Chaos and ergodicity in an entangled two-qubit Bohmian system

Abstract

We study in detail the onset of chaos and the probability measures formed by individual Bohmian trajectories in entangled states of two-qubit systems for various degrees of entanglement. The qubit systems consist of coherent states of 1-d harmonic oscillators with irrational frequencies. In weakly entangled states chaos is manifested through the sudden jumps of the Bohmian trajectories between successive Lissajous-like figures. These jumps are succesfully interpreted by the ‘nodal point-X-point complex’ mechanism. In strongly entangled states, the chaotic form of the Bohmian trajectories is manifested after a short time. We then study the mixing properties of ensembles of Bohmian trajectories with initial conditions satisfying Born’s rule. The trajectory points are initially distributed in two sets S1 and S2 with disjoint supports but they exhibit, over the course of time, abrupt mixing whenever they encounter the nodal points of the wavefunction. Then a substantial fraction of trajectory points is exchanged between S1 and S2, without violating Born’s rule. Finally, we provide strong numerical indications that, in this system, the main effect of the entanglement is the establishment of ergodicity in the individual Bohmian trajectories as : different initial conditions result to the same limiting distribution of trajectory points.

The role of chaotic and ordered trajectories in establishing Born's rule

“…This was shown in Fig. 7 of our paper [16] in the case of maximum entanglement. The same happens for other values of the entanglement.…”

“…In our previous papers [16,17] we considered the trajectories in the cases c 2 = √ 2/2 0.707 (maximally entangled state), c 2 = 0.5 (strongly entangled state), c 2 = 0.2 (weakly entangled state) and c 2 = 0 (product state) and checked whether a distribution reaches the Born rule in the long run, by comparing the final pattern of the points of the trajectories of the initial distribution with that of the Born rule. These patterns are formed by collecting all the points of the trajectories inside the cells of a 360 × 360 grid covering the space [x, y] ∈ [−9, 9] at times equal to t = n∆t(n = 0, 1, 2, ...) and up to a sufficiently large time t f , with a step ∆t = 0.05 and plotting them by use of a spectral color plot.…”

“…The line of nodes rotates clockwise and counterclockwise from time to time, thus covering most areas of the configuration space (see Fig. 1 of [16]). When the blobs of |Ψ| 2 are far from the line of nodes the value of |Ψ| 2 between the nodes is very small (less than 10 −11 ).…”

Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators

“…As a technical detail, we mention that, as expected, Bohmian trajectories do not cross into the space (not plotted) but they cross in the subspace x of Figure 4 a. In addition, one can expect some chaotic behavior in 2D systems [ 39 , 40 ] that is not present in the 1D system that is shown in the Figure 4 a.…”

Chaos in Bohmian Quantum Mechanics: A short review

“…If, however, initially P 0 is different from |Ψ 0 | 2 , then it is expected that in the long run P will tend to |Ψ| 2 . This process is called dynamical relaxation towards a quantum equilibrium, which is exactly Born's rule and has been studied by several authors [47,48,49,50,51,52,53,40,54,55,56,57]. However, there are cases where there is no such relaxation.…”