Simulating and detecting artificial magnetic fields in trapped atoms

A Bose-Einstein condensate exhibiting a nontrivial phase induces an artificial magnetic field in immersed impurity atoms trapped in a stationary, ring-shaped optical lattice. We present an effective Hamiltonian for the impurities for two condensate setups: the condensate in a rotating ring and in an excited rotational state in a stationary ring. We use Bogoliubov theory to derive analytical formulas for the induced artificial magnetic field and the hopping amplitude in the limit of low condensate temperature where the impurity dynamics is coherent. As methods for observing the artificial magnetic field we discuss time-of-flight imaging and mass current measurements. Moreover, we compare the analytical results of the effective model to numerical results of a corresponding two-species Bose-Hubbard model. We also study numerically the clustering properties of the impurities and the quantum chaotic behavior of the two-species Bose-Hubbard model.

Figure 1

Induced phase in the lattice with $30$ lattice sites for (a) a ring rotating with angular speed $\Omega$ and (b) the BEC in a lattice with phase term ${\alpha }_c$. We assume a BEC of ${}^{87}\mathrm{Rb}$ with a linear density $n_0=5\times {10}^6\hspace{0.5em}{\mathrm{m}}^{-1}$. The impurities are ${}^{87}\mathrm{Rb}$ atoms in a different hyperfine level. The lattice spacing is $d=400\hspace{0.5em}\mathrm{nm}$, which means that the ring has a circumference of $L=12\hspace{0.5em}\mathrm{\mu }\mathrm{m}$. In (a) the interspecies coupling strength is $\kappa /2{\mathit{dE}}_R=0.045$, where $E_R=\hslash {}^2{\pi }^2/2m_a{d}^2$ is the recoil energy of the impurity lattice with impurity mass $m_a$, and the curves indicate different BEC couplings $g/2{\mathit{dE}}_R=0.01,\hspace{0.5em}0.015,\hspace{0.5em}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}}0.02$ (solid, dashed, and dashed-dotted lines, respectively). In (b) we assume that both lattices have the same depth, interspecies coupling is $U_I/J_c=1.5$, and BEC couplings $U_c/J_c=0.5,\hspace{0.5em}1,\hspace{0.5em}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}}1.5$ (solid, dashed, and dashed-dotted lines, respectively).

Figure 2

Induced phase with a BEC in an excited state. The parameters are the same as in Fig. 1a and we abbreviate $\tilde{g}=g/2{\mathit{dE}}_R$. The critical momenta for the different interaction strengths are given by $k_{\mathrm{crit}}L/2\pi \simeq 2.1$, $2.6$, and $3.0$ (for $\tilde{g}=0.01,\hspace{0.5em}0.015,\hspace{0.5em}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}}0.02$, respectively). The lines are only to guide the eye.

Figure 3

(a) Relative error ${\overline{k}}_{\mathrm{ana}}/{\overline{k}}_{\mathrm{num}}-1$ of Eq. (19) compared to the numerically exact ground-state quasimomentum for different interactions at $N_s=12$, $6$ BEC atoms, and one impurity with ${\tilde{J}}_a=J_c$. Line styles indicate different BEC interactions $U_c$ (see legend) and increasing lightness (from top to bottom) indicates decreasing interspecies coupling $U_I/J_c\in \{0.2,0.15,0.14,\dots ,0.05,0.01\}$ (dashed-dotted lines), $\{0.9,0.8,\dots ,0.1,0.05,0.01\}$ (dashed), $\{1,0.7,0.6,\dots ,0.1,0.05,0.01\}$ (solid). (b)–(c) Comparison of analytical induced phase (solid), numerically exact mean quasimomentum (dashed) and the rescaled analytical mean quasimomentum $-2\overline{k}/\pi$ (dashed-dotted) for (b) $U_c=0.5J_c$, $U_I=0.4J_c$ and (c) $U_c=J_c$, $U_I=0.2J_c$. The rescaling factor $-2/\pi$ results from Eq. (20). The dotted vertical line indicates ${\alpha }_c=1/N_s$.

Figure 4

Time-of-flight distribution ${\rho }_{\mathrm{ToF}}$ of a single impurity released from a ring lattice with $N_s=12$ at $T=0$ in the single-species model [Eq. (4)]. The central peak in (a) indicates a zero quasimomentum ground state for ${\alpha }_a=0.04$. In (b) the vanishing density in the center indicates that the ground state exhibits nonzero momentum (${\alpha }_a=0.05$). The jump between these two momentum states occurs at ${\alpha }_a=1/2N_s\simeq 0.042$. In (c) we show the corresponding density profiles indicated with horizontal lines in (a) and (b). The ToF expansion time is $t=50\hspace{0.5em}\hspace{1em}\mathrm{ms}$ for ${}^{87}\mathrm{Rb}$ atoms in a ring of radius $R=12\hspace{0.5em}\mu \mathrm{m}$.

Figure 5

Influence of temperature on the ToF images in the two-species Bose-Hubbard model. A single impurity is submerged into a BEC with six atoms in $N_s=12$ lattice sites, hopping ${\tilde{J}}_a=J_c$, interspecies coupling $U_I=0.5J_c$, BEC interaction $U_c=0.4J_c$, and BEC phase ${\alpha }_c=0.042$. In (a) $k_{B}T=0$ and in (b) $k_{B}T=2J_c$.

Figure 6

(a) Correlations of impurities for $N_s=16$ lattice sites, three atoms, and $U_a=-0.125{\tilde{J}}_a$. The two lines represent the two families of correlations depending on the momentum state of the system (i.e., the phase ${\alpha }_a$). Each family is a set of correlations for many values of ${\alpha }_a$. (b) The spectrum exhibits crossings below and above the critical phases ${\alpha }_{\mathrm{crit}}$ (the first critical phase is indicated with a vertical dashed line), where the impurities switch their momentum ground state (thick solid and dashed lines). The thin gray lines indicate higher lying energy levels.

Figure 7

Cumulative level spacing for different disorder strengths $e/J_c=0,1,20$ (solid blue, dashed red, and dashed-dotted yellow lines, respectively). The edge of the dark shaded area indicates the Poisson prediction and the light shaded area the GOE prediction. The curves describe a crossover from regular behavior to chaotic, and back to regular for increasing disorder. This is for three BEC atoms and $2$ impurities in lattices with five sites averaged over $100$ independent disorder realizations. The other parameters are $U_c=1.5J_c$, $U_a=0.1J_c$, $U_I=J_c$, ${\alpha }_c=0.057$.

Figure 8

${\chi }^2$ tests of the Poisson (solid line) and GOE predictions (dashed line). For intermediate values of $u$ ($0.2\lesssim u\lesssim 0.9$) the energy distribution closely follows the GOE prediction. Below and above these values we observe a more regular behavior as the system is in either the hopping-only or the no-hopping state, respectively. As parameters we have chosen $N_s=5$ lattice sites, 3 BEC atoms, 2 impurities, ${\alpha }_c=0.057$. The results have been averaged over $100$ independent disorder realizations. The inset shows the individual cumulative level distributions (thin lines, which appear as a gray area at this resolution) together with the Poisson (solid line) and GOE predictions (dashed line).

Figure 9

Contours of the relative condensate density ${\overline{n}}_0/\overline{n}$ at $T=0$ for (a) $N_s=10$ and $\overline{n}=0.5$, and (b) $N_s=30$ and $\overline{n}=2$. The values in the plot indicate the relative density of the respective contour.