Quantum trajectories for the variational description of closed systems: A case study with Gaussian states

In this paper, it is proposed to improve the quality of a variational description of a closed quantum system by adding fictitious dissipation that reduces the entanglement. The proposal is implemented for a small Bose-Hubbard chain in the mean field regime, which shows chaotic behavior and associated fast growth of quantum fluctuations. For appropriately chosen dissipation, good agreement with the truncated Wigner approximation is found. This shows that, when chaos leads to the breakdown of a variational method, adding fictitious dissipation can be a cure.

Figure 1

Illustration of the chaos in the Bose-Hubbard system. The TWA average strongly differs from the GPE because the trajectories with slightly different initial conditions evolve very differently. Black dots visualize the end points of the TWA trajectories, and the gray dot indicates the initial condition. Parameters: total particle number $N=3090, U/J=0.01$, and initial condition ${\vec{\alpha }}^{\left(0\right)}=(30-6.6i,9.7-7.3i,40,20)$.

Figure 2

Growth of fluctuations on the first site computed with the TWA (thick black line), with the Gaussian variational wave function (red dashed line), and with the TGA with $\delta N_{\max }=20$ (blue line). Parameters are the same as in Fig. 1.

Figure 3

Sketch of the different representations of the state in the TWA, HFB, and TGA. In the TWA, points in phase space are sampled, illustrated by the black dots. In the HFB, a single squeezed state is evolved, as represented by the red ellipse. In the TGA, the state is represented by a mixture of squeezed states.

Figure 4

Comparison of the TWA, GPE, and TGA with various maximal fluctuation numbers for a four-site Bose-Hubbard model. Parameters: (a) total particle number $N=3090$ and $U/J=0.01$ and (b) $N=30900$ and $U/J=0.001$.

Figure 5

Comparison of the TWA (red dashed line) with exact diagonalization (black full line) for a three-site Bose-Hubbard model. Parameters: (a) total particle number $N=30$ and $U/J=0.5$ and (b) $N=60$ and $U/J=0.25$.