Generation of entanglement and non-stationary states via competing coherent and incoherent bosonic hopping

TL;DR

The paper investigates how competing coherent and incoherent bosonic hopping in a two-mode system can generate robust quantum correlations and lead to non-stationary, time-crystal-like phases. Using a Bose-Hubbard dimer with coherent hopping $\Omega$, on-site interaction $U$, and incoherent hopping mediated by a third mode (rates set by $n_{th}$ and $\kappa$), the authors combine mean-field analysis, Liouvillian spectral theory, and Gaussian fluctuations to map out dissipative phases. A sequence of time-crystal phases TC1, TC2, and TC3 emerge, with a second-order phase transition at $U=0$ where entanglement diverges at $\Omega_c/\kappa=1$, and a bistable, first-order transition between TC2 and TC3 for $U>U_c$. The results reveal a dissipative mechanism to realize on-demand entanglement and non-stationary dynamics, with implications for quantum metrology and the control of open quantum systems using simple incoherent processes, underpinned by a PT-symmetric Liouvillian structure and a spin-equivalent BTC mapping in certain limits.

Abstract

Incoherent stochastic processes added to unitary dynamics are typically deemed detrimental since they are expected to diminish quantum features such as superposition and entanglement. Instead of exhibiting energy-conserving persistent coherent motion, the dynamics of such open systems feature, in most cases, a steady state, which is approached in the long-time limit from all initial conditions. This can, in fact, be advantageous as it offers a mechanism for the creation of robust quantum correlations on demand without the need for fine-tuning. Here, we show this for a system consisting of two coherently coupled bosonic modes, which is a paradigmatic scenario for the realization of quantum resources such as squeezed entangled states. Rather counterintuitively, the mere addition of incoherent hopping, which results in a statistical coupling between the bosonic modes, leads to steady states with robust quantum entanglement and enables the emergence of persistent coherent non-stationary behavior.

Figures (7)

• Figure 1: Model and entanglement properties.$(a)$ Illustration of a Bose-Hubbard dimer (BHD) showing coherent ($\Omega$) and incoherent hopping ($\gamma_{R,L}$), where each mode can also be subject to on-site interactions (not shown here) as described by Eq. (\ref{['eq:BH_themal']}). The rate $\gamma_{R}$ ($\gamma_{L}$) parametrizes the incoherent hopping strength to the right (left). (b) Sketch of the steady-state entanglement as a function of parameters. Notably, stronger incoherent hopping favors entanglement generation.
• Figure 2: Non-stationary phases. (a-c) Mean-field dynamics of three different time-crystal phases. (d-f) The exact dynamics of Eq. (\ref{['eq:BH_themal']}) approaches the corresponding mean-field dynamics (a-c) when increasing the system size. (g-i) Finite-size scaling shows that the dominant eigenvalues of $\mathcal{L}$ (left panel) approach the frequencies of the mean-field variables $p_\alpha$ described by their normalized Fourier transform $\mathcal{F}(p_\alpha)$ (right panel). Color intensities of markers increase with increasing $N$. We consider $N=\{20,25,\ldots, 75\}$. The parameter values for three different time-crystal phases (TC1, TC2, TC3) described by sub-figures (a,d,g), (b,e,h), and (c,f,i) are ($\Omega/\kappa=1.45, U/\kappa=0$), ($\Omega/\kappa=0.8, U/\kappa=0.25$), and ($\Omega/\kappa=1.45, U/\kappa=0.25$), respectively.
• Figure 3: Entanglement and bistability. (a) The time-averaged logarithmic negativity $\varepsilon$ exhibits a maximum at the critical point $\Omega_c/\kappa=1$ for $U/\kappa=0$, and it decreases with increasing $n_{th}$. (b) As time $t$ increases, the entanglement $\varepsilon$ grows at a rate proportional to $\ln(\kappa t)$ when tuned to the critical point $\Omega_c/\kappa$. Increasing $n_{th}$ does not change the slope but merely shifts the onset of entanglement generation. The divergence of correlations indicates a second-order phase transition. (c) For a fixed $U/\kappa \neq 0$, entanglement grows with increasing $\Omega/\kappa$ and decreases rapidly for $\Omega/\kappa$ values beyond the solid (black) line, which represents $U_c/\kappa$ (here $n_{th}=0$). The two time-crystal phases are separated by a bistable region between the dashed (white) and solid (black) lines. (d) The order parameter $\Delta N$ displays bistability when adiabatically sweeping $\Omega/\kappa$ in forward and backward directions, keeping $U/\kappa=0.25$ fixed [the parameter regime is indicated by the horizontal (green) dashed line in panel (c)]. The average entanglement in (a) and (c) is calculated for $\kappa t=4 \times 10^3$.
• Figure S1: (a) Order parameter $\Delta \bar{R} = |\bar{R}_a - \bar{R}_b|/\sqrt{2}$, showing the phase transition from TC2 ($U>U_c$) to TC3 ($U<U_c$) as $\Omega/\kappa$ increases for fixed $U/\kappa$. The transition occurs at $U_c/\kappa$ (dashed white line) obtained from the fixed-point analysis. (b) The transition from TC3 to TC2 occurs as $\Omega/\kappa$ decreases, with higher $U/\kappa$ values exhibiting bistability beyond the dashed line.
• Figure S2: Dependence of time-averaged entanglement ($\varepsilon$), quantum discord ($\mathcal{D}^{a \leftarrow b}$) and classical discord ($\mathcal{J}^{a \leftarrow b}$) on $n_{th}$ for a range of $\Omega/\kappa$ values with $U=0$. Entanglement (a) and quantum discord (b) exhibit maxima at $\Omega_c/\kappa$ and decrease with increasing $n_{th}$. Panels (d) and (e) show that $\varepsilon$ and $\mathcal{D}^{a \leftarrow b}$ grow at a rate proportional to $\ln(\kappa t)$. An increase in $n_{th}$ delays the onset of quantum correlations in (d,e) but does not change their rate. (c) Classical correlations ($\mathcal{J}^{a \leftarrow b}$) also exhibit a sharp transition at $\Omega/\kappa=1$, while they increase with increasing $n_{th}$, as shown in (f).