4. The PT -symmetric double well 29
5.3. Numerical test of estimates for singular perturbations
5.3.2. Eigenvalue spectra
Since the system (5.12) is one-dimensional, the eigenvaluesµcan be obtained using the integration method presented in Sec. 3.1. Especially the singular perturbations can be modeled easily by introducing a jump of the first derivative in the integration of the differential equation. Figure 5.2 shows the lower section of the harmonic oscillator spectrum. The eigenvalues are shown for the linear system,g = 0, and increasing perturbation strengthsγ. The perturbations are used at three different positions, close to the center of the harmonic oscillator,b = 0.2(a),(b), at the turning point of the ground state, b = 1(c), and at that of the third excited state,b =√
7 (d).Observing the real (a) and imaginary (b) parts of the eigenvalues in Fig. 5.2, one recognizes that successive pairs of eigenvalues coalesce in exceptional points, from where onwards they turn into complex conjugate pairs. This is the expected behavior for a PT-symmetric system as the double well presented in Chap. 4.
Going up in the spectrum, the branch points are shifted to larger values ofγ. For the most part, the real parts of the eigenvalues of the PT-broken states remain approximately constant beyond the branch points. The imaginary parts quickly tend to zero after their initial growth. What is surprising is that the eighth and ninth excited state experience huge shifts in both their real and imaginary parts of the eigenvalue shortly after the bifurcation. Eventually they saturate at a value above the 30th excited state.
For perturbations further away from the center, shown in Figs. 5.2 (c),(d), only the real part of the eigenvalues is shown. The delta functions reside outside the
0 20 40 60
0 5 10 15 20 25 30
(a)
0 20 40 60
0 5 10 15 20 25 30
(c)
−3
−2
−1 0 1 2 3
0 5 10 15 20 25 30
(b)
0 20 40 60
0 5 10 15 20 25 30
(d)
Reµ
γ
Reµ
γ
Imµ
γ
Reµ
γ
Figure 5.2.: Eigenvaluesµevolving from the unperturbed levels of the harmonic oscillator with n = 0, . . . ,29in dependence of the coupling strength γ of the non-Hermiticity for the linear (g = 0) Gross-Pitaevskii equa-tion (5.12). For singular perturbaequa-tions near the originb= 0.2, the real (a) and imaginary (b) parts are shown. Every two adjacent eigenvalues bifurcate and twoPT-broken states withImµ6= 0emerge. For higher distances between the center and the perturbations, b = 1 (c), the perturbation lies at the classical turning point of the ground state. Ac-cordingly, the ground state is not involved in any bifurcation scenario.
Atb=√
7(d), i.e., the turning point of the state withn = 3, all states up to the third excited state remainPT symmetric.
5.3. Numerical test of estimates for singular perturbations
1 2 3 4 5
0 1 2 3 4 5
(a)
−2
−1 0 1 2
0 1 2 3 4 5
(b)
Reµ
γ
Imµ
γ b= 0.897 b= 0.915 b= 0.925
Figure 5.3.: Transition of the bifurcation partner of the unperturbed energy level n = 1, i.e., the first excited state, from the ground (n = 0) to the second excited state (n = 2) by varying the position of the delta functions in the vicinity ofb = 0.9. The real (a) and imaginary (b) parts of the eigenvalues are shown.
classically allowed region of the unperturbed ground state n= 0(c) or all states up to n = 3 (d) respectively. Thus, it is no surprise that for b = 1 the ground state no longer “feels” the delta functions, and therefore no longer unites with the first excited state at a branch point. Rather its eigenvalue remains real for any perturbation strengthγ. The same can be observed for the states withn= 0,1,2,3 in Fig. 5.2(d). However, these states are not the only robust [84] states of this system. If the delta functions reside near a node of the wave function, their effect is also too weak for the state to form a branch point. For the case b = √
7 this behavior is found for many states, namely n= 6,7,10,11,14,19,22, and 27.
Comparison of Figs. 5.2(a) and (c) shows, that forb= 0.2the first excited state forms a branch point with the ground state, while forb = 1it branches with the second excited state. The transition from one coalescence behavior to the other is illustrated in Fig. 5.3. The real and imaginary parts of the eigenvalues emerging from the lowest lying three eigenstates are shown as functions ofγ and the three positions of the delta functions b = 0.897, b = 0.915, andb = 0.925 are compared.
It is evident that at b = 0.897 the ground and first excited state still coalesce, giving rise to a pair of complex conjugate eigenvalues. Following the real part of these eigenvalues reveals a crossing with the second excited state. It has to be emphasized that the states are not degenerate due to the finite imaginary part of the broken states, i.e., only the real parts cross. After the crossing the state remainsPT broken.
The behavior for b = 0.915 is similar at first, since the first branch point still exists. However the eigenvalues resulting from this coalescence of the ground and
0 10 20 30 40 50 60
0 1 2 3 4 5 6 7 8 9
Reµ
γ
Figure 5.4.: Real parts of the eigenvalues of (5.12) with a nonvanishing nonlin-earityg = 5perturbed by the two delta functions with distanceb = 1 from the trap center. Eigenvalues evolving from the unperturbed levels up ton = 29 are shown. Magenta lines depict purely real, i.e., PT-symmetric states, whereas the real parts of every two complex conjugate eigenvalues are shown by single green lines. These complex eigenvalues no longer start at the branch points of the real eigenvalues but bifurcate at previously unspecific perturbation strengthsγ from anyPT-symmetric state’s eigenvalue.
first excited state does not stayPT symmetric. Instead they again split into two real eigenvalues, the lower of which remains real for anyγ. The upper eigenvalue again collides with the second excited state, this time forming a branch point and giving birth to a new pair of complex eigenvalues. In a last step forb = 0.925, the transition is completed. The two branch points have merged and the intermediate real eigenvalue has vanished. The ground state is represented by a single real energy level while the first two excited states coalesce. A similar behavior, the emergence as well as the transition betweenrobustandfragilestates, has already been discussed for singular perturbations in a square well [84, 85].
The present work, however, deals with Bose-Einstein condensates whose interac-tion renders the characterizing differential equainterac-tion nonlinear. Eq. (5.12) already implements the appropriate term for contact interactions, the strength of which is controlled by the parameter g. Figure 5.4 shows the real part of the eigenvalue spectrum for a nonlinearity g = 5 with the delta functions placed at b = 1. A direct comparison with the linear case in Fig. 5.2(c) shows major differences. Even though the perturbation lies at its classical turning point of the linear system, the
5.3. Numerical test of estimates for singular perturbations
1 10 100
(a)
10−4 10−3 10−2
0.5
10−1
1 10 100
(b)
n
∆µ ∆µ
n
∝ n−0.5
∝ log(n)/n3/2 b= 1 b= 2 b= 3 b= 5
Figure 5.5.: Comparison between rigorous estimates (fully drawn curves) and numerical data. The shift of the eigenvalues for a perturbation with strengthγ = 1at different positionsbfrom the unperturbed levels as a function of the quantum numbernis shown. The linear case (a) shows a perfect agreement between data and estimate while the nonlinear case (b) shows a slower decay behavior of approximatelyn−0.37.
ground state still coalesces with the first excited state revealing an increase in the wave function’s extension due to the repulsive interaction. However, at the branch point both states vanish with noPT-broken state emerging. One should emphasize that an analytic continuation could once more reveal new states, which would still arise from the tangent bifurcation [21, 68].
Like in the double-well system presented in Chap. 4, the broken states are shifted away due to the nonlinear contribution and emerge at new previously unspecific points in the spectrum. As a matter of fact, the lowest lying broken state arises from the robust third excited state that does not break symmetry for any parameterγ. In general, thus, it is not sufficient to account for γ to pinpoint where complex eigenvalues emerge. However, the calculations in the nonlinear double well indicate that such an emergence of symmetry-broken states does not necessarily lead to a stability transition in states not involved in the bifurcation.