Atomic matter of nonzero-momentum Bose-Einstein condensation and orbital current order

The paradigm of Bose-Einstein condensation has been associated with zero momentum to which a macroscopic fraction of bosons condense. Here we propose a new quantum state where bosonic alkali-metal atoms condense at nonzero momenta, defying the paradigm. This becomes possible when the atoms are confined in the $p$-orbital Bloch band of an optical lattice rather than the usual $s$-orbital band. The new condensate simultaneously forms an order of transversely staggered orbital currents, reminiscent of orbital antiferromagnetism or $d$-density wave in correlated electronic systems but different in fundamental ways. We discuss several approaches of preparing atoms to the $p$-orbital band and propose an “energy blocking” mechanism by Feshbach resonance to protect them from decaying to the lowest $s$-orbital band. Such a model system seems very unique and novel to atomic gases. It suggests a new concept of quantum collective phenomena of no prior example from solid state materials.

Figure 1

(Color online) The overlap between $s$ and $p$ density clouds (wave functions squared) is smaller than between two $s$ clouds.

Figure 2

(Color online) Energy-band shift due to Feshbach resonance between bosons and fermions. (a) The bosonic $1s$ band is moved close to the $1p$ band while the $2s$ and $1d$ bands (not shown) are both pushed higher. (b) The fermions fill up their own lowest $s$ band as an band insulator. The relative shift of the fermionic band energy is less significant than that of the bosonic case, for ${\omega }_f$ can be made far larger than both ${\omega }_b$ and the (Feshbach tunable) energy shift.

Figure 3

(Color online) Anisotropic hopping matrix elements of $p$-orbital bosons on a cubic lattice. The longitudinal $t_{\Vert }$ is in general far greater than $t_{\perp }$ because the overlap integral for the latter is exponentially suppressed. The “±” symbols indicate the sign of two lobes of $p$-orbital wave function.

Figure 4

(Color online) Illustration of $p$-band dispersions. (a) The energy dispersion of the $p_x$ orbital band (as an example) on the $k_x$ axis in the first Brillouin zone, in comparison with the $s$ band. Inset: its dispersion along the line $(\frac{\pi }{a},-\frac{\pi }{a},0)$—$(\frac{\pi }{a},\frac{\pi }{a},0)$; the energy zero line is shifted arbitrarily for display. Note that the $s$ and $p$ bands have different locations for minimum energy. The band gap between the two levels is tuned small by the Feshbach resonance. (b) The $p$-band minima in the $k_x-k_y$ plane (the $p_z$’s minima are not shown but can be obtained by rotating 90° out of the plane). Orbital BEC should occur at a subset of the minima, spontaneously breaking the lattice translational and (orbital) rotational symmetry, in addition to the U(1).

Figure 5

(Color online) The real-space configuration of the TSOC state, which exhibits a staggered and uniform orbital current pattern in the $xy$ plane and along the $z$ axis, respectively.

Figure 6

(Color online) Prediction of density of atoms for time-of-flight experiment. (a) Density integrated over the $z$ axis is shown in a quarter of $k_x-k_y$ momentum plane, with other quarters obtained by reflection symmetry. (b) Density shown along the $k_x$ axis. Assuming a free expansion, the atom density at distance $\mathbf{r}$ is ${⟨n^b\left(\mathbf{r}\right)⟩}_{\tau }\propto {\mid {\varphi }_{p_x}^b\left(\mathbf{k}\right)\mid }^2{\delta }^3(\mathbf{k}-\mathbf{G}-{\mathbf{Q}}_x)+{\mid {\varphi }_{p_y}^b\left(\mathbf{k}\right)\mid }^2{\delta }^3(\mathbf{k}-\mathbf{G}-{\mathbf{Q}}_y)$ where $\mathbf{k}=m\mathbf{r}∕\hslash \tau$, $\tau$ is the time of flight, and $\mathbf{G}$ runs over all three-dimensional reciprocal lattice vectors. The absence of peak at $\mathbf{k}=0$ distinguishes the $p$-OBEC from paradigmatic BEC. The highest peak is not necessarily the closest to zero-momentum origin, a unique feature of orbital BEC, depending on the size $l_b$ of the bosonic Wannier function ${\varphi }_{\mu }^b\left(\mathbf{r}\right)$. Parameters are $l_b∕a=0.1$; the $\delta$ function is replaced by a Lorentzian line for display.

Figure 7

Proposed phase diagram for a system of anisotropic lattice potentials, sketched as a function of the mean interaction energy $nU$ and the anisotropic ratio $\gamma ={\omega }_{x,y}∕{\omega }_z$.