日時: 2023年4月19日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 小久保 治哉(物性理論)
題目: Size dependence of the critical velocity for quantum vortex formation by the superfluid wake with a plate obstacle
概要: The wake generated by an object moving through a fluid depends on the size and velocity of the obstacle and is associated with various physical phenomena such as vortex formation and turbulent transition. The wake has been studied experimentally and theoretically in superfluids. In cold atomic systems, optical potentials have been used in superfluid wake experiments [1-2]. In numerical simulations [3-4], Gaussian potential is often assumed as obstacles. However, the dynamics of the wake and the critical velocity for vortex formation depend on the specifics of the obstacle potential, complicating the universal discussion. Plate obstacle is comparatively easy to handle both theoretically and numerically. The size of the obstacle can be characterized only by the width of the plate, making it easy to study its effects.In this study, the size dependence of the critical velocity for quantum vortex formulation in the wake of a 2D Bose-Einstein condensate with the plate obstacle is investigated.
[1]Woo Jin Kwon, Joon Hyun Kim, Sang Won Seo, and Y. Shin, Phys. Rev. Lett. 117, 245301 (2016)
[2]Younghoon Lim, et al, New J. Phys. 24, 083020 (2022)
[3]Kazuki Sasaki, Naoya Suzuki, and Hiroki Saito, Phys. Rev. Lett. 104, 150404 (2010)
[4]M. T. Reeves, et al, Phys. Rev. Lett. 114, 155302 (2015)
日時: 2023年4月26日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 笠松 健一(物性理論)
題目: Recent understanding and problems on vortex dynamics in binary Bose-Einstein condensates
概要: I review some recent progress and remaining problems on the properties of quantized vortices in two-component (binary) Bose-Einstein condensates (BECs). A vortex in binary BECs takes a rich variety of structures, forming a half-quantized vortex, as a result of the presence of multiple order parameters. This vortex exhibits nontrivial structure and dynamics when we consider the system of several or lots of vortices. In this talk, we will discuss (i) Interactions and dynamics of two separated vortices [1,2], (ii) Structures of vortex lattices in a rotating potential [3], (iii) Quantum turbulence by stirring of a localized potential [4].
[1] K. Kasamatsu, M. Eto, and M. Nitta, Phys. Rev. A, 93, 013615 (2016).
[2] J. Han, K. Kasamatsu, and M. Tsubota, J. Phys. Soc. Jpn. 91, 024401 (2022).
[3] K. Kasamatsu, M. Tsubota and M. Ueda, Int. J. Mod. Phys. B 19, 1835 (2005).
[4] T. Mithun, K. Kasamatsu, B. Dey, and P. G. Kevrekidis, Phys. Rev. A 103, 023301 (2021).
日時: 2023年5月10日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 鏡原 大地(量子多体)
題目:Classical simulation of non-Hermitian boson sampling dynamics using matrix product states
概要:Sampling complexity means the difficulty of sampling from a distribution close to the desired (e.g., quantum system) probability distribution and is discussed in the context of quantum supremacy. A famous example where sampling complexity is often discussed is boson sampling [1], which consists of sampling from the distribution of identical bosons generated by a linear interferometer. An interferometer is characterized by a unitary matrix. It is shown that boson sampling is hard by classical computers for random unitary matrices. Recently, by considering the sampling problem of identical bosons time-evolved by a quadratic Hamiltonian from a product state, a transition from the state where sampling can be done easily to one where it is difficult has been proposed [2]. Furthermore, it has been pointed out that when time evolution is extended to be non-unitary by considering open systems, there is a transition in sampling complexity that has not been observed in unitary systems [3].
In the previous QPT seminar, we studied the Rényi entanglement entropy in the non-Hermitian boson sampling dynamics discussed in Ref. [3] and found that low-entangled states appear in some parameter regions. In this talk, we consider a classical simulation using matrix product states (MPS) with the expectation that low-entangled states would be well described using MPS and therefore an efficient classical simulation of the sampling would be possible. We investigate the dynamics of the bond dimension, which corresponds to the number of Schmidt states kept in MPS, because it is closely related to the efficiency of sampling [4]. Our results clarify the region where an efficient classical simulation can be performed using MPS.
[1] S. Aaronson and A. Arkhipov, in Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 2011), pp. 333–342.
[2] A. Deshpande, B. Fefferman, M. C. Tran, M. Foss-Feig, and A. V. Gorshkov, Phys. Rev. Lett. 121, 030501 (2018).
[3] K. Mochizuki and R. Hamazaki, Phys. Rev. Res. 5, 013177 (2023).
[4] H.-L. Huang, W.-S. Bao, and C. Guo, Phys. Rev. A 100, 032305 (2019); C. Oh, K. Noh, B. Fefferman, and L. Jiang, Phys. Rev. A 104, 022407 (2021).
日時: 2023年5月24日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: Polo, Juan(Quantum Research Center, Technology Innovation Institute)
題目:Fractionalization of the angular momentum in SU(N) atomtronic circuits
概要:Neutral atoms guided in ring-shaped atomtronic circuits present quantized values of the angular momentum per particle. Depending on the specific parameters characterizing the system (eg: nature of the particle statistics, interactions), the winding number present different quantizion properties. In this talk, I will showcase how such a phenomenon occurs in an atomtronic circuit with a quantum fluid consisting of strongly interacting N component fermions, the so-called SU(N) fermions. For repulsive interactions a specific emerging phenomenon of attraction from repulsion appears, providing a quantization with similar properties to the attracting bosonic case. For attractive interactions, the quantization is determined by the number of components N . The suggested implementation of our work is provided by cold atoms therefore, I will also present how the fractionalization of the winding number can be read-out experimentally.
日時: 2023年5月31日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 段下 一平(量子多体)
題目:Correlation-spreading dynamics after a quantum quench in low-dimensional Ising models with transverse field
概要:Recent advances in controlling and manipulating Rydberg atoms trapped with an optical tweezer array have made it possible to utilize this platform as a quantum simulator of quantum Ising models, where dynamics of spin-spin correlations can be measured at a single-site resolution. Motivated by such experimental development, we theoretically study dynamics of spatial spreading of equal-time spin-spin correlations in the transverse-field Ising model subjected to a sudden change of the transverse field [1]. We assume that the initial state is a state polarized completely along the transverse direction, which is the ground state in the large-field limit. In one spatial dimension (1D), we use the exact analytical method and the linear spin-wave approximation (LSWA). We compare the outputs of the former with those of the latter in order to show that the latter can quantitatively capture the exact group velocity of the correlations as long as the transverse-field after the quench is sufficiently large compared to the spin-spin interaction. However, it fails to capture the detailed time dependence of the correlation functions. In 2D, with use of LSWA, we give a specific estimate of the group velocity in the large-field region. We also utilize the projected entangled pair states algorithm in order to provide quantitatively accurate time-evolution of the correlation functions for a relatively short time.
[1] R. Kaneko and I. Danshita, arXiv:2301.01407 [cond-mat.quant-gas].
日時: 2023年6月14日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: Mikkelsen, Mathias(量子多体)
題目:Correlation spreading dynamics in SU(N) Fermi-Hubbard models
概要:For local Hamiltonians the propagation front of correlations is generally bounded by a light-cone like structure, where an initially localized correlation spreads to distant sites with a maximum velocity [1,2]. Loose bounds on this velocity can be found in a range of models using the Lieb-Robinson bound [1], but the actual speed is usually much slower. These concepts have been studied [3] and experimentally confirmed [2] quenching from a Mott-state to a finitely-interacting Bose-Hubbard (BH) model. These investigations have shown that a dip in the density-density correlations will propagate with a velocity which depends on the interaction strength U. For large U the physics can be effectively described in terms of doublon-holon excitations. Similarly SU(N) Fermi-Hubbard (FH) models will have excitations corresponding to doublon pairs with different flavors.
In this work we calculate SU(2)-SU(4) equal time density correlations and compare with the BH results using the time-evolving block decimation (based on the Suzuki-Trottter decomposition) method for tensor network representations of the system [4]. Preliminary results suggest that the propagation velocity is generally lower than the BH case, particularly for the standard SU(2) FH model, but increases with N asymptotically towards the BH case (in line with Lieb-Robinson bounds which are similar to bosons in the large N limit [1]). Our study is motivated by the successful experimental implementation of SU(N) Fermi-Hubbard models using Alkaline earth(-like) atoms [5,6]. While site-resolved measurement of dynamics remains an experimental challenge our investigation shows that potentially interesting physics can be found and provides a theoretical/numerical comparison for future experiments.
[1] Z. Wang and K. R.A. Hazzard, PRX Quantum 1, 010303 (2020)
[2] M. Cheneau et al., Nature 481, 484–487 (2012)
[3] P. Barmettler, D. Poletti, M. Cheneau and C. Kollath, Phys.Rev. A 85, 053625 (2012)
[4] E. M. Stoudenmire and Steven R White, New J. Phys. 12, 055026 (2010)
[5] S. Taie, R. Yamazaki, S. Sugawa, and Y. Takahashi, Nature Physics 8, 825 (2012).
[6] C. Hofrichter, et al. Phys. Rev. X 6, 021030 (2016).
日時: 2023年6月21日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 安藤 京介(物性理論)
題目:Machine learning analysis of XY models in two-dimensional square lattice
概要: The recent remarkable development of artificial neural networks in image recognition, image classification, and natural language processing has influenced many scientific fields, and the search for new discoveries by applying this technology to any problem has begun. In the field of classical statistical physics, machine learning algorithms were introduced to identify symmetry-broken phase [1-3], and in some of these cases neural networks were shown to be able to learn order parameters and other thermodynamic parameters [1,3]. Having been able to apply machine learning techniques to conventional phase transitions, it is natural to ask whether the algorithm can be applied to unconventional phase transitions. An example of such a system is a two-dimensional XY model that exhibits the Kosterlitz-Thouless transition (KT transition) [4].
The purpose of this study is to check whether machine learning algorithms can reproduce the known results for the well understood two-dimensional square lattice XY model. We confirmed that the spin configurations obtained by Monte Carlo simulations reproduce the known behavior of the correlation function, and trained the model with an all-associative neural network and a convolutional neural network. Using the trained model, we confirm the accuracy and size dependence of the transition temperature at which the KT transition occurs from spin configuration.
[1] J. Carrasquilla and R. G. Melko, Nat. Phys. 13, 431 (2017).
[2] E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Nat. Phys.13, 435 (2017).
[3] S. J. Wetzel and M. Scherzer, Phys. Rev. B 96, 184410 (2017).
[4] A. Krizhevsky, I. Sutskever, and G. Hinton, Commun. ACM 60, 84 (2017).
日時: 2023年6月29日13:15-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 武上 響生(京都大学)
題目:Numerical analysis on anomalous tunneling of Bogoliubov excitations through a Gaussian potential barrier
概要:Elementary excitations of a weakly interacting Bose gas, which are referred to as Bogoliubov excitations [1], exhibit an anomalous tunneling property in that the transmission probability approaches unity as the excitation energy decreases towards zero [2,3]. Despite more than 20 years having passed since its first prediction [2], the anomalous tunneling has not been observed in experiments. We plan to observe the anomalous tunneling of Bogoliubov excitations using the cloud-based remote experimental apparatus of ultracold atomic Bose-Einstein condensates, called Albert [4]. For identifying the anomalous tunneling in experiments, it is necessary to have theoretical values for the transmission probability of Bogoliubov excitations through the Gaussian potential barrier used in the experiment, as a reference for comparison.
The purpose of this study is to provide theoretical values for the transmission probability in this case by numerically solving the Bogoliubov equation that describes the Bogoliubov excitations. First, we review the case of a rectangular potential, which allows for an approximate analytical expression of the transmission probability [3], as an example to illustrate the important properties of the anomalous tunneling. Next, we explain the finite element method that is used to numerically calculate the transmission probability [5]. We reproduce the calculation of the transmission probability for Bogoliubov excitations in a rectangular potential [3]. Finally, we calculate the transmission probability for a Gaussian potential and discuss optimal values of the barrier width and height for observing the anomalous tunneling with Albert.
[1] L. P. Pitaevskii and S. Stringari, Bose–Einstein Condensation., (Oxford University Press, Oxford, 2003).
[2] D. L. Kovrizhin, Phys. Lett. A 46 , 392 (2001).
[3] Yu. Kagan, D. L. Kovrizhin, and L. A. Maksimov, Phys. Rev. Lett. 90, 130402 (2003).
[4] https://bec.coldquantaapis.com
[5]計算物理Ⅱ, 夏目雄平および植田毅, 朝倉書店, 2002年.
日時: 2023年7月5日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 木屋 晴貴(量子制御)
題目:Robust Single-Qubit Gates and Polygons on Sphere
概要:In single-qubit gates, two typical inevitable systematic errors, Pulse Length Error (PLE) and Off-Resonance Error (ORE) exist. PLE is often caused by a control field strength error, while ORE is by a resonance frequency calibration error of a qubit. These errors can be compensated by a method called Composite Quantum Gate (CQG): A single gate is replaced with a sequence of gates such that their errors are canceled with each other [1]. We discuss a symmetric CQG (sCQG) which is robust against PLE from the viewpoint of a corresponding polygon on a sphere.
We are interested in the gate times of CQGs [2] and Geometric Quantum Gates [3,4]. We hope that the graphical understanding of the PLE robust condition helps to find the shortest gate time and to deepen the understanding of CQGs.
[1] M. H. Levitt and R. Freedman, J. Magn. Reson.33,472(1979).
[2] S. Kukita, H. Kiya and Y. Kondo, submitted.
[3] Y. Kondo and M. Bando, J. Phys. Soc. Jpn.80,054002(2011).
[4] E.Sjöqvist, Int. J. Quantum Chem. 2015,115,1311-1326
日時: 2023年7月12日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 數田 裕紀(量子多体)
題目:Quantum and classical simulations of non-ergodic behavior in a disorder-free Bose-Hubbard system
概要:In this work, following the protocol proposed in Ref. [1], we analyze non-ergodic dynamics in a one-dimensional Bose-Hubbard system by means of a quantum simulator built with ultracold Bose gases in optical lattices. Specifically, we analyze dynamics starting from two different initial states. In the first (second) state, one particle occupies (two particles occupy) each odd-numbered site while no particle occupies each even-numbered site. We observe the time evolution of the atomic density in order to show that in the case of the second state the atomic density is not relaxed to equilibrium even after long-time evolution, exhibiting non-ergodic behavior. In order to provide quantitative references to the quantum-simulation outputs, we present numerical simulations of the experiments with use of matrix product states.
[1] M. Kunimi and I. Danshita, Phys. Rev. A 104, 043322 (2021).
日時: 2023年7月19日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 近藤 康(量子制御)
題目:Composite Quantum Gates and Geometric Phase Gates
概要:After reviewing geometric phases, I shall introduce two examples of geometric phase gates applied (proposed) for Rydberg atoms (PhysRevA.96.052316, PhysRevResearch.2.043130). Then, geometric properties of composite quantum gates are discussed (arXiv:2301.05627, JPSJ.80.054002). Some questionable points (= what I cannot understand) in the above papers are also discussed.
日時: 2023年10月4日10:00-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者1: 植田 健太(量子多体)
題目:長距離相互作用を持つ2次元XY模型で記述されるRydberg原子系における量子相転移
概要:絶対零度の系で量子的な効果によって起こる相転移は量子相転移と呼ばれる。観測するためには熱ゆらぎの効果が無視できるほどの低温でかつ物理パラメータを制御可能な系を人工的に用意する必要がある。人工量子系の一つである光ピンセットで配列されたRydberg原子系は十分に低温で制御性の高い物理系であるため、特に量子スピン摸型での量子相転移の観測に向いている[1,2]。
この発表は、実験論文[3]の紹介パートと解析計算による考察パートに分かれる。文献[3]の実験では、原子間距離の3乗に反比例して減衰するスピン間相互作用を持つスピン1/2のXY模型で記述されるRydberg原子系において、連続対称性(具体的にはU(1)対称性)の自発的な破れを伴う量子相転移が観測された。模型と実験の対応関係、実験のプロトコル、実際の観測結果についての詳細を説明する。発表の後半では、実験結果の妥当性や量子相転移の性質について調べるために、平均場近似[4,5]に基づく解析計算を行う。具体的には、基底状態エネルギー、秩序変数、励起の分散関係を計算し、これらの量から量子相転移の性質を明らかにする。
[1] A. Browaeys et al., J. Phys. B: At. Mol. Opt. Phys. 49, 152001 (2016).
[2] A. Browaeys and T. Lahaye, Nat. Phys. 16, 132 (2020).
[3] C. Chen et al. Nat. 616, 691–695 . 27 (2023).
[4] R. T. Scalettar et al., Phys. Rev. B 51, 8467(1995).
[5] I. Danshita and D. Yamamoto, Phys. Rev. A 82, 013645(2010).
発表者2: 渡部 元輝(量子多体)
題目:Rydberg原子配列からなる量子計算機における量子ゲート操作のエラーの要因
概要:光ピンセットを用いて配列された冷却リドベルグ原子は量子計算機の分野において有望なプラットフォームの1つとして研究されている[1,2]。このリドベルグ原子配列を用いた量子計算機は、近年目覚ましい発展を遂げているものの、基底状態とリドベルグ状態の間のコヒーレント制御のフィデリティが低いことにより、さらなる進歩が制限されている。本研究は、そのフィデリティを下げる様々な要因について、その減衰過程をシミュレーションし、量子計算機の開発を理論面からサポートできる理論模型およびその数値解析を提供することを目的とする。まず初めに単一量子ビット操作に注目して、リドベルグ状態と基底状態の間のラビ振動に生じる主要なエラー要因について先行研究[3]を学習し、それぞれのエラー要因に起因する減衰過程の数値シミュレーションを再現する。また、エラー要因の一つであるレーザー位相ノイズについて新規な理論模型を考案し、従来の模型の結果と比較する。次に2量子ビットをエンタングルさせる操作の一つであるCZゲート操作についての先行研究[4]学習し、そこで示されている実験結果を再現する理論模型を説明する。
[1]M.Saffman, J. Phys. B: At. Mol. Opt. Phys. 49, 202001 (2016).
[2] X. Wu et al., Chinese Phys. B 30, 020305 (2021).
[3] S. de Léséleuc et al. Phys. Rev. A 97, 053803 (2018).
[4] H. Levine et al. Phys. Rev. Lett. 123, 170503 (2019).
日時: 2023年10月11日9:30-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者1: 木村 漣(量子多体)
題目:Rydberg原子系での異常トンネル効果の理論解析に向けて
概要:2023年に光ピンセットで配列されたRydberg原子系でスピン1/2のXY模型のU(1)対称性の自発的破れに伴う量子相転移現象が観測された[1]。同じくU(1)対称性の自発的破れに起因する相転移現象の一つに、弱く相互作用するBose気体系のBose-Einstein凝縮(BEC)がある。この系のU(1)対称性を自発的に破った秩序相であるBose凝縮相での低エネルギー励起には、よく知られているトンネル効果とは対照的な性質を持つ異常トンネル効果という現象が見られる[2,3]。本研究では、長距離相互作用が特徴的なRydberg原子系において異常トンネル効果を理論解析することを目的とする。一般的には、U(1)対称性を自発的に破った秩序相の低エネルギー励起は音波的な線形分散を持つが、文献[1]のRydberg原子系では長距離相互作用の効果で運動量の平方根に比例する分散関係を持ちうることが分かっている[4]。低エネルギー励起の分散関係の違いが異常トンネル効果にどのような影響を与えるかという点に特に注目する。今回の発表ではその目的への準備としてRydberg原子系の概要を説明する。そして、弱く相互作用するBose気体のBEC状態の素励起を記述するBogoliubov理論を導入し、実際にδ関数型障壁ポテンシャルの場合における低エネルギー励起の異常トンネル効果を確認する[2,3]。
[1] C. Chen et al., Nature 616, 691 (2023).
[2] 段下一平, 物性研究, 85, 96 (2005).
[3] 土屋俊二, 物性研究, 94, 1 (2010).
[4] D. Peter et al., Phys. Rev. Lett. 109, 025303 (2012).
発表者2: 森本 陸斗(量子多体)
題目:制限BoltzmannマシンによるBose-Hubbard模型の数値解析に向けて
概要:2017年にCarleoとTroyerが制限Boltzmannマシン(RBM)型のニューラルネットワークを用いた変分波動関数を量子スピン系の数値解析に応用して以来[1]、RBM変分波動関数を用いた量子多体系の理論研究は急速に発展している[2]。一方で、光格子中の冷却気体系からなる量子シミュレータを用いた量子多体系の非平衡ダイナミクスの研究が近年精力的になされている[3]。研究の背景を簡潔かつ適切に記述する必要があった。本研究では、RBM変分波動関数を用いて、光格子中の冷却気体系を記述する典型的な模型であるBose-Hubbard模型の非平衡ダイナミクスを解析することを目的とする。現在そのための準備としてRBM変分波動関数を用いたスピン1/2 Heisenberg模型の基底状態解析[1,4]の再現に取り組んでおり、本発表では、その進捗状況を報告する。
まず初めに、文献[4]に基づいて、近年発展を続けているニューラルネットワークを用いた機械学習について概説する。つづいて、RBM変分波動関数の最適化に用いたマルコフ連鎖型モンテカルロ法と最急降下法を説明する。それらを用いて、4スピンの場合に対して一つの変分パラメータを有するRBM波動関数を最適化し、基底状態を求め、正しい基底状態エネルギーが得られることを確認する。より大きなスピン数(Lとする)の場合に適用するために、(L2+2L)個の変分パラメータを有するRBM波動関数を準備した場合でも同様に正しい基底状態エネルギーが得られることを確認する。
[1] G. Carleo and M. Troyer, Science 355, 602 (2017).
[2] R. G. Melko et al., Nat. Phys. 15, 887-892 (2019).
[3] F. Schäfer, et al., Nat. Rev. Phys. 2, 411 (2020).
[4] ―Pythonで実践― 基礎からの物理学とディープラーニング入門,福島 健二および桂 法称, 科学情報出版株式会社, 2022年.
日時: 2023年10月25日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 籔内 雄大(大阪公立大学)
題目:Collective Excitations in Relativistic Quantum Droplets of Two-Component Bose-Hubbard Model
概要:Considerable progresses in cooling and controlling ultracold atoms have led to the realization of quantum droplets of two-component [1] or dipolar [2] Bose gases in continuum. More recently, Machida et al. have analyzed two-component Bose gases in optical lattices in order to show that a quantum droplet state can be stabilized at the first-order quantum phase transition between the superfluid and Mott-insulating states [3].
In this work, we focus on the fact that the Ginzburg-Landau (GL) equation describing the quantum droplets of the two-component lattice bosons acquires a Lorentz invariance near the effective particle-hole symmetric point [4]. We specifically investigate properties of some low energy excitations of such relativistic quantum droplets within the GL approximation. It is well known that the amplitude and phase fluctuations of the order parameter of a relativistic superfluid are perfectly decoupled [5]. In the case of amplitude modes, we find that the monopole and quadrupole modes are surface modes and that the dipole mode is the Nambu-Goldstone zero mode that emerges because the presence of the droplet spontaneously breaks the translational symmetry of space. In the case of phase modes, we find that monopole, dipole and quadrupole modes of phase are a bulk modes. We show that these results are in clear contrast with those of non-relativistic quantum droplets obtained previously in Ref. [6].
[1] C. R. Cabrera et al., Science 359, 301 (2018).
[2] I . Ferrier et al., Phys. Rev. Lett. 116, 215301 (2016).
[3] Y. Machida et al., Phys. Rev. A 105, L031301 (2022).
[4] I. Danshita, D. Yamamoto, and Y. Kato, Phys. Rev. A 91, 013603 (2015).
[5] D. Pekker and C. M. Varma, Annu. Rev. Condens. Matter Phys. 6, 269 (2015).
[6] H. Hu and X.-J. Liu, Phys. Rev. A 102, 053303 (2020).
日時: 2023年11月8日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 宮崎 優希(青山学院大学)
題目:Evaluation of Quantum Entanglement via Permutationally Invariant Quantum State Tomography
概要:Quantum state tomography (QST), in which the density matrix of a quantum many-body system is reconstructed by the expectation values of a set of observables, is experimentally hard due to (i) the exponential increase of degrees of freedom with system size and, in the case of cold atomic systems in optical lattice, (ii) the practical problem of local quantization axis rotation. In a previous work [1], permutationally invariant (PI) QST was introduced as the reconstruction of the part of a density matrix that is invariant under permutations of lattice sites. It has been reported not only that PIQST can avoid the above issues, but also that the PI part of a density matrix can encode some important properties of the original density matrix [2]. In this work, we investigate the relation between the PI part of a density matrix and some entanglement measures and obtain some benchmarks for the relation.
[1] G. Tóth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Phys. Rev. Lett. 105, 250403 (2010).
[2] T. Gao, F. Yan, and S.J. van Enk, Phys. Rev. Lett. 112, 180501 (2014).
日時: 2023年11月15日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 鏡原 大地(量子多体)
題目:Entanglement entropies in free boson systems
概要:Entanglement is one of the indispensable concepts in modern quantum physics. It describes non-classical and non-local correlations and is often quantified by entanglement entropies. While the von Neumann entanglement entropy is a standard measure of entanglement, its variant, Rényi entanglement entropy, was experimentally measured in highly controllable systems [1]. In this context, we theoretically studied the second-order Rényi entanglement entropy of free boson systems and revealed an interesting formula for it [2]. However, our formula is specific to the (second-order) Rényi entropy, and it is more desirable to access the von Neumann entanglement entropy directly.
In this talk, we generalize our result on the second Rényi entanglement entropy of free boson systems [2] to arbitrary order ones, including the von Neumann entanglement entropy. We also discuss possible applications of this result.
[1] R. Islam et al., Nature 528, 77 (2015); A. M. Kaufman et al., Science 353, 794 (2016); T. Brydges et al., Science 364, 260 (2019); D. Bluvstein et al., Nature 604, 451 (2022).
[2] D. Kagamihara, R. Kaneko, S. Yamashika, K. Sugiyama, R. Yoshii, S. Tsuchiya, I. Danshita, Phys. Rev. A 107, 033305 (2023).
日時: 2023年11月22日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 小久保 治哉(物性理論)
題目:Critical velocity for quantized vortex formation in a superfluid with a plate-shaped obstacle
概要:Wake is the flow that occurs behind an obstacle moving through fluids, the dynamics of which is determined by the size and velocity of the obstacle, and is associated with various fluid phenomena such as vortex formation and turbulent transition. Wakes in superfluids have been studied both experimentally [1-2] and theoretically [3-4], and it has been shown that the critical velocity depends on the shape of the obstacle [5]. In numerical simulations, Gaussian potentials are often used to simulate an optical laser obstacle. However, it is difficult to measure the dependence of the critical velocity on the shape of the obstacle due to the unclear effects of the tail in the Gaussian potential. We consider the wake with a plate-shaped obstacle to evaluate the dependence of the critical velocity on the size of the obstacle. Plate-shaped obstacles have no shape profile other than width and can be easily measured both theoretically and numerically. In this talk, we describe the size dependence of the critical velocity by numerical simulations for a 2-dimensional Bose-Einstein condensate and present a method for quantitative evaluation of the critical velocity using the complex potential flow.
[1] Woo Jin Kwon, et al., Phys. Rev. Lett. 117, 245301 (2016)
[2] Younghoon Lim, et al., New J. Phys 24, 083020 (2022)
[3] Kazuki Sasaki, Naoya Suzuki, and Hiroki Saito, Phys. Rev. Lett 104, 150404 (2010)
[4] M. T. Reeves, et al., Phys. Rev. Lett. 114, 155302 (2015)
[5] Kwon, Woo Jin, et al., Phys. Rev. A 91, 053615 (2015)
日時: 2023年11月29日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: Rammohan, Sidharth(量子多体)
題目:Tailoring the Phonon Environment of Embedded Rydberg Aggregates
概要:State-of-the-art experiments can controllably create Rydberg atoms inside a Bose-Einstein condensate (BEC) [1]. The large Rydberg electron orbital volume contains many neutral atoms, resulting in electron-atom scattering events. The number of atoms within the orbit, and hence the Rydberg-BEC interaction, can be tuned by choice of principal quantum number or condensate density [1]. This makes the hybrid system a fascinating platform for quantum simulation. We studied the physics of the interaction and corresponding dynamics of single or multiple Rydberg atoms in two internal electronic states embedded inside a BEC, to assess their utility for controlled studies of decoherence and quantum simulations of excitation transport similar to photosynthetic light-harvesting.
We initially developed a theoretical framework to calculate the open quantum system input parameters like the bath correlation function and the spectral density, initially for a single Rydberg atom, possibly in two internal states with angular momentum quantum numbers l = 0 (|s⟩) and l = 1 (|p⟩) [2], in BEC and then for a chain of Rydberg atoms, forming an aggregate. The electron-atom contact interactions lead to Rydberg-BEC coupling, which creates Bogoliubov excitations (phonons) in the BEC.
Using this spin-boson model with the calculated parameters, we examine the decoherence dynamics of a Rydberg atom in a superposition of |s⟩ and |p⟩ states, resulting from the interaction with its condensate environment. Further, we investigated the emergence of the Non-Markovian features in the system in the presence of a microwave external drive of the Rydberg atom using a stochastic computational technique for Non-Markovian open quantum systems [3].
Finally, we extend this to the aggregate case, where one of the atoms in the aggregate is in the state |p⟩, while the rest are in the state |s⟩, resulting in excitation transport via dipole-dipole interaction [4]. We investigate the effects of Non-Markovinity and decoherence on the excitation transport based on an effective model described by a Holstein Hamiltonian, allowing us to set up the dynamics similar to those found in light-harvesting complexes, but at a different time and energy scales.
References:
- J. B. Balewski, A. T. Krupp, A. Gaj, D. Peter, H. P. Büchler, R. Löw, S.
Hofferberth and T. Pfau; Nature 502 664 (2013).
- S. Rammohan, A. K. Chauhan, R. Nath, A. Eisfeld, and S. Wüster; Phys. Rev.
A 103, 063307 (2021).
- S. Rammohan, S. Tiwari, A. Mishra, A. Pendse, A. K. Chauhan, R. Nath, A.
Eisfeld, and S. Wüster; Phys. Rev. A 104, L060202 (2021).
- D. W. Schönleber, A. Eisfeld, M. Genkin, S. Whitlock and S. Wüster; Phys. Rev. Lett. 114 123005 (2015).
日時: 2023年12月6日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 數田 裕紀(量子多体)
題目:Quantum simulation of nonergodic behavior in a disorder-free Bose-Hubbard system
概要:Thanks to their high controllability and near-perfect isolation from environment, cold-atom systems serve as a unique platform for studying non-equilibrium dynamics of isolated quantum systems. To understand how an isolated quantum system reaches thermal equilibrium through unitary time evolution, it is important to investigate mechanisms of nonergodic systems, such as integrability [1], many-body localization [2], quantum many-body scar states [3], and the Hilbert space fragmentation (HSF) [4]. A previous theoretical study [5] proposed that non-ergodic dynamics due to HSF originated from strong interparticle interactions can occur in a one-dimensional Bose-Hubbard system with weak trapping potentials when in the initial state of the dynamics the odd-numbered (even-numbered) sites are doubly occupied (empty). In this presentation, following the proposed protocol, we experimentally analyze this nonergodic dynamics by means of a quantum simulator built with ultracold Bose gases in optical lattices. To cross-check the quantum-simulation results, we numerically simulate the real-time evolution with use of matrix product states combined with the local density approximation.
[1] T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440 900 (2006).
[2] J. Choi et al., Science 352, 1547 (2016).
[3] H. Bernien, et al., Nature 551, 579 (2017)
[4] S. Scherg et al., Nat. Comm. 12, 4490 (2021)
[5] M. Kunimi and I. Danshita, Phys. Rev. A 104, 043322 (2021).
日時: 2023年12月13日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: Gietka, Karol(University of Innsbruck)
題目:Combining quantum with critical metrology, and temperature enhanced critical metrology
概要:In the talk, I will present two ways of improving quantum critical metrology. First relies on a combination of conventional metrology protocols, like Ramsey interferometry, with critical metrology protocols. Second relies on showing how finite temperature can be, in principle, used to increase the quantum Fisher information and the sensitivity of estimating physical parameters.
日時: 2023年12月13日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 木屋 晴貴(量子制御)
題目:Isoholonomic problem and composite quantum gate robust against pulse length error
概要:One confronts two typical systematic errors in one-qubit control: Pulse Length Error (PLE) and Off Resonance Error (ORE). To prevent the loss of control accuracy due to these systematic errors, a technique called Composite Quantum Gate (CQG) [1] exists, replacing a single operation with a sequence of several operations. In this talk, we will focus on PLE robust CQGs. The geometry of the PLE robust CQG is related to the Aharonov-Anandan phase [3], an extension of the Berry phase [2]. We argued the shortest operation time by minimizing the length of the path drawn on the Bloch sphere while satisfying the robustness condition against PLE [4]. However, the bound in [4] is not tight and needs more discussions: This is the subject of this talk and is related to the isoholonomic problem [5,6].
[1] Levitt, M. H. (1986). Progress in Nuclear Magnetic Resonance Spectroscopy, 18(2), 61-122.
[2] Berry, M. V. (1984). Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392(1802), 45-57.
[3] Aharonov, Y., & Anandan, J. (1987). Physical Review Letters, 58(16), 1593.
[4] Kukita, S., Kiya, H., & Kondo, Y. (2023). Journal of Physics A: Mathematical and Theoretical, 56(48), 485305.
[5] Montgomery, R. (1990). Communications in Mathematical Physics, 128, 565-592.
[6] Tanimura, S., Nakahara, M., & Hayashi, D. (2005). Journal of mathematical physics, 46(2).
日時: 2024年1月10日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 安藤 京介(物性理論)
題目:Machine learning analysis of Fully Frustrated XY Model (FFXY) in two-dimensional square lattice
概要:The recent remarkable development of artificial neural networks in image recognition, image classification, and natural language processing has influenced many scientific fields, and the search for new discoveries by applying this technology to any problem has begun. In the field of classical statistical physics, machine learning algorithms were introduced to identify symmetry-broken phase [1-3], and in some of these cases neural networks were shown to be able to learn order parameters and other thermodynamic parameters [1,3]. Having been able to apply machine learning techniques to conventional phase transitions, it is natural to ask whether the algorithm can be applied to unconventional phase transitions. As an example of such a system, we focus on the 2D Fully Frustrated XY Model (FFXY) [4]. The FFXY model has an Ising model-like transition and an XY model-like transition.
The purpose of this study is to see if machine learning algorithms trained by the XY and Ising models can detect phase transitions for the two-dimensional square lattice FFXY model. The two neural networks used were trained from the spin configuration of the Ising model and the vortex configuration of the XY model. The FFXY model detects vortices from spin configurations obtained from Monte Carlo simulations, and inputs them to the learning model.
[1] J. Carrasquilla and R. G. Melko, Nat. Phys. 13, 431 (2017).
[2] E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Nat. Phys.13, 435 (2017).
[3] S. J. Wetzel and M. Scherzer, Phys. Rev. B 96, 184410 (2017).
[4] Stephen Teitel, 40 Years of Berezinskii–Kosterlitz–Thouless Theory (World Scientific), pp. 201-235 (2013).
日時: 2024年1月17日9:00-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者: 宮井 誠一郎(量子多体)
題目:Toward an analysis of correlation propagation in the Bose-Hubbard model with dipole-dipole interactions
概要:Rapid technological advances in preparing and manipulating cold atoms have offered unique opportunities for studies of non-equilibrium dynamics of quantum many-body systems. One of the fundamental questions to be addressed is how correlations propagate in these systems. Specifically, such correlation propagation dynamics have been analyzed in experiments with Bose gases in optical lattices [1,2], which can be well described by the Bose-Hubbard model. Moreover, recent experiments using atoms with strong dipole-dipole interactions [3,4] have opened up new possibilities for studying effects of the long-range interactions on correlation propagation dynamics. In this work, we aim to theoretically investigate correlation spreading in the Bose-Hubbard model with dipole-dipole interactions. In preparation for analyzing this model, this presentation will introduce the auxiliary bosonic operators, which are the basic approximations used in the analysis of the 1D Bose-Hubbard model, and present a part of what has been learned about the approximation of unconstrained fermions (UF approximation) [5].
References:
[1] Marc Cheneau, Peter Barmettler, Dario Poletti, et al., nature10748, NATURE vol481,487,(2012).
[2] Y. Takasu, et al., Science Advances 6, eaba9255 (2020).
[3] S. Baier, M. J. Mark, D. Petter, et al., Science 352, 201 (2016).
[4] Lin Su, Alexander Douglas, Michal Szurek, et al., Nature 622, 724 (2023).
[5] Peter Barmettler, Dario Poletti, et al., Phys. Rev. A 85, 053625 (2012).
日時: 2024年1月19日10:45-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者1: 渡部 元輝(量子多体)
題目:Rydberg原子配列からなる量子コンピュータの量子ゲート操作に生じる主要なエラー
概要:光ピンセットを用いて配列された冷却Rydberg原子は量子計算機の分野において有望なプラットフォームの1つとして研究されており[1,2]、近年目覚ましい発展を遂げている。しかし、基底状態とRydberg状態の間のコヒーレント制御のフィデリティが低いことにより、さらなる進歩が制限されている。本研究は、そのフィデリティを下げる様々な要因について、その減衰過程をシミュレーションし、量子計算機の開発を理論面からサポートできる理論模型およびその数値解析を提供することを目的とする。本発表では、まず量子ゲート操作の中でも重要なCZゲート操作について先行研究[3]で提案されている実験操作を紹介する。理想的なCZゲート操作では完全な Rydbergブロッケードが仮定されているが、現実にはこのブロッケードは完全ではなく、二原子ともにRydberg状態を取る確率が有限に存在する。このことがCZゲートのフィデリティにどの程度影響するかを検証するために、Rydbergブロッケードを仮定しない場合のCZゲート操作を数値シミュレーションする。その後、Rydberg原子のラビ振動に生じる主要なエラー要因[4]を全て含めた数値計算コードを作成する準備として、それらのエラー要因についての概要を説明する。
[1] M. Saffman, J. Phys. B: At. Mol. Opt. Phys. 49, 202001 (2016).
[2] X. Wu et al., Chinese Phys. B 30, 020305 (2021).
[3] H. Levine et al. Phys. Rev. Lett. 123, 170503 (2019).
[4] S. de Léséleuc et al. Phys. Rev. A 97, 053803 (2018).
発表者2: 植田 健太(量子多体)
題目:双極子・双極子相互作用を持つRydberg原子系におけるスピン二量体系の量子相転移
概要:絶対零度の系で量子的な効果によって起こる相転移は量子相転移と呼ばれる。観測するためには熱ゆらぎの効果が無視できるほどの低温でかつ物理パラメータを制御可能な系を人工的に用意する必要がある。人工量子系の一つである光ピンセットで配列されたRydberg原子系は十分に低温で制御性の高い物理系であるため、特に量子スピン摸型での量子相転移の観測に向いている[1,2]。実際に文献[3]の実験では、原子間距離の3乗に反比例して減衰するスピン間相互作用(長距離相互作用)を持つスピン1/2のXY模型で記述されるRydberg原子系において、連続対称性(具体的にはU(1)対称性)の自発的な破れを伴う量子相転移が観測された。この実験の設定を発展させることで、これまでにRydberg原子系で実現されていない種類の量子相転移が実現できることを提示することが、この研究の目的である。具体的には、双極子・双極子相互作用を持つRydberg原子系[3]において、文献[4]でなされているように二原子対を周期的に配列することで、スピン二量体系を実現することを提案する。この系において、二量体内相互作用に対する二量体間相互作用の強さを増大させると、無秩序なスピン一重項状態からXY反強磁性状態への量子相転移が起こることを、一次元の場合を例にして示す。この相転移はTlCuCl3などの固体物質のスピン二量体系[5]で実現されているものと同種のものであると予想される。今後の研究でこの量子相転移近傍における準粒子励起の性質を解析するための準備として、平均場近似による励起の分散関係の導出方法[6,7]をおさらいする。
[1] A. Browaeys et al., J. Phys. B: At. Mol. Opt. Phys. 49, 152001 (2016).
[2] A. Browaeys and T. Lahaye, Nat. Phys. 16, 132 (2020).
[3] Cheng Chen et al. Nat. 616, 691–695 . 27 (2023).
[4] Y. Chew et al., Nat. Photonics 16, 724 (2022).
[5] Ch. Rüegg et al., Phys. Rev. Lett. 100, 205701 (2008).
[6] R. T. Scalettar et al., Phys. Rev. B 51, 8467(1995).
[7] I. Danshita and D. Yamamoto, Phys. Rev. A 82, 013645(2010).
発表者3: 射手矢 将寿(物性理論)
題目:エネルギーバンド構造とバンドマッピング
概要:調和ポテンシャルと格子ポテンシャルに束縛された冷却原子の物性を調べる手段として、気体のエネルギーバンド構造を指標として用いるバンドマッピングと呼ばれる手法がある。この方法では格子ポテンシャルをなくし、時間発展するガスの運動を分析することで、元々の気体の特性を逆算的に調べるという手法である。この時間発展を調べることが本研究の目的である。
発表者4: 渕 彰悟(物性理論)
題目:2成分Bose-Einstein凝縮体における渦輪の伝播速度
概要:本研究では、2成分Bose-Einstein凝縮体(BEC)の渦輪の伝播速度を調べる。渦輪とは、量子渦の渦芯が円形に繋がったものである。本発表では、初めに1成分BECにおける渦輪の伝播速度を数値計算で求め、解析的に知られている表式と比較する。その後、同様の手法で2成分BECでの渦輪の伝播速度を計算したのでその結果について紹介する。
日時: 2024年1月24日9:00-
教室: 31号館3階シミュレーション実験室 + Zoomでのオンライン配信
発表者1: 木村 漣(量子多体)
題目:Rydberg原子系の強磁性XY状態におけるスピン波励起のトンネル効果の理論解析に向けて
概要:2023年にInstitut d’Optiqueの実験グループがRydberg原子系でスピン1/2のXY模型を実現し、U(1)対称性の自発的な破れを伴う量子相転移を観測した[1]。このU(1)対称性の自発的な破れを伴う相転移現象の一つにBose-Einstein凝縮(BEC)があり、そのBECの特徴的な物性として異常トンネル効果がある。異常トンネル効果とは入射エネルギーが小さいほど透過確率が上昇し、0極限では完全に透過するというSchrödinger方程式から導かれる一般的なトンネル効果とは対照的な性質を持つトンネル現象である[2]。本研究はU(1)対称性の自発的な破れが見られるという共通点から、Rydberg原子系でトンネル効果を数値計算で理論的に解析することを目的とする。本発表では、まず本研究の根幹となる要素であるRydberg原子系と異常トンネル効果について解説する。特に異常トンネル効果の解説においては、理解を深めるためにBose凝縮体系での異常トンネル効果を記述することに適したBogoliubov理論を導入する[3][4]。次にRydberg原子系での低エネルギー励起の分散関係を先行研究に則って確認する[5][6]。それからRydberg原子系での異常トンネル効果を検証する準備として、最近接スピン間相互作用のみを持つスピン1/2のXY模型において、スピン波励起の障壁ポテンシャルによる散乱の問題を考える[7]。この場合において、スピン波励起が異常トンネル効果を示すことを確認する。
[1] C. Chen et al., Nature, 616, 691 (2023).
[2] Yu. Kagan et al., PRL 90, 130402 (2003).
[3] 段下一平, 物性研究, 85, 96 (2005).
[4] 土屋俊二, 物性研究, 94, 2 (2010).
[5] D. Peter et al., Phys. Rev. Lett. 109, 025303 (2012)
[6] I. Danshita and D. Yamamoto, Phys. Rev. A 82, 013645 (2010).
[7] Y. Kato and S. Watabe, Journal of Physics: Conferencce Series 400 , 032036 (2010).
発表者2: 高田 伊織(物性理論)
題目:質量比の異なる2成分ボース・アインシュタイン凝縮体における量子渦格子の構造
概要:2成分間の質量が等しい2成分ボース・アインシュタイン凝縮体における回転ポテンシャル中の量子渦状態では様々な構造を示すことが知られている。本研究では2成分間の質量比が異なる2成分ボース・アインシュタイン凝縮体がつくる回転ポテンシャル中の量子渦状態を原子間相互作用を変数とし、回転系における時間に依存しない2次元GP方程式の数値シミレーションに基づき量子渦格子の構造を調査した。そこで質量比や異成分原子間相互作用δの値によって量子渦格子の配置に違いが出ることが明らかとなった。
発表者3: 松尾 洋孝(物性理論)
題目:Physics-informed Neural Networkによる運動方程式の推定
概要:近年、機械学習と物理学の理論的手法群の統合によって基礎物理学の課題を解決しようとする動きが高まっている。しかし、そのような背景がある一方で、機械学習にデータを直接、物理情報を失わずに取り入れることはいまだに難しい難問題である。また、現実の物理情報を含んだデータを生成することは極めて難しく、より洗練されたコードが必要である。最近ではディープニューラルネットワークを使用し、トレーニングする手法が編み出されてきたがトレーニングするには大量のデータが必要である。そこで代わりに登場したのがPhysics-informed Neural Network (PINN)を使用した手法である。
本研究では、PINNを使用し、減衰振動の運動方程式を予測する。PINNの性質をよく理解し、別の物理系にも実装することを目的とする。
日時: 2024年2月15日13:15-
教室: 31号館301教室 + Zoomでのオンライン配信
発表者: Cazalilla, Miguel(Donostia International Physics Center)
題目:Quantum Dissipation in One-Dimensional Quantum Many-Particle Systems
概要:I will review several old [1,2] and new results concerning one-dimensional (1D) quantum dissipative systems. Beginning with a brief review of some early studies on the effect of dissipation in Tomonaga-Luttinger liquids [1,2] and spin systems [3], I will conclude by describing some recent results [5] that demonstrate some interesting analogies between the superconductor-metal transition in 1D superconducting wires or Josephson junction arrays and the de confinement of bosons in an anisotropic optical lattice [6,7]. Finally, considering dissipation in out of equilibrium systems, I will describe our recent results on the non-equilibrium dynamics induced by two-body losses of strongly interacting bosons in one-dimensional optical lattices.
[1] M. A. Cazalilla, F. Sols, and F. Guinea, Phys. Rev. Lett. 97, 076401 (2006).
[2] E. Malatsetxebarria, Z. Cai, U. Schollwöck, and MAC, Phys. Rev. A 88, 063630 (2013)
[3] A. M. Lobos, M. A. Cazalilla, and P. Chudzinski, Phys.Rev. B 86 035455 (2012).
[4] C.-H. Huang, T. Giamarchi, and MAC, Phys. Rev. Res. 5, 043192 (2023).
[5] M. A. Cazalilla, to be published.
[6] A. F. Ho, M. A. Cazalilla, and T. Giamarchi, Phys. Rev. Lett. 92, 130405 (2004).
[7] M. A. Cazalilla, A. F. Ho, and T. Giamarchi, New J. Phys. 8, 158 (2006).