Josephson oscillations of two weakly coupled Bose-Einstein condensates

TL;DR

The paper addresses whether Josephson currents between two weakly coupled Bose-Einstein condensates emerge deterministically or remain random due to initial phase distributions. It develops a number-conserving quantum field theory with non-local order parameters and a Boltzmann-equilibrium projection to model two initially independent condensates at finite temperature, using Monte-Carlo sampling to quantify phase correlations. The results show the initial relative phase is quantized around multiples of $2\pi$, and coherent coupling yields deterministic Josephson dynamics that depend on the initial phase, including phase-locked current shifts. The approach provides ab initio predictions for phase correlations and Josephson frequencies in double-well BECs and links thermalization to an internal measurement-like projection that explains the observed nonrandom initial phases.

Abstract

A numerical experiment based on a particle number-conserving quantum field theory is performed for two initially independent Bose-Einstein condensates that are coherently coupled at two temperatures. The present model illustrates ab initio that the initial phase of each of the two condensates doesn't remain random at the Boltzmann equilibrium, but is distributed around integer multiple values of $2π$ from the interference and thermalization of forward and backward propagating matter waves. The thermalization inside the atomic vapors can be understood as an intrinsic measurement process that defines a temperature for the two condensates and projects the quantum states to an average wave field with zero (relative) phases. Following this approach, focus is put on the original thought experiment of Anderson on whether a Josephson current between two initially separated Bose-Einstein condensates occurs in a deterministic way or not, depending on the initial phase distribution.

Figures (7)

• Figure 1: (Color online) Figure highlights realizations of a local average quantum field in complex number representation for a Bose-Einstein condensate confined in an external trapping potential with trap parameters $\omega^{(1,2)}_x = 2\pi\times250 {\rm~Hz}$, $\omega^{(1,2)}_y = 2\pi\times150 {\rm~Hz}$ und $\omega^{(1,2)}_z = 2\pi\times50 {\rm~Hz}$ at temperature $T = 10$ nK obtained from a Monte-Carlo simulation of Eq. (\ref{['eq.1']}).
• Figure 2: (Color online) Figure highlights interference effects with reduced volume in the generalized phase space (${\rm Re}(\chi), {\rm Im}(\chi)$). Numerical realizations of local average quantum fields are obtained from Eq. (\ref{['eq.2']}) for two coherently coupled Bose-Einstein condensates in complex number representation confined in two different external trapping potentials with the same trap parameters as in Fig. \ref{['fig.1']} at temperatures $T_1 = 10$ nK (left well) and $T_2 = 125$ nK (right well).
• Figure 3: (Color online) Shown are random fluctuations of the quantum field for different sample realizations of two coupled Bose-Einstein condensates at the same parameters as in Fig. \ref{['fig.1']}. As indicated by the simulations, the generalized phase space volume, as well as the intensity of the two interfering wave fields, varies in each sample realization, i.e. as a function of the initial conditions for the quantum field.
• Figure 4: (Color online) Figure highlights different numerical model realizations of the correlation function in Eq. (\ref{['eq.3']}) for always the same parameters of Fig. \ref{['fig.1']} with temperatures $T_1 = 10$ nK (left well) and $T_2 = 25$ nK (right well). As the results indicate, the initial phase $\Delta \phi_0$ is always distributed around multiples of the circle number $2\pi$. This illustrates non-deterministic correlations with deterministic and well-defined initial phase distributions around $\Delta \phi_0 = k\times2\pi$ from constructive and destructive interference of partial matter waves that follow complex spectra $\Omega^{(1,2)}(\mu^{(1,2)}) = \Omega^{(1,2)}(\omega) + \Omega^{(1,2)}(\Gamma) = \lbrace\omega^{(1,2)}_{\bf{k}} + i\Gamma^{(1,2)}_{\bf{k}} \rbrace$. Small offset values of some realizations of the correlation function indicate that the phase is weakly randomized by non-deterministic background fluctuations (from temperature). The scaling behavior is universal for different temperatures and trap geometries.
• Figure 5: (Color online) Shown are Josephson oscillations as a function of initial particle imbalance (orange line) and phase difference (blue line) for two weakly coupled Bose-Einstein condensates in a double-well potential. Parameters are set to coupling strength of $J = 0.5$ and initial particle number imbalance $\Delta z = 0$ with initial phase difference $\Delta \phi_0 = 0.05$ (upper figures) and $\Delta \phi_0 = -0.05$ (lower figures). Hence, for an initial quantum field realization of zero particle number imbalance and arbitrary small initial phase around the value of $\Delta\phi_0 = 0$ at ideally the same trap geometry, the oscillating phase distribution is shifted by a positive quarter oscillations period ($+\tau/2$), and a Joesphon current flows at any trial.