Tilted optical lattices with defects as realizations of $\mathcal{PT}$ symmetry in Bose-Einstein condensates

A $\mathcal{PT}$-symmetric Bose-Einstein condensate can theoretically be described using a complex optical potential, however, the experimental realization of such an optical potential describing the coherent incoupling and outcoupling of particles is a nontrivial task. We propose an experiment for a quantum-mechanical realization of a $\mathcal{PT}$-symmetric system, where the $\mathcal{PT}$-symmetric currents are implemented by an accelerating Bose-Einstein condensate in a titled optical lattice. A defect consisting of two wells at the same energy level then acts as a $\mathcal{PT}$-symmetric double well if the tilt in the energy offsets of all further wells in the lattice is varied in time. We map the time dependence of the amplitudes of a frozen Gaussian variational ansatz to a matrix model and increase the system size step by step starting with a six-well setup. In terms of this simple matrix model, we derive conditions under which two wells of the Hermitian multiwell system behave exactly as the two wells of the $\mathcal{PT}$-symmetric system.

Figure 1

(a) The $\mathcal{PT}$-symmetric two-well system as discussed in Sec. 2. The imaginary part of the external potential induces currents from the environment to the left well and from the right well to the environment. (b) By embedding the two-well system in two additional outer wells, the tunneling currents may be used as an implementation of $\mathcal{PT}$ symmetry [21]. (c) With more outer wells the reservoir is larger, which can be advantageous. (d) In the limit of an infinite number of reservoir wells, the system may be interpreted as the two-well system embedded in an optical lattice.

Figure 2

(a) Population of the outer wells for the four-mode model, where an adiabatic current ramp is simulated with a final time of $t_{\text{f}}=20\hslash /J_{m,m+1}$. (b) The time that is available for realizing $\mathcal{PT}$ symmetry should be extended. On the one hand, one can simply increase the initial number of particles in the reservoir wells. (c) On the other hand, the six-mode model can be used to effectively enlarge the reservoir. Now the difference of the particle numbers between the embedded and reservoir wells is smaller, which could help measure the well population in an experiment. The nearly constant population of the embedded wells $n_m\left(t\right)=n_{m+1}\left(t\right)\approx 0.5$ is not shown.

Figure 3

(a) Tunneling elements $J_{m-1,m}$ and $J_{m+1,m+2}$ and (b) on-site energies $E_m$ and $E_{m+1}$. The curves (i)–(iii) correspond to the scenarios discussed in Figs. 2–2, respectively. Comparing the curves (ii) and (iii), for the tunneling elements there is not much difference. The on-site energies, however, differ by some amount. The magnitude for the six-mode model is smaller, which could be an advantage in an experiment.

Figure 4

Adiabatic current ramp with two reservoir wells on each side with $N_{\text{w}}=6$ wells. Shown are (a) particle numbers in the reservoir wells, (b) interwell currents, (c) tunneling elements, and (d) on-site energies. We start with the ground state for the parameters given in the text. The target currents are chosen in such a way that the left reservoir wells become empty at nearly the same time. This allows us to effectively use all particles in the reservoir wells and thus to maximize the time that is available for $\mathcal{PT}$ symmetry.

Figure 5

Nonquasistationary solutions for the parameters $\mathrm{\Gamma }/J_{56}=0.6$ and $g/J_{56}=10$ with $N_{\text{w}}=10$ wells. See the text for initial conditions. Shown are (a) particle numbers, (b) interwell currents, (c) tunneling elements, and (d) on-site energies. The currents are chosen such that the reservoir wells are emptying with a rate proportional to the initial population. With this choice the reservoir can be used in an optimal way. Here we give up the leveled out on-site energies.

Figure 6

(a) The $\mathcal{PT}$ parameter $\mathrm{\Gamma }$, (b) tunneling elements, (c) energy slop $\mathrm{\Delta }E$, and (d) energy offset $E^{\left(0\right)}$ of the energy distribution (37). For $t=0$ a Gaussian wave function has been prepared and the $\mathcal{PT}$ symmetry is realized using an adiabatic current ramp for ${\mathrm{\Gamma }}_{\text{f}}/J_{m,m+1}=0.3$ and $t_{\text{f}}=40\hslash /J_{m,m+1}$. The calculation is repeated for different values of the interaction strength. For $t<t_{\text{f}}$ the slope $\mathrm{\Delta }E$ is used to accelerate the wave packet, whereas for $t>t_{\text{f}}$ the slope must be changed to compensate for the decrease in population of the moving wave packet. The offset $E^{\left(0\right)}$ is the necessary perturbation so that $\mathcal{PT}$ symmetry can be realized.

Figure 7

Wave function in position space of the scenario of Fig. 6 for different times (a)–(d) and a vanishing interaction strength. The initial Gaussian wave packet is accelerated to the right due to the negative energy slope $\mathrm{\Delta }E$. At the same time the purely Gaussian shape is perturbed due to the energy offset. In combination, these effects contribute to the realization of $\mathcal{PT}$ symmetry in the embedded wells $m=150$ and $m+1=151$ with a constant number of particles $n_{150}$ and $n_{151}$. The dashed curves give the wave function of the semiclassical approximation, where the energy offset is ignored.

Figure 8

Same as Fig. 7, but for the wave function in momentum space. The dashed line gives the wave function of the semiclassical approximation. The last row (e) and (f) shows an enlarged plot of the previous row (c) and (d). In momentum space, the wave function is dominated by the sharp peak in $q_0$, of which the time dependence can be calculated according to the semiclassical approximation and which gives a good description for the dominating peak. Only in the enlarged views (e) and (f) one notices the perturbation due to the energy offset. This perturbation is getting stronger during the simulation time, however, it does not reach the magnitude of the momentum peak.