Numerical solutions without gain and loss

The starting point for this discussion is the caseγ = 0of an isolated system. For the first part of this analysis, the strength of the nonlinearity is fixed toN a= 1. The formation of the first vortex in a ground state of this system can be demonstrated using the finite-element method from Sec. 3.2. Figure 7.1 shows the wave function of the original ground state of the harmonic oscillator and the first excited state

1.4 1.5 1.6 1.7 1.8

0.7 0.75 0.8 0.85 0.9 0.95

EMF

Ω no vortex

1 vortex 2 vortices 3 vortices 2-4 vortices 4 vortices

Figure 7.2.: Mean-field energy of the ground states of the rotating harmonic oscillator in two dimensions. The borders of each region, at which the ground state changes, are shown by black bars. Figure 7.3 shows the corresponding wave functions for each red dot.

withl_z = 1. The state without vortices (v0, Fig. 7.1(a)) is not affected by the rotation.

Since it has no angular momentum, the mean-field energy stays constant for all values ofΩ. The first excited state (v₁, Fig. 7.1(b)) with one vortex andLˆ_z|v₁i=|v₁i also stays nearly constant in its shape. However, due to the angular momentum, this results in a linear decay of the mean-field energy. Already forΩ≈0.7356the statev₁ becomes the new ground state of the system.

Increasing the rotation frequency further gives rise to additional vortices, as the mean-field energies of even higher excited states cross the current ground state.

To reveal these states, various vortex configurations were created to act as initial values for the nonlinear root search used in the finite-element method. Figure 7.2 shows the mean-field energy of all states that act as ground state anywhere in the parameter regime discussed ofΩ< 0.95. The number of vortices increases from zero to four vortices in five distinguishable areas. While the areas in which no or one vortex exists are relatively large, fromΩ≈0.88on the number of vortices increases very fast.

One additional remark has to be made on the full eigenvalue spectrum: Due to a high number of bifurcations, portraying every stationary state would be incredibly complicated. For a growing rotation frequencyΩ, the number of vortices in the ground state increases. To this end, new states bifurcate from the existing

7.2. Two-dimensional calculations

−3 0 3

y

(a) (b) (c)

−3 0 3 x

−3 0 3

y

(d)

−3 0 3 x (e)

−3 0 3 x (f)

Figure 7.3.: The particle densities of the ground states of the rotating two dimen-sional harmonic oscillator are shown as color maps, in which darker regions correspond to higher densities. For each region distinguished in Fig. 7.2 one wave function is shown. The original ground state (a) has no vortex (v₀) and does not change for any value ofΩ. ForΩ = 0.85 (b) the ground state has one central vortex (v₁). The states with two vortices (v₂, (c), Ω = 0.88), three vortices (v₃, (d), Ω = 0.91) and four vortices (v₄, (f), Ω = 0.94) have their vortices equally spaced around the center of coordinates. AtΩ = 0.9315the four vortices already exist (v₂₋₄, (e)) but only a 2-fold rotational symmetry is present.

branches featuring higher number of vortices. However, the number of vortices is not even constant for a given branch of solutions. If the rotation frequencies increase, new vortices may appear in the stationary state although no bifurcation occurs. In the region from Ω = 0.7 to Ω = 0.95 presented in Fig. 7.2 a total of five different branches of solutions contribute to the six different ground states.

Figure 7.3 shows the wave functions corresponding to the different ground states that are labeled by red dots in Fig. 7.2.

Three classes of wave functions can be distinguished. The statesv₀ andv₁ have full cylindrical symmetry. One should emphasize that these two states are the only ground states shown in Fig. 7.3 that exist even forN a= 0andΩ = 0. The statesv₂, v₃, andv₄have emerged from the statev₀at some point featuring off-center vortices that break the rotational symmetry spontaneously. Instead, they exhibit a 2-, 3-and 4-fold rotational symmetry according to their number of vortices. The last class

−6 −3 0 3 6 x

−3 0 3

y

(a)

−6 −3 0 3 6 x

(b)

Figure 7.4.: The particle densities of the statev₂ for the rotation frequency Ω = 0.88 andN a = 1 (a) and N a = 2 (b) of the rotating two dimensional harmonic oscillator are shown as color maps, in which darker regions correspond to higher densities. Due to the higher interaction strength, two new vortices appear.

is represented by the statev₂₋₄ which has only a 2-fold symmetry although four vortices are present. Indeed, this configuration of vortices has not been created in an bifurcation from the statev₀ but developed continuously from the statev₂, i.e., the ground state at Ω = 0.88. While ramping up the rotation frequency to Ω = 0.9315, two additional vortices drift into the wave function. However, since these vortices are farther away from the rotation center, only the 2-fold symmetry remains. This behavior is a typical example for a branch of solutions with a varying number of vortices.

Up to now, the interaction strength was locked to N a = 1. However, since the occurrence of non-central vortices is only possible in nonlinear systems, the precise strength of the interaction should strongly influence the stationary states.

Figure 7.4 compares the wave functions of the statev₂ atΩ = 0.88andN a= 1 (a) and after increasing the nonlinearity toN a= 2(b). Two additional vortices appear and the original vortices are shifted slightly outwards as if the vortices reject each other. Moreover, a comparison with Fig. 7.3(c) reveals that the ground state for Ω = 0.9315and N a = 1, i.e. v₂₋₄, and the vortex state discovered at Ω = 0.88 andN a = 2are nearly the same. This conclusion is again in agreement with the results from Butts and Rokhsar [31]. In case of weak interaction strengths, they found that the transition from a non-rotating ground state to the first vortex state happens atΩ = 1−c awherecis some potential-dependent constant andais the scattering length. A stronger nonlinearity has the same effect on the choice of the ground state as if the rotation frequency is increased.

Since the states with two and more vortices break the cylindrical symmetry of the potential, the system exhibits an intrinsic infinite degeneracy. Each wave

7.2. Two-dimensional calculations function can be rotated by an arbitrary angle and still presents itself as a solution of the Gross-Pitaevskii equation. To select onlyPT-symmetric solutions one must require the modulus square of the wave functions to be symmetric with respect to the space reflectiony→ −y. This extracts a finite number of wave functions from the infinitely degenerate ground states.