Ground state and excitations of a Bose gas: From a harmonic trap to a double well

We determine the low-energy properties of a trapped Bose gas split in two by a potential barrier over the whole range of barrier heights and asymmetry between the wells. For either weak or strong coupling between the wells, our two-mode theory yields a two-site Bose-Hubbard Hamiltonian with the tunneling, interaction, and bias parameters calculated simply using an explicit form of two mode functions. When the potential barrier is relatively low, most of the particles occupy the condensate mode and our theory reduces to a two-mode version of the Bogoliubov theory, which gives a satisfactory estimate of the spatial shape and energy of the lowest collective excitation. When the barrier is high, our theory generalizes the standard two-site Bose-Hubbard model into the case of asymmetric modes, and correctly predicts a full separation of the modes in the limit of strong separation of the wells. We provide explicit analytic forms for the number squeezing and coherence as a function of particle number and temperature. We compare our theory to other two-mode theories for bosons in a double well and discuss their validity in different parameter regimes.

Figure 1

Energies of single particle (left) and collective (right) excitations in a double-well potential of the form (15): a harmonic trap of frequency $\omega =2\pi \times 100$ Hz, plus a Gaussian barrier of width $d=5$ $\mu$m and variable height $V_0$. Here the particle-particle interaction strength reduced to one dimension is $g_{1D}N=69.7l_{\mathrm{ho}}\hslash \omega$, where $l_{\mathrm{ho}}=\sqrt{\hslash /m\omega }$, and $g_{1D}N=4.98\times {10}^{-36}$ J m for ${}^{87}$Rb atoms. The dashed vertical line shows the barrier height for which $V_0=\mu$. (a) and (b) are for a symmetric potential ($x_0=0$): when the barrier grows, pairs of symmetric and antisymmetric modes become nearly degenerate. (c) and (d) are for a slightly asymmetric potential ($x_0=0.1$ $\mu$m), whereas (e) and (f) have a stronger asymmetry ($x_0=0.5\mu$m): pairs of modes sustain a constant energy splitting when the barrier grows, which indicates the suppression of tunneling between higher modes in the two wells. Yet, due to collisional repulsion, the first excited-state single-particle energy still tends to zero for a high barrier. A two-mode approximation is justified under these circumstances.

Figure 2

Comparison of the multimode Bogoliubov theory to the two-mode approximation. Top panels show the deviation $1-|\langle {\phi }_{u,v}|{\phi }_1\rangle {|}^2$ of ${\phi }_{u,v}$ (normalized $u_1$ and $v_1$ quasiparticle functions) in the multimode theory from the single-particle modes ${\phi }_1$ used for the two-mode approximation: (a) for a symmetric trap and (b) for an asymmetric trap ($x_0=0.5\mu$m). The bottom panels compare the first excitation energy in the two-mode theory to the multimode Bogoliubov theory for the two cases: (c) symmetric and (d) asymmetric. The maximum deviation is $\sim$19$\%$ in the harmonic case ($V_0=0$).

Figure 3

First excitation energy following from the Hamiltonian (19) as a function of the interaction energy for a harmonic trap. For relatively low particle numbers ($N=500$), the excitation energy grows linearly with the interaction energy. The excitation energy stays close to the correct value $E_1=\hslash \omega$ only in the limit $N\to \infty$. Neglecting the last term in Hamiltonian (19) leads to an excitation energy that stays close to the correct value, $\hslash \omega$, with less than 19$\%$ deviation (curve with solid circles).

Figure 4

Parameters of the Bose-Hubbard Hamiltonian (41) as a function of barrier height $V_0$, asymmetric case ($x_0=0.1\mu$m), with $N=1000$, and other parameters as used in Fig. 1. The ratio $UN/J$ determines the degree of number squeezing (see Sec. 5). The ratio $U/NJ$ determines ground-state coherence, which drops significantly when $J\sim U/N$. The parameter $G$, which measures the overlap between the left and right modes, was neglected in the Hamiltonian. At low barrier heights, its neglect is justified by compatibility with the Bogoliubov theory when most of the particles are in the condensate, which satisfies requirement (a) in Sec. 2b. At high barrier heights, $G$ drops quickly to negligible values. The dashed line denotes the energy difference between the two minima of the double well, which does not coincide with the bias parameter $\epsilon$.

Figure 5

(a) Parameters of the Bose-Hubbard Hamiltonian as in Fig. 4 for a gas of ${}^{87}$Rb atoms in a three-dimensional harmonic potential with longitudinal frequency ${\omega }_x=2\pi \times 500$ Hz and transverse frequencies ${\omega }_y={\omega }_z=2\pi \times 1.5$ kHz. The barrier parameters are as in Fig. 1. (b) Single-particle (dashed) and collective (solid) excitation energies as a function of barrier height. As in the one-dimensional example, the excitation energy for the harmonic potential ($V_0=0$) differs from the expected value $E_1=\hslash {\omega }_x$ by $\sim$20$\%$.

Figure 6

Energy of the lowest excitation as a function of barrier height for a slightly asymmetric double well, as in Fig. 1, and the Bose-Hubbard parameters of Fig. 4. The Josephson excitation energy $\hslash {\omega }_J$ of Eq. (52) (thin solid curve) is similar to the two-mode Bogoliubov energy in Fig. 2d. It is higher, by $\sim$$18\%$ at $V_0=0$ and $\sim$10$\%$ at high $V_0$, than the multimode Bogoliubov energy (dashed curve), which predicts the lowest oscillation frequency of a slightly perturbed initial particle distribution around the equilibrium state, as measured in Ref. [40]. The thick solid curves give the excitation energy obtained numerically by averaging over the eigenvalues of the Hamiltonian (41) over few values of $N$ around $\overline{N}=100,200,500$, and 1000 (top to bottom), such that $UN$ is fixed. In the low tunneling regime $J<U/N$, the ground state becomes a Fock (number) state with average excitation energy ${\overline{E}}_1\approx \frac{1}{2}U$. In this case, oscillations around ground-state populations of the two wells are suppressed because all number states are eigenstates of the Hamiltonian.

Figure 7

Coherence $\langle e^{i\widehat{\varphi }}\rangle$ between the left and right modes as follows from the parameters of the Bose-Hubbard Hamiltonian in Fig. 4 according to the quadratic approximation for the Josephson model [Eq. (62)]. (a) At zero temperature, for different numbers of particles, such that $g_{1D}N$ is kept constant. (b) For $N=1000$ for different temperatures (in units of $\hslash \omega /k_B$). At low barrier height, most of the particles are in a superposition of the left and right mode and the coherence is close to 1. The deviation from 1 indicates the existence of single-particle excitations in the ground state (condensate depletion). When the barrier grows, the ground state in (a) becomes incoherent due to the growth of the population of the excited single-particle mode in the ground state or due to thermal excitation of this mode in (b).

Figure 8

Tunneling rate $J$ in a symmetric double-well potential as in Fig. 1: The result of our theory (solid curve) is similar to the results of the semiclassical approximation [15] (circles) over most of the range where that approximation is applicable, namely, a symmetric potential with two minima. The theory based on eigenmodes of the GPE (dashed curve) coincides with the last two methods at the deep tunneling limit, while the HF calculation (dots) produces different results in both the deep tunneling regime and the low barrier regime.

Figure 9

Comparison between the shape of the first excited mode ${\phi }_1$ within different theories in two regimes: (a) no barrier and (b) high barrier. (a) In a harmonic potential, our theory yields a mode function ${\phi }_1$ fairly close to the normalized Bogoliubov quasiparticle function $v_1\left(\mathbf{r}\right)$, while the eigenmode of the GPE and the Hartree-Fock mode are different. (b) For a high barrier, the eigenmode of the GPE coincides with our theory and $v_1$, whereas the HF mode is still different.

Figure 10

Spatial modes obtained with the two-mode HF method for a strongly separated double-well potential with a slight asymmetry as in Fig. 1. (a) In the condensate-excitation representation, the excitation has a narrower profile compared to the condensate mode at each well, due to an effective attractive potential [Eq. (74)]. (b) This gives rise to considerable tails of the left mode ${\phi }_L$ in the right well, and vice versa, so that even in the limit of complete separation, the overlap between ${\phi }_L$ and ${\phi }_R$ does not vanish.