Josephson Oscillation and Nonlinear Self-Trapping in Quasi-one-dimensional Quantum Liquid

arXiv:2602.00674 · cond-mat.quant-gas · Submitted 2026-01-31 · Read on arXiv

Listen

Radio episode about this paper

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Hi, I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Josephson Oscillation and Nonlinear Self-Trapping in Quasi-one-dimensional Quantum Liquid".

Mira: Here's the gist of the paper. The gist The study investigates Josephson oscillation and nonlinear self-trapping in quasi-one-dimensional quantum liquids by analyzing two-mode dynamics under mean-field, beyond mean-field, and three-body interactions.

Kai: Theoretical Framework The analysis employs the two-mode approximation to understand the dynamics of macroscopic wave functions separated by a barrier <ref:2602.00674#pg2> The condensate wave function can be written as ψ(x, t) = ϕL(x)cL(t) + ϕR(x)cR(t)<ref:2602.00674#pg5> The time-dependent complex coefficients ci(t) describe the population and phase dynamics in each mode<ref:2602.00674#pg6> The effective interaction energy Ueff f includes the usual two-body mean field interaction along with BMF and three-body interaction such that Ueff f = −g3ψ2 + g4ψ3 + g5ψ4<ref:2602.00674#pg3> The system is described by the extended Gross-Pitaevskii equation i∂tψ = h−∂2x2/2 + V(x) + Ueff f i ψ<ref:2602.00674#pg7> Dynamics and Regimes The dynamics in a double well potential can lead to three distinct regimes based on the population imbalance between the wells <ref:2602.00674#pg2> At low particle imbalance between the wells, one can observe plasma oscillation (PO), while at moderate imbalance, Josephson oscillation (JO) and at high imbalance leads to self trapping (ST)<ref:2602.00674#pg2> The shift in frequencies serves as a clear indication of the BMF effect<ref:2602.00674#pg4> Under the two mode approximation, the condensate wave function can be written as ψ(x, t) = ϕL(x)cL(t) + ϕR(x)cR(t)<ref:2602.00674#pg7> The shifts in particle number imbalance (change in z(t)) and relative phase shift (change in ϕ(t)) play a prominent role<ref:2602.00674#pg8> Interaction Effects and Dimensionality The role of dimensionality is systematically studied by examining the Q1D and 1D systems<ref:2602.00674#pg4> In mathematical terms, a Q1D system comprises of an NLSE with cubic and quartic nonlinearity with attractive MF interaction (cubic nonlinearity) and repulsive BMF interaction (quartic nonlinearity) while in a 1D system the nonlinear exponents are cubic (MF) and quadratic (BMF)<ref:2602.00674#pg5> The effective interaction energy includes the mean-field term g3, BMF term g4, and three-body term g5 such that Ueff f = −g3ψ2 + g4ψ3 + g5ψ4<ref:2602.00674#pg3> The competition between cubic MF and quintic 3B interaction has been studied in the context of flat-top solitons<ref:2602.00674#pg5> Symmetric and Asymmetric Potentials For a symmetric double well (SDW) potential, the critical imbalance for the JO to ST transition is found to be lowest at z0 = 0.038 when only MF interaction is considered<ref:2602.00674#pg8> In asymmetric double well (ADW) potential, an asymmetry or a tilted lattice may yield a more pronounced effect of the interaction competition<ref:2602.00674#pg2> The critical imbalance increases as the system is subjected to more interaction competition in Q1D and 1D systems<ref:2602.00674#pg11> Bogoliubov Analysis and Instabilities The Bogoliubov quasiparticle method provides a complementary description, suggesting that a frequency match of JO and roton-like minima in the Bogoliubov spectrum indicates resonant coupling leading to an enhancement of tunnelling, damping or instabilities in Josephson dynamics<ref:2602.00674#pg13> The dispersion relation for Q1D and 1D systems is given by Kd − ωMd Kd + ω, uq vq = 0<ref:2602.00674#pg4> The interplay of different interaction components leads to a nontrivial dispersion as displayed in Fig.

Mira: 9 and 10<ref:2602.00674#pg13> The kink in the Bogoliubov frequency is considered as the precursor for generation of super-solid like phase<ref:2602.00674#pg13> The analysis concludes that the Josephson frequency is non-monotonic as a function of particle number in Q1D, and addition of small asymmetry pronounces the interaction competition effect The critical imbalance is lowest in quasi one dimension when the particles are trapped in a symmetric double well potential, while it is relatively higher in asymmetric double well potential<ref:2602.00674#pg11> The Bogoliubov excitation calculation qualitatively corroborates with TMM result but importantly it allows us to explicate on the region of instabilities in the dispersion The roton-like mode might suggest the onset of super-solid phase in the Bose system<ref:2602.00674#pg13> The critical imbalance is lowest in quasi one dimension when the particles are trapped in a symmetric double well potential, while it is relatively higher in asymmetric double well potential<ref:2602.00674#pg11> How it works The two-mode approximation characterizes the dynamics by defining the fractional relative population as z(t) = NL(t) − NR(t) / NL(t) + NR(t)<ref:2602.00674#pg5> The number of particles in each well is calculated as NL(t) = R0−∞ ψ(x, t)2dx (NR(t) = R∞0 ψ2dx<ref:2602.00674#pg6> The coupled equations for z(t) and θ(t) are derived from the Hamiltonian H<ref:2602.00674#pg13> The critical value of Λc is determined by solving the coupled equations, and the system transitions between JO and ST regimes when Λ overcomes this value<ref:2602.00674#pg8> How it works The Josephson frequency in Q1D is given by ωJ Q1D = s2J0 / 2J0 + ¯g3N + 9/4√2 g¯4N3/2 + ¯g5N2/2 The critical imbalance for the symmetric case is z0c(Q1D) = MF, which is 0.039 This critical imbalance increases as the system is subjected to more interaction competition in Q1D and 1D systems<ref:2602.00674#pg11> The Josephson frequency can be noted as non-monotonic as a function of particle number in Q1D, with three-body effect dominating for relatively large particle number<ref:2602.00674#pg3> How it works The Bogoliubov dispersion relation in Q1D is given by ωQ1D = r/q4 + Aq2/2 + B<ref:2602.00674#pg4> The critical momentum qc corresponding to the dispersion kink is qc = 1.84<ref:2602.00674#pg13> The region of instability is at higher frequency in Q1D system compared to the 1D system<ref:2602.00674#pg13> The roton-like mode might suggest the onset of super-solid phase in the Bose system<ref:2602.00674#pg13> How it works The critical imbalance beyond which one can observe a transition between Josephson regime and self trapping is lowest in quasi one dimension when the particles are trapped in a symmetric double well potential, while it is relatively higher in asymmetric double well potential<ref:2602.00674#pg11> The variation is relatively small in the symmetric cases while a small asymmetry allows relatively significant variation in the frequency The oscillation frequency is actually gets suppressed when there is more competition between different interactions How it works The critical imbalance beyond which one can observe a transition between Josephson regime and self trapping is lowest in quasi one dimension when the particles are trapped in a symmetric double well potential, while it is relatively higher in asymmetric double well potential<ref:2602.

Paper summary: Kai: So we're looking at this paper, "Josephson Oscillation and Nonlinear Self-Trapping in Quasi-one-dimensional Quantum Liquid." The main idea here is using a two-mode method to analyze how Josephson oscillation happens in a trapped binary Bose-Einstein condensate while including those beyond the mean-field and three body interactions.

Mira: Basically, they use the archetypal double well potential to look at the Josephson oscillation and self-trapping phases specifically in quasi one dimension. It’s important because it shows how these extra interactions affect what we expect to see in a simple system.

Lev: What matters is that they're not just looking at mean field stuff; they're including beyond mean field and three body interactions, which complicates the picture a lot for any real simulation or experiment.

Kai: They also do a quantitative discussion on how asymmetry and dimension affect the results, which is key because we often think about perfect symmetry in these setups.

Mira: And to back up their claims, they check their findings against the Bogoliubov quasi-particle method and actually notice some things like a roton-like mode.

Lev: That roton-like mode suggests there are some instabilities in the system that could lead to something else developing, which is important for understanding the phase transitions here.

Kai: So, we see they’re taking a standard setup—a trapped binary BEC—and pushing it with these extra interaction terms to see how those dynamics change in one dimension. That sets up what they find next about the different regimes of oscillation.

Conclusion: Mira: Thinking about the title, "Josephson Oscillation and Nonlinear Self-Trapping in Quasi-one-dimensional Quantum Liquid," it really captures the core physics they're exploring here, focusing on those two specific dynamic phases in a one dimensional quantum fluid.

Kai: And it’s interesting that the authors focused so much on how asymmetry plays a role, showing that this isn't just about a perfectly symmetric setup. They found that the critical imbalance for the transition between Josephson oscillation and self-trapping is lowest when you have a symmetric double well potential in quasi one dimension.

Lev: That’s interesting because it means if we were building hardware, the setup where we observe this transition most clearly would be in a symmetric configuration in that specific dimensional setting.

Mira: And they also found that asymmetry actually tends to increase the critical imbalance when you look at Q1D and 1D systems, which is a bit counter-intuitive depending on how you look at the competition between those interactions <ref:2602.00674#pg1>.

Kai: So, it’s not always a simple trade-off; sometimes adding more interaction competition can actually make the system need a bigger imbalance to get that transition happening.

Lev: It just tells us that when we model these systems, we can't rely on one simple picture of how interactions compete across different dimensions.

Department of Physics, School of Engineering and Applied Sciences, Bennett University

cond-mat.quant-gas

Submitted: 2026-01-31

Updated: 2026-10-21

Comments: 15 Pages, 11 figures; To appear in European Physical Journal Plus

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 62/100

The gist: The gist The study investigates Josephson oscillation and nonlinear self-trapping in quasi-one-dimensional quantum liquids by analyzing two-mode dynamics under mean-field, beyond mean-field, and

Key concepts

Two-Mode Approximation
This approximation simplifies the complex dynamics of a trapped condensate by treating it as two separate modes, one in each well of a double well potential. It allows researchers to track how the population imbalance and relative phase evolve over time, which is crucial for understanding oscillation and trapping.
Josephson Oscillation (JO) vs. Self-Trapping (ST)
These are two distinct dynamical regimes observed in the system based on particle imbalance. At low imbalance, the system exhibits Josephson oscillation where particles tunnel back and forth between wells. At high imbalance, it transitions to self-trapping, where the dynamics change significantly due to strong nonlinear interactions.
Beyond Mean-Field (BMF) and Three-Body Interactions
These terms describe higher-order interactions beyond simple mean-field theory. The BMF interaction is repulsive and affects the system's dynamics differently than standard mean-field. The three-body term introduces further complexity, influencing phenomena like flat-top solitons and the overall stability of the condensate.
Roton-like Mode
This mode appears in the Bogoliubov quasiparticle analysis. Its existence suggests a resonant coupling between Josephson oscillation and other modes, which can lead to enhanced tunneling or instabilities in the Josephson dynamics, potentially signaling a transition to a super-solid phase.

Summary

The gist The study investigates Josephson oscillation and nonlinear self-trapping in quasi-one-dimensional quantum liquids by analyzing two-mode dynamics under mean-field, beyond mean-field, and three-body interactions.

Theoretical Framework

The analysis employs the two-mode approximation to understand the dynamics of macroscopic wave functions separated by a barrier <ref:2602.00674#pg2> The condensate wave function can be written as ψ(x, t) = ϕL(x)cL(t) + ϕR(x)cR(t)<ref:2602.00674#pg5> The time-dependent complex coefficients ci(t) describe the population and phase dynamics in each mode<ref:2602.00674#pg6> The effective interaction energy Ueff f includes the usual two-body mean field interaction along with BMF and three-body interaction such that Ueff f = −g3ψ2 + g4ψ3 + g5ψ4<ref:2602.00674#pg3> The system is described by the extended Gross-Pitaevskii equation i∂tψ = h−∂2x2/2 + V(x) + Ueff f i ψ<ref:2602.00674#pg7>

Dynamics and Regimes

The dynamics in a double well potential can lead to three distinct regimes based on the population imbalance between the wells <ref:2602.00674#pg2> At low particle imbalance between the wells, one can observe plasma oscillation (PO), while at moderate imbalance, Josephson oscillation (JO) and at high imbalance leads to self trapping (ST)<ref:2602.00674#pg2> The shift in frequencies serves as a clear indication of the BMF effect<ref:2602.00674#pg4> Under the two mode approximation, the condensate wave function can be written as ψ(x, t) = ϕL(x)cL(t) + ϕR(x)cR(t)<ref:2602.00674#pg7> The shifts in particle number imbalance (change in z(t)) and relative phase shift (change in ϕ(t)) play a prominent role<ref:2602.00674#pg8>

Interaction Effects and Dimensionality

The role of dimensionality is systematically studied by examining the Q1D and 1D systems<ref:2602.00674#pg4> In mathematical terms, a Q1D system comprises of an NLSE with cubic and quartic nonlinearity with attractive MF interaction (cubic nonlinearity) and repulsive BMF interaction (quartic nonlinearity) while in a 1D system the nonlinear exponents are cubic (MF) and quadratic (BMF)<ref:2602.00674#pg5> The effective interaction energy includes the mean-field term g3, BMF term g4, and three-body term g5 such that Ueff f = −g3ψ2 + g4ψ3 + g5ψ4<ref:2602.00674#pg3> The competition between cubic MF and quintic 3B interaction has been studied in the context of flat-top solitons<ref:2602.00674#pg5>

Symmetric and Asymmetric Potentials

For a symmetric double well (SDW) potential, the critical imbalance for the JO to ST transition is found to be lowest at z0 = 0.038 when only MF interaction is considered<ref:2602.00674#pg8> In asymmetric double well (ADW) potential, an asymmetry or a tilted lattice may yield a more pronounced effect of the interaction competition<ref:2602.00674#pg2> The critical imbalance increases as the system is subjected to more interaction competition in Q1D and 1D systems<ref:2602.00674#pg11>

Bogoliubov Analysis and Instabilities

The Bogoliubov quasiparticle method provides a complementary description, suggesting that a frequency match of JO and roton-like minima in the Bogoliubov spectrum indicates resonant coupling leading to an enhancement of tunnelling, damping or instabilities in Josephson dynamics<ref:2602.00674#pg13> The dispersion relation for Q1D and 1D systems is given by Kd − ωMd Kd + ω, uq vq = 0<ref:2602.00674#pg4> The interplay of different interaction components leads to a nontrivial dispersion as displayed in Fig. 9 and 10<ref:2602.00674#pg13> The kink in the Bogoliubov frequency is considered as the precursor for generation of super-solid like phase<ref:2602.00674#pg13>

Conclusion

The analysis concludes that the Josephson frequency is non-monotonic as a function of particle number in Q1D, and addition of small asymmetry pronounces the interaction competition effect The critical imbalance is lowest in quasi one dimension when the particles are trapped in a symmetric double well potential, while it is relatively higher in asymmetric double well potential<ref:2602.00674#pg11> The Bogoliubov excitation calculation qualitatively corroborates with TMM result but importantly it allows us to explicate on the region of instabilities in the dispersion The roton-like mode might suggest the onset of super-solid phase in the Bose system<ref:2602.00674#pg13> The critical imbalance is lowest in quasi one dimension when the particles are trapped in a symmetric double well potential, while it is relatively higher in asymmetric double well potential<ref:2602.00674#pg11>

How it works

The two-mode approximation characterizes the dynamics by defining the fractional relative population as z(t) = NL(t) − NR(t) / NL(t) + NR(t)<ref:2602.00674#pg5> The number of particles in each well is calculated as NL(t) = R0−∞ ψ(x, t)2dx (NR(t) = R∞0 ψ2dx<ref:2602.00674#pg6> The coupled equations for z(t) and θ(t) are derived from the Hamiltonian H<ref:2602.00674#pg13> The critical value of Λc is determined by solving the coupled equations, and the system transitions between JO and ST regimes when Λ overcomes this value<ref:2602.00674#pg8>

How it works

The Josephson frequency in Q1D is given by ωJ Q1D = s2J0 / 2J0 + ¯g3N + 9/4√2 g¯4N3/2 + ¯g5N2/2 The critical imbalance for the symmetric case is z0c(Q1D) = MF, which is 0.039 This critical imbalance increases as the system is subjected to more interaction competition in Q1D and 1D systems<ref:2602.00674#pg11> The Josephson frequency can be noted as non-monotonic as a function of particle number in Q1D, with three-body effect dominating for relatively large particle number<ref:2602.00674#pg3>

How it works

The Bogoliubov dispersion relation in Q1D is given by ωQ1D = r/q4 + Aq2/2 + B<ref:2602.00674#pg4> The critical momentum qc corresponding to the dispersion kink is qc = 1.84<ref:2602.00674#pg13> The region of instability is at higher frequency in Q1D system compared to the 1D system<ref:2602.00674#pg13> The roton-like mode might suggest the onset of super-solid phase in the Bose system<ref:2602.00674#pg13>

How it works

The critical imbalance beyond which one can observe a transition between Josephson regime and self trapping is lowest in quasi one dimension when the particles are trapped in a symmetric double well potential, while it is relatively higher in asymmetric double well potential<ref:2602.00674#pg11> The variation is relatively small in the symmetric cases while a small asymmetry allows relatively significant variation in the frequency The oscillation frequency is actually gets suppressed when there is more competition between different interactions

How it works

The critical imbalance beyond which one can observe a transition between Josephson regime and self trapping is lowest in quasi one dimension when the particles are trapped in a symmetric double well potential, while it is relatively higher in asymmetric double well potential<ref:2602.

Improvements for AI systems

  1. Enhanced simulation of quantum liquid dynamics by incorporating beyond mean-field and three body interactions allows AI systems to predict specific regimes like Josephson oscillation, self-trapping, or plasma oscillation based on interaction strengths.

  2. Ability to quantitatively discuss the effect of asymmetry and dimension enables AI to model how a tilted lattice in an asymmetric double well potential can yield a pronounced effect of the interaction competition.

  3. Implementation of the two mode approximation allows AI systems to calculate time dependent complex coefficients cL(t) and cR(t), enabling prediction of population imbalance shift (change in z(t)) and relative phase shift (change in phi(t)).

  4. Calculation based on the Hamiltonian H, which includes terms for MF, BMF, and 3B interactions, allows AI to determine critical imbalances like Λc from Eq.(16) and predict the transition between Josephson oscillation and self trapping phases.

  5. Application of the Bogoliubov quasiparticle method enables AI to corroborate Time-Dependent Matrix Method (TMM) results by analyzing the dispersion relation, specifically identifying regions of instability and roton-like mode in frequency-momentum curves.

  6. Capability to analyze the Bogoliubov dispersion relation shows that the interplay of different interaction components leads to a nontrivial dispersion as displayed in Fig. 9 and 10, helping AI identify the precursor for a super-solid like phase.

  7. Modeling of the Josephson frequency allows AI systems to predict how it is non-monotonic as a function of particle number in Q1D and how the addition of three body effect can straighten the Josephson frequency at large particle numbers.

Abstract

In this article, we study the two-mode method to analyze the Josephson oscillation for a trapped binary Bose-Einstein condensate while taking into account the beyond mean-field and three body interactions. For this purpose, we use the archetypal model of double well potential and study the Josephson oscillation and self-trapping phases in quasi-one dimension. Additionally, our analysis provides quantitative discussion on the effect of asymmetry and dimension. We further corroborate our findings with Bogoliubov quasi-particle method and notice the existence of roton-like mode.

Related papers