Results for the repulsive configuration

The results presented in this section have been obtained with the condensates placed in the repulsive configuration. Although we performed calculations for a wide range of the parametersa and Γ we will concentrate here on a domain, where we found interesting phenomena. Still, keep in mind that the following effects and characteristics can be present in different parameter regions for a different dipole strength a_dd. In figure 6.4 results for a_dd = 0.3 and several values of the scattering length a are given. We show the mean-field energy and the chemical potential both as functions of the gain-loss parameter Γ in figure 6.4b and figure 6.4a, respectively. For a =−0.03 at Γ = 0 four different states are present. The two states with the higher mean-field energyS_1,1 andS_1,2 break thePT symmetry as it can be seen in figures 6.4e–f. This leads to complex energy eigenvalues and thus to a finite imaginary part of the mean-field energy and the chemical potential.

By the application of the PT operator to these PT-broken states S_1,α, (α = 1,2) two new states S_1,α⁰ which have complex conjugate E_mf and µ can be generated.

These new states S_1,α⁰ are also stationary solutions and therefore altogether six states exist. However, we omit the statesS_1,α⁰ in figure6.4a,band in the following discussion as they exhibit the same behavior as the states S_1,α. The lower states S_0,1 andS_0,2 preserve thePT symmetry and thus yield real E_mf andµ. Both pairs of states S_0,α and S_1,α disappear in two separate tangent bifurcations T₀ and T₁, respectively (see figure 6.4b).

Decreasing the scattering lengthacausesE_mf^r of thePT-broken states to become smaller and approach the values of the PT-symmetric states. In this process the mean-field energy of the state S_1,1 crosses both of the PT-symmetric states S_0,α (see e.g. dashed blue curve for a = −0.035 in figure 6.4a), yet the crossing point is no exceptional point i.e. the wave functions are diverse. A further decrease of

(a)

-297.5 -297 -296.5 -296 -295.5 -295

Er mf

S0,1

S_0,2

S1,2 S1,1

0 0.5 1 1.5

0 0.5 1 1.5 2 2.5

Ei mf

Γ a=−0.0425 a=−0.038 a=−0.035 a=−0.03

(b)

-297 -296 -295 -294 -293 -292 -291

µr

T0

T1

0 0.5 1 1.5

0 0.5 1 1.5 2 2.5

µi

Γ a=−0.0425 a=−0.038 a=−0.035 a=−0.03

(c) S_0,1 (d) S_0,2 (e) S_1,1 (f) S_1,2

Figure 6.4.: (a) Mean-field energy and (b) chemical potential as functions of the gain-loss parameter Γ, where the upper panel shows the real and the lower one the imaginary part. The dipole strength is set toa_dd = 0.3. For a=−0.03 the arrows in (a) label the states for which in (c)-(f) the absorption images are shown. Thex-axis is plotted as abscissa and thez-axis as ordinate, corresponding to figure6.3b(the dipoles are therefore aligned into the y-direction of view). The field of view is 1×1. For a detailed description see the text.

a has an effect similar to that observed in section 6.2. Here, we include figure 6.5 for detailed discussion. The stateS_1,2 separates in a pitchfork bifurcation from the PT-symmetric state S_0,2 (see red dashed-dotted line for a=−0.038 in figure 6.4a or solid blue line in figure 6.5). The pitchfork bifurcation is labeled P in figure 6.5b, whereas the tangent bifurcations of the S_0,α and S_1,α states are denoted T₀ and T₁, respectively. For the scattering length a ≈ −0.03985 (dashed light-blue line in figure 6.5) the tangent bifurcation T₀ of the PT-symmetric states and the pitchfork bifurcation P merge in one point, marked with a green dot, labeled E

6.3. Dipolar BECs in a PT-symmetric double-well potential

(a)

-297 -296.8 -296.6 -296.4 -296.2 -296

Er mf

0 0.1 0.2 0.3

0 0.1 0.2 0.3 0.4 0.5

Ei mf

Γ a=−0.0415 a=−0.03985 a=−0.038

(b)

-295 -294 -293

µr

P

T1

T0

E

0 0.1 0.2 0.3

0 0.1 0.2 0.3 0.4 0.5

µi

Γ a=−0.0415 a=−0.03985 a=−0.038

Figure 6.5.: (a) Mean-field energy and (b) chemical potential as functions of the gain-loss parameter Γ, where the upper panel shows the real and the lower one the imaginary part. The dipole strength is set to a_dd = 0.3.

in figure 6.5b. The singular point E is present without the DDI for vanishing nonlinearity g = 0 (see figure 6.1) as well and it has been shown that it possesses the properties of an exceptional point of order 4 [117]. However, here this point appears at nontrivial parameters and particularly for a finite nonlinearity. At E the energy eigenvalues obviously become real and the wave function preserves the PT symmetry. Note that the tangent bifurcationT₁has not merged with this point and remains separate at a slightly higher value of Γ. Yet this might be different for another value of the dipole strength.

Further decreasing the scattering length leads P to move down the other PT -symmetric state S_0,1, see e.g. the dashed-dotted magenta line for a = −0.0415 in figure6.5. There, the tangent bifurcationT₁ has already merged with the pitchfork bifurcation so that the state S_1,1 has disappeared and the state S_1,2 is the only remaining PT-broken state. If the scattering length is tuned lower even more, P is shifted to smaller Γ until the remaining PT-broken state eventually disappears and only PT-symmetric states are left (see dashed-double-dotted green line for a = −0.0425 in figure 6.4). If we decrease a from there on, both PT-symmetric solutions would disappear as well. This behavior is well-known for a condensate in a real double-well potential [42].

(a) S_0,1

-1 -0.5 0 0.5 1

Λr

-2 -1 0 1

0 0.1 0.2 0.3 0.4

Λi

Γ

(b) S_0,2

-0.5 -0.25 0 0.25 0.5

Λr

-0.5 -0.25 0 0.25

0 0.1 0.2 0.3 0.4

Λi

Γ (c) S_1,1

-2 -1 0 1 2

Λr

-60 -30 0 30

0 0.1 0.2 0.3 0.4

Λi

Γ

(d) S_1,2

-1 -0.5 0 0.5 1

Λr

-1 -0.5 0 0.5

0 0.1 0.2 0.3 0.4

Λi

Γ

Figure 6.6.: Eigenvalues Λ = Λ^r+ iΛⁱ of the Jacobian (3.108) as functions of the gain-loss parameter Γ for the states with labels given above each plot, a = −0.038, and a_dd = 0.3. In (a) the stability eigenvalues of the PT-symmetric state S_0,1 are shown. All eigenvalues are imaginary with vanishing real parts implying stable fixed points. The upper state in (b) is stable up to the pitchfork bifurcation P. The PT -broken states S_1,α shown in (c) and (d) are unstable in the whole range of Γ, thereby in (d) a pitchfork bifurcation can be seen (for ΓP . 0.24 we get S_1,2 = S_0,2, S_0,2 is plotted in light green here).

Stability and dynamics

The linear stability of the stationary points is investigated by the use of the method presented in section 3.4.3. Vanishing real parts of the stability eigenvalues of the Jacobian (3.108) correspond to stable fixed points. In figure 6.6 the stability eigenvalues for a = −0.038 (blue solid line in figure 6.5) are shown for all four

6.3. Dipolar BECs in a PT-symmetric double-well potential

(a)

0.10 0.20.3 0.40.5 0.60.7

Ik

0.40.6 0.81

0 200 400 600 800 1000 1200 P k,lIkl 0

t

I¹ (left) I² (right)

(b)

Figure 6.7.: Real-time evolution of the state S_0,2 for a_dd = 0.3, a = −0.038, and Γ = 0.2. (a) The upper panel shows the populations of the left and the right well (see definition of quantities in section 5.3.1). The lower panel shows the overall norm of the wave function. In (b) absorption images during the first oscillation are plotted with the parameters given in figure 6.4. It can be seen that during the oscillation the wave function takes the shape of the different PT-broken states.

states (the states obtained by the application of the PT-operator show the same behavior as the statesS_1,α). The lowest-lying stateS_0,1 is stable in the whole range of Γ as it is not involved in any pitchfork bifurcation with a PT-breaking state.

This is different for the state S_0,2 (see figure6.6b) which looses its stability in the pitchfork bifurcation at ΓP ≈0.24. The PT-broken states shown in figures 6.6c,d are unstable in the whole range, where they exist.

The real-time evolution of the stable state S_0,1 provides no further insight as all parameters stay constant. Yet, the corresponding evolution of the state S_0,2 reveals some very interesting effects. In figure 6.7 the real-time evolution of this PT-symmetric state with a higher mean-field energy thanS_0,1is shown. At first the state remains approximately constant. Then a nonlinear oscillation between the wells sets in. The shape of the wave function which is illustrated by the absorption images in figure 6.7b passes through states similar to the states shown in figures

6.4c–f. This behavior can be interpreted in the following way. As Γ < ΓP the PT-broken state has not separated from the state S_0,2. Although the PT-broken state is not present, it influences the dynamics of the system.

Interestingly, the oscillations observed in figure6.7 can not be classified as small oscillations corresponding to the smallest finite Λⁱ(Γ = 0.2)

min ≈ 1 as the cor-responding time scale would be one magnitude smaller than the time scale of the oscillations in figure 6.7a. This confirms the fact that a strong nonlinear coupling between thePT-symmetric and thePT-broken states takes place. The importance of this coupling is impressively demonstrated in figure6.8a, where the presence of the other states prevents the collapse of the condensate by a dynamical stabi-lization mechanism. The larger the energetic distance between the states is, the smaller the coupling gets. For Γ = 0.4 (see figure6.8b) the dynamics are no longer stabilized and the local collapse is induced by small perturbations.

Im Dokument Variational approaches to dipolar Bose-Einstein condensates (Seite 103-108)