Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Scientific Reports

Fragility of the Schrödinger Cat in thermal environments

Sandip Bera, Kenny L. S. Yip, Sajeev John

Published Oct 31, 2023• 1 citation
Publisher

Abstract

We describe the decoherence instability of Schrödinger Cat states in the two-site Bose-Hubbard model with an attractive on-site interaction between particles. For N particles with onsite attractive energy U and hopping amplitude between sites t, Cat states exist for [Formula: see text] at zero temperature. However, they are increasingly unstable to small thermal fluctuations as the Cat itself is increasingly well-defined and its components become well-separated. For any given [Formula: see text], the decoherence temperature becomes smaller for large N. The loss of off-diagonal coherence peaks in the equilibrium density matrix is dominated by the thermal admixture of the first excited state of the many-body system with its ground state. Particle number fluctuations, described in the grand canonical ensemble also reduce coherence, but to a lesser degree than thermal fluctuations. The full density matrix of the Schrödinger Cat is obtained by exact numerical diagonalization of the many-body Hamiltonian and a narrow regime in the parameter space of the particle number, temperature, and U/t is identified where small Cat states may survive decoherence in a physical environment.

DOI
10.1038/s41598-023-45701-3

Related papers

Analytical approach to the two-site Bose-Hubbard model: From Fock states to Schrödinger cat states and entanglement entropy

Luca Dell’Anna

We study the interpolation from occupation number Fock states to Schr\"odinger cat states on systems modeled by a two-mode Bose-Hubbard Hamiltonian, like, for instance, bosons in a double well or superconducting Cooper pair boxes. In the repulsive interaction regime, by a simplified single particle description, we calculate analytically energy, number fluctuations, stability under coupling to a heat bath, entanglement entropy, and Fisher information, all in terms of hypergeometric polynomials of the single particle overlap parameter. Our approach allows us to find how those quantities scale with the number of bosons. In the attractive interaction regime we calculate the same physical quantities in terms of the imbalance parameter, and find that the spontaneous symmetry breaking, occurring at interaction ${U}_{c}$, predicted by a semiclassical approximation, is valid only in the limit of infinite number of bosons. For a large but finite number we determine a characteristic strength of interaction ${U}_{c}^{*}$, which can be promoted as the crossover point from coherent to incoherent regimes and can be identified as the threshold of fragility of the cat state. Moreover, we find that the Fisher information is always in direct ratio to the variance of on-site number of bosons, for both positive and negative interactions. We finally show that the entanglement entropy is maximum close to ${U}_{c}^{*}$ and exceeds its coherent value within the whole range of interaction between $2{U}_{c}$ and zero.

arXiv:1108.6188•Physical Review A•May 9, 2012•15 citations

Coherence and entanglement in the ground state of a bosonic Josephson junction: From macroscopic Schrödinger cat states to separable Fock states

G. Mazzarella, Luca Salasnich, Alberto Parola +1 more

We consider a bosonic Josephson junction made of $N$ ultracold and dilute atoms confined by a quasi-one-dimensional double-well potential within the two-site Bose-Hubbard model framework. The behavior of the system is investigated at zero temperature by varying the interatomic interaction from the strongly attractive regime to the repulsive one. We show that the ground state exhibits a crossover from a macroscopic Schr\"odinger-cat state to a separable Fock state through an atomic coherent regime. By diagonalizing the Bose-Hubbard Hamiltonian we characterize the emergence of the macroscopic cat states by calculating the Fisher information $F$, the coherence by means of the visibility $\ensuremath{\alpha}$ of the interference fringes in the momentum distribution, and the quantum correlations by using the entanglement entropy $S$. Both Fisher information and visibility are shown to be related to the ground-state energy by employing the Hellmann-Feynman theorem. This result, together with a perturbative calculation of the ground-state energy, allows simple analytical formulas for $F$ and $\ensuremath{\alpha}$ to be obtained over a range of interactions, in excellent agreement with the exact diagonalization of the Bose-Hubbard Hamiltonian. In the attractive regime the entanglement entropy attains values very close to its upper limit for a specific interaction strength lying in the region where coherence is lost and self-trapping sets in.

arXiv:1103.1976•Physical Review A•May 5, 2011•60 citations

Quantum and thermal decoherence in two weakly coupled Bose-Einstein condensates

Chaohong Lee, Libin Fu, Yuri S. Kivshar

We study the coherence fluctuations in a Josephson junction created by two weakly coupled Bose-Einstein condensates. We apply the two-mode approximation in the full quantum theory analysis, and so that the system is described by a two-site Bose-Hubbard Hamiltonian. The coherence fluctuations are determined by the total number of particles N, charging energy EC, junction energy EJ, asymmetry δ and temperature T. For zero-temperature symmetric systems, the coherence fluctuations increase with EC/EJ, however, the number squeezing increases with N2EC/EJ. For zero-temperature asymmetric systems, the coherence fluctuations are suppressed by the balance effects between asymmetry and interparticle interaction and the single-atom resonance effects. The coherence fluctuations grow with temperature, and the formation of a self-trapping state will cause abnormal behaviors at high temperatures.

Quantum-Atom Optics Downunder (QAO)•Jan 1, 2007•0 citations

Strong thermalization of the two-component Bose-Hubbard model at finite temperatures

J. M. Zhang, Chao Shen, W. M. Liu

We study thermalization of a two-component Bose-Hubbard model by exact diagonalization. Initially, the two components do not interact and are both at equilibrium but with different temperatures. As the onsite intercomponent interaction is turned on, perfect thermalization occurs. Remarkably, not merely those simple ``realistic'' physical observables thermalize but even the density matrix of the whole system---the time-averaged density matrix of the system can be well approximated by that of a canonical ensemble. We also discuss how symmetry of the quenched Hamiltonian affects the level spacing distribution and thermalization of the system.

arXiv:1102.2469•Physical Review A•Jan 27, 2012•11 citations

Relaxation and thermalization in the one-dimensional Bose-Hubbard model: A case study for the interaction quantum quench from the atomic limit

Stefan Sorg, Lev Vidmar, Lode Pollet +1 more

Motivated by recent experiments, we study the relaxation dynamics and thermalization in the one-dimensional Bose-Hubbard model induced by a global interaction quench. Specifically, we start from an initial state that has exactly one boson per site and is the ground state of a system with infinitely strong repulsive interactions at unit filling. Using exact diagonalization and the density-matrix renormalization-group method, we compute the time dependence of such observables as the multiple occupancy and the momentum distribution function. Typically, the relaxation to stationary values occurs over just a few tunneling times. The stationary values are identical to the so-called diagonal ensemble on the system sizes accessible to our numerical methods and we further observe that the microcanonical ensemble describes the time averages of many observables reasonably well for small and intermediate interaction strength. The expectation values of observables in the canonical ensemble agree quantitatively with the time averages obtained from the quench at small interaction strengths, and qualitatively provide a good description even in parameter regimes where the microcanonical ensemble is not applicable due to finite-size effects. We discuss our numerical results in the framework of the eigenstate thermalization hypothesis. Moreover, we also observe that the diagonal and the canonical ensembles are practically identical for our initial conditions already on the level of their respective energy distributions for small interaction strengths. Finally, we discuss implications of our results for the interpretation of a recent sudden expansion experiment [Phys. Rev. Lett. 110, 205301 (2013)], in which the same interaction quench was realized.

arXiv:1405.5404•Physical Review A•Sep 8, 2014•129 citations

Quantum and thermal fluctuations in bosonic Josephson junctions

Bruno Juliá-Díaz, Alex D. Gottlieb, Joan Martorell +1 more

We use the Bose-Hubbard Hamiltonian to study quantum fluctuations in canonical equilibrium ensembles of bosonic Josephson junctions at relatively high temperatures, comparing the results for finite particle numbers $N$ to the classical limit that is attained as $N$ approaches infinity. We consider both attractive and repulsive atom-atom interactions, with special focus on the behavior near the $T=0$ quantum phase transition that occurs, for large enough $N$, when attractive interactions surpass a critical level. Differences between Bose-Hubbard results for small $N$ and those of the classical limit are quite small even when $N\ensuremath{\sim}100$, with deviations from the limit diminishing as $1/N$.

arXiv:1307.4595•Physical Review A•Sep 3, 2013•10 citations
We describe the decoherence instability of Schrödinger Cat states in the two-site Bose-Hubbard model with an attractive on-site interaction between particles. For N particles with onsite attractive energy U and hopping amplitude between sites t, Cat states exist for [Formula: see text] at zero temperature. However, they are increasingly unstable to small thermal fluctuations as the Cat itself is increasingly well-defined and its components become well-separated. For any given [Formula: see text], the decoherence temperature becomes smaller for large N. The loss of off-diagonal coherence peaks in the equilibrium density matrix is dominated by the thermal admixture of the first excited state of the many-body system with its ground state. Particle number fluctuations, described in the grand canonical ensemble also reduce coherence, but to a lesser degree than thermal fluctuations. The full density matrix of the Schrödinger Cat is obtained by exact numerical diagonalization of the many-body Hamiltonian and a narrow regime in the parameter space of the particle number, temperature, and U/t is identified where small Cat states may survive decoherence in a physical environment.
We study the interpolation from occupation number Fock states to Schr\"odinger cat states on systems modeled by a two-mode Bose-Hubbard Hamiltonian, like, for instance, bosons in a double well or superconducting Cooper pair boxes. In the repulsive interaction regime, by a simplified single particle description, we calculate analytically energy, number fluctuations, stability under coupling to a heat bath, entanglement entropy, and Fisher information, all in terms of hypergeometric polynomials of the single particle overlap parameter. Our approach allows us to find how those quantities scale with the number of bosons. In the attractive interaction regime we calculate the same physical quantities in terms of the imbalance parameter, and find that the spontaneous symmetry breaking, occurring at interaction Uc{U}_{c}Uc​, predicted by a semiclassical approximation, is valid only in the limit of infinite number of bosons. For a large but finite number we determine a characteristic strength of interaction Uc∗{U}_{c}^{*}Uc∗​, which can be promoted as the crossover point from coherent to incoherent regimes and can be identified as the threshold of fragility of the cat state. Moreover, we find that the Fisher information is always in direct ratio to the variance of on-site number of bosons, for both positive and negative interactions. We finally show that the entanglement entropy is maximum close to Uc∗{U}_{c}^{*}Uc∗​ and exceeds its coherent value within the whole range of interaction between 2Uc2{U}_{c}2Uc​ and zero.
We consider a bosonic Josephson junction made of NNN ultracold and dilute atoms confined by a quasi-one-dimensional double-well potential within the two-site Bose-Hubbard model framework. The behavior of the system is investigated at zero temperature by varying the interatomic interaction from the strongly attractive regime to the repulsive one. We show that the ground state exhibits a crossover from a macroscopic Schr\"odinger-cat state to a separable Fock state through an atomic coherent regime. By diagonalizing the Bose-Hubbard Hamiltonian we characterize the emergence of the macroscopic cat states by calculating the Fisher information FFF, the coherence by means of the visibility $\ensuremath{\alpha}$ of the interference fringes in the momentum distribution, and the quantum correlations by using the entanglement entropy SSS. Both Fisher information and visibility are shown to be related to the ground-state energy by employing the Hellmann-Feynman theorem. This result, together with a perturbative calculation of the ground-state energy, allows simple analytical formulas for FFF and $\ensuremath{\alpha}$ to be obtained over a range of interactions, in excellent agreement with the exact diagonalization of the Bose-Hubbard Hamiltonian. In the attractive regime the entanglement entropy attains values very close to its upper limit for a specific interaction strength lying in the region where coherence is lost and self-trapping sets in.
We study the coherence fluctuations in a Josephson junction created by two weakly coupled Bose-Einstein condensates. We apply the two-mode approximation in the full quantum theory analysis, and so that the system is described by a two-site Bose-Hubbard Hamiltonian. The coherence fluctuations are determined by the total number of particles N, charging energy EC, junction energy EJ, asymmetry δ and temperature T. For zero-temperature symmetric systems, the coherence fluctuations increase with EC/EJ, however, the number squeezing increases with N2EC/EJ. For zero-temperature asymmetric systems, the coherence fluctuations are suppressed by the balance effects between asymmetry and interparticle interaction and the single-atom resonance effects. The coherence fluctuations grow with temperature, and the formation of a self-trapping state will cause abnormal behaviors at high temperatures.
We study thermalization of a two-component Bose-Hubbard model by exact diagonalization. Initially, the two components do not interact and are both at equilibrium but with different temperatures. As the onsite intercomponent interaction is turned on, perfect thermalization occurs. Remarkably, not merely those simple ``realistic'' physical observables thermalize but even the density matrix of the whole system---the time-averaged density matrix of the system can be well approximated by that of a canonical ensemble. We also discuss how symmetry of the quenched Hamiltonian affects the level spacing distribution and thermalization of the system.
Motivated by recent experiments, we study the relaxation dynamics and thermalization in the one-dimensional Bose-Hubbard model induced by a global interaction quench. Specifically, we start from an initial state that has exactly one boson per site and is the ground state of a system with infinitely strong repulsive interactions at unit filling. Using exact diagonalization and the density-matrix renormalization-group method, we compute the time dependence of such observables as the multiple occupancy and the momentum distribution function. Typically, the relaxation to stationary values occurs over just a few tunneling times. The stationary values are identical to the so-called diagonal ensemble on the system sizes accessible to our numerical methods and we further observe that the microcanonical ensemble describes the time averages of many observables reasonably well for small and intermediate interaction strength. The expectation values of observables in the canonical ensemble agree quantitatively with the time averages obtained from the quench at small interaction strengths, and qualitatively provide a good description even in parameter regimes where the microcanonical ensemble is not applicable due to finite-size effects. We discuss our numerical results in the framework of the eigenstate thermalization hypothesis. Moreover, we also observe that the diagonal and the canonical ensembles are practically identical for our initial conditions already on the level of their respective energy distributions for small interaction strengths. Finally, we discuss implications of our results for the interpretation of a recent sudden expansion experiment [Phys. Rev. Lett. 110, 205301 (2013)], in which the same interaction quench was realized.
We use the Bose-Hubbard Hamiltonian to study quantum fluctuations in canonical equilibrium ensembles of bosonic Josephson junctions at relatively high temperatures, comparing the results for finite particle numbers NNN to the classical limit that is attained as NNN approaches infinity. We consider both attractive and repulsive atom-atom interactions, with special focus on the behavior near the T=0T=0T=0 quantum phase transition that occurs, for large enough NNN, when attractive interactions surpass a critical level. Differences between Bose-Hubbard results for small NNN and those of the classical limit are quite small even when $N\ensuremath{\sim}100$, with deviations from the limit diminishing as 1/N1/N1/N.