Excitation spectra and hard-core thermodynamics of bosonic atoms in optical superlattices

A generalized double-well-basis coupled representation is proposed to investigate excitation spectra and thermodynamics of bosonic atoms in double-well optical superlattices. In the hard-core limit and with a filling factor of one, excitations describing the creation of pairs of a doubly occupied state and a simultaneous empty state, and those from a symmetric singly occupied state to an antisymmetric state are carefully analyzed and their excitation spectra are calculated within mean-field theory. Based on the hard-core statistics, the equilibrium properties such as heat capacity and particle populations are studied in detail. The cases with other filling factors are also briefly discussed.

Figure 1

Geometry of the double-well superlattice. It is generated by superposition of laser standing waves along the $x$ and $y$ axes: $V(x,y)=-V_0{sin}^2\left(k_{1}x\right)-V_0{sin}^2\left(k_{2}x\right)-2V_0{sin}^2\left(k_{3}y\right)$, where $2k_1=k_2=2k_3=2\pi /\lambda$ and $\lambda$ is the laser wavelength. $t_{\sigma }(\sigma =1,2,3$) are tunneling amplitudes, and $U_{\sigma }$ are intra- ($\sigma =1$) and inter-double-well ($\sigma =2,3$) interactions.

Figure 2

Bosonic atoms in the double-well superlattice with filling factor one. (a) Simultaneous generation of a doubly occupied ($|d\rangle$) and empty ($|e\rangle$) states from the ground state. (b): Mixing between the symmetric ($|s\rangle$) and antisymmetric ($|a\rangle$) states resulting in two degenerate bases $|0_{\mathrm{L}}{1}_{\mathrm{R}}\rangle$ and $|1_{\mathrm{L}}{0}_{\mathrm{R}}\rangle$. (c) Ground state composed of symmetric state ($|s\rangle$). (d) State that mixes $|d\rangle$, $|e\rangle$ and $|s\rangle$. (e) Checker-board-like insulator state characterized by wave vectors $(\pi ,0)$ or $(0,\pi )$.

Figure 3

Excitation spectra of pseudoparticle $\tilde{d}$ (associated with the double-well basis with two atoms) in the double-well superlattice with filling factor $\rho =1$. The spectra are obtained by regarding the wave function made up of $|s\rangle$ as the ground state. Tunnelling amplitudes $t_2/t_1=t_3/t_1=0.1$, and the ratio $\lambda$ of repulsive interactions ($U_{\sigma }$) to the corresponding tunneling amplitudes ($t_{\sigma }$) is set to be $\lambda =U_{\sigma }/t_{\sigma }=-0.4$ with $\sigma =1,2,3$.

Figure 4

Excitation spectra of pseudoparticle $\tilde{e}$ (associated with the double-well basis without atoms) in the double-well superlattice with filling factor $\rho =1$. Tunnelling amplitudes $t_2/t_1=t_3/t_1=0.1$, and the ratio $\lambda$ is set to be $\lambda =U_{\sigma }/t_{\sigma }=-0.4$ with $\sigma =1,2,3$.

Figure 5

Excitation spectra of pseudoparticle $\tilde{a}$ (associated with the antisymmetric double-well basis with one atom) in the double-well superlattice with filling factor $\rho =1$. Tunnelling amplitudes $t_2/t_1=t_3/t_1=0.1$ and the ratio $\lambda$ is set to be $\lambda =U_{\sigma }/t_{\sigma }=-0.4$ with $\sigma =1,2,3$.

Figure 6

$\lambda$ dependence of excitation gaps ${\Delta }_{\alpha }$, middle values $M_{\alpha }$, and half band widths $H{W}_{\alpha }$ of excitation spectra ($\alpha =a,d,e$ corresponding to pseudoparticles $\tilde{a}$, $\tilde{d}$, $\tilde{e}$, respectively) in the double-well superlattice with filling factor $\rho =1$, temperature $T/t_1=0.1$, and tunnelling amplitudes $t_2/t_1=t_3/t_1=0.2$. ${\Delta }_d={\Delta }_e$, $M_d=M_e$, and $H{W}_d=H{W}_e$ because $\tilde{d}$ and $\tilde{e}$ are simultaneously created in excitation processes.

Figure 7

$\lambda$ dependence of excitation gaps ${\Delta }_{\alpha }$ ($\alpha =a,d,e$) in the double-well superlattice with filling factor $\rho =1$ and temperature $T/t_1=0.1$, at different tunneling amplitudes $t_2/t_1=t_3/t_1=0.0,0.1,0.2$. ${\Delta }_d$ (equal to ${\Delta }_e$) decreases with increasing $t_2$ or decreasing $\left|\lambda \right|$ and ${\Delta }_a$ decreases with increasing $t_2$ or $\left|\lambda \right|$, which suggest the assumed ground state becomes unstable upon enlargement of tunneling amplitudes.

Figure 8

Temperature ($T/t_1$) dependence of critical tunneling amplitudes, $t_2/t_1$ (set $t_3=t_2$) in the double-well superlattice with filling factor $\rho =1$, at $\lambda =U_{\sigma }/t_{\sigma }=-0.4(\sigma =1,2,3$). The critical tunneling, which is determined by setting the lowest excitation gap to be zero, increases with temperature due to populations in excitation levels led by thermal fluctuations suppressing the influence of tunnellings.

Figure 9

$\lambda$ dependence of the population numbers $n_{\alpha }(\alpha =s,a,e,d$) of the ground state ($s$) and excitation levels ($a,e,d$) in the double-well superlattice with filling factor $\rho =1$ at temperature $T/t_1=0.1$ and different tunneling amplitudes $t_2/t1=t_3/t_1=0.0,0.1,0.2$. The population numbers satisfy the constraint condition $n_s+n_a+n_e+n_d=1$.

Figure 10

Temperature ($T/t_1$) dependence of the population numbers $n_{\alpha }(\alpha =s,a,e,d$) in the double-well superlattice with filling factor $\rho =1$ at tunneling amplitudes $t_2/t_1=t_3/t_1=0.2$ and $\lambda =U_{\sigma }/t_{\sigma }=-0.4$. The difference of $n_{\alpha }(\alpha =s,a,e,d$) with one another decreases with increasing temperature due to the hard-core nature of the system.

Figure 11

Temperature ($T/t_1$) dependence of the heat capacity of the system in the double-well superlattice with filling factor $\rho =1$ at tunneling amplitudes $t_2/t_1=t_3/t_1=0.1$, and $\lambda =U_{\sigma }/t_{\sigma }=-0.4(\sigma =1,2,3$). A wide peak occurs in the curve. When temperature exceeds the peak value, the capacity decreases slowly with increasing temperature due to the hard-core nature; while temperature lower than the peak value, the capacity falls rapidly with decreasing temperature because of finite excitation gaps.