Thermal Entanglement and Out-of-Equilibrium Thermodynamics in 1D Bose gases

Abstract

We investigate entanglement in and out of equilibrium in a one-dimensional Bose gas in its low-energy Bogoliubov regime. In this Gaussian setting, the state is fully characterized by its covariance matrix, which allows us to detect and quantify entanglement using a covariance-based framework and associated entanglement monotones. For thermal states, we determine the optimal entanglement witness arising from the covariance matrix criterion and show that it has a remarkably simple mode-resolved structure: it is diagonal in the normal-mode basis and admits a simple analytic form that can be expressed as a product of only two normal-mode uncertainties. We then study out-of-equilibrium dynamics induced by unitary compression and show that entanglement can be generated even from initially separable thermal states. When the evolution is fully adiabatic, the optimal witness retains the same two-mode structure as in the thermal case. Departing from this regime, i.e., performing increasingly rapid compression, the optimal witness becomes genuinely more intricate. Our methods and results provide a unified and physically intuitive picture of how entanglement emerges and evolves in 1D quantum Bose gases, and identify an optimal witness structure relevant more broadly to the analysis of entanglement in quadratic bosonic models and its role in thermodynamic cycles.

Figures (6)

• Figure 1: Discretizing a 1D BEC: Starting from the Bogoliubov Hamiltonian in Eq. \ref{['eq:H_Bogoliubov']} that describes a degenerate Bose gas in the limit of weak interactions and low density fluctuations, the discretization involves several steps. Introducing discretized density and phase operators that satisfy a rescaled commutation relation, following the procedure from Ref. MoraCastin, and neglecting the second term, we obtain a low-energy approximation that describes the Bose gas as a free phonon gas.
• Figure 2: Entanglement, quantified by the optimal witness and the logarithmic negativity, as a function of temperature for thermal states of a $50~\mu \rm m$ 1D BEC of $\approx 6000$ atoms with a fixed weak-coupling Hamiltonian. Positive values indicate the presence of entanglement. The temperature at which entanglement vanishes is marked by the dashed vertical line and coincides for both quantifiers, i.e., $T^* = 3.7896 ~ \rm nK$ (for $\Delta = 125 ~\text{nm} \approx \xi/2$).
• Figure 3: Numerical results for a $50~ \mu \rm m$ 1D BEC of $\approx 6000$ atoms at $T = 30~ \rm nK$: Optimal entanglement witness as a function of time (in seconds) for a quasi-adiabatic compression protocol going from $L_0 \to L = 0.9$ in $0.13s$ using around $10^5$ Trotter steps. Positive values indicate the presence of entanglement.
• Figure 4: Optimal witness over time (in seconds) during compression in the case of (quasi-)adiabatic evolution (blue) and relatively fast evolution (red).
• Figure 5: Optimal witness over time (in seconds) during compression $L/L_0 = 0.9$ and thermalization to a hot bath $T_h = 60 ~ \rm nK$ for a BEC with initial temperature $T_h = 30 ~ \rm nK$. Positive values indicate the presence of entanglement. (Other BEC parameters same as Fig. \ref{['fig:ent_over_compression']}.)

Theorems & Definitions (1)

• Lemma 1: Slater's theorem for SDP Watrous2018