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arXiv:2403.15154 · Physical review. B./Physical review. B

Spread complexity and dynamical transition in multimode Bose-Einstein condensates

Bozhen Zhou, Shu Chen

Published Aug 21, 2024• 9 citations
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Abstract

We study the spread complexity in two-mode Bose-Einstein condensates and unveil that the long-time average of the spread complexity ${\overline{C}}_{K}$ can probe the dynamical transition from self-trapping to Josephson oscillation. When the parameter $\ensuremath{\omega}$ increases over a critical value ${\ensuremath{\omega}}_{c}$, we reveal that the spread complexity exhibits a sharp transition from lower to higher value, with the corresponding phase space trajectory changing from self-trapping to Josephson oscillation. Moreover, we scrutinize the eigenspectrum and uncover the relation between the dynamical transition and the excited state quantum phase transition, which is characterized by the emergence of singularity in the density of states at critical energy ${E}_{c}$. In the thermodynamical limit, the cross point ${E}_{c}(\ensuremath{\omega})$ and the initial energy ${E}_{0}(\ensuremath{\omega})$ determine the dynamical transition point ${\ensuremath{\omega}}_{c}$. Furthermore, we show that the different dynamical behavior for the initial state at a fixed point can be distinguished by the long-time average of the spread complexity, when the fixed point changes from unstable to stable. Finally, we also examine the sensitivity of ${\overline{C}}_{K}$ for the triple-well bosonic model which exhibits the transition from chaotic dynamics to regular dynamics.

DOI
10.1103/physrevb.110.064318

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We study the spread complexity in two-mode Bose-Einstein condensates and unveil that the long-time average of the spread complexity C‾K{\overline{C}}_{K}CK​ can probe the dynamical transition from self-trapping to Josephson oscillation. When the parameter \ensuremathω\ensuremath{\omega}\ensuremathω increases over a critical value \ensuremathωc{\ensuremath{\omega}}_{c}\ensuremathωc​, we reveal that the spread complexity exhibits a sharp transition from lower to higher value, with the corresponding phase space trajectory changing from self-trapping to Josephson oscillation. Moreover, we scrutinize the eigenspectrum and uncover the relation between the dynamical transition and the excited state quantum phase transition, which is characterized by the emergence of singularity in the density of states at critical energy Ec{E}_{c}Ec​. In the thermodynamical limit, the cross point Ec(\ensuremathω){E}_{c}(\ensuremath{\omega})Ec​(\ensuremathω) and the initial energy E0(\ensuremathω){E}_{0}(\ensuremath{\omega})E0​(\ensuremathω) determine the dynamical transition point \ensuremathωc{\ensuremath{\omega}}_{c}\ensuremathωc​. Furthermore, we show that the different dynamical behavior for the initial state at a fixed point can be distinguished by the long-time average of the spread complexity, when the fixed point changes from unstable to stable. Finally, we also examine the sensitivity of C‾K{\overline{C}}_{K}CK​ for the triple-well bosonic model which exhibits the transition from chaotic dynamics to regular dynamics.
We investigate the coherence dynamics of a two-mode Bose-Einstein condensate coupled with the environment in the mean-field approximation. We give an analytical result of the time-average coherence in the absence of system-environment coupling, and find that the time-average coherence attains its maximum value at the critical point which corresponds to the boundary between the self-trapping regime and the Josephson oscillation regime. The effect of noise on dynamical coherence is also considered by analyzing the couplings of the condensate to the environment. With the first kind of coupling, the coherence finally stabilizes at a fixed value. Meanwhile, we show that the presence of the noise can even enhance the coherence for some particular interaction. With the second kind of coupling as collisional dephasing, the noise leads a sudden transition of the coherence which then subjects to an exponential decay.
We investigate the tunneling dynamics of Bose-Einstein condensates (BECs) in a symmetric as well as in a tilted triple-well trap within the framework of mean-field treatment. The eigenenergies as the functions of the zero-point energy difference between the tilted wells show a striking entangled star structure when the atomic interaction is large. We then achieve insight into the oscillation solutions around the corresponding eigenstates and observe several new types of Josephson oscillations. With increasing the atomic interaction, the Josephson-type oscillation is blocked and the self-trapping solution emerges. The condensates are self-trapped either in one well or in two wells but no scaling law is observed near transition points. In particular, we find that the transition from the Josephson-type oscillation to the self-trapping is accompanied with some irregular regime where tunneling dynamics is dominated by chaos. The above analysis is facilitated with the help of the Poincar\'e section method that visualizes the motions of BECs in a reduced phase plane.
Dynamics of the repulsive Bose–Einstein condensate (BEC) in a double-well trap is explored within the 3D time-dependent Gross–Pitaevskii equation. The model avoids numerous common approximations (two-mode treatment, time-space factorization, fixed values of the chemical potential and barrier penetrability, etc) and thus may provide a realistic description of BEC dynamics, including both weak-coupling (sub-barrier) and strong-coupling (above-barrier) regimes and their crossover. The strong coupling regime is achieved by increasing the number N of BEC atoms and thus the chemical potential. The evolution with N of Josephson oscillations (JO) and macroscopic quantum self-trapping (MQST) is examined and the crucial impact of the BEC interaction is demonstrated. At weak coupling, the calculations well reproduce the JO/MQST experimental data. At strong coupling with a significant overlap of the left and right BECs, we observe a remarkable persistence of the Josephson-like dynamics: the JO and MQST converge to a high-frequency JO-like mode where both population imbalance and phase difference oscillate around the zero averages.
Quantum Fisher information, which plays a central role in parameter estimation theory, is a useful quantity for metrological applications. We use quantum Fisher information to distinguish and characterize behaviors of the evolved state for Bose-Einstein condensates in a symmetric double well, which display a classical bifurcation and a transition from Josephson oscillation to self-trapping. We find that the precision of the parameter estimation on the evolved state in the unstable regime is much higher than that in the stable regime, which lies at the shot-noise level. We also show that the parameter sensitivity of the evolved state approaches the Heisenberg limit in the Josephson oscillation regime, while it is just scaled as the shot-noise limit in the self-trapping regime. Finally, we discuss the relation between the parameter sensitivity of the evolved state and the entanglement entropy.
In this work, we investigate the quantum dynamics of a model for two single-mode Bose–Einstein condensates which are coupled via Josephson tunnelling. Using direct numerical diagonalization of the Hamiltonian, we compute the time evolution of the expectation value for the relative particle number across a wide range of couplings. Our analysis shows that the system exhibits rich and complex behaviours varying between harmonic and non-harmonic oscillations, particularly around the threshold coupling between the delocalized and self-trapping phases. We show that these behaviours are dependent on both the initial state of the system and regime of the coupling. In addition, a study of the dynamics for the variance of the relative particle number expectation and the entanglement for different initial states is presented in detail.
We present a complete analysis of the dynamics of a Bose–Einstein condensate trapped in a symmetric triple-well potential. Our classical analogue treatment, based on a time-dependent variational method using SU(3) coherent states, includes the parameter dependence analysis of the equilibrium points and their local stability, which is closely related to the condensate collective behaviour. We also consider the effects of off-site interactions, and how these 'cross-collisions' may become relevant for a large number of trapped bosons. Even in the presence of cross-collisional terms, the model still features an integrable sub-regime, known as the twin-condensate dynamics, which corresponds to invariant surfaces in the classical phase space. However, the quantum dynamics preserves the twin-condensate defining characteristics only partially, thus breaking the invariance of the associated quantum subspace. Moreover, the periodic geometry of the trapping potential allowed us to investigate the dynamics of finite angular momentum collective excitations, which can be suppressed by the emergence of chaos. Finally, using the generalized purity associated with the su(3) algebra, we were able to quantify the dynamical classicality of a quantum evolved system, as compared to the corresponding classical trajectory.