Lifshitz-like metastability and optimal dephasing in dissipative bosonic Lattices

Abstract

In dissipative bosonic systems, dephasing is typically expected to accelerate relaxation and suppress coherent dynamics. However, we show that in networks of coherently coupled bosonic modes with non-uniform local dissipation, the presence of quasi-dark states leads to a nontrivial response to dephasing: while weak dephasing facilitates equilibration, moderate to strong dephasing induces a pronounced slowdown of relaxation, revealing the existence of an optimal dephasing rate that enhances equilibration. Using exact dynamical equations for second-order moments, we demonstrate that dephasing suppresses coherent transport and gives rise to long-lived collective modes that dominate the system’s late-time behavior. This phenomenon bears striking similarities to Lifshitz-tail states, which are known in disordered systems to cause anomalously slow relaxation. Our results uncover a counterintuitive mechanism by which dephasing, rather than promoting equilibration, can dynamically decouple specific modes from dissipation, thereby protecting excitations. These findings highlight how non-Hermitian physics in open bosonic systems can give rise to unexpected dynamical regimes, paving the way for new strategies to control relaxation and decoherence in bosonic quantum systems, with broad implications for both experimental and theoretical quantum science.

Graphical abstract

Keywords

non-Hermitian physics / open quantum systems / bosonic networks

1 Introduction

Bosonic lattices and networks — such as photonic lattices, optomechanical arrays, ultracold atoms in optical lattices and superconducting quantum circuits — have emerged as powerful platforms for simulating nonequilibrium quantum dynamics in engineered environments [1-6]. These systems are not only key platforms for quantum simulation but also play a central role in the rapidly developing field of non-Hermitian physics, where effective non-Hermitian Hamiltonians are used to model the interplay between coherent dynamics, loss and gain [7, 8]. This framework has led to the discovery of a range of exotic phenomena, including exceptional points and parity−time symmetry breaking [9-16], the non-Hermitian skin effect [17-26], non-Hermitian topological phases and phase transitions [20, 21, 27-34], anomalous transport behavior [35-39], and new mechanisms for controlling localization [40]. However, to fully capture quantum effects such as decoherence, entanglement, syncrhonization and quantum-to-classical transition, a complete open quantum systems description [41, 42] is required (see e.g., [43-52]). In this broader framework, relaxation dynamics can display intriguing features [46, 48, 53-56], which are of major importance in understanding thermalization, metastability, and coherence preservation in open quantum systems. In models with spatially localized dissipation or when dissipation occurs within the same environment, certain collective excitations— such as quasi-dark states or metastable modes [57]— can become weakly coupled to the environment, giving rise to long-lived metastable dynamics [53, 57-61]. These modes often preserve nonzero coherences that decay only at long timescales, leading to a pronounced separation of relaxation times [53]. Dephasing, which suppresses phase coherence between system eigenstates, is generally expected to accelerate relaxation by destroying quantum interference responsible for dark or quasi dark-states, allowing the system to more quickly explore its available states. This behavior has been observed in various contexts, where dephasing assists thermalization and can enhance energy transport (see e.g., [62-68]). In particular, dephasing typically drives a breakdown of genuine quantum coherence and induces a transition from coherent quantum dynamics to effectively classical transport behavior, although certain features of non-Hermitian coherent dynamics may persist under specific conditions [69].

In this research article, we show that in networks of coherently coupled bosonic modes with non-uniform local dissipation, the presence of quasi-dark states leads to a nontrivial response to dephasing: while weak dephasing facilitates equilibration, moderate to strong dephasing induces a pronounced slowdown of relaxation, reminiscent of the behavior observed in the Zeno regime of strong dissipation [70]. We demonstrate that dephasing suppresses coherent transport and dynamically decouples certain collective modes from the dissipative channels. These long-lived modes, dominating the late-time dynamics and resulting into a significantly slower relaxation than in the purely dissipative case, exhibit features reminiscent of Lifshitz tail states [71-78], which are known to govern anomalously slow dynamics in disordered systems. This leads to an optimal dephasing rate that facilitates faster equilibration, balancing the suppression of quasi-dark states while avoiding the slow dynamics associated with isolated metastable Lifshitz tail states. Our results uncover a counterintuitive mechanism by which dephasing, instead of accelerating equilibration, can under certain conditions protect excitations and inhibit relaxation. This mechanism offers new opportunities for engineering metastable dynamics in open quantum systems, with broad relevance for bosonic quantum technologies.

2 Dissipative bosonic networks: Theoretical model

We consider a rather arbitrary linear network of bosonic modes comprising N sites or nodes, such as a bosonic lattice, with local dissipative terms describing single-particle loss and gain processes, with rates {\gamma }_n and g_n, respectively (n is node number), and with uniform dephasing at a rate \mathrm{\Gamma } [Fig.1(a)]. Under Markovian dynamics, the evolution of the density operator \rho (t) of the bosonic field is governed by the Lindblad master equation [41, 42, 79-81] (\mathrm{d}\rho /\mathrm{d}t)=\mathcal{L}\rho (t) with Liouvillian

(1)\begin{array}{rl}\mathcal{L}\rho =& -\mathrm{i}[H,\rho ]+\sum _n\left({\gamma }_n\mathrm{\Lambda }[a_n]+g_n\mathrm{\Lambda }[a_n^{†}]+\mathrm{\Gamma }\mathrm{\Lambda }[a_n^{†}{a}_n]\right)\\ \equiv & -\mathrm{i}[H,\rho ]+\mathcal{D}\rho ,\end{array}

where H=\sum _{n,m}{J}_{n,m}{a}_n^{†}{a}_m describes the coherent Hamiltonian dynamics, with J_{n,m}=J_{m,n}^{\ast }, a_n, a_n^{†} are the destruction and creation bosonic operators at network node n satisfying the usual commutation relations

[a_n,a_m^{†}]={\delta }_{n,m},[a_n,a_m]=[a_n^{†},a_m^{†}]=0

and

\mathrm{\Lambda }[o]=o\rho o^{†}-(1/2)(o^{†}o\rho +\rho o^{†}o)

are the dissipative superoperators, with the jump operators o=a_n, o=a_n^{†} and o=a_n^{†}{a}_n describing local particle loss, gain and dephasing at rates {\gamma }_n, g_n and \mathrm{\Gamma }, respectively. The Liouvillian superoperator can be equivalently written as

(2)\begin{array}{rl}\mathcal{L}\rho =& -\mathrm{i}\left(H_{eff}\rho -\rho H_{eff}^{†}\right)+\sum _n{\gamma }_n{a}_n\rho a_n^{†}\\ & +\sum _n{g}_n{a}_n^{†}\rho a_n+\mathrm{\Gamma }\sum _n{a}_n^{†}{a}_n\rho a_n^{†}{a}_n,\end{array}

where

H_{eff}=H-{\displaystyle \frac{\mathrm{i}}{2}}{\displaystyle \sum _n\left({\gamma }_n{a}_n^{†}{a}_n+g_n{a}_n{a}_n^{†}+\mathrm{\Gamma }{a}_n^{†}{a}_n{a}_n^{†}{a}_n\right)}

is an effective non-Hermitian Hamiltonian, and the last three terms on the right hand side of Eq. (2), \sum _k{L}_k\rho L_k^{†}, describes quantum jumps. To avoid instabilities, we assume . We also assume that there is a single stationary (equilibrium) state {\rho }_0, such that \mathcal{L}{\rho }_0=0. When \mathrm{\Gamma }=0, i.e., in the absence of dephasing, the Liouvilllian is quadratic in the bosonic operators, the equilibrium state is Gaussian and its existence and unicity can be proved under rather general conditions [82-85]. For example, taking g_n/{\gamma }_n=\exp (-\beta ) independent of site index n, it can be readily shown that the stationary state is given by {\rho }_0=Z^{-1}\exp (-\beta \sum _n{a}_n^{†}{a}_n), where Z is the normalization factor and the parameter \beta provides the strength of thermal excitation. Remarkably, this Gaussian state is also the equilibrium state when considering dephasing (\mathrm{\Gamma }\ne 0), even though the Liouvillian is not anymore quadratic. We consider the interesting situation where, besides the stationary state, the system admits of quasi dark states [57], i.e., slowly-decaying (metastable) states, when dephasing is switched off (\mathrm{\Gamma }=0). A dark state is a pure state {\rho }_d=|{\psi }_d⟩⟨{\psi }_d| where |{\psi }_d⟩ is an eigenstate of H such that \mathcal{D}{\rho }_d=0. {\rho }_d is clearly an eigenstate of the Liouvillian \mathcal{L} as well. For a quasi-dark state, instead, \mathcal{D}{\rho }_d\simeq 0 and {\rho }_d becomes a resonance [57]. Dark and quasi-dark states arise rather generally in networks under certain topology or symmetry constraints, such that some eigenstates of H have not excitations in dissipative nodes of the network (see for instance Refs. [52, 57]). Indicating by {\xi }_n (n=1,2,...,N) an eigenstate of the N\times N matrix J_{n,m} with {\xi }_n=0 at some nodes n\in \mathcal{S}, let us assume {\gamma }_n=g_n=0 for n\notin \mathcal{S}; then any Fock state |{\psi }_d⟩, obtained from the vacuum state by application of the creation operator b^{†}=\sum _n{\xi }_n{a}_n^{†}, is an eigenstate of H that does not dissipate and is thus a dark state. A quasi-dark state is obtained by introducing a small loss rate at an additional site n=n_0\notin \mathcal{S}, which yields metastability in the relaxation process toward the equilibrium state. For example, in a linear lattice with open boundary conditions and nearest-neighbor hopping rate J [Fig.1(b)], comprising an odd number of sites N with the bosonic modes possessing the same frequency, the eigenstates of the tridiagonal matrix J_{n,m} are given by the standing waves {\xi }_n\propto \sin [\pi ln/(N+1)] (l,n=1,2,...,N). For l=(N+1)/2, {\xi }_n vanishes at every even site of the lattice, \mathcal{S}=\{2,4,...,N-1\}, and thus dark states arise whenever {\gamma }_n=g_n=0 at odd sites.

3 Relaxation dynamics

The central question is to elucidate the role of dephasing in the relaxation process toward equilibrium, which is notably hindered by the presence of quasi-dark states. In our model, the Lindbladian and effective non-Hermitian approaches exhibit significantly different dynamical behaviors, primarily because the jump operators representing dephasing effects are quadratic in the bosonic operators [38]. To accurately capture the impact of dephasing on the emergence of long-lived Lifshitz tail states, it is therefore essential to explicitly incorporate quantum jumps into the dynamical description.

When the relaxation dynamics does not display strong non-normal features [50, 51, 86], the relaxation time is basically established by the inverse of the Liouvillian spectral gap. For \mathrm{\Gamma }=0, the Liouviallian \mathcal{L} is quadratic in the bosonic creation/destruction operators and the model is fully integrable via the third-quantization method [82, 83]. In particular, the entire dynamics is fully described by the equations of motion of first-order and second-order moments [49, 79, 85], and the spectral gap of the Liouvillian \mathcal{L} is given by the decay rate of the slowest-decaying mode in the relaxation dynamics of the covariance matrix [83]. Therefore, to characterize the relaxation dynamics toward the equilibrium state we can limit considering the relaxation dynamics of the mean values A_n(t)=\mathrm{T}\mathrm{r}(\rho (t)a_n) and of the two-point correlation matrix C_{n,m}(t)=\mathrm{T}\mathrm{r}(\rho (t)a_n^{†}{a}_m). In the presence of dephasing, the Liouvillian is quartic in the bosonic operators, however an extension of the third-quantization method is still possible [84, 85] and a closed-form equation for A_n(t) and C_{n,m}(t) can be obtained as well. For our network model defined by the Liouvillian given in Eq. (1), the evolution equations for A_n(t) and C_{n,m}(t) read explicitly

(3)\begin{array}{rl}\frac{\mathrm{d}{A}_n}{\mathrm{d}t}=& -\mathrm{i}\sum _m{J}_{n,m}{A}_m(t)-\frac{1}{2}({\gamma }_n+\mathrm{\Gamma }-g_n)A_n\\ \equiv & \sum _m{\mathcal{A}}_{n,m}{A}_m(t),\end{array}

(4)\begin{array}{rl}\frac{\mathrm{d}{C}_{n,m}}{\mathrm{d}t}=& -\mathrm{i}\sum _l\left(J_{m,l}{C}_{n,l}-J_{l,n}{C}_{l,m}\right)-\mathrm{\Gamma }(1-{\delta }_{n,m})C_{n,m}\\ & -\frac{1}{2}\left({\gamma }_n+{\gamma }_m-g_n-g_m\right)C_{n,m}+g_n{\delta }_{n,m}\\ \equiv & \sum _{l_1,l_2}{\mathcal{B}}_{n,m;l_1,l_2}{C}_{l_1,l_2}+G_{n,m},\end{array}

where we have set G_{n,m}\equiv g_n{\delta }_{n,m} and where \mathcal{A} and \mathcal{B} are N\times N and N^2\times N^2 matrices. The equilibrium values are readily obtained by letting \mathrm{d}{A}_n/\mathrm{d}t=0 and \mathrm{d}{C}_{n,m}/\mathrm{d}t=0 in the above equations. Clearly, the equilibrium value (A_n{)}_{eq} for the mean values A_n is (A_n{)}_{eq}=0. On the other hand, the equilibrium value (C_{n,m}{)}_{eq} for the second moments is non-vanishing owing to the forcing term G_{n,m} due to incoherent pumping. When g_n/{\gamma }_n=\exp (-\beta ), (C_{n,m}{)}_{eq} is diagonal, independent of \mathrm{\Gamma } and n, and given by (C_{n,m}{)}_{eq}=g_n{\delta }_{n,m}/({\gamma }_n-g_n)={\delta }_{n,m}/(\exp (\beta )-1) [see as an example Fig.2(a)], showing the characteristic divergence of the correlations as the instability point \beta \to 0^{+} is approached. On the other hand, when the ratio g_n/{\gamma }_n depends on n, the stationary correlation matrix (C_{n,m}{)}_{eq} is not anymore diagonal and its elements depend on \mathrm{\Gamma } [an example is given in Fig.3(a)].

The eigenvalues {\lambda }_{\alpha } and {\mu }_{\alpha ,\beta } of the two matrices \mathcal{A} and \mathcal{B}, with \alpha ,\beta =1,2,...,N, determine the relaxation rates toward the equilibrium. In the stable regime , all eigenvalues have negative real part. We indicate by \mathrm{\Phi }(\mathrm{\Gamma })={\mathrm{m}\mathrm{i}\mathrm{n}}_{\alpha }|\mathrm{R}\mathrm{e}({\lambda }_{\alpha })| and \mathrm{\Theta }(\mathrm{\Gamma })={\mathrm{m}\mathrm{i}\mathrm{n}}_{\alpha ,\beta }|\mathrm{R}\mathrm{e}({\mu }_{\alpha ,\beta })| the eigenvalues of the two matrices with the smallest value (in absolute value) of the real part, corresponding to the slowest decaying rates in the system. When the dephasing is switched off, i.e., for \mathrm{\Gamma }=0, one has {\mu }_{\alpha ,\beta }={\lambda }_{\alpha }+{\lambda }_{\beta }, \mathrm{\Theta }=2\mathrm{\Phi } and \mathrm{\Phi } is very small, owing to the existence of long-lived quasi-dark states. As \mathrm{\Gamma } is increased above zero, clearly from the form of \mathcal{A} it is expected that \mathrm{\Phi }(\mathrm{\Gamma }) increases monotonously and secularly with \mathrm{\Gamma }, i.e., the mean values A_n(t) undergo a faster damping toward the equilibrium value (A_n{)}_{eq}=0 as a result of the additional dephasing [Fig.1(d)]. Likewise, \mathrm{\Theta }(\mathrm{\Gamma }) is expected to increase as well since dephasing destroys phase coherence and thus the quasi-dark (metastable) states in the system. However, for large dephasing rates the decay rate \mathrm{\Theta }(\mathrm{\Gamma }) is expected to decrease and vanish as \mathrm{\Gamma }\to \mathrm{\infty }, as sketched in Fig.1(d). The underlying physical mechanism is that strong dephasing induces diffusive (incoherent) transport within the network, which can be effectively described by a classical Markov master equation. In the limit of large dephasing strength \mathrm{\Gamma }, the hopping rate vanishes, and the system enters the so-called Zeno regime [70]. In this regime, new metastable states can emerge within dissipation-free regions, reminiscent of Lifshitz tail states, which are known to govern anomalously slow dynamics in disordered systems [71-73, 88, 92]. This leads to an optimal dephasing rate [Fig.1(d)] that facilitates faster equilibration, balancing the suppression of quasi-dark states while avoiding the slow dynamics associated with isolated metastable Lifshitz tail states.

To clarify this point, let us consider the dynamical equations (4) in the large dephasing regime, where \mathrm{\Gamma } is much larger than the coherent hopping rates |J_{n,m}| and dissipative rates {\gamma }_n,g_n, i.e., (|J_{n,m}|,{\gamma }_n,g_n)/\mathrm{\Gamma }\sim ϵ\ll 1. In this regime, the correlations C_{n,m}(t) for n\ne m rapidly decay to small values, at least of order \sim ϵ, and they can be eliminated adiabatically from the dynamics using standard perturbative methods (see for example [89-91]). The relevant equation for the diagonal elements P_n(t)=C_{n,n}(t)=\mathrm{T}\mathrm{r}(\rho (t)a_n^{†}{a}_n), i.e., of mean excitation numbers (populations), is given by the classical master equation

(5){\displaystyle \frac{\mathrm{d}{P}_n}{\mathrm{d}t}}={\displaystyle \sum _{m\ne n}{J}_{n,m}^{(inc)}(P_m-P_n)-({\gamma }_n-g_n)P_n+g_n,}

where J_{n,m}^{(inc)}=J_{m,n}^{(inc)}=2|J_{n,m}{|}^2/\mathrm{\Gamma } is the effective incoherent hopping rate between nodes n and m of the network. After letting P_n(t)=P_n^{(eq)}+p_n(t), where P_n^{(eq)} is the equilibrium state, Eq. (5) reads

(6){\displaystyle \frac{\mathrm{d}{p}_n}{\mathrm{d}t}}={\displaystyle \sum _{m\ne n}{J}_{n,m}^{(inc)}(p_m-p_n)-({\gamma }_n-g_n)p_n.}

The above equation formally describes a continuous-time classical random walk on the network with traps placed at the dissipative nodes, n\in \mathcal{S} [Fig.1(c)]: whenever the particle sits at the dissipative node n, it is annihilated at the rate ({\gamma }_n-g_n) per unit time. Trapping problems of this kind have been studied extensively (see e.g., [73, 88, 92]), and the asymptotic longtime falloff of the survival probability, establishing the time scale of relaxation toward equilibrium, is known to be related to Lifshitz tail states [73, 76, 77, 92]. These are long-lived metastable states where the classical particle resides for long times in wide dissipation-free regions of the network before being annihilated by the traps.

4 Illustrative examples and discussion

4.1 Linear chain with deterministic loss/gain

As a first illustrative example, let us consider a linear chain (a lattice) of N sites with equal resonance frequencies, nearest-neighbor hopping amplitude J and open boundary conditions [Fig.1(b)], where dissipation takes uniform values {\gamma }_n=\gamma, g_n=g in a fraction p of sites n\in \mathcal{S}, while {\gamma }_n=g_n=0 in the other fraction (1-p) of sites n\notin \mathcal{S}. In this case, in the large N limit the decay rate \mathrm{\Theta }(\mathrm{\Gamma }) of the most long-lived Lifshitz tail state is given by [77]

(7)\mathrm{\Theta }(\mathrm{\Gamma })\simeq {\displaystyle \frac{2{\pi }^2{J}^2}{\mathrm{\Gamma }(l_m+1{)}^2}},

where l_m is longest island of consecutive dissipation-free sites in the lattice. As one can see from Eq. (7), the relaxation rate of P_n(t) vanishes as 1/\mathrm{\Gamma } in the larger dephasing (classical) regime. To obtain a quasi-dark state, it is sufficient to apply the gain and loss terms in some even sites of the lattice, with an additional weak loss/gain term in one site of the lattice with odd number. For example, let us assume a lattice comprising N=21 sites, and let us assume \mathcal{S}=\{2,6,8,20\} as the set of dissipative sites, with {\gamma }_n/J=1.2 and g_n/J=0.2 at such sites. This lattice sustains an exact dark state. To obtain a quasi-dark state, an additional small dissipation is introduced at the odd site n_0=1, with {\gamma }_{n_0}/J=0.06 and g_{n_0}/J=0.01, which transforms dark states into long-lived quasi-dark states at \mathrm{\Gamma }=0. For such parameter values, one has l_m=11 for the longest island of consecutive dissipation-free sites in the lattice. Since g_n/{\gamma }_n=\exp (-\beta )=1/6 is independent of n, the stationary values of correlations (C_{n,m}{)}_{eq} are diagonal and independent of \mathrm{\Gamma } and n, as depicted in Fig.2(a). Fig.2(b) and (c) show the numerically-computed behavior of the relaxation rates \mathrm{\Phi } and \mathrm{\Theta } versus \mathrm{\Gamma }, as obtained by diagonalization of the matrices \mathcal{A} and \mathcal{B} entering in Eqs. (3) and (4). According to the theoretical predictions, while the relaxation rate \mathrm{\Phi } for the mean values A_n(t) monotonously increases as \mathrm{\Gamma } is increased [Fig.2(b)], the relaxation rate \mathrm{\Theta } of correlations undergoes an initial increase with \mathrm{\Gamma }, with a largest decay rate at around \mathrm{\Gamma }/J\simeq 0.25 [inset in Fig.2(c)], corresponding to the optimal dephasing value for fastest equilibration. Above this value, \mathrm{\Theta } decreases with \mathrm{\Gamma } and vanishes as \mathrm{\Gamma }/J\to \mathrm{\infty }. The dashed curve in Fig.2(c) depicts the theoretically-predicted value of the decay rate \mathrm{\Theta } in the large dephasing regime of classical random walk [Eq. (7)], which fits very well with the exact numerical results.

Since the formation of long-lived Lifshitz tail states requires solely the existence of islands of dissipation-free nodes, the above scenario is observed even when the ratio g_n/{\gamma }_n depends on index n, i.e., when the equilibrium correlation matrix (C_{n,m}{)}_{eq} is not diagonal and its elements depend on \mathrm{\Gamma }. An illustrative example is shown in Fig.3. In this example we consider the same linear chain of Fig.2 with the same parameter values, except than the condition g_n/{\gamma }_n=\exp (-\beta )=1/6 is broken by letting g_8/{\gamma }_8=1/3 at the dissipative node n=8. Fig.3(a) depicts three indicative behaviors of the stationary correlation matrix (C_{n,m}{)}_{\mathrm{e}\mathrm{q}} for increasing values of dephasing rate \mathrm{\Gamma }, showing that for vanishing or small dephasing the correlation matrix is not diagonal. The behavior of decay rates \mathrm{\Phi } and \mathrm{\Theta } versus dephasing rate \mathrm{\Gamma } is plotted in Fig.3(b) and (c), clearly indicating the slow down of relaxation for the correlations at large dephasing rates due to the emergence of long-lived Lifshitz tail states.

4.2 Linear chain with stochastic gain/loss

The previous example focused on deterministic distributions of dissipation that support quasi-dark states, however similar phenomena can also emerge when dissipation is applied to a subset \mathcal{S} of nodes selected stochastically within the network. A notable example is the Bernoulli-Anderson model, where the local dissipation rates {\gamma }_n are randomly assigned, taking the value {\gamma }_n=\gamma with probability p, and {\gamma }_n=0 with probability (1-p). The gain rates are taken as g_n={\gamma }_n\exp (-\beta ). In this case, due to the stochastic placement of dissipative nodes within the network, the formation of long-lived quasi dark states arising from destructive interference in the \mathrm{\Gamma }=0 limit is generally inhibited. Nevertheless, in the weak dephasing regime and for small concentrations of dissipative nodes, i.e., for p\ll 1, equilibration can still be accelerated by dephasing, resulting in an optimal dephasing rate that provides the fastest relaxation to equilibrium. As dephasing increases above such an optimal value, the system crosses over into a classical hopping regime, where a pronounced slowdown of relaxation emerges, driven by the appearance of long-lived Lifshitz-tail-like states. This behavior is illustrated in Fig.4(a, b), which show the numerically computed decay rates \mathrm{\Phi } and \mathrm{\Theta } as functions of \mathrm{\Gamma } for a Bernoulli−Anderson distribution of dissipation in the same linear lattice considered in Fig.2. The results are obtained by averaging over 100 stochastic realizations of dissipative node configurations, with parameter values \gamma =1.2J, p=0.2, and g_n={\gamma }_n/6. In the classical limit of large dephasing, \mathrm{\Gamma }/J\gg 1, the decay rate \mathrm{\Theta } is well approximated by Eq. (7), where the characteristic length l_m is replaced by its statistical average {\overline{l}}_m, computed using extreme value theory for stochastic Bernoulli processes [92]

(8){\overline{l}}_m=-{\displaystyle \frac{\ln (\sigma pN)}{\ln (1-p)}},

where \sigma \sim 1 is a form factor. As the concentration p of dissipative nodes in the network increases toward p=1, corresponding to the case where all nodes are dissipative, dephasing in the weak regime becomes progressively less effective in accelerating equilibration. In the limit p\to 1, the decay rate \mathrm{\Theta } turns into a monotonically decreasing function of \mathrm{\Gamma }. This behavior is illustrated in Fig.4(c) and (d).

4.3 Experimental realizations and platforms

The theoretical framework developed in this work can be explored experimentally using several state-of-the-art platforms that support engineered bosonic modes with controlled dissipation and dephasing. Promising candidates include nanophotonic systems such as coupled microcavities or photonic crystal cavities, optical waveguide lattices, synthetic photonic lattices, ultracold atomic gases, and superconducting quantum circuits.

A particularly accessible realization of the linear network in Fig.1(b) is through arrays of evanescently coupled optical waveguides [93], coupled microcavities or photonic crystal cavities [94, 95], or coupled microring resonators [29, 96]. In such systems, individual control of optical gain and loss can be achieved via spatially selective pumping, absorption, or scattering, enabling site-resolved engineering of dissipative processes [29]. Dephasing can be introduced by modulating the refractive index (and thus the propagation constants) with noise in time or space, or by incoherent probing using broadband light sources [97, 98]. These photonic systems provide excellent control and measurement capabilities, particularly suited for monitoring non-Hermitian transport dynamics.

Synthetic photonic lattices – based on modulated ring resonators, fiber loops, or time-multiplexed quantum walk architectures [99-101] – offer another versatile platform. In these systems, temporal degrees of freedom emulate spatial lattices, and gain/loss elements and phase modulators can be inserted to realize tailored Lindbladian evolution, including both local dissipation and dephasing [48, 69, 102]. These setups enable dynamic control over system parameters and have been used to simulate non-Hermitian physics and topological effects.

Ultracold atomic systems in optical lattices also present a viable setting. Localized loss and gain processes can be engineered using tightly focused electron beams or optical potentials, while dephasing can be introduced via controlled noise in lattice depths or by coupling to auxiliary thermal reservoirs [103, 104]. Such systems inherently feature bosonic statistics and tunable interactions, providing a natural platform for realizing dissipative many-body quantum dynamics. Finally, superconducting quantum circuits offer chip-based realizations of coupled bosonic modes (e.g., microwave resonators or transmon qubits in the dispersive regime) with programmable loss, gain, and dephasing channels. Using quantum reservoir engineering techniques, these platforms have demonstrated precise control over open-system dynamics and have been used to explore non-Hermitian Hamiltonians and dissipative phase transitions [66, 105, 106].

4.4 Fermionic networks

While the analysis presented in this work focuses on non-interacting bosons, similar considerations apply to networks of spinless, non-interacting fermions undergoing coherent hopping dynamics and subject to local particle gain/loss and dephasing processes [107]. In this case, the Lindblad master equation remains of the same general form as Eq. (1), but the creation and annihilation operators obey fermionic anti-commutation relations:

(9)\left\{a_n,a_m^{†}\right\}={\delta }_{n,m},\left\{a_n,a_m\right\}=\left\{a_n^{†},a_m^{†}\right\}=0.

Owing to these different statistics, the evolution equation for the two-point correlation matrix, defined as C_{n,m}(t)=\mathrm{T}\mathrm{r}[\rho (t)a_n^{†}{a}_m], takes a form distinct from its bosonic counterpart [84, 85, 107]. Specifically, one obtains:

(10)\begin{array}{rl}\frac{\mathrm{d}{C}_{n,m}}{\mathrm{d}t}=& -\mathrm{i}\sum _l\left(J_{m,l}{C}_{n,l}-J_{l,n}{C}_{l,m}\right)-\mathrm{\Gamma }(1-{\delta }_{n,m})C_{n,m}\\ & -\frac{1}{2}\left({\gamma }_n+{\gamma }_m+g_n+g_m\right)C_{n,m}+g_n{\delta }_{n,m}\\ \equiv & \sum _{l_1,l_2}{\mathcal{B}}_{n,m;l_1,l_2}^{(\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{m})}{C}_{l_1,l_2}+G_{n,m},\end{array}

where {\mathcal{B}}^{(\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{m})} denotes the generator of the dissipative dynamics and G_{n,m} is the gain source term. A key difference with respect to the bosonic case lies in the structure of the gain terms: for fermions, they appear with a damping sign in the relaxation matrix {\mathcal{B}}^{(\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{m})}. This reflects the Pauli exclusion principle, which forbids more than one particle per site and thus prevents the occurrence of bosonic-like instabilities (i.e., unbounded population growth due to gain). As a consequence, the fermionic system is dynamically stable regardless of the gain/loss balance. Aside from this statistical distinction – which becomes negligible in the limit of weak or vanishing gain – the overall relaxation dynamics of the covariance matrix, governed by the spectral properties of {\mathcal{B}}^{(\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{m})}, exhibits behavior qualitatively similar to the bosonic case. In particular, weak dephasing suppresses quasi-dark modes and accelerates relaxation, while strong dephasing suppresses coherent hopping and leads to the emergence of long-lived, collective decay modes – the Lifshitz tail states– that dominate the system’s late-time dynamics.

5 Conclusions

To conclude, this study uncovers a non-monotonic role of dephasing in the relaxation dynamics of bosonic networks with non-uniform dissipation, driven by the formation of metastable Lifshitz tail states. For weak dephasing rates, decoherence accelerates equilibration by disrupting the coherence in the system and destroying long-lived quasi-dark modes. However, at strong dephasing rates, a pronounced slowdown of relaxation is observed due to the emergence of long-lived metastable modes, reminiscent of Lifshitz tail states, which become decoupled from dissipation. This leads to an optimal dephasing rate that facilitates faster equilibration, balancing the suppression of quasi-dark states while avoiding the slow dynamics associated with isolated metastable Lifshitz tail states. Similar results should occur for the relaxation dynamics of spinless non-interacting fermions in dissipative networks. The findings demonstrate that dephasing, rather than always accelerating relaxation, can in some regimes inhibit decay and protect excitations, offering a new route for controlling relaxation and decoherence in open quantum systems. The ability to tune dephasing could thus play a crucial role in engineering metastable states and controlling relaxation dynamics in a range of bosonic quantum systems, with significant implications for the development of robust quantum technologies and advancing the understanding of decoherence in modern quantum science.

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