Nonlinear quantum dynamics in a $\mathcal{PT}$-symmetric double well

We investigate the mean-field dynamics of a Bose-Einstein condensate described by the Gross-Pitaevskii equation (GPE) in a double-well potential with particle gain and loss, rendering the system $\mathcal{PT}$ symmetric. The stationary solutions of the system show a change from elliptically stable behavior to hyperbolically unstable behavior caused by the appearance of $\mathcal{PT}$-broken solutions of the GPE and influenced by the nonlinear interaction. The dynamical behavior is visualized using the Bloch sphere formalism. However, the dynamics is not restricted to the surface of the sphere due to the nonlinear and non-Hermitian nature of the system.

Figure 1

(a) The chemical potential $\mu$ of the eigenvalues without gain or loss, i.e., $\gamma =0$, as a function of the particle number scaled scattering length. The symmetric (blue solid branch) and antisymmetric (yellow dashed branch) stationary solutions arise at two almost identical bifurcation points at $Na\approx -1.4$, and the doubly degenerate symmetry-breaking eigenstates (green dot-dashed branch) at $Na\approx -0.7$. (b) Introducing the gain-loss $\gamma =0.02$ hardly changes $Re\mu$ of the symmetry-breaking ($\mathcal{PT}$-broken) eigenstates, but the two $\mathcal{PT}$-symmetric branches now arise at a tangent bifurcation. (c) The tangent bifurcation of the two $\mathcal{PT}$-symmetric states in the vicinity of the bifurcation point for a small value of $\gamma$. (d) The difference between the chemical potential of the $\mathcal{PT}$-symmetric states and their mean value $\overline{\mu }$. Different values of $\gamma$ are compared to the case $\gamma =0$ (black solid line).

Figure 2

(a) Real and (b) imaginary part of the chemical potential of the eigenstates for small values of $Na$. From top to bottom, the four different gain-loss contributions $\gamma =0$, $0.02$, $0.04$, and $0.042$ are used. The eigenvalues of the $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken solutions in three dimensions are in excellent agreement with the numerically exact one-dimensional solutions. Stable branches are highlighted as thick lines (see Sec. 3).

Figure 3

Real (solid lines) and imaginary (dotted lines) part of the wave function of the $\mathcal{PT}$-symmetric (a) ground and (b) first excited state and the two $\mathcal{PT}$-broken states with a (c) negative and (d) positive imaginary part of the chemical potential. For $\gamma =0$, the ground state is parity symmetric and the first excited state is antisymmetric. For finite values of $\gamma$, the real part of both $\mathcal{PT}$-symmetric wave functions is even and the imaginary part is odd. The $\mathcal{PT}$-broken states are asymmetric, thus breaking the $\mathcal{PT}$ symmetry of the Hamiltonian.

Figure 4

(a),(c) Real and (b),(d) imaginary parts of the eigenvalues ${\omega }_{g/e}$ of the Bogoliubov-de Gennes equations for (a),(b) the ground state and (c),(d) the first excited state. Only the first nontrivial eigenvalue with the smallest absolute real part is shown. Due to the symmetries of the equations, the real and the imaginary parts of the eigenvalues both occur in pairs. Both states are stable for weak interactions but become unstable in the vicinity of the pitchfork bifurcations.

Figure 5

(a),(c) Real and (b),(d) imaginary part of the Bogoliubov-de Gennes eigenvalues ${\omega }_{\mathrm{i}\pm }$ for the $\mathcal{PT}$-broken states with (a),(b) positive and (c),(d) negative imaginary part of the chemical potential $\mu$ at $\gamma =0.03$. The four eigenvalues with the smallest real parts are shown. In addition to each eigenvalue $\omega$, a solution with negative real part $-{\omega }^{*}$ exists. All but one eigenvalue of the $\mathcal{PT}$-broken state with $Im\mu >0$ ($Im\mu <0$) have a negative (positive) imaginary part.

Figure 6

Imaginary part of the Bogoliubov-de Gennes eigenvalues of all states involved in (a) the pitchfork bifurcation at $\gamma =0.03$ and (b) the tangent bifurcation at $\gamma =0.04$. The ground state becomes unstable before the pitchfork bifurcation at which the $\mathcal{PT}$-broken solutions arise. For decreasing $N_{0}a$, the excited state becomes unstable before the tangent bifurcation at which the two $\mathcal{PT}$-symmetric states coalesce and vanish. At the bifurcations, the stability eigenvalues of the states involved coalesce since the states themselves become equal. The bifurcation points are marked by black vertical lines.

Figure 7

Imaginary part of the stability eigenvalues $\omega$ of the Bogoliubov-de Gennes equations (8) for the norm-independent nonlinearity (7) in the vicinity of (a) the pitchfork bifurcation and (b) the tangent bifurcation. A constant gain-loss parameter (a) $\gamma =0.03$ and (b) $\gamma =0.04$ is used. The ground state becomes unstable at the pitchfork bifurcation and the excited state is stable until it vanishes at the tangent bifurcation. The $\mathcal{PT}$-broken states are now a pure sink or source. The bifurcation points are marked by black vertical lines.

Figure 8

The Bloch sphere representation using the coordinates defined in Eq. (10). The basis vectors $|e_1\rangle$ and $|e_2\rangle$ correspond to the north and south pole, respectively. On the front side of the plotted great cycle (black line), the azimuth angle is $\phi =0$, and on the back side, $\phi =\pi$. All states residing on the plane in which this great cycle lies are $\mathcal{PT}$ symmetric and the time evolution in the system is symmetric with respect to this plane. Furthermore, the great cycle is used as the starting point for trajectories discussed in Sec. 4c. Additionally, the four eigenstates are plotted for $\gamma =0.03$ and the value of $Na$ corresponding to the current location (cf. Sec. 4b).

Figure 9

The dynamics of wave packets projected to the space spanned by the $\mathcal{PT}$-symmetric ground and excited state illustrated on a Bloch sphere. The interaction strength $Na=-0.05$ on the surface of the sphere is constant in all figures, whereas the gain-loss parameters are varied, with (a) $\gamma =0$, (b) $\gamma =0.0025$, (c) $\gamma =0.01$, (d) $\gamma =0.02$, (e) $\gamma =0.03$, and (f) $\gamma =0.04$. The solutions of the time-independent GPE are plotted for orientation (pink lines, cf. Fig. 8). The ground state starts in the center of the sphere and goes through the north pole, whereas the penetration point of the excited state is at the south pole for $\gamma =0$ and wanders on the meridian $\phi =0$ to the north pole for increasing values of $\gamma$. The $\mathcal{PT}$-broken solutions emerge from the ground state. All wave packets shown start on a great cycle through the north pole, south pole, and the excited state. Wave packets starting on the meridian between the ground state at the north pole and the excited state on the front side of the sphere evolve to a smaller norm, thus diving into the sphere (closed thick blue lines inside the sphere). Wave packets starting in the remaining region of the great circle either show oscillations with a larger norm outside the sphere (closed red lines outside the sphere) or diverge while encircling the $\mathcal{PT}$-broken solutions (green lines departing from the left and the right side).