Spin-orbit-coupled Bose-Einstein condensates in a cavity: Route to magnetic phases through cavity transmission

We study the spin-orbit-coupled ultracold Bose-Einstein condensate placed in a single-mode Fabry-Pérot cavity. The cavity introduces a quantum optical lattice potential which dynamically couples with the atomic degrees of freedom and realizes a generalized modified Bose-Hubbard model whose zero-temperature phase diagram can be controlled by tuning the cavity parameters. In the noninteracting limit, where the atom-atom interaction is set to 0, the resulting atomic dispersion shows interesting features such as a bosonic analog of Dirac points, a cavity-controlled Hofstadter spectrum which bears the hallmark of pseudospin-1/2 bosons in the presence of Abelian and non-Abelian gauge fields (the latter due to spin-orbit coupling) in a cavity-induced optical lattice potential. In the presence of atom-atom interaction, using a mapping to a generalized Bose-Hubbard model of spin-orbit-coupled bosons in a classical optical lattice, we show that the system realizes a host of quantum magnetic phases whose magnetic order can be detected from the cavity transmission. This provides an alternative approach to detecting quantum magnetism in ultracold atoms. We discuss the effects of cavity-induced optical bistability on these phases and their experimental consequences.

Figure 1

${}^{87}\mathrm{Rb}$ BEC inside an optical cavity: SOC is created by two counterpropagating Raman lasers with frequencies ${\omega }_L$ and ${\omega }_L+\Delta {\omega }_L$ that are applied along $\widehat{x}$. The Raman beams are polarized along $\widehat{z}$ and $\widehat{y}$ (gravity is along $-\widehat{z}$). A bias field $B_0$ is applied along $\widehat{y}$ to generate the Zeeman shift. Inset: Level diagram of the ${}^{87}\mathrm{Rb}$ atom. Internal states are denoted $|1\rangle ,|2\rangle ,|3\rangle$. The coupling of these states is shown schematically.

Figure 2

Variation of the two coefficients used in (24) with effective detuning. Experimental parameters are set to be $\{\eta ,\kappa ,U_0,J_0\}=\{10,1,0.2,2\}{\omega }_r$.

Figure 3

Three-dimensional view of the energy spectrum plotted for a purely ($\Phi =0$ non-Abelian gauge field. The strength of SOC is (a) $\alpha =\pi /2=\beta$ and (b) $\alpha =\pi /2+0.25$, $\beta =\pi /2-0.25$. The surface plot is an intensity map of the energy difference between $E_{+}$ and $E_{-}$. The four green spots on the surface correspond to the four (bosonic) Dirac points (at the zone centers) where the energy gap between the two bands vanishes. The red band and blue band correspond to $E_{+}$ and $E_{-}$, respectively. $W$ is the maximum band gap, which occurs at the zone boundaries. (c, d) Location of the Dirac points on the momentum space are shown for $2m\pi \Phi =0.75$ and 1.5, respectively. With increasing $\Phi$ the Dirac points move along the positive $k_y$ axis.

Figure 4

(a) Variation of overlap integral elements with potential depth. We study the variation in two regions, separated at $V=25E_r$: cavity spectrum for a deep lattice (region I) and for a $6\times 6$ lattice, $\{U_0,\kappa \}=\{12,1\}{\omega }_r$. (b) Variation with pump amplitude $\eta$ for ${\Delta }_c=5000{\omega }_r$. (c) Variation with detuning ${\Delta }_c$ for $\eta =6{\omega }_r$. Dotted (red) lines are unstable regions of the photon count.

Figure 5

(a) Spin vectors in internal spin spaces of two neighboring sites. (b–d) Cavity spectrum for different phases in the MI region for different non-Abelian flux insertions. The SOC strength for all phases is $(\alpha ,\beta )/\pi$ = (0.01,0.01) zFM, (0.2, 0.2) 4-spiral, (0.3,0.3) 3-spiral, (0.5,0.5) stripe, and (0.34, 0.34) VX. Note that the turning points are highly dependent on the phases. The dotted parts of curves show the unstable region of the spectrum. Red and blue legends correspond to the magnetic order, shown in insets.

Figure 6

(a) Variation of the grating function $f(K,\Phi )$ with the inserted Abelian flux. Legends indicate the size of the lattice. In a large lattice limit the grating function does not sense the variation of $\Phi$. (b, c) Cavity spectrum for different (b) Abelian fields (with a fixed non-Abelian field, $\alpha =-\beta =\pi /2-0.15$) and (c) non-Abelian fields (with a fixed Abelian field, $\Phi =0.08{\Phi }_0$). The negative-slope region is the unstable [shaded (gray)] part of the spectrum.

Figure 7

Schematic of an optical lattice. The phase operator $U_x$ determines the phase acquired by an atom when it hops from site $(m,n)$ to site $(m+1,n)$. Similarly, the operator $U_y$ determines the phase acquired by hopping along the positive $y$ axis. The operators $U_x^{†}$ and $U_y^{†}$ determine the phase acquired in hopping along the negative $x$ and $y$ axes, respectively.