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Characterizing the Dynamical Phase Diagram of the Dicke Model Via Classical and Quantum Probes

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Characterizing the Dynamical Phase Diagram of the Dicke Model Via Classical and Quantum Probes PHYSICAL REVIEW RESEARCH 3, L022020 (2021) Letter Characterizing the dynamical phase diagram of the Dicke model via classical and quantum probes R. J. Lewis-Swan ,1,2 S. R. Muleady ,3,4,* D. Barberena ,3,4,* J. J. Bollinger ,5 and A. M. Rey 3,4 1Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, Norman, Oklahoma 73019, USA 2Center for Quantum Research and Technology, The University of Oklahoma, Norman, Oklahoma 73019, USA 3JILA, NIST, Department of Physics, University of Colorado, Boulder, Colorado 80309, USA 4Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA 5National Institute of Standards and Technology, Boulder, Colorado 80305, USA (Received 8 February 2021; accepted 11 May 2021; published 4 June 2021) We theoretically study the dynamical phase diagram of the Dicke model in both classical and quantum limits using large, experimentally relevant system sizes. Our analysis elucidates that the model features dynamical critical points that are strongly influenced by features of chaos and emergent integrability in the model. Moreover, our numerical calculations demonstrate that mean-field features of the dynamics remain valid in the exact quantum dynamics, but we also find that in regimes where quantum effects dominate signatures of the dynamical phases and chaos can persist in purely quantum metrics such as entanglement and correlations. Our predictions can be verified in current quantum simulators of the Dicke model including arrays of trapped ions. DOI: 10.1103/PhysRevResearch.3.L022020 Introduction. Advances in atomic, molecular, and optical freedom such that the model remains amenable to analytic and (AMO) quantum simulators are driving a surge in the investi- numerical treatments. Moreover, the model is already stud- gation of dynamical phase transitions (DPTs) and associated ied in state-of-the-art AMO quantum simulators, including nonequilibrium phases of matter [1–6]. In a closed system, a trapped-ion arrays [38,39] and cavity QED [40–43]. Here, we DPT is a critical point separating distinct dynamical behaviors investigate the DPT in analytically tractable spin- and boson- that emerge after a sudden quench of a control parameter dominated limits, as well as a nonintegrable regime where [7–13], and can be defined via nonanalytic behavior in a near-resonant coupling of spin and bosons leads to chaotic time-averaged order parameter [14–17]. dynamical phases also seen in other non-integrable systems To date, the majority of experimental investigations in [7,8]. Following recent works linking DPTs to coexisting this direction have been tailored towards integrable models, excited-state QPTs (EQPTs) [20,36,44], we find different featuring effective infinite-range interactions between spins, dynamical critical points in the spin and boson-dominated which admit analytical treatments [18–21]. Richer noninte- regimes that reflect the presence of distinct EQPTs in these grable models have been pursued in trapped ion systems limits. By studying large, experimentally relevant system sizes [22], but the associated complexity of the quantum dynamics using efficient numerical methods we are able to demon- limited the theoretical analysis of the DPT to small system strate that mean-field features of the dynamics remain valid sizes and prevented a rigorous scaling analysis. Hence, it is in the exact quantum dynamics. Conversely, in regimes where highly desirable to find and study DPTs in nonintegrable mod- quantum effects dominate we find that signatures of the DPT els featuring novel nonequilibrium phenomena that are both and chaos persist in purely quantum metrics such as entangle- implementable in tunable quantum simulators and theoreti- ment and correlations. cally tractable under controllable approximations. Model. The Dicke Hamiltonian for N spin-1/2 particles is We advance this direction by studying a DPT in the iconic given by [30] Dicke model [23–25], which describes the collective coupling of many spins to a single harmonic oscillator. The model 2g † † HˆD = √ (ˆa + aˆ )Sˆz + δaˆ aˆ + Sˆx. (1) is attractive as it features an array of phenomena, such as N nonintegrability [26–28], chaos [29–31], and equilibrium † quantum phase transitions (QPTs) in both ground and excited Here,a ˆ (ˆa ) is the bosonic annihilation (creation) operator of δ ˆ = / N σ α states [32–37], but involves only a pair of disparate degrees of an oscillator with frequency , Sα 1 2 j=1 ˆ j are collec- α = , , σ α tive spin operators for x y z, and ˆ j Pauli matrices of the jth spin. The spins are subject to a transverse field of strength and the spin-boson coupling is characterized by *These two authors contributed equally. g. The Hamiltonian exhibits a Z2 parity symmetry associated π ˆ + † + / ˆ = i (Sx aˆ aˆ N 2) ˆ , ˆ = Published by the American Physical Society under the terms of the with the operator e such that [HD ] 0. Creative Commons Attribution 4.0 International license. Further The equilibrium phase diagram of the Dicke model = distribution of this work must maintain attribution to the author(s) √features a ground-state QPT at a critical coupling gQPT and the published article’s title, journal citation, and DOI. δ/2[32,33], which delineates a superadiant (g gQPT) 2643-1564/2021/3(2)/L022020(6) L022020-1 Published by the American Physical Society R. J. LEWIS-SWAN et al. PHYSICAL REVIEW RESEARCH 3, L022020 (2021) (a) (b) FIG. 2. (a) Typical time trace near the transition in the resonant region,g ˜ ≈ 1.299 and δ/ = 1. (b) Characterization of phase space via Lyapunov exponent λL. The star indicates parameter regime of the time trace. (a) FIG. 1. Dynamical phase diagram for time-averaged order parameter Sz (center) and typical time traces (surrounding) for the Dicke model. Parameters of time traces are (clockwise from top left) (δ/,g˜) = (0.1, 2), (δ/,g˜) = (4, 2), (δ/,g˜) = (4, 1), and δ/, = . , ( g˜) (0 1 1). Time-averaged Sz is obtained up to a maximum (b) (c) time gT = 104. phase, with degenerate ground states associated with different parity sectors that in the limit → 0 take the form |ψs= √1 [|(−N/2) ⊗|α ±|(−N/2) ⊗|−α ], and a normal 2 z s z s (g gQPT) phase, with |ψn≈|(N/2)x⊗|0 in the limit of large [38]. Here, we have defined collective spin states via ˆ | = | |±α Sx,y,z mx,y,z mx,y,√z mx,y,z and s is the bosonic coher- ent state for αs = g N/δ. Additionally, an excited-state QPT (EQPT) exists in the spectrum of the superradiant phase, g > gQPT, defined by a critical energy Ec =−N/2[34,36,37]. The EQPT is signaled by a nonanalyticity in the density of states [35] and the existence of pairs of degenerate eigenstates with different parity for E < Ec. Dynamical phase diagram. We study the dynamical phase FIG. 3. (a) In the integrable limits the DPT can be described as diagram that arises after a quench of the transverse field. a particle (green markers) in a 1D potential with coordinate ξ = Sz Concretely, the system is initialized in the ground state of HˆD (δ/ 1) or ξ = X (δ/ 1). At smallg ˜ g˜DPT (untrapped at fixed g and δ with = 0, such that |ψ(0)=|(−N/2)z⊗ |α phase, left) the particle has sufficient energy to traverse both wells of s , and the transverse field is then quenched to a final value the potential, whereas for largeg ˜ g˜ (trapped phase, right) the = DPT 0. To garner insight into the dynamics we first study particle remains energetically confined to a single well. (b) Typical the classical model (Figs. 1–3) before probing the role of trajectories in phase space for trapped (red) and untrapped (blue) quantum fluctuations. The classical limit of the Dicke model phases at δ/ = (0.1, 1, 10) (top to bottom). For the spin phase- is equivalent to solving the Heisenberg equations of motion space we use coordinates r = 1 + 2Sx/N and ϕ = arctan(Sy/Sz ) for operators under a mean-field approximation, wherein ex- (c) Dynamical phase diagram as a function of initial state. At small pectation values are factorized according to Oˆ 1(t )Oˆ 2(t )= detuning δ/ = 0.1 we vary the boson amplitude α (top panel), and Oˆ Oˆ O ≡ 1(t ) 2(t ) [45]. For brevity we adopt the notation at large detuning δ/ = 4 we vary the tipping angle θ0 relative to the Oˆ (t ) herein. south pole of the Bloch sphere (Sz =−N/2) (bottom panel). L022020-2 CHARACTERIZING THE DYNAMICAL PHASE DIAGRAM OF … PHYSICAL REVIEW RESEARCH 3, L022020 (2021) It is convenient√ to study dynamics in terms of two model due to the coupling of the spin and boson degrees of variables:g ˜ = 2g/ δ ≡ g/gQPT and δ/. The former freedom. characterizes the effective strength of the spin-boson inter- The dynamical phase diagram of SDR and BDR is captured action relative to the single-particle terms, while the latter by an effective model involving only the dominant degree describes the relative energy scales of spin and bosonic exci- of freedom [45,48–52]. Specifically, the mean-field dynamics tations and loosely expresses the relative importance of each are reduced to an equivalent picture of a classical particle degree of freedom to the dynamics. In the limit δ/ 1 with coordinate ξ = Sz (spin dominated) or ξ = X (boson andg ˜ ∼ 1 the dynamics are equivalent to a spin model de- dominated) confined within a one-dimensional (1D) potential scribed by the Lipkin-Meshkov-Glick (LMG) Hamiltonian, V (ξ ) that depends only ong ˜, δ/, and the initial state. Near ˆ = χ/ ˆ2 + ˆ ξ Heff ( N)Sz Sx with the boson-mediated interaction the DPT V ( ) is well approximated by a double well potential characterized by χ ≡ 4g2/δ [17,46].
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