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

arXiv:2605.04598

Two-site Bose-Hubbard hopping and Schrödinger cat states

Madeline Berezowski, Artur Sowa, Jonas Fransson

quant-phmath-phSubmitted May 6, 2026• Announced May 7, 2026• 0 citations
arXivHTMLPDF

Abstract

The Bose-Hubbard Hamiltonian can be simplified to have only two lattice sites, in which case the system being described is referred to as a dimer. Due to its structure, the hopping term of the dimer Hamiltonian enjoys invariance in a family of subspaces indexed by a whole number $k$, each subspace corresponding to a system of only $k$ particles. We have invented an inductive argument using the bosonic canonical commutation relations to find the eigenvalues and eigenvectors of the dimer hopping Hamiltonian in its $k$-particle subspaces. In particular, this Hamiltonian, when restricted to one of the $k$-particle subspaces, is exactly the spin projection operator along the $x$-axis, where the number of particles $k$ in the dimer system yields the projection matrix for spin quantum number $s=k/2$. Thus, a new method for computing the eigenvalues and eigenvectors of the $x$-axis spin projector has been unearthed. We use the explicit construction to study the dynamics of coherent states induced by the square of the dimer hopping hamiltonian. We find that it generates Schrödinger cat states in the two-site setting.

Comments
7 pages, 2 figures

Related papers

Fragility of the Schrödinger Cat in thermal environments

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

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.

Scientific Reports•Oct 31, 2023•1 citation

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

Comparison of classical and quantum properties in an extended Bose-Hubbard model

Albaro Vega Gutierrez de Pineres

In order to explore a quantum version of a discrete nonlinear Schrodinger equation (DNLS), we quantize one nonlinear Schrodinger model, which is used to study different physical systems, e.g. coupled Bose-Einstein condensates. We will focus on small systems, like Dimer and Trimer.In our efforts to solve this quantum problem, we develop a Mathematica routine that implements the Number State Method and solves the corresponding Schrodinger equation. We calculate analytically and numerically the energy spectrum of the Dimer and Trimer systems. Those eigenenergies depend on the parameter set Q=Q1, Q2, Q3, Q4, Q5 and by adjusting this set Q, we can obtain the desired results and examine their effects. After the quantization of the extended DNLS we obtain a quantum DNLS, also known as an extended Bose-Hubbard (BH) model. The aim of this Master's thesis is to study the differences and similarities between the classical DNLS and the extended BH model, and what happens when we approach from the quantum regime to the classical one. Taking into account that the Hamiltonian has an important conserved quantity, the number operator, enables the total Hamiltonian to be block-diagonalized. This can be accomplished by taking advantage of additional symmetries, such as translational symmetry, which will simplify the analysis of the Hamiltonian matrix. In our results we discuss several effects that break the lattice symmetry, as the intersection between symmetric and antisymmetric states. We also compare our results with those obtained in previous works for the classical model, and we find some similarities, e.g. the transition of the highest-energy state from a one-site solution to a two-site solution depending on which Q parameters we vary, but also differences, as the appearance of a three-site solution, in a Trimer system.

•Jan 1, 2011•1 citation

Lieb's Theorem for Bose Hubbard Models

Zhongchao Wei, Chong Ke Zhao

Using a cone-theoretical method, we prove the uniqueness of the ground state for two Bose Hubbard models. The first model is the usual Bose Hubbard model with real hopping coefficients and attractive interactions. The second model is a two-component Bose Hubbard model. Under certain conditions, we show that the ground state in the subspace with particle number $N=2n$($n$ is a positive integer) is unique for both models. For the second model, we show that the ground state has spin along the z-axis $S^{z}=0$. When the hopping coefficients are real, it has zero spin quantum number, i.e., it is a singlet. Our proofs work equally well for any arbitrary finite-size lattice.

arXiv:2506.10319•Jun 12, 2025•0 citations

Fast-forward ground-state preparation in finite two-site Bose–Hubbard systems

Iwan Setiawan, Dedy Hamdani, Riska Ekawita +2 more

Abstract We present a method to accelerate the adiabatic quantum dynamics of two-site Bose–Hubbard systems by using the fast-forward method proposed by Masuda and Nakamura. The accelerated scheme is constructed by adding a driving Hamiltonian to the original Hamiltonian and introducing a large time-scaling factor together with an adiabatic parameter, enabling adiabatic quantum dynamics within a shortened timescale. We apply this protocol to a finite-time ground-state preparation in the two-site Bose–Hubbard model. For two bosons, the dynamics reduces to a two-level problem in the symmetric Fock subspace, which yields an analytic current-like auxiliary coupling that suppresses diabatic transitions during an initial ground-state configuration to a desired final ground-state configuration. For three bosons, the Hamiltonian separates into symmetric and antisymmetric parity blocks; the symmetric block governs ground-state preparation from the normalized state 1 2 | 2 , 1 ⟩ + 1 2 | 1 , 2 ⟩ ), while the antisymmetric correction is relevant only for transitionless driving in the full-parity sector. For four bosons, the exchange-symmetric sector is three dimensional. This three-level structure constitutes the principal multilevel extension presented in this work. Numerical solutions of the time-dependent Schrödinger equation show that the fast-forward Hamiltonian maintains high instantaneous ground-state fidelity and reproduces the target population distribution within the prescribed finite protocol time, whereas evolution under the bare Hamiltonian produces appreciable diabatic excitations. The protocol provides an analytically tractable example of fast-forward control in an interacting bosonic dimer.

Physica Scripta•Oct 15, 2026•0 citations

Exact solution of the extended dimer Bose–Hubbard model with multi-body interactions

Feng Pan, Dongkang Li, Sai Cui +3 more

Abstract It is shown that the extended one-dimensional dimer Bose–Hubbard model with multi-body interactions can be solved exactly by using the algebraic Bethe ansatz mainly due to the site-permutation S 2 symmetry. The solution for the model with up to three-particle hopping and three-body on-site interaction is explicitly shown. As an example of the application, lower part of the excitation energy levels and the ground-state entanglement measure of the standard Bose–Hubbard Hamiltonian with the attractive two-body on-site interaction plus the three-body on-site interaction for 100 bosons with variation of the control parameter are calculated by using the exact solution. It is shown that the attractive three-body on-site interaction reinforces the critical point entanglement of the system, which may be helpful for design of an optical lattice for ultracold atoms or a tuneable superconducting quantum interference device with maximal entanglement.

Journal of Statistical Mechanics Theory and Experiment•Apr 1, 2020•3 citations
The Bose-Hubbard Hamiltonian can be simplified to have only two lattice sites, in which case the system being described is referred to as a dimer. Due to its structure, the hopping term of the dimer Hamiltonian enjoys invariance in a family of subspaces indexed by a whole number kkk, each subspace corresponding to a system of only kkk particles. We have invented an inductive argument using the bosonic canonical commutation relations to find the eigenvalues and eigenvectors of the dimer hopping Hamiltonian in its kkk-particle subspaces. In particular, this Hamiltonian, when restricted to one of the kkk-particle subspaces, is exactly the spin projection operator along the xxx-axis, where the number of particles kkk in the dimer system yields the projection matrix for spin quantum number s=k/2s=k/2s=k/2. Thus, a new method for computing the eigenvalues and eigenvectors of the xxx-axis spin projector has been unearthed. We use the explicit construction to study the dynamics of coherent states induced by the square of the dimer hopping hamiltonian. We find that it generates Schrödinger cat states in the two-site setting.
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.
In order to explore a quantum version of a discrete nonlinear Schrodinger equation (DNLS), we quantize one nonlinear Schrodinger model, which is used to study different physical systems, e.g. coupled Bose-Einstein condensates. We will focus on small systems, like Dimer and Trimer.In our efforts to solve this quantum problem, we develop a Mathematica routine that implements the Number State Method and solves the corresponding Schrodinger equation. We calculate analytically and numerically the energy spectrum of the Dimer and Trimer systems. Those eigenenergies depend on the parameter set Q=Q1, Q2, Q3, Q4, Q5 and by adjusting this set Q, we can obtain the desired results and examine their effects. After the quantization of the extended DNLS we obtain a quantum DNLS, also known as an extended Bose-Hubbard (BH) model. The aim of this Master's thesis is to study the differences and similarities between the classical DNLS and the extended BH model, and what happens when we approach from the quantum regime to the classical one. Taking into account that the Hamiltonian has an important conserved quantity, the number operator, enables the total Hamiltonian to be block-diagonalized. This can be accomplished by taking advantage of additional symmetries, such as translational symmetry, which will simplify the analysis of the Hamiltonian matrix. In our results we discuss several effects that break the lattice symmetry, as the intersection between symmetric and antisymmetric states. We also compare our results with those obtained in previous works for the classical model, and we find some similarities, e.g. the transition of the highest-energy state from a one-site solution to a two-site solution depending on which Q parameters we vary, but also differences, as the appearance of a three-site solution, in a Trimer system.
Using a cone-theoretical method, we prove the uniqueness of the ground state for two Bose Hubbard models. The first model is the usual Bose Hubbard model with real hopping coefficients and attractive interactions. The second model is a two-component Bose Hubbard model. Under certain conditions, we show that the ground state in the subspace with particle number N=2nN=2nN=2n(nnn is a positive integer) is unique for both models. For the second model, we show that the ground state has spin along the z-axis Sz=0S^{z}=0Sz=0. When the hopping coefficients are real, it has zero spin quantum number, i.e., it is a singlet. Our proofs work equally well for any arbitrary finite-size lattice.
Abstract We present a method to accelerate the adiabatic quantum dynamics of two-site Bose–Hubbard systems by using the fast-forward method proposed by Masuda and Nakamura. The accelerated scheme is constructed by adding a driving Hamiltonian to the original Hamiltonian and introducing a large time-scaling factor together with an adiabatic parameter, enabling adiabatic quantum dynamics within a shortened timescale. We apply this protocol to a finite-time ground-state preparation in the two-site Bose–Hubbard model. For two bosons, the dynamics reduces to a two-level problem in the symmetric Fock subspace, which yields an analytic current-like auxiliary coupling that suppresses diabatic transitions during an initial ground-state configuration to a desired final ground-state configuration. For three bosons, the Hamiltonian separates into symmetric and antisymmetric parity blocks; the symmetric block governs ground-state preparation from the normalized state 1 2 | 2 , 1 ⟩ + 1 2 | 1 , 2 ⟩ ), while the antisymmetric correction is relevant only for transitionless driving in the full-parity sector. For four bosons, the exchange-symmetric sector is three dimensional. This three-level structure constitutes the principal multilevel extension presented in this work. Numerical solutions of the time-dependent Schrödinger equation show that the fast-forward Hamiltonian maintains high instantaneous ground-state fidelity and reproduces the target population distribution within the prescribed finite protocol time, whereas evolution under the bare Hamiltonian produces appreciable diabatic excitations. The protocol provides an analytically tractable example of fast-forward control in an interacting bosonic dimer.
Abstract It is shown that the extended one-dimensional dimer Bose–Hubbard model with multi-body interactions can be solved exactly by using the algebraic Bethe ansatz mainly due to the site-permutation S 2 symmetry. The solution for the model with up to three-particle hopping and three-body on-site interaction is explicitly shown. As an example of the application, lower part of the excitation energy levels and the ground-state entanglement measure of the standard Bose–Hubbard Hamiltonian with the attractive two-body on-site interaction plus the three-body on-site interaction for 100 bosons with variation of the control parameter are calculated by using the exact solution. It is shown that the attractive three-body on-site interaction reinforces the critical point entanglement of the system, which may be helpful for design of an optical lattice for ultracold atoms or a tuneable superconducting quantum interference device with maximal entanglement.