Interaction blockade for bosons in an asymmetric double well

The interaction blockade phenomenon isolates the motion of a single quantum particle within a multiparticle system, in particular for coherent oscillations in and out of a region affected by the blockade mechanism. For identical quantum particles with Bose statistics, the presence of the other particles is still felt by a bosonic stimulation factor $\sqrt{N}$ that speeds up the coherent oscillations, where $N$ is the number of bosons. Here we propose an experiment to observe this enhancement factor with a small number of bosonic atoms. The proposed protocol realizes an asymmetric double-well potential with multiple optical tweezer laser beams. The ability to adjust bias independently of the coherent coupling between the wells allows the potential to be loaded with different particle numbers while maintaining the resonance condition needed for coherent oscillations. Numerical simulations with up to three bosons in a realistic potential generated by three optical tweezers predict that the relevant avoided level crossing can be probed and the expected bosonic enhancement factor observed.

Figure 1

Level structure within the Bose–Hubbard model for (left) $N=1$ and (right) $N=2$. Rapid quench to the position of the anticrossing will induce coherent oscillations with a frequency proportional to the inverse of the energy splitting. The parameters are $U_L/J=7, U_R/J=4$.

Figure 2

Energy splittings at tunneling resonances involving single boson tunneling for $N=2$ (left) and $N=3$ (right). Markers correspond to numerical results from exact diagonalization of the full Hamiltonian. Solid lines correspond to the expected $\sqrt{N}$ scaling in the interaction blockade limit. Broken lines denote the result of first-order perturbation in Eq. (8). Other parameter: $U_R=4U_L/7$.

Figure 3

Asymmetric double-well optical tweezer at the $x-z$ plane. Shown is the potential $V\left(r\right)$ of Eq. (12) formed with $P=3$ laser beams for parameter values: $\{V_i/h\}\left(\mathrm{kHz}\right)=\{4659.775,4137.962,4585.886\}$ and $\{x_i/w_0\}=\{-0.658,0.264,1.176\}$.

Figure 4

Single-particle energy splitting $\mathrm{\Delta }E$ for $N=1$ and intrawell interaction energy $U_{\ell }=U_L=U_R$ of a symmetric double with two optical tweezer beams as a function of the beam separation $d$ with $V_0/h=5\mathrm{MHz}$. The horizontal dashed-dotted line represents the minimum energy splitting for experimentally observable single-particle tunneling.

Figure 5

Single-particle energies in the asymmetric double-well trap as a function of the left-most laser beam depth $V_1$: (left) First four single-particle energies and (right) zoom-in of the ground and first-excited energies. The single-particle resonance identified from an avoided level crossing is indicated by the circle.

Figure 6

Ground, first-, and second-excited states as a function of $V_3$ for the asymmetric double well for (left) $N=2$ and (right) $N=3$. The reference ground-state energies are $E_1^{\mathrm{AC}}/h=-11.653\mathrm{MHz}$ for $N=2$ and $E_1^{\mathrm{AC}}/h=-17.480\mathrm{MHz}$ for $N=3$. Solid lines correspond to MCTDHB results. Broken lines correspond to diagonalization of the Bose–Hubbard Hamiltonian. Resonances are indicated in circles.

Figure 7

Slice of the two-particle wave function for $N=2$ at $y_1=y_2=y_3=z_1=z_2=z_3=0$ as a function of $x_1$ and $x_2$. Left panel shows the ground state, right panel shows the first-excited state.

Figure 8

Three-particle wave function for $N=3$ at $y_1=y_2=y_3=z_1=z_2=z_3=0$. The blue surfaces correspond to positive values while the red surfaces correspond to negative values of the wave function. Left panel shows the ground state and right panel shows the first-excited state.

Figure 9

Dynamics for $N=2$ following a quench to the position of the avoided crossing from the ground state at the largest value of $V_3$ in Fig. 6. Dashed curve (left $y$ axis) denotes the dynamics of the expectation value of mode occupation number in the right (wide) well $\langle {\widehat{n}}_R\rangle$ within the Bose–Hubbard model. Solid curve (right $y$ axis) depicts the dynamics of the average center-of-mass position in the axis of asymmetry $\langle x\rangle /w_0$ calculated from MCTDHB.

Figure 10

Multiple particle energy splittings as a function of $N$. Solid line corresponds to the expected $\sqrt{N}$ dependence. The dashed line corresponds to the first-order approximated splitting given by Eq. (8) for an effective Bose–Hubbard parameter of $U_L/J\approx 7$ and $U_R/J\approx 4$. Blue crosses correspond to the numerical results at the positions of resonance indicated by the circles in Figs. 5 and 6.

Figure 11

Relaxation using imaginary time propagation within MCTDHB for $N=2\left({}^{87}\mathrm{Rb}\right)$ at the avoided crossing. The left panels shows the energy difference between the ground and first-excited state. The right panel shows the natural occupancies of the ground-state single-particle density matrix.

Figure 12

Slice of the natural orbitals for the ground state at the avoided crossing of $N=2$ and $M=4$ at $z=0$ as a function of $x$ and $y$. Top-left panel shows the highest-occupied orbital; top-right panel shows the second-highest occupied orbital; bottom-right panels shows the third-highest occupied orbital; bottom-right panel shows the least-occupied orbital.