattice. The black dot denotes the condensate in the first band at $k=0$. The solid lines show first and second with no transverse excitations. Arrows denote an energy and momentum conserving collision. The resonances in Figs. 4 and 5 correspond to the situation where the final states have $\left|ka\right|=\pi$.

Figure 7

Stability phase diagram for a BEC in a driven 1D lattice for a fixed value of the forcing amplitude, $F_0=F_c$. Here, $\omega -{\omega }_0$ is the detuning of the shaking frequency $\omega$ from the zero-momentum band gap ${\omega }_0$.

Figure 8

Adimensional scattering rate $\mathrm{\Gamma }$ as a function of the forcing amplitude $F_0$ in the limit of weak confinement ($\hslash \mathrm{\Omega }=0.08E_R$) for a 2D lattice [Fig. 1].

Figure 9

Logarithm of the adimensional scattering rate $\mathrm{\Gamma }$ in a 2D lattice as a function of the transverse confinement $\mathrm{\Omega }$ for a fixed value of the forcing amplitude, $F_0=F_c$. The black dashed line shows the value of ln $\left(\mathrm{\Gamma }\right)$ for different values of the transverse confinement for which the BEC lifetime is greater than 10 s [assuming the parameters quoted after Eq. (16)].