Analytical description of collisional decoherence in a BEC double-well accelerometer

TL;DR

This paper develops an analytic density-matrix framework for collisional decoherence in a Bose-Einstein condensate confined to a double-well, treating a weakly interacting, closed Bose gas and deriving how interparticle collisions damp Josephson oscillations while shifting their frequency under external acceleration. The approach starts from a two-state single-particle model in a 3D cubic double-well, extends to a many-particle Hamiltonian with energy-conserving collisions, and yields an effective 2x2 density matrix that captures population imbalance and coherence through a time-dependent degree of coherence $f(t)$. It shows that decoherence can be connected to phase fluctuations via a frequency distribution $P(
)$ and relates the density-matrix description to pure-state Josephson dynamics, recovering standard results in the limit $f(t) o1$ and weak decoherence. A central result is a closed-form expression for the acceleration-induced shift of the Josephson frequency, enabling an estimate of accelerometer sensitivity: the period shift scales with the interaction strength $V$, particle number $N$, trap size $L$, and acceleration component along the trap axis, with practical limits set by the decoherence timescale $ au_{ ext{dec}} oughly rac{ ilde{oldsymbol{h}}}{V ext{√}(N-1)}$ and by the regime $|oldsymbol{ ext{Λ}}|<1$. Overall, the work provides analytic benchmarks for decoherence in BEC interferometers, clarifies when a pure-state description suffices, and offers explicit guidance on how trap geometry and interactions control both coherence and accelerometer performance.

Abstract

BEC-based quantum sensors offer a huge, yet not fully explored potential in gravimetry and ac- celerometry. In this paper, we study a possible setup for such a device, which is a weakly interacting Bose gas trapped in a double-well potential. In such a trap, the gas is known to exhibit Josephson oscillations, which rely on the coherence between the potential wells. Applying the density matrix approach, we consider transitions between the coherent, partially incoherent, and fully incoherent states of the Bose gas. We provide an analytical description of the collisional decoherence due to weak interactions, causing the Josephson oscillations to decay with time. In particular, we give the mathematical link between that decay in the density matrix approach and its interpretation in terms of phase fluctuations. To investigate the potential of the double-well setup as a quantum sensor we apply additional external acceleration to the system. The interplay of collisional interaction and ac- celeration leads to an additional shift of the oscillation frequency. We give the analytical expression for this shift and estimate the sensitivity of a hypothetical BEC double-well accelerometer based on that effect.

Figures (7)

• Figure 1: Potential geometry and the single particle energy eigenstates: the ground state $\phi_0$ (solid line) and the excited state $\phi_1$ (dashed line).
• Figure 2: Collisional processes in Bose gas: (a) collisions of type $\hat{a}_i^\dagger \hat{a}_i^\dagger \hat{a}_i \hat{a}_i$, where both ingoing and outgoing bosons belong to the same energy eigenstate $\epsilon_i$ (depicted by same color); (b) Hartree (direct) collisions and (c) Fock (exchange) collisions $\hat{a}_i^\dagger \hat{a}_j^\dagger \hat{a}_i \hat{a}_j$ of bosons in different states (depicted by different colors).
• Figure 3: Population imbalance $Z$ (top) and degree of coherence $f$ (bottom) as functions of time $t$. Here the number of bosons is $N = 10^4$ and $g_0 = 6.86 \times 10^{-55}$ J$\times$m$^3$.
• Figure 4: Top: the pairs of cubes depict the trap geometry (\ref{['Fig: trap geometry']}), while the blue cloud illustrates the initial condition (\ref{['eq: left state']}). The arrows show the three cases of the direction of the applied acceleration $\mathbf{a}$. Middle and bottom: Population imbalance $Z$ as a function of $t$ for different time scales. Here the number of bosons is $N = 10^4$ and self-interaction strength $g = 15 g_0 = 1.03 \times 10^{-53}$ J$\times$m$^3$ is the same for all graphs. The purple (solid), green (dashed) and red (dot-dashed) lines correspond to the cases of acceleration $\mathbf{a}$ directed along $\theta =0$, $\pi/2$, $\pi$ respectively, cf. upper figure.
• Figure 5: Energies $\tilde{E}_0^z$ and $\tilde{E}_1^z$ depending on the depth of a double-well $\tilde{U}_0$