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Weakly interacting Bose gas with two-body losses
by Chang Liu, Zheyu Shi, Ce Wang
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Submission summary
| Authors (as registered SciPost users): | Ce Wang |
| Submission information | |
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| Preprint Link: | scipost_202305_00032v2 (pdf) |
| Date submitted: | 2024-02-01 11:43 |
| Submitted by: | Wang, Ce |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the many-body dynamics of weakly interacting Bose gases with two- particle losses. We show that both the two-body interactions and losses in atomic gases may be tuned by controlling the inelastic scattering process between atoms by an optical Feshbach resonance. Interestingly, the low-energy behavior of the scattering amplitude is governed by a single parameter, i.e. the complex s-wave scattering length a_c. The many-body dynamics are thus described by a Lindblad master equation with complex scattering length. We solve this equation by applying the Bogoliubov approximation in analogy to the closed systems. Various peculiar dynamical properties are discovered, some of them may be regarded as the dissipative counterparts of the celebrated results in closed Bose gases. For example, we show that the next-order correction to the mean-field particle decay rate is to the order of |n a_c^3|^1/2, which is an analogy of the Lee-Huang-Yang correction of Bose gases. It is also found that there exists a dynamical symmetry of symplectic group Sp(4,C) in the quadratic Bogoliubov master equation, which is an analogy of the SU(1,1) dynamical symmetry of the corresponding closed system. We further confirmed the validity of the Bogoliubov approximation by comparing its results with a full numerical calculation in a double-well toy model. Generalizations of other alternative approaches such as the dissipative version of the Gross-Pitaevskii equation and hydrodynamic theory are also discussed in the last.
Author comments upon resubmission
In the following, we address his/her comments and suggestions.
(1) “Providing a physical explanation of c_theta in Eq (20) would greatly assist readers in comprehending the physical implications of a complex scattering length, particularly when comparing it to a negative scattering length in closed system. This clarification would enhance the understanding of the phenomenon described.”
In dilute Bose gas without dissipation, the LHY correction of the ground state energy becomes ill-defined for a negative scattering length. This corresponds to an instability in the dynamic evolution, where the quantum depletion grows to a non-perturbative value. In comparison, the loss rate for dissipative Bose gas (eq. (20) in the manuscript) also becomes ill-defined for arg(1/a_c)>\pi/6. This also reflects the same instability of exponentially growing Bogoliubov modes because of the attractive interaction. In addition, the fact that the transition argument of 1/a_c is not \pi/2 illustrates that the two-particle loss also helps suppress the increase of these Bogoliubov modes.
We thank the referee for the kind suggestion and added an explanation paragraph in the revised manuscript (page 9, the end of chapter 3).
(2)“A clarification is required regarding the lack of change in gamma_b while the bare interaction g_b becomes g, as observed from Eq. (16) to (17). There must be some underlying considerations for this discrepancy. Hence, an explanation is necessary to address this inconsistency.”
The inconsistency arises from the subtlety of the renormalization of g_b and gamma_b. Recall that in the conventional derivation of the LHY correction of non-dissipative Bose gas (see e.g. C. J. Pethick, H. Smith, Bose–Einstein condensation in dilute gases), the momentum summation of the zero-point energies of Bogoliubov modes also diverges while incorporating a renormalized interacting strength. The divergence arises from the fact that the renormalized interaction is only valid for small momenta. To resolve this issue, one can introduce an intermediate momentum cutoff \Lambda and then consider an effective interaction with second-order processes in the mean-field energy component. Consequently, we can expand the bare interaction strength g_b to the second order of g \Lambda, resulting in a finite LHY correction.
A similar method is implemented in the calculation of the loss rate formula. Instead of an expansion of g_b, we further introduce an expansion of gamma_b (eq.(18) & (19) in the revised version). This cures the divergent momenta summation of eq. (17), resulting in a finite loss rate.
This subtlety in the renormalization calculation is not stated clearly in the first version of our manuscript. We thank the referee for pointing out this which helps us improve the manuscript’s readability.
(3) “By employing the dynamical symmetry, the system dynamics can be described by a set of first-order differential equations, as illustrated in Eq. (37). It would be valuable to investigate whether it is possible to analytically solve this set of equations. Exploring potential analytical solutions would further enhance the understanding of the system's behavior and contribute to the depth of the analysis.”
Indeed, it is possible to find analytical solutions of the first-order differential equations (eq. (37)), which would greatly help to understand the many-body dynamical behavior of the dissipative Bose gases. Yet since it is a challenging task to find the analytical solution for 7 coupled ODEs, we regretfully decide that this direction is beyond the scope of the current work. In the revised manuscript, we have briefly mentioned this interesting possibility and hope it will stimulate future studies on the exact many-body dynamics of dissipative Bose gases.
List of changes
1. In page 7 and page 8, we have replaced the bare interaction parameters \gamma_b and g_b in Eq. (15-17) with the renormalized interaction \gamma and g.
2. In page 8, we have added a paragraph to explain the reason for replacing the decay rate \gamma in the mean field term with the effective decay rate \tilde{\gamma} which is corresponding to the second order process from Eq17 to Eq 20.
3. In page 9, we have added a paragraph to explain the physical effect of the particle number decay rate in Eq 20 at the end of section 3.
4. In page 14, we have added a paragraph to illustrate that this closed algebra can provide great convenience in numerical operations and a new reference [59] “ C. Weedbrook, S. Pirandola, R. Garc ́ıa-Patr ́on, N. J. Cerf, T. C. Ralph, J. H. Shapiro and S. Lloyd, Gaussian quantum information, Reviews of Modern Physics 84(2), 621 (2012)” has been cited.
5. We have also updated our funding information.
Current status:
Reports on this Submission
Report
In the present manuscript the authors explore many-body dynamics in open quantum systems by introducing atomic losses to a ultracold bosonic gas near a optical Feshbach resonance. Essentially, this approach derived by introducing a non-Hermitian absorbing potential that ultimately led to an complex scattering length. Using this model the authors explore a number of characterizations of the open quantum system and performed analysis that led to a number of very interesting results, in particular the derivation of the dissipative GP equation. This manuscript easly satisfy the criterial of journals like Phys. Rev. A and I would like to recommend this manuscript for publication in SciPost. I would like, however, to suggest the authors consider the following points:
1) Please, revise the manuscript for grammar issues
2) In the first sentence of section 2 the words “two-body losses” and “inelastic collisions” should be swapped
3) Please, cite Refs [39-41] after Eq. (1)
4) In figure 1 (and perhaps throughout the manuscript) Im(1/a_c) is shown as a positive quantity. The imaginary part of the complex scattering length is always negative.
5) Below Eq. (8), I’m unsure what the authors meant by “decaying to the environment”. Please, be more precise.
6) Near Eq. (20), please, define explicitly the relation a_c=a_r-i*a_i for the complex scattering length
Report
The authors study the many-body dynamics of a weakly interacting Bose gas characterized by dissipation in the form of two-body losses. They discuss how inelastic collisions caused by two-body losses lead to a complex scattering length, which can be experimentally tuned via an optical Feshbach resonance. They generalize the Bogoliubov approximation to such open systems. They confirm the validity of this approximation by comparing its results to those of a full numerical calculation based on a toy model which has the same main features (interactions and dissipation). Furthermore, they explore the quench dynamics and dynamical symmetry of these systems. This work is theoretically novel and topically relates to current cold-atom experiments on open systems. The writing is clear and the derivations are comprehensive. I believe the newest version ("v2") of the manuscript meets the publishing criteria of SciPost Physics. Thus, I recommend it for publication.
Strengths
1- Given the recent advances in non-Hermitian and open quantum systems, the results presented in the paper are timely.
2- The results in the paper are robust and comprehensive.
3- Their findings provide valuable insights. Subsequent works focusing on three-body and four-body systems would be straightforward.
Report
The authors have addressed my questions and concerns in a scientific manner. I believe the current manuscript meets the criteria for publication in SciPost. Therefore, I recommend its publication.