Microscopic finite-temperature theory of thermal-atom conductance and damping in trapped bosonic Josephson junctions

Zapata, Sols and Leggett (Phys. Rev. A 57, R28 (1998)) derived the Josephson coupling energy, capacitive energy and plasma frequency for two weakly linked trapped Bose-Einstein condensates, but treated dissipation from the incoherent exchange of thermal (uncondensed) atoms only through two qualitative limiting pictures: Arrhenius-Kramers activated tunneling for barriers high compared with $k_BT$, and a ballistic contact-resistance estimate when the thermal cloud lies mostly above the barrier. They explicitly deferred a quantitative microscopic finite-temperature theory of the junction's normal conductance $G$, together with quantitative damping rates, the overdamped/underdamped crossover, and observability thresholds for realistic geometries. Later work resolved parts of this program: stochastic projected Gross-Pitaevskii simulations (a classical-field method that explicitly simulates the low-energy thermal modes with stochastic noise) now extract $G(T)$ at high temperature, Zaremba-Nikuni-Griffin kinetic theory (coupling a dissipative Gross-Pitaevskii equation for the condensate to a quantum Boltzmann equation for the thermal cloud) quantifies collisional damping at subcritical temperature, and experiments have measured damping rates. However, the intermediate regime where $k_BT$ is comparable to the barrier height above the chemical potential remains outside the stated validity of all available treatments, with only order-of-magnitude agreement with the measured self-trapping decay. The surviving open problem is a quantitatively validated microscopic theory of thermal-atom conductance and the resulting damping of the trapped double-well bosonic Josephson junction across this crossover and the low-temperature discrete-level regime.

Background

When a dilute gas of bosonic atoms is cooled below the Bose-Einstein condensation temperature, a macroscopic fraction of atoms occupies a single quantum state described by a macroscopic wave function $\Psi=\rho^{1/2}e^{i\varphi}$. Two condensates held in a double-well trap and separated by a tunable barrier form a bosonic Josephson junction (BJJ), the cold-atom analogue of a superconducting Josephson junction: atoms tunnel coherently through the weak link, and the population imbalance between the wells and their relative phase play the roles of the transferred charge and the superconducting phase difference. In the standard two-mode description (Raghavan, Smerzi, Fantoni and Shenoy, Phys. Rev. A 59, 620 (1999)), the system reduces to a non-rigid pendulum with Hamiltonian $H=E_C\delta N^2/2+E_J(1-\cos\chi)$, where $\delta N$ is half the population imbalance, $\chi$ the relative phase, $E_J$ the Josephson coupling energy and $E_C$ the 'capacitive' energy cost of an imbalance; small oscillations occur at the Josephson plasma frequency $\omega_{JP}=\sqrt{E_JE_C}/\hbar$, while a sufficiently large imbalance drives full pendulum rotations, i.e. macroscopic quantum self-trapping. Zapata, Sols and Leggett (Phys. Rev. A 57, R28 (1998)) gave a three-dimensional semiclassical treatment: exploiting an electrostatic analogy, they expressed $E_J=(\hbar^2/m)C[\rho]$, where $C[\rho]$ is the mutual capacitance of two conductors shaped like the condensates embedded in an inhomogeneous dielectric of 'permittivity' equal to the condensate density; within a WKB (semiclassical tunneling) approximation $E_J$ carries the exponential factor $e^{-S_0}$, with $S_0$ the barrier action $\int p\,dz/\hbar$; and $E_C$ was estimated in the Thomas-Fermi regime (interactions strong enough that the trapped density profile is an inverted parabola). The unresolved part of their analysis concerns dissipation at nonzero temperature. Each well then also contains thermal ('normal', uncondensed) atoms whose spatial distribution in a harmonic trap is much broader than the condensate's. Incoherent exchange of these atoms through the barrier adds an Ohmic current $I_n=-G\,\delta\mu$ (with $\delta\mu$ the chemical-potential difference) on top of the Josephson supercurrent — the analogue of a resistively shunted superconducting junction — and damps the pendulum motion. Zapata, Sols and Leggett gave only two limiting estimates of the conductance $G$: for barriers high compared with $k_BT$, an Arrhenius-Kramers activated-tunneling rate $P_n=(\kappa\omega_0/2\pi)\exp(-V_1/k_BT)$ with $G\simeq P_nN_n/k_BT$, where $V_1=V_0-\mu$ is the barrier height above the chemical potential and $N_n$ the number of normal atoms; and for $k_BT$ well above the barrier top, a ballistic Landauer-type contact-resistance estimate $G=N_{ch}/h$ with $N_{ch}\sim(k_BT/\hbar\omega_0)^2$ effective transverse channels, where $\hbar\omega_0$ is the harmonic-trap level spacing. They explicitly declined a quantitative finite-temperature discussion, noting that in a trap the problem is 'much less trivial' than for junctions between homogeneous superconductors or superfluids because the thermal cloud's spatial scale differs from the condensate's. They further noted that (i) their WKB estimate of $E_J$ is controlled only for $S_0\gtrsim1$; (ii) below temperatures of order $\hbar\omega_0/k_B$ the single-particle energy-level spacing of the trap becomes comparable to $k_BT$, so 'level-crossing' (resonant, discrete-level) effects can no longer be ignored; and (iii) when the thermal cloud lies mostly above the barrier, 'a radically different approach is needed' for its transport properties. Quantitative damping rates, the temperature at which the junction crosses from overdamped to underdamped Josephson dynamics, and parameter thresholds for observing coherent Josephson oscillations in specific experimental geometries were all left open, as reviewed in Gati and Oberthaler, J. Phys. B 40, R61 (2007).

Problem Statement

Develop a quantitative microscopic theory of dissipation due to the incoherent exchange of thermal atoms in a bosonic Josephson junction formed by two weakly linked condensates in a three-dimensional double-well trap, and use it to compute — not merely estimate in limiting cases — the junction's normal conductance $G$ as a function of temperature and barrier parameters, accounting for the trapped geometry, the full energy dependence of the transmission through the barrier, and the thermal population of the excited trap modes. The theory must remain valid through the intermediate regime $k_BT\sim V_0-\mu$ (thermal energy comparable to the barrier height above the chemical potential), where neither the Arrhenius-Kramers high-barrier picture nor the ballistic/contact picture — nor currently available classical-field (c-field) treatments that explicitly simulate the low-energy thermal modes — is quantitatively reliable, and through the low-temperature regime $k_BT\lesssim\hbar\omega_0$ where the discrete single-particle level structure of the trap controls the transport. From this theory, derive quantitative predictions of all three of the following, each of which is a required part of a complete resolution: (i) the damping rates of Josephson plasma oscillations and of self-trapped dynamics as functions of temperature, barrier height and width, and atom number; (ii) the crossover temperature between overdamped and underdamped Josephson dynamics in these parameters; and (iii) the parameter thresholds for observing coherent Josephson dynamics in realistic experimental geometries. The common objective unifying these deliverables is a validated finite-temperature dissipative theory of the trapped bosonic Josephson junction that replaces the two qualitative limiting pictures of Zapata, Sols and Leggett across the whole parameter range; the theory's predictions must be compared quantitatively with existing published experimental damping data (e.g. the single-junction self-trapping decay measurements), and this quantitative validation is a required part of the resolution.

References

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