Double barrier potential

In this first example we study a one-dimensional system with a double barrier poten-tial. On the one hand this allows for a comparison to results known from Ref. [69] and to confirmation of our numerical method. On the other side we give further interpre-tation for the observed results, and especially relate the approach from the scattering description to the problem of a decaying quasi bound state in the nonlinear regime. To this end we consider the one-dimensional Gross-Pitaevskii equation:

i~∂ where ˜g = 2as~ω⊥ is the one-dimensional interaction strength. We especially focus on a one-dimensional wave guide with a harmonic transverse confinement and a corre-sponding harmonic oscillator width of a_⊥ =p

~/(mω_⊥) in the following. We consider a double barrier potential given by:

V(x, y) =V0

The transmission is treated as a function of the chemical potential of the incoming wave with wave vector k = √

2mµ/~ and fixed incident current. For definiteness we specify a double barrier potential with a height of V0 = 1.1~ω and a barrier width of σ= 1.0a⊥. The two barriers are separated by a distance of L= 4.25a⊥.

For the linear Schr¨odinger equation the double barrier potential can also be seen as a Fabry-P´erot interferometer where we get the well-known Breit-Wigner distribution for the transmission from.

T(µ) = (~γ₀/2)²

(µ−µ0)²+ (~γ0/2)² (3.43)

A transmission of T = 1 is observed if the energy µ of the incoming monochromatic wave has a resonance frequency with energyµ0. The width of the resonance is deter-mined by the decay rate γ0. Such a transmission spectrum is shown with the black line in Fig. 3.7. Of course the above Breit-Wigner distribution is only a good approx-imation for well separated resonances that do not overlap, which is the case for our

Figure 3.7: The transmission through a double barrier potential is studied. The solid line refers to the time-dependent integration of the Gross-Pitaevskii equation. The dotted line shows an alternative approach where the stationary Gross-Pitaevskii equation is solved for a fixed incident current (see Ref. [65, 73] for the methods used). Parameters: V₀ = 1.1~ω, σ= 1.0a_⊥,L= 4.25a_⊥,j_in= 1ω.

parameters. The energy of the lowest resonance is found at µ0 = 0.389~ω. And the next resonance is already above the potential maxima.

In Ref. [69] the transmission spectrum was studied for the nonlinear Gross-Pitaevskii equation then. It was shown there that the resonant peak bends to the right for re-pulsive atom-atom interaction and to the left for attractive interaction. This bending results in a bi-stability phenomenon: For a fixed incident current more than one sta-tionary scattering solution to the stasta-tionary Gross-Pitaevskii equation is found, which is shown in the dotted lines of Fig. 3.7. The solution with high transmission results in a high population of the quasi bound state in the resonator, whereas for the solution with low transmission also a low density inside the resonator is found. Additionally we carried out the time-dependent integration process of the Gross-Pitaevskii equation, where the results are presented in the solid lines of Fig. 3.7. This shows that only the stationary solution with the lowest transmission is populated.

In the following we explain the origin of the bi-stability, and especially draw a connec-tion between the scattering states and the corresponding decaying quasi bound states.

We recognize that in the high transmitting state the corresponding high density inside the resonator shifts the resonance energy µ0 due to the interaction energy in the non-linear case µ0 → µ0(N), where N = RL/2

−L/2|Ψ(x)|²dx is the condensate density inside the resonator. Similarly, the decay rate depends on the densityγ0 →γ0(N). Thus, the transmission function has to be modified in the following way:

T(µ) = (~γ0(N)/2)²

(µ−µ0(N))²+ (~γ0(N)/2)² (3.44)

This ansatz assumes that the important modifications arise due to the interaction in

3.8. SELECTED EXAMPLES 33 the resonator, and the nonlinear effects outside the double barrier can be neglected.

This is justified near a resonance with a small decay rate and correspondingly with a high density inside the double barrier. The open question is now the calculation of the dependence of the densityN on the chemical potentialµand the incoming current jin. In the linear case it can be shown that the density is given by [73]:

N = ~γ0/2

(µ−µ0)²+ (~γ0/2)² ~jin (3.45)

In order to account for the nonlinearity in the above equation the resonance energy and the decay rate depend again on the density itself. Therefore we have to solve the resulting equation self-consistently:

N = ~γ0(N)/2

(µ−µ0(N))²+ (~γ0(N)/2)² ~jin (3.46) Since we want to compare the new formulas Eq. 3.44 and Eq. 3.46 in the nonlinear regime with the stationary solution of the Gross-Pitaevskii equation shown in Fig. 3.7, we need the density dependent resonance energy and decay rate:

This information can be extracted now from the corresponding decaying quasi bound state [75–78]. We choose the method of time integration of the Gross-Pitaevskii equa-tion with density renormalizaequa-tion. To this end we perform a numerical simulaequa-tion of the time-dependent Gross-Pitaevskii equation without source term. Instead of this we use a Gaussian wave packet in the double barrier as initial condition for the propaga-tion process close to the resonance state. After each propagapropaga-tion step we renormalize the wave function to the density N in the resonator according to:

N = Z L/2

−L/2|Ψ(x)|² dx (3.47)

With this description convergence to the energetically lowest quasi bound state is realized with the normalization N (see Fig. 3.8 a). Finally we extract the energy of the quasi bound state by:

The decay rate of the quasi bound state is given by:

γ₀(N) =−∆N

Nτ , (3.49)

where ∆N is the change of the density during the integration by a time step τ. The results are shown in Fig. 3.8 b and c. With this information that the intensity is

a) b) c)

Figure 3.8: a) The black line shows the double barrier potential. The red line (in arbitrary units) shows the density of the wave function of the decaying quasi bound state, which is obtained by time evolution with density renormalization in the resonator. Absorbing boundaries are at the left and right boundary of the simulation region. b) Resonance energy depending on the intensity. c) Decay rate depending on the intensity.

dependent on the resonance energy and decay rate we can solve the equations Eq. 3.44 and Eq. 3.46 self-consistently, and obtain the related transmission spectrum from the information of the decay problem. The results are shown in Fig. 3.9. The solid line shows the transmission received from the information of the decaying state. The dotted line shows the stationary solutions of the Gross-Pitaevskii equation. For the methods used to solve the stationary Gross-Pitaevskii equation see Ref. [65, 73]. The good agreement besides a small overestimation of the line width for higher nonlinearity -shows the one to one correspondence between nonlinear scattering states and nonlinear resonances. Especially it reveals the bistability phenomenon.

With the density depending on the resonance energy we can now provide an

inter-Figure 3.9: The dotted line shows the transmission corresponding to all stationary solutions of the stationary Gross-Pitaevskii equation for a fixed incident current (see Ref. [65, 73] for the methods used). The solid line shows the transmission calculated from the density dependent resonance energy and decay rate for the corresponding quasi bound state.

3.8. SELECTED EXAMPLES 35 pretation, why only the lowest transmission state is populated in the time-dependent transmission process with source (see solid line in Fig. 3.7). To illustrate this let us consider a nonlinearity strength and a chemical potential of the incoming wave, where a bistability exists near the nonlinear resonance. Since the starting point of the time-dependent integration is an empty wave guide we are in the low density regime or, similarly expressed, in the almost linear case, which implies that the incoming wave is off-resonance. To this end no high density in the resonator can build up and the high transmitting state can not be realized. Because of this a necessary condition to populate the high transmitting state is an initial high density in the double barrier.

This can for example be realized by an adiabatic increase of the chemical potential, from the linear resonance energy to the nonlinear resonance energy.

If the high transmitting state is populated in this way it is found that this state is dynamically unstable [69] and decays over a characteristic time to the low transmit-ting state. This extracted information is a further advantage of the time-dependent approach to the Gross-Pitaevskii equation, namely that we can extract information about the dynamic stability of stationary solutions.