Collective Excitations and Nonequilibrium Phase Transition in Dissipative Fermionic Superfluids

PHYSICAL REVIEW LETTERS 127, 055301 (2021) Collective Excitations and Nonequilibrium Phase Transition in Dissipative Fermionic Superfluids Kazuki Yamamoto ,1,* Masaya Nakagawa,2 Naoto Tsuji ,2,3 Masahito Ueda,2,3,4 and Norio Kawakami1 1Department of Physics, Kyoto University, Kyoto 606-8502, Japan 2Department of Physics, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan 3RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan 4Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan (Received 11 June 2020; accepted 17 June 2021; published 29 July 2021) We predict a new mechanism to induce collective excitations and a nonequilibrium phase transition of fermionic superfluids via a sudden switch on of two-body loss, for which we extend the BCS theory to fully incorporate a change in particle number. We find that a sudden switch on of dissipation induces an amplitude oscillation of the superfluid order parameter accompanied by a chirped phase rotation as a consequence of particle loss. We demonstrate that when dissipation is introduced to one of the two superfluids coupled via a Josephson junction, it gives rise to a nonequilibrium dynamical phase transition characterized by the vanishing dc Josephson current. The dissipation-induced collective modes and nonequilibrium phase transition can be realized with ultracold fermionic atoms subject to inelastic collisions. DOI: 10.1103/PhysRevLett.127.055301 Introduction.—Collective excitations of superconductors [43–45], quantum transport [46,47], and superfluidity and superfluids have been widely studied in condensed [48,49]. In particular, high controllability of parameters matter physics [1–20]. Recent experimental progress in in ultracold atoms has enabled investigations of nonequili- ultracold atoms has enabled studies of out-of-equilibrium brium quantum dynamics induced by dissipation [49–63], dynamics of superfluids [21–24]. For example, a periodic and studies of fermionic superfluidity in ultracold atoms modulation of the amplitude of the order parameter excites undergoing inelastic collisions have achieved remarkable the Higgs amplitude mode, which has been observed with progress [48,49,64–71]. The effect of particle loss in ultracold fermions [23] and in solid-state systems by light fermionic superfluids has been studied in the framework illumination on BCS superconductors [25–33]. As for of the non-Hermitian BCS theory [49]; however, it ignores collective phase modes, the Nambu-Goldstone mode exists a significant change in particle number due to quantum in neutral superfluids, and the relative-phase Leggett mode jumps. It is crucially important to go beyond the non- has been predicted for multiband superfluids [2,33–38].In Hermitian framework to describe the long-time dynamics particular, ultracold atoms allow for a dynamical control of of a superfluid and associated collective modes of the order various system parameters, offering an ideal playground to parameter. investigate collective modes. However, they suffer from In this Letter, we theoretically investigate collective atom loss due to inelastic scattering, which has received excitations and a nonequilibrium phase transition of fer- little attention in literature. mionic superfluids driven by a sudden switch on of two- In dissipative open quantum systems, the dynamics, after particle loss due to inelastic collisions between atoms. By environmental degrees of freedom are traced out, is formulating a dissipative BCS theory that fully incorporates nonunitary and described by a completely positive and a change in particle number, we find that dissipation trace-preserving map [39,40]. Such nonunitary dynamics is fundamentally alters the superfluid order parameter and relevant for atomic, molecular, and optical systems, dras- induces collective oscillations in its amplitude and phase. In tically changing various aspects of physics such as quan- particular, we elucidate that a coupling between the order tum critical phenomena [41,42], quantum phase transitions parameter and dissipation leads to a chirped phase rotation, in sharp contrast to the case of an interaction quench in closed systems [see Fig. 1(a)]. To experimentally observe the collective phenomena Published by the American Physical Society under the terms of induced by dissipation, we propose introducing a particle the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to loss in one of two coupled superfluids to induce a relative- the author(s) and the published article’s title, journal citation, phase oscillation analogous to the Leggett mode [2,33–38] and DOI. [see Fig. 1(b)]. The phase mode causes an oscillation of a 0031-9007=21=127(5)=055301(8) 055301-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 127, 055301 (2021) Z ρ D ¯ ¯ iS 1 Z ¼ tr ¼ ½c−; c−;cþ; cþe ¼ ; ð3Þ with an action Z ∞ X ¯ ∂ − ¯ ∂ − S ¼ dt ðckσþi tckσþ ckσ−i tckσ−Þ Hþ FIG. 1. (a) Schematic illustration of the amplitude and phase −∞ modes in a Mexican-hat free-energy potential as a function of the kσ Δ γ X complex order parameter , when either the interaction UR or the i ¯ ¯ − 2 ¯ dissipation γ is suddenly switched on. A sudden quench of the þH− þ 2 ðLiþLiþ þ Li−Li− LiþLi−Þ ; ð4Þ i interaction UR and that of the dissipation γ kick Δ in a direction parallel and perpendicular to the radial direction, respectively. − Note that a finite change of γ excites both the phase and amplitude where the subscriptsP þ and denoteP forward and back- ϵ ¯ − ¯ ¯ modes. (b) Two superfluids coupled via a Josephson junction, ward paths, Hα ¼ kσ kckσαckσα UR i ci↑αci↓αci↓αci↑α, ¯ where one superfluid (system 2) is subject to two-body loss. Liα ¼ ci↓αci↑α, and Liα ¼ c¯i↑αc¯i↓α ðα ¼þ; −Þ. Note that iθ the action has U(1) symmetry under ciσα → e ciσα though Josephson current around a nonvanishing dc component. the particle number of the system is not conserved [79,80]. Remarkably, when dissipation becomes strong, the coupled By introducing auxiliary fields via the Hubbard- system undergoes a nonequilibrium phase transition char- Stratonovich transformation, we rewrite the action in a acterized by the vanishing dc Josephson current, which can quadratic form of fermionic Grassmann fields as [49,81] be regarded as a generalization of a dynamical phase Z X i∂ − ϵk −Δ transition [11,12,72,73] to dissipative quantum systems. t t S dt ψ¯ ψ k ¼ kþ Ã þ Our findings can experimentally be tested with ultracold k −Δ −i∂t þ ϵk atoms through introduction of dissipation via a photo- i∂ − ϵk −Δ t t association process [50,59]. − ψ¯ ψ k− ; ð5Þ k− −ΔÃ − ∂ ϵ Dissipative BCS theory.—We consider ultracold fer- i t þ k mionic atoms described by the three-dimensional attractive t t Hubbard model where ψ¯ kα ¼ðc¯k↑α;c−k↓α Þ and ψ kα ¼ðck↑α; c¯−k↓α Þ X X ðα ¼þ; −Þ, with t denoting transposition. Here Δ is the ϵ † − † † superfluid order parameter which can be determined from H ¼ kckσckσ UR ci↑ci↓ci↓ci↑; ð1Þ kσ i the requirement that the action be extremal as [81] X X where UR > 0, ϵk is the single-particle energy dispersion, U U Δ ¼ − trðc−k↓ck↑ρÞ ≡ − hc−k↓ck↑i; ð6Þ and ckσ (c σ) denotes the annihilation operator of a spin-σ i N0 k N0 k fermion with momentum k (at site i). When the system is subject to inelastic collisions, scattered atoms are lost to a where U ¼ UR þ iγ=2 is an effective complex coupling surrounding environment, resulting in dissipative dynamics constant including a contribution from the atom loss [49], as observed experimentally [50,55,56,58]. Here, we study ρ and N0 is the number of sites. Importantly, the order the time evolution of the density matrix which is parameter includes the loss rate γ, which leads to dis- described by the Lindblad equation [39,40] sipation-induced collective modes as discussed below. The ρ γ X action (5) describes the mean-field time-evolution equation d Lρ − ρ − † ρ − 2 ρ † ¼ ¼ i½H; 2 ðfLi Li; g Li Li Þ; ð2Þ of the density matrix as dt i ρ d − ρ L c ↓c ↑ ¼ i½Heff; ; ð7Þ where i ¼ i i is a Lindblad operator that describes dt two-body loss with loss rate γ > 0. We note that the kinetic X energy of lost atoms is large because of large internal † ϵk Δ H ¼ Ψ Ψk; ð8Þ energy of atoms before inelastic collisions. Under such eff k ΔÃ −ϵ situations, atoms after inelastic collisions are quickly lost k k into the surrounding environment and the Born-Markov Ψ † t approximation is justified [74–76]. where k ¼ðck↑;c−k↓ Þ is the Nambu spinor. In the We first study how the standard BCS theory is Supplemental Material [81], we show that Eq. (7) can be generalized in open dissipative systems by formulating a derived from two different methods, i.e., the mean-field time-dependent mean-field theory in terms of a closed- theory for the Lindblad equation and the time-dependent time-contour path integral [77,78]. We start with a gen- Bogoliubov–de Gennes analysis. While Eq. (7) describes erating functional defined as unitary evolution, it is consistent with the original Lindblad 055301-2 PHYSICAL REVIEW LETTERS 127, 055301 (2021) equation (2) as a consequence of the time-dependent BCS ansatz [81]. We use Anderson’s pseudospin representation – σ 1 Ψ† τ Ψ [1,7P 12,14,32] defined by k ¼ 2 k · · k and Heff ¼ 2 k bk · σk, where τ ¼ðτx; τy; τz Þ is the vector of the Pauli matrices. The pseudospins satisfy the commutation FIG. 2. Dynamics of a superfluid after the atom loss with γ ¼ j k l 2 81Δ 12 2Δ relations ½σk; σk¼iϵjklσk.

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