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arXiv:2405.01223 · Physical Review A

Dissipative phase transition: From qubits to qudits

Lukas Pausch, François Damanet, Thierry Bastin, John Martin

Published Dec 10, 2024• 3 citations
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Abstract

We investigate the fate of dissipative phase transitions in quantum many-body systems when the individual constituents are qudits ($d$-level systems) instead of qubits. As an example system, we employ a permutation-invariant $XY$ model of $N$ infinite-range interacting $d$-level spins undergoing individual and collective dissipation. In the mean-field limit, we identify a dissipative phase transition, whose critical point is independent of $d$ after a suitable rescaling of parameters. When the decay rates between all adjacent levels are identical and $d\ensuremath{\ge}4$, the critical point expands, in terms of the ratio between dissipation and interaction strengths, to a critical region in which two phases coexist and which increases as $d$ grows. In addition, a larger $d$ leads to a more pronounced change in spin expectation values at the critical point. Numerical investigations for finite $N$ reveal symmetry-breaking signatures in the Liouvillian spectrum at the phase transition. The phase transition is furthermore marked by maximum entanglement negativity and a significant purity change of the steady state, which become more pronounced as $d$ increases. Considering qudits instead of qubits thus opens new perspectives on accessing rich phase diagrams in open many-body systems.

DOI
10.1103/physreva.110.062208

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We investigate the fate of dissipative phase transitions in quantum many-body systems when the individual constituents are qudits (ddd-level systems) instead of qubits. As an example system, we employ a permutation-invariant XYXYXY model of NNN infinite-range interacting ddd-level spins undergoing individual and collective dissipation. In the mean-field limit, we identify a dissipative phase transition, whose critical point is independent of ddd after a suitable rescaling of parameters. When the decay rates between all adjacent levels are identical and d\ensuremath≥4d\ensuremath{\ge}4d\ensuremath≥4, the critical point expands, in terms of the ratio between dissipation and interaction strengths, to a critical region in which two phases coexist and which increases as ddd grows. In addition, a larger ddd leads to a more pronounced change in spin expectation values at the critical point. Numerical investigations for finite NNN reveal symmetry-breaking signatures in the Liouvillian spectrum at the phase transition. The phase transition is furthermore marked by maximum entanglement negativity and a significant purity change of the steady state, which become more pronounced as ddd increases. Considering qudits instead of qubits thus opens new perspectives on accessing rich phase diagrams in open many-body systems.
We investigate the fate of dissipative phase transitions in quantum many-body systems when the individual constituents are qudits (d-level systems) instead of qubits. As an example system, we employ a permutation- invariant XY model of N infinite-range interacting d-level spins (Lipkin-Meshkov-Glick model) undergoing individual and collective dissipation. In the mean-field limit, we identify a dissipative phase transition, whose critical point is independent of d after a suitable rescaling of parameters. When the decay rates between all adjacent levels are identical and d ≥ 4, the critical point expands, in terms of the ratio between dissipation and interaction strengths, to a critical region in which two phases coexist and which increases as d grows. In addition, a larger d leads to a more pronounced change in spin expectation values at the critical point. Numerical investigations for finite N reveal symmetry breaking signatures in the Liouvillian spectrum at the phase transition. The phase transition is furthermore marked by maximum entanglement negativity and a significant purity change of the steady state, which become more pronounced as d increases. Considering qudits instead of qubits thus opens new perspectives on accessing rich phase diagrams in open many-body systems.
In open quantum systems undergoing phase transitions, the intricate interplay between unitary and dissipative processes leaves many information-theoretic properties opaque. We are here interested in interparticle correlations within such systems, specifically examining quantum entanglement, quantum discord, and classical correlation within the steady state of a driven-dissipative collective spin model. This model is renowned for, counterintuitively, its transition from a high-purity to a low-purity state with decreasing dissipation. Our investigation, rooted in numerical analysis using PPT criteria, underscores that entanglement reaches its zenith precisely at the phase transition juncture. Intriguingly, even within the mesoscopic scale near the transition point, entanglement endures across both phases, despite the open nature of the model. Notably, when employing traditional pairwise entanglement measures tailored to this model, detecting entanglement in the low-purity phase has proven elusive. In stark contrast, quantum discord and its variations chart an alternate trajectory, ascending monotonically as the system progresses into the low-purity phase. Consequently, lowered dissipation amplifies quantum correlation, yet it engenders entanglement solely in proximity to the transition point.
The interplay of coherent dynamics and dissipation in many body sysems give rise to a rich class of non-equilibrium phenomena with the emergence of non-trivial phases. In this paper, we investigate this interplay of dynamics in a model of interacting spins with infinite range interactions described by Heisenberg interaction. Using Holestien-Primarkoff transformations, the spin model is bosonized to a collective mode with self interaction at 1N\frac{1}{N}N1​ order. Employing Keldysh field theoretic technique with saddle point approximation, we see that the system breaks Z2\mathcal{Z}_2Z2​-symmetry at the transition point. An effective temperature arises due to dissipation which depends linearly on effective system-bath coupling, and is independent of the dissipation rate and cutoff frequency of the bath with Drude-Lorentz cutoff in wide class of bath spectral densities. Furthermore, fluctuations over mean field are studied and is shown that the dissipative spectrum is modified by O(1N){\rm O}(\frac{1}{N})O(N1​) correction term which results change in various physically measurable quantities.
Open driven quantum systems have defined a powerful paradigm of nonequilibrium phases and phase transitions; however, quantum phase transitions are generically not expected in this setting due to the decohering effect of dissipation. In this Letter, we consider a quantum Ising model subject to bulk dissipation (at rate Γ) and show that, although the correlation length remains finite (hence no phase transition), it develops a pronounced peak close to the ground-state quantum critical point. While standard techniques fail in this regime, we develop a versatile analytical approach that becomes exact with vanishing dissipation (Γ→0 but finite Γt). On a technical level, our approach builds on previous work where the state of the system is described by a slowly evolving generalized Gibbs ensemble that accounts for the integrability of the Hamiltonian while treating dissipation perturbatively. Finally, we demonstrate a kind of universality in that integrability-breaking perturbations of the Hamiltonian lead to the same behavior. To this end, we first show that the steady state of a chaotic Ising Hamiltonian under local Markovian dissipation that preserves the Ising symmetry, and in the limit Γ→0, is identical to that of quench dynamics in the absence of dissipation. This intriguing connection then allows us to draw on recent findings about quantum phase transition signatures in quench dynamics.
In specific open systems with collective dissipation the Liouvillian can be mapped to a non-Hermitian Hamiltonian. We here consider such a system where the Liouvillian is mapped to an XXZ Richardson-Gaudin integrable model and detail its exact Bethe ansatz solution. While no longer Hermitian, the Hamiltonian is pseudo-Hermitian/PT\text{pseudo-Hermitian}/\mathcal{P}\mathcal{T}pseudo-Hermitian/PT symmetric, and as the strength of the coupling to the environment is increased, the spectrum in a fixed symmetry sector changes from a broken pseudo-Hermitian phase with complex conjugate eigenvalues to a pseudo-Hermitian phase with real eigenvalues, passing through a series of exceptional points and associated dissipative quantum phase transitions. The homogeneous limit supports a nontrivial steady state, and away from this limit this state gives rise to a slow logarithmic growth of the decay rate (spectral gap) with system size. Using the exact solution, it is furthermore shown how at large coupling strengths the ratio of the imaginary to the real part of the eigenvalues becomes approximately quantized in the remaining symmetry sectors.
Open quantum many-body systems with controllable dissipation can exhibit novel features in their dynamics and steady states. A paradigmatic example is the dissipative transverse field Ising model. It has been shown recently that the steady state of this model with all-to-all interactions is genuinely non-equilibrium near criticality, exhibiting a modified time-reversal symmetry and violating the fluctuation-dissipation theorem. Experimental study of such non-equilibrium steady-state phase transitions is however lacking. Here we propose realistic experimental setups and measurement schemes for current trapped-ion quantum simulators to demonstrate this phase transition, where controllable dissipation is engineered via a continuous weak optical pumping laser. With extensive numerical calculations, we show that strong signatures of this dissipative phase transition and its non-equilibrium properties can be observed with a small system size across a wide range of system parameters. In addition, we show that the same signatures can also be seen if the dissipation is instead achieved via Floquet dynamics with periodic and probabilistic resetting of the spins. Dissipation engineered in this way may allow the simulation of more general types of driven-dissipative systems or facilitate the dissipative preparation of useful many-body entangled states.