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:1011.3517 · Physical Review A

Quantum rotor theory of spinor condensates in tight traps

Ryan Barnett, Hoi-Yin Hui, Chien-Hung Lin, Jay Deep Sau, S. Das Sarma

Published Feb 22, 2011• 18 citations
arXivHTMLPDF

Abstract

In this work, we theoretically construct exact mappings of many-particle bosonic systems onto quantum rotor models. In particular, we analyze the rotor representation of spinor Bose-Einstein condensates. In a previous work [R. Barnett et al., Phys. Rev. A 82, 031602(R) (2010)] it was shown that there is an exact mapping of a spin-one condensate of fixed particle number with quadratic Zeeman interaction onto a quantum rotor model. Since the rotor model has an unbounded spectrum from above, it has many more eigenstates than the original bosonic model. Here we show that for each subset of states with fixed spin ${F}_{z}$, the physical rotor eigenstates are always those with the lowest energy. We classify three distinct physical limits of the rotor model: the Rabi, Josephson, and Fock regimes. The last regime corresponds to a fragmented condensate and is thus not captured by the Bogoliubov theory. We next consider the semiclassical limit of the rotor problem and make connections with the quantum wave functions through the use of the Husimi distribution function. Finally, we describe how to extend the analysis to higher-spin systems and derive a rotor model for the spin-two condensate. Theoretical details of the rotor mapping are also provided here.

DOI
10.1103/physreva.83.023613

Related papers

Quantum rotor theory of systems of spin-2 bosons

Matjaž Payrits, Ryan Barnett

We consider quantum phases of tightly confined spin-2 bosons in an external field under the presence of rotationally invariant interactions. Generalizing previous treatments, we show how this system can be mapped onto a quantum rotor model. Within the rotor framework, low-energy excitations about fragmented states, which cannot be accessed within standard Bogoliubov theory, can be obtained. In the spatially extended system in the thermodynamic limit there exists a mean field ground-state degeneracy between a family of nematic states for appropriate interaction parameters. It has been established that quantum fluctuations lift this degeneracy through the mechanism of order by disorder and select either a uniaxial or square-biaxial ground state. On the other hand, in the full quantum treatment of the analogous single-spatial-mode problem with finite-particle number, it is known that, due to symmetry-restoring fluctuations, there is a unique ground state across the entire nematic region of the phase diagram. Within the established rotor framework, we investigate the possible quantum phases under the presence of a quadratic Zeeman field, a problem which has previously received little attention. By investigating wave-function overlaps, we do not find any signatures of the order-by-disorder phenomenon which is present in the continuum case. Motivated by this, we consider an alternative external potential which breaks less symmetry than the quadratic Zeeman field. For this case, we do find the phenomenon of order by disorder in the fully quantum system. This is established within the rotor framework and with exact diagonalization.

arXiv:1604.05639•Physical Review A•Aug 2, 2016•2 citations

Antiferromagnetic spinor condensates are quantum rotors

Ryan Barnett, Jay Deep Sau, S. Das Sarma

We establish a theoretical correspondence between spin-one antiferromagnetic spinor condensates in an external magnetic field and quantum rotor models in an external potential. We show that the rotor model provides a conceptually clear picture of the possible phases and dynamical regimes of the antiferromagnetic condensate. We also show that this mapping simplifies calculations of the condensate's spectrum and wave functions. We use the rotor mapping to describe the different dynamical regimes recently observed in $^{23}\mathrm{Na}$ condensates [Phys. Rev. Lett. 102, 125301 (2009); Phys. Rev. Lett. 102, 225301 (2009)]. We also suggest a way to experimentally observe quantum-mechanical effects (collapse and revival) in spinor condensates.

arXiv:1003.2634•Physical Review A•Sep 27, 2010•33 citations

Exact versus mean-field description of the Bose-Einstein condensate: a model study

Magdalena A. Załuska–Kotur, Mariusz Gajda, Arkadiusz Orłowski +1 more

We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.

arXiv:cond-mat/0004469•Apr 27, 2000•4 citations

Quantum states of a binary mixture of spinor Bose-Einstein condensates

Zhi-Fang Xu, Jie Zhang, Yunbo Zhang +1 more

We study the structure of quantum states for a binary mixture of spin-1 atomic Bose-Einstein condensates. In contrast to collision between identical bosons, the $s$-wave scattering channel between interspecies does not conform to a fixed symmetry. The spin-dependent Hamiltonian thus contains noncommuting terms, making the exact eigenstates more challenging to obtain because they now depend more generally on both the intra- and interspecies interactions. We discuss two limiting cases, where the spin-dependent Hamiltonian reduces respectively to sums of commuting operators. All eigenstates can then be directly constructed, and they are independent of the detailed interaction parameters.

arXiv:1001.5335•Physical Review A•Mar 9, 2010•19 citations

Dynamics of spinor Bose-Einstein condensates close to spin-spatial resonances

W. Wills, D. Blume, Q. Guan

We develop a coupled-channel framework to describe the dynamics of spinor Bose-Einstein condensates (BECs), with particular emphasis on the behavior near resonances between spin dynamics and spatial excitations. Taking advantage of the disparity between the spin-dependent and spin-independent scattering lengths in typical spinor BECs, the Bogoliubov modes of the spin-independent part of the full system Hamiltonian provide an efficient set of basis functions for describing the system dynamics in a coupled-channel framework. For quadratic Zeeman shifts far from any resonance, the system can be described by a single spatial wavefunction during the spin dynamics, i.e., the so-called single-mode approximation holds. By tuning the quadratic Zeeman shift, we find resonant excitations of the Bogoliubov modes, which can be classified into two categories: those with particle-hole correlations and those without particle-hole correlations. We show that the beyond-quadratic-order terms that are neglected in standard Bogoliubov theories become increasingly important for capturing the long-time dynamics of the system near resonances. The coupled-channel framework is benchmarked against results from 1D Gross-Pitaevskii equation simulations. The framework developed in this work not only provides a numerically efficient tool for describing spinor BEC dynamics governed by different length scales, but also provides a clean physical interpretation of resonance phenomena in spinor BECs. Applications of this approach to other systems and extensions to the beyond-mean-field regime are also discussed.

arXiv:2604.16732•Physical Review A•Oct 20, 2026•0 citations

Quantum Rotor Models: Large-N Limit

Subir Sachdev A

This chapter turns to the models obtained by the quantum–classical mapping QC on the D -dimensional, N -component, classical ferromagnets in (3.3) with N ≥ 2. These are the O( N ) quantum rotor models in d = D − 1 dimensions, originally written down in (1.23). The quantum Ising model studied previously had a discrete Z 2 symmetry. An important new ingredient in the rotor models will be the presence of a continuous symmetry: The physics is invariant under a uniform, global O( N ) transformation on the orientation of the rotors, which is broken in the magnetically ordered state. We will introduce the important concept of the spin stiffness , which characterizes the rigidity of the ordered state and determines the dispersion spectrum of the low energy “spin-wave” excitations. Apart from this, much of the technology and the physical ideas introduced earlier for the d = 1 Ising chain will generalize straightforwardly, although we will no longer be able to obtain exact results for crossover functions. The characterization of the physics in terms of three regions separated by smooth crossovers, the high- T and the two low- T regions on either side of the quantum critical point, will continue to be extremely useful and will again be the basis of our discussion. Because we will consider models in spatial dimensions d > 1, it will be possible to have a thermodynamic phase transition at a nonzero temperature. We shall be particularly interested in the interplay between the critical singularities of the finite-temperature transition and those of the quantum critical point.

Cambridge University Press eBooks•Jan 13, 2000•0 citations
In this work, we theoretically construct exact mappings of many-particle bosonic systems onto quantum rotor models. In particular, we analyze the rotor representation of spinor Bose-Einstein condensates. In a previous work [R. Barnett et al., Phys. Rev. A 82, 031602(R) (2010)] it was shown that there is an exact mapping of a spin-one condensate of fixed particle number with quadratic Zeeman interaction onto a quantum rotor model. Since the rotor model has an unbounded spectrum from above, it has many more eigenstates than the original bosonic model. Here we show that for each subset of states with fixed spin Fz{F}_{z}Fz​, the physical rotor eigenstates are always those with the lowest energy. We classify three distinct physical limits of the rotor model: the Rabi, Josephson, and Fock regimes. The last regime corresponds to a fragmented condensate and is thus not captured by the Bogoliubov theory. We next consider the semiclassical limit of the rotor problem and make connections with the quantum wave functions through the use of the Husimi distribution function. Finally, we describe how to extend the analysis to higher-spin systems and derive a rotor model for the spin-two condensate. Theoretical details of the rotor mapping are also provided here.
We consider quantum phases of tightly confined spin-2 bosons in an external field under the presence of rotationally invariant interactions. Generalizing previous treatments, we show how this system can be mapped onto a quantum rotor model. Within the rotor framework, low-energy excitations about fragmented states, which cannot be accessed within standard Bogoliubov theory, can be obtained. In the spatially extended system in the thermodynamic limit there exists a mean field ground-state degeneracy between a family of nematic states for appropriate interaction parameters. It has been established that quantum fluctuations lift this degeneracy through the mechanism of order by disorder and select either a uniaxial or square-biaxial ground state. On the other hand, in the full quantum treatment of the analogous single-spatial-mode problem with finite-particle number, it is known that, due to symmetry-restoring fluctuations, there is a unique ground state across the entire nematic region of the phase diagram. Within the established rotor framework, we investigate the possible quantum phases under the presence of a quadratic Zeeman field, a problem which has previously received little attention. By investigating wave-function overlaps, we do not find any signatures of the order-by-disorder phenomenon which is present in the continuum case. Motivated by this, we consider an alternative external potential which breaks less symmetry than the quadratic Zeeman field. For this case, we do find the phenomenon of order by disorder in the fully quantum system. This is established within the rotor framework and with exact diagonalization.
We establish a theoretical correspondence between spin-one antiferromagnetic spinor condensates in an external magnetic field and quantum rotor models in an external potential. We show that the rotor model provides a conceptually clear picture of the possible phases and dynamical regimes of the antiferromagnetic condensate. We also show that this mapping simplifies calculations of the condensate's spectrum and wave functions. We use the rotor mapping to describe the different dynamical regimes recently observed in 23Na^{23}\mathrm{Na}23Na condensates [Phys. Rev. Lett. 102, 125301 (2009); Phys. Rev. Lett. 102, 225301 (2009)]. We also suggest a way to experimentally observe quantum-mechanical effects (collapse and revival) in spinor condensates.
We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.
We study the structure of quantum states for a binary mixture of spin-1 atomic Bose-Einstein condensates. In contrast to collision between identical bosons, the sss-wave scattering channel between interspecies does not conform to a fixed symmetry. The spin-dependent Hamiltonian thus contains noncommuting terms, making the exact eigenstates more challenging to obtain because they now depend more generally on both the intra- and interspecies interactions. We discuss two limiting cases, where the spin-dependent Hamiltonian reduces respectively to sums of commuting operators. All eigenstates can then be directly constructed, and they are independent of the detailed interaction parameters.
We develop a coupled-channel framework to describe the dynamics of spinor Bose-Einstein condensates (BECs), with particular emphasis on the behavior near resonances between spin dynamics and spatial excitations. Taking advantage of the disparity between the spin-dependent and spin-independent scattering lengths in typical spinor BECs, the Bogoliubov modes of the spin-independent part of the full system Hamiltonian provide an efficient set of basis functions for describing the system dynamics in a coupled-channel framework. For quadratic Zeeman shifts far from any resonance, the system can be described by a single spatial wavefunction during the spin dynamics, i.e., the so-called single-mode approximation holds. By tuning the quadratic Zeeman shift, we find resonant excitations of the Bogoliubov modes, which can be classified into two categories: those with particle-hole correlations and those without particle-hole correlations. We show that the beyond-quadratic-order terms that are neglected in standard Bogoliubov theories become increasingly important for capturing the long-time dynamics of the system near resonances. The coupled-channel framework is benchmarked against results from 1D Gross-Pitaevskii equation simulations. The framework developed in this work not only provides a numerically efficient tool for describing spinor BEC dynamics governed by different length scales, but also provides a clean physical interpretation of resonance phenomena in spinor BECs. Applications of this approach to other systems and extensions to the beyond-mean-field regime are also discussed.
This chapter turns to the models obtained by the quantum–classical mapping QC on the D -dimensional, N -component, classical ferromagnets in (3.3) with N ≥ 2. These are the O( N ) quantum rotor models in d = D − 1 dimensions, originally written down in (1.23). The quantum Ising model studied previously had a discrete Z 2 symmetry. An important new ingredient in the rotor models will be the presence of a continuous symmetry: The physics is invariant under a uniform, global O( N ) transformation on the orientation of the rotors, which is broken in the magnetically ordered state. We will introduce the important concept of the spin stiffness , which characterizes the rigidity of the ordered state and determines the dispersion spectrum of the low energy “spin-wave” excitations. Apart from this, much of the technology and the physical ideas introduced earlier for the d = 1 Ising chain will generalize straightforwardly, although we will no longer be able to obtain exact results for crossover functions. The characterization of the physics in terms of three regions separated by smooth crossovers, the high- T and the two low- T regions on either side of the quantum critical point, will continue to be extremely useful and will again be the basis of our discussion. Because we will consider models in spatial dimensions d > 1, it will be possible to have a thermodynamic phase transition at a nonzero temperature. We shall be particularly interested in the interplay between the critical singularities of the finite-temperature transition and those of the quantum critical point.