<figure><img src="image_32.png" /><figcaption>Figure 29: Role of short-range correlations on the phase diagram of the dissipative XYZ model, comparing (a) mean-field predictions vs (b) cluster mean-field theory in two dimensions. Adapted from Ref. [283].</figcaption></figure>

Further works on the dissipative XYZ model have used other methods to investigate the DPT in one or two dimensions, beyond mean-field. The Keldysh approach combined with renormalization group was used in [582] to study the critical state of an Ising model with long-range losses. References [331, 407] used a tensor network technique: infinite projected entangled pair operators (iPEPO) while Ref. [330] used the corner-space renormalisation group. In addition, Ref. [579] discussed the dissipative transition using quantum trajectories by extracting the Lindbladian gap and discussing signatures of critical slowing down. All these studies have highlighted how the DPT is washed out in one dimensional systems while it survives in two-dimensional lattices, although a conclusive results on the location of the phase boundaries is yet to be found.

Another interesting limit of the model we have discussed corresponds to $J_z = 0$, where mean-field theory predicts a staggered XY phase for any finite value of the anisotropy and a featureless paramagnetic phase when the couplings $J_{x,y}$ have the same sign. The effects of beyond-mean-field fluctuations on this result have been explored using Keldysh field-theory techniques [450] which predict a washing out of the quasi-long range order in two-dimensions. In addition to the features seen for the model discussed above, other models of dissipative spin chains have been discussed, which show a wider variety of phases. In particular, several such models [237, 340, 342, 371, 564, 583] observed transitions to a state which, in mean-field theory, corresponds to a periodic dynamics. As discussed further below, such a state corresponds to the occurrence of a time crystal, see Sec. 6.2.3.

### 6.2.2 Coherent drive and bistability

We now discuss the cases in which there are first-order, i.e. discontinuous, DPTs without symmetry breaking. As noted at the start of this section, semiclassical analysis suggests bistability in such cases, but the full treatment reveals a unique steady state at each point in the thermodynamic limit. Here we show some concrete examples of such behaviour.

**Kerr nonlinear oscillator.** A paradigmatic example of a first order DPT occurs in the context of a driven-dissipative non-linear oscillator, as discussed in Ref. [203]. This is described by Hamiltonian and loss terms:

$$ \hat{H} = -\Delta \hat{a}^{\dagger} \hat{a} + \frac{U}{2} \hat{a}^{\dagger} \hat{a}^{\dagger} \hat{a} \hat{a} + F (\hat{a}^{\dagger} + \hat{a}), \quad \hat{L} = \sqrt{\kappa} \hat{a}. \qquad (183) $$

While this is a single-mode problem without an immediately apparent thermodynamic limit, it has been shown that a well-defined thermodynamic limit can be still defined by taking $F \to \infty$
<figure><img src="image_33.png" /><figcaption>Figure 30: Dissipative phase transition in a Kerr non-linear resonator, as defined in Eq. (183). Left panels, (a,b) shows the system-size dependence (i.e. N dependence) of the evolution vs rescaled pump strength $\tilde{F}$. Panel (a) shows the number of photons $\langle \hat{a}^\dagger \hat{a} \rangle$ in the cavity, and panel (b) shows the Lindbladian gap $\lambda_1 \equiv \lambda_{ADR}$ plotted on a logarithmic scale. As discussed above, there is an extended range of $\tilde{F}/\kappa$ over which $\lambda_1$ is strongly dependent on system size, and for which one can anticipate $\lambda_1 \to 0$ in the limit $N \to \infty$. The right panels, (c,d) compare the true steady state solution to the two solutions $\hat{\rho}^{<,>}$ extracted from finite-size calculations according to Eq. (173). Panel (c) shows the evolution of the photon number in comparison to the semiclassical solution, which is shown as a dashed line. One may see that over some range of $\tilde{F}$, $\hat{\rho}^{<,>}$ follow the two branches of the semiclassical solution. Panel (d) shows the fidelity between the steady-state density matrix $\hat{\rho}_{ss}$ and the states $\hat{\rho}^{<,>}$ and $\hat{\xi} = (\hat{\rho}^< + \hat{\rho}^>) / 2$, using the fidelity $f(\hat{\rho}_A, \hat{\rho}_B) = \text{Tr}[\sqrt{\hat{\rho}_A}\hat{\rho}_B\sqrt{\hat{\rho}_A}]$. From Ref. [203] [Copyright (2018) by the American Physical Society].</figcaption></figure>

at fixed $UF^2$. In this limit one finds that the number of bosons has a discontinuity as one tunes the ratio of driving and loss.

To explore the approach to this thermodynamic limit, one can define $U = \tilde{U}/N, F = \tilde{F}\sqrt{N}$. The behaviour of the photon number and eigenvectors of the Lindbladian is shown in Fig. 30, showing how a sharp transition emerges as N increases. One important feature of this model is that it shows an extended range of F over which $\lambda_{ADR}$ decreases with increasing system size. While $\lambda_{ADR}$ is minimum at $\eta = \eta_c$, it shows strong dependence on system size over an extended region. This figure strongly suggests that $\lambda_{ADR}$ vanishes over an extended region.

**First-order transitions in spin lattices.** Classical bistability, and the associated first-order transition in the full dynamics, has also been explored in models of spins on lattices, which more directly provide a many-body context and thermodynamic limit. Here we discuss two examples. The first is a coherently driven XY quantum spin model on a d-dimensional hypercubic lattice, with Hamiltonian

$$ \hat{H} = -J \sum_{\langle i,j \rangle} (\hat{\sigma}_i^+ \hat{\sigma}_j^- + \text{H.c.}) + \sum_i \left( \frac{\Delta}{2} \hat{\sigma}_i^z + \Omega \hat{\sigma}_i^x \right), \quad (184) $$

where $\Omega, \Delta$ are respectively the strength of the local drive and the detuning, and local jump operators describing dissipation $\hat{L}_{i-} = \sqrt{\gamma} \hat{\sigma}_i^-$. This model was studied in Ref. [426] using matrix product operator (MPO) simulations, in d = 1, 2, and an approximation scheme that
<figure><img src="image_34.png" /><figcaption>Figure 31: Bistability in coherently driven spin chains. Left Panel: XY model with longitudinal and transverse field. Average steady-state magnetisation $\mu_x$ for mean-field (MF), mean-field plus quantum fluctuations (MFQF) in $d = 2, 3$, and matrix product operator (MPO) on small two-dimensional lattices of size $12 \times 4$. From Ref. [426] [Copyright (2018) by the American Physical Society]. Right Panel: Ising Model in a Transverse Field. In both cases MF bistability is reduced by the addition of quantum fluctuations. From Ref. [585] [Copyright (2018) by the American Physical Society].</figcaption></figure>

accounts for fluctuations beyond mean-field (MFQF) in $d = 1, 2, 3$, equivalent to the cumulant expansion discussed in Sec. 4. Both methods were compared with mean-field theory, which predicts a bistable region for the steady-state magnetisation $\mu_\alpha = \frac{1}{N_s} \sum_i \text{Tr}[\hat{\rho}_{ss} \hat{\sigma}_i^\alpha]$, as a function of the detuning $\Delta$. In one dimension both the MPO and the MFQF results show that bistability is destroyed by quantum fluctuations, a result which is consistent with that found in other coherently driven-dissipative models (see below). In two dimensions however bistability is found to survive in MFQF, although in a reduced region of parameter space, as shown in the left panel of Fig. 31. The MPO results are obtained on small cylinders, mapping the two-dimensional problem to a one-dimensional chain. These compare well with MFQF over a large region of detunings, however they show a smooth crossover instead of true bistability, see left panel of Fig. 31.

A second example is the transverse-field Ising chain with Hamiltonian and jump operators

$$ \hat{H} = \frac{V}{4} \sum_{\langle i,j \rangle} \hat{\sigma}_i^z \hat{\sigma}_j^z + \frac{g}{2} \sum_i \hat{\sigma}_i^x, \quad \hat{L}_{i-}^z = \frac{\sqrt{\gamma}}{2} (\hat{\sigma}_i^x - i\hat{\sigma}_i^y), \quad \hat{L}_{i+}^x = \frac{\sqrt{\gamma}}{2} (\hat{\sigma}_i^y + i\hat{\sigma}_i^z). \quad (185) $$

This model was studied in Ref. [585], comparing mean-field results to cluster-mean-field theory in two dimensions, as shown in the right panel of Fig. 31. Once again mean-field theory predicts bistability, while the cluster-mean-field theory shows either a crossover in place of a transition, or a sharp first-order transition.

As discussed in Sec. 6.1, the survival of bistability beyond mean-field theory can be understood as a consequence of considering the large system size limit before the long time limit. That is, when the large time limit is taken first, the smooth crossover at finite size turns into a sharp first order phase transition in the thermodynamic limit as discussed in [203]. However, in the opposite case, where the large system size limit is taken first, the lifetime of the two metastable states diverges, leading to bistability. Signatures of such bistable behaviour appear in the unique steady state at finite $N$, which display a bimodal distribution of local
observables [426, 585].

There have been a large number of works, using a variety of methods, studying other examples of spin lattices (as well as other lattice based problems). Such problems provide a natural context to compare the predictions of real-space mean-field theory to beyond-mean-field treatments. In one-dimensional lattices with nearest-neighbour interactions, the mean-field bistability is found to be replaced by a crossover driven by large quantum fluctuations [554, 586–589]. That is, in one-dimensional spin-chains, there generally is often no DPT (however see Sec. 6.2.4 for a notable counter example). For two-dimensional lattices, a DPT does exist beyond mean field, but (as discussed above), mean-field bistability is replaced by a first-order transition. Examples of this include nonlinear bosons using a truncated Wigner approximation [554, 589], Ising spins using a variational Ansatz accounting for short-range correlations [586], cluster mean-field approaches [585], and two-dimensional tensor networks (iPEPO) [331, 407, 408]. In addition to finding the steady state, one can also study the dynamics, and it has been seen that around the DPT the rate of convergence towards the steady state slows down [554, 589], as expected from the spectral theory outlined above [203, 578, 590].

### 6.2.3 Boundary time-crystal and dissipative limit-cycles

We next discuss the DPT to a state that breaks time-translation symmetry, i.e. a *time crystal* or *limit cycle state*. As explained in Ref. [563], time-crystal states in a dissipative system can be considered as corresponding to a “boundary time crystal”, a name referring to the fact that the open quantum system can be thought of as the “boundary” of the environment. Such a consideration is important because there is a well established no-go theorem [562] that prevents a time crystal arising in the ground state of a quantum system. This no-go theorem should apply to the combined system and environment, but the reason why a time crystal is nonetheless possible in the open system is that the no-go theorem prevents periodic dynamics for macroscopic observables in the system; the boundary however corresponds to a vanishing fraction of the combined system and environment. As such, periodic dynamics of the boundary does not violate the no-go theorem. For a complete discussion of this subtle point see Ref. [563].

In order to make a connection with the discussion presented in Sec. 5, time-crystals prepared via dissipation can be considered as an example of state preparation where in the steady state a dark-subspace (rather than a dark state) is selected. This fact, that we are now going to extensively discuss, opens very interesting research perspectives due to the possible relations of boundary time crystals with decoherence-free subspaces and noiseless subsystems [591]. Recent experimental detection of dissipative time-crystals has been reported in [592, 593]

**Collective spin model.** As a first example we consider the following Hamiltonian and jump operator:

$$ \hat{H} = \omega_0 \hat{S}^x + \frac{\omega_x}{S} (\hat{S}^x)^2 + \frac{\omega_z}{S} (\hat{S}^z)^2, \quad \hat{L} = \sqrt{\frac{\kappa}{S}} \hat{S}^{-}, \qquad (186) $$

describing a collection of two-level systems, with $\hat{S}^\alpha = \sum_{j=1}^N (1/2) \hat{\sigma}_j^\alpha$ with $\alpha = x, y, z$. The total spin $S$ is conserved by the master equation and we fix it equal to the maximum value $S = N/2$. For simplicity we focus here on the case $\omega_x = \omega_z = 0$ where there is only one control parameter in the problem, namely the ratio $\eta \equiv \omega_0/\kappa$. The general case is discussed in the supplement of Ref. [563].

Figure 32 shows the Lindbladian eigenspectrum, and the real-time dynamics of the spin. The spectrum (calculated for finite $N$) shows signatures of what will become a phase transition at $\eta = \eta_c \equiv 1$. Below the transition, a unique steady state exists, corresponding to an isolated zero mode in the spectrum. Above the transition, there is instead persistent oscillations. At finite $N$, this appears at modes at finite frequency (finite imaginary part of $\lambda$) and small real
<figure><img src="image_35.png" /><figcaption>Figure 32: Boundary Time-Crystal. Top: Lindblad spectrum across a dissipative time crystal phase transition, where time-translation invariance is spontaneously broken. In panel (a), for $\omega_0/\kappa = 0.5$ we see a finite dissipative gap, corresponding to a well defined steady state and a dynamics which leads to damping. In panel (b), for $\omega_0/\kappa = 1.5$ the Lindbladian spectrum becomes dense near $\lambda = 0$, acquiring also a finite imaginary part. In the thermodynamic limit this gives rise to spontaneous oscillations. Bottom: Time evolution of $\langle \hat{S}^z \rangle$ showing long-lived oscillations at finite $N$, and the corresponding thermodynamic limit, plotted for $\omega_0/\kappa = 1.5$. From Ref. [563]. [Copyright (2018) by the American Physical Society].</figcaption></figure>

part. As $N \to \infty$ the real part vanishes and the oscillations become long lived. The same behaviour can be seen by viewing the dynamics in the time domain as shown in the lower panel of Fig. 32.

One can understand what happens in the thermodynamic limit, corresponding to the limit of large spin $S \to \infty$, by using a semiclassical approximation, replacing expectations of products of operators by products of expectations. Writing $S_\alpha = \langle \hat{S}^\alpha \rangle$ we obtain classical equations of motion:

$$ \frac{d}{dt} S_x = \frac{\kappa}{S} S_x S_z, \qquad (187a) $$

$$ \frac{d}{dt} S_y = -\omega_0 S_z + \frac{\kappa}{S} S_y S_z, \qquad (187b) $$

$$ \frac{d}{dt} S_z = \omega_0 S_y - \frac{\kappa}{S} (S_x^2 + S_y^2). \qquad (187c) $$
In addition to the conserved magnitude of spin (as discussed above), there is a second conserved quantity $M = S_x/S_y - \omega_0/\kappa$. This conservation law has important consequences on the dynamics of the system.

Considering the stationary states, i.e. $\frac{d}{dt}S_\alpha = 0$ we find that for $\eta = \omega_0/\kappa \le 1$ the steady states are

$$ (S_x, S_y, S_z) = S(0, \eta, \pm\sqrt{1-\eta^2}). $$

Linear stability analysis shows that the solution $S_z = -\sqrt{1-\eta^2}$ is stable. Such stability analysis also gives the time to reach the stationary state, $\tau = 1/\kappa\sqrt{1-\eta^2}$. These results suggest a transition at $\eta = 1$. For $\eta > 1$ one can find fixed point solutions

$$ (S_x, S_y, S_z) = S(\pm\sqrt{1-1/\eta^2}, 1/\eta, 0), $$

however these are not stable. Linear stability analysis show that fluctuations about these have imaginary eigenvalues! This corresponds to a Hopf bifurcation, with a finite-frequency instability. Typically such Hopf bifurcations lead to dynamics with a limit cycle as an attractor. While we indeed see a periodic attractor, it is not a limit cycle [594], in that there is not a unique periodic attractor, but instead multiple periodic attractors. Nonetheless, as seen in Fig. 32(c), there is periodic behaviour seen at late times (which survives to longer and longer times as $N \to \infty$), hence this state can be understood as a time crystal.

**Dissipative Bose-Hubbard model.** A different example of DPT with breaking of time-translation invariance occurs for models in which the semiclassical equations display a limit cycle phase, and exploring how this phase is affected by quantum fluctuations. Examples that fall into this class include semiclassical models of lasing (see Sec. 2.6) and optical parametric oscillators (See for example [595, 596]). More recently the focus has shifted on lattice models of driven-dissipative systems, such as the driven-dissipative Bose-Hubbard model which we discuss now. There have been many forms of driven-dissipative Bose-Hubbard models studied, see for example Refs. [464, 597, 598]. In this section we focus on a particular form of driving and dissipation introduced in Ref. [284]. As discussed previously the Bose-Hubbard model has the Hamiltonian:

$$ \hat{H} = -t_H \sum_{\langle i,j \rangle} (\hat{a}_i^\dagger \hat{a}_j + \text{H.c.}) + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1), \quad (188) $$

and the driven-dissipative version we consider here is formed by considering two kinds of jump operators for each lattice site $i$,

$$ \hat{L}_{i+(2)} = \sqrt{\kappa_{-}^{(2)}} (\hat{a}_i)^2, \quad \hat{L}_{i-} = \sqrt{\kappa_{+}^{(1)}} \hat{a}_i^\dagger. \quad (189) $$

Such a model has both time translation invariance and a weak $U(1)$ symmetry $\hat{S} = \prod_i e^{i\theta_i \hat{n}_i}$.

The semiclassical equations of motion for this model—which should hold for large numbers of bosons per site—correspond to a lattice version of the Gross-Pitaevskii equation. The semiclassical equations predict a coherent phase for any non-zero pumping, $\kappa_+^{(1)} > 0$, independent of $t_H/U$. The coherent phase corresponds to phase locked oscillations of the coherent field on each site, i.e. $\langle \hat{a}_i \rangle = |\psi_0|e^{i\mu t}$. As such, this coherent phase breaks both time translation and the $U(1)$ symmetry, and so this model is a candidate to understand how quantum fluctuations modify this phase.

Figure 33 shows results from Ref. [284], calculating the phase diagram of this model using DMFT as introduced in Sec. 4.5. This shows a phase transition from a normal state at small $t_H$, characterized by a well defined steady-state which is incoherent $\langle a_i \rangle = 0$, to a nonequilibrium superfluid state which breaks time translation symmetry at large $t_H$, $\langle a_i \rangle \sim e^{i\Omega_* t}$, i.e. a dissipative time-crystal or limit cycle phase with a frequency $\Omega_*$ which is renormalized by quantum
<figure><img src="image_36.png" /></figure>

Figure 33: Phase Diagram of the driven-dissipative Bose-Hubbard model found using DMFT, as a function of pump/loss ratio $r = \kappa_+^{(1)} / \kappa_-^{(2)}$ and hopping/interaction ratio $\tilde{t}_H / U$ where $t_H = \tilde{t}_H / z$, for different values of the lattice connectivity $z$. These results are compared to the Gutzwiller mean-field theory that should hold at $z = \infty$. DMFT results are shown only for large values of $z$, as for smaller $z$, the computation becomes costly at large pumping, due to the large Hilbert space dimension. Adapted from Ref. [284].

fluctuations with respect to the semiclassical value [284]. As discussed in Sec. 4.5, we rescale the hopping as $t_H = \tilde{t}_H / z$, and show results as a function of the coordination number $z$. In addition to DMFT results, the Gutzwiller mean-field phase boundary (i.e. assuming real-space factorisation of the density matrix) is also shown. As expected, one sees that the DMFT results converge on this result in the $z = \infty$ limit. The incoherent normal state at small $t_H$ is consistent with what is known [306, 599, 600] for a individual nonlinear bosonic modes with one-photon pumping and two-photon loss, including a negative density of states at positive frequency which signals the onset of gain [444]. The DMFT results for the phase boundary show a strong dependence on the connectivity $z$ indicating a strong effect of fluctuations; one may note that this dependence is considerably stronger than has been seen for the Bose-Hubbard model in equilibrium. [354, 365, 366]. The role of fluctuations is to shrink the ordered phase via a mechanism of hopping-induced decoherence [284]: hopping processes to neighbouring sites, included within DMFT via the coupling to the non-Markovian bath, increase the local effective dissipation and destroy the limit cycle phase.

### 6.2.4 Absorbing state phase transitions

An intriguing class of dissipative phase transitions is that involving absorbing states [142, 560, 601]. This idea, first discussed in classical statistical mechanics [602], describes models where there is a state that, once reached, can never be left. Such states have close connections to dark states as discussed above, and in some cases are equivalent. We discuss below an example extending this concept to dissipative quantum systems.

Reference [560] introduced a quantum model that extends a standard classical model of absorbing state transitions. The classical model involves a lattice of sites in a ground or excited state, with processes where an excitation may decay, as well as “activated” processes where an excitation may split in two (“branching”) or two neighbouring excitations may merge into one (“coagulation”). Activated processes refer to changes to the state of one site conditional on the site of other sites.

A quantum extension of this model can be written in terms of a lattice of two-level systems represented by Pauli operators. It is convenient to define $\hat{n}_i = \hat{\sigma}_i^+ \hat{\sigma}_i^-$, the projector onto the
<figure><img src="image_37.png" /><figcaption>Figure 34: Illustration of absorbing state phase transition. Panel (a) illustrates the processes involved in the model, as defined in Eq. (190). Panel (b) shows the phase diagram of the resulting model. The axis $\Omega = 0$ corresponds to the purely classical model. As illustrated this shows a critical point in which the phase diagram changes from first order to second order. Panels (c,d,e) show the order parameter (excitation density) as a function of the tuning parameter $\eta$ (i.e. the analogue of Fig. 25) for three scenarios corresponding to (c) discontinuous transition, (d) critical point, and (e) continuous transition. As discussed in the text, only positive values of $n$ are physical [Panels (a,b) from Ref. [560] - Copyright (2016) by the American Physical Society.].</figcaption></figure>

excited state of site $i$, so one may then write the Hamiltonian and jump operators as:

$$ \hat{H} = \Omega \sum_{\langle i,j \rangle} \hat{n}_j \hat{\sigma}_i^x, \quad \hat{L}_{i,-} = \sqrt{\gamma} \hat{\sigma}_i^-, \quad \hat{L}_{\langle i,j \rangle, C+} = \sqrt{\kappa} \hat{n}_j \hat{\sigma}_i^-, \quad \hat{L}_{\langle i,j \rangle, C-} = \sqrt{\kappa} \hat{n}_j \hat{\sigma}_i^+. \quad (190) $$

The Hamiltonian describes a spin flip on site $i$, conditioned on the states of its neighbours $j$, i.e. a quantum version of an activated process. The three types of jump operators describe single-site decay $\hat{L}_{i,-}$ and two activated processes that act on pairs of nearest-neighbour sites. Process $\hat{L}_{\langle i,j \rangle, C+}$, is the classical branching process, creating an excitation on site $i$ conditioned on there being excitations on a neighbouring site $j$. The inverse coagulation process $\hat{L}_{\langle i,j \rangle, C-}$ destroys an excitation conditioned on an excitation on neighbouring site. These processes are illustrated in Fig. 34(a).

The model in Eq. (190) includes the classical model [602] in the limit $\Omega = 0$. It is clear to see that in general this model has a dark state $\hat{\rho} = \prod_i |\downarrow\rangle\langle\downarrow|_i$, which is equivalent to the absorbing state for the corresponding classical model. For some parameters, there can also be a second non-trivial state with a finite density of excitations, so $n = \langle\hat{n}_i\rangle$ serves as order parameter for this phase transition. It is notable that $\prod_i |\downarrow\rangle\langle\downarrow|_i$ is always a dark state of the model for any parameters. This therefore differs from the generic spectral theory of phase transitions discussed earlier.

As discussed in Ref. [560], the phase transition at general $\Omega$ can be understood from a saddle-point analysis of a Keldysh action. This reveals an effective action that can be written in the form $\Gamma(n) = -\eta n^2 + u_3 n^3 + u_4 n^4$ with $\eta, u_3, u_4$ being combinations of the model parameters $\Omega, \kappa, \gamma$ along with the lattice coordination number $z$. When $u_3 = 0$, this model has a continuous
phase transition with a critical point at $\eta = 0$, of the form as shown in Fig. 25(b). There is however a difference in that only $n \ge 0$ corresponds to physical states, hence the model does not show any symmetry breaking, even in this apparently symmetric case. When $u_3 > 0$ the transition remains continuous at $\eta = 0$, but for $u_3 < 0$ multiple saddle points appear indicating a discontinuous transition. These three scenarios are illustrated in Fig. 34(c,d,e). Using the form of the coefficients $\eta, u_3$ in terms of the bare parameters (as given in Ref. [560]), one finds that in the classical limit $u_3 > 0$, but increasing $\Omega$ and decreasing $\kappa$ can make $u_3$ change sign, leading to the phase boundary shown in Fig. 34(b).

A remarkable feature of absorbing state phase transitions is that they can still show phase transitions in one-dimensional systems. This was studied in detail in Ref. [142], which considered the $\kappa = 0$ limit of the model defined above. The saddle-point analysis summarised above suggests a first-order transition would occur in this limit for any dimension. Numerical analysis in Ref. [142] using matrix product state methods (specifically infinite time evolving block decimation [399]) showed the existence of a transition even in one dimension. Remarkably, the numerical results indicate a continuous transition with critical exponents matching those at the critical point [601] corresponding to $u_3 = 0$. This was interpreted as a result of strong fluctuation effects in one dimension driving the system toward the critical point.

The ability to realise such transitions even in one dimensional systems makes these promising models for realisation in experiments. Indeed, as discussed above, results have already been presented for a similar model in a system of Rydberg atoms [145].

# 7 Dynamics of approach to the steady state

In this Section we will go beyond the simple interest in the steady state of the dissipative many-body process and discuss several of its dynamical aspects. For this reason, we will focus on situations in which the interplay between system and environment leads to a simple and featureless stationary state (i.e. without any particular quantum or classical correlation) and the interesting behaviour is mainly found in the way the system approaches its stationary properties.

## 7.1 Heating dynamics under dephasing

We begin our discussion by focusing on the heating dynamics under dephasing, which is for instance what happens when ultracold-atomic gases evolve under controlled spontaneous emission (see Sec. 2.2). More generically, in this subsection we will study Lindblad master equations on a lattice with Hermitian site-local jump operators $\hat{L}_i^\dagger = \hat{L}_i$, so that we can write:

$$ \frac{d}{dt}\hat{\rho} = -i[\hat{H},\hat{\rho}] + \sum_i (\hat{L}_i\hat{\rho}\hat{L}_i - \frac{1}{2}\{\hat{L}_i^2,\hat{\rho}\}) = -i[\hat{H},\hat{\rho}] - \frac{1}{2}\sum_i[\hat{L}_i, [\hat{L}_i,\hat{\rho}]]. \quad (191) $$

The second expression makes very explicit that in the presence of Hermitian jump operators the identity matrix, normalised in order to have unit trace, is a steady state of the evolution. This state can be identified with the infinite-temperature state, corresponding to the limit $\lim_{\beta \to 0^+} e^{-\beta \hat{H}/Z}$: the dynamics therefore describes an heating process towards infinite temperature. The subject that we will explore is how such a stationary state is approached and the role of interactions and many-body effects on this transient dynamics. Among the many works that have studied this problem, in Sec. 7.1.1 we will address dephasing problems in spin chains, whereas bosonic systems will be the focus of Sec. 7.1.2.

It is however important to stress that the dynamics of heating towards infinite temperature has been the subjects of many other works that we cannot review in details here. The heating dynamics of dissipative Luttinger liquids and their lattice realisations have been studied
in a number of works [603–605]. A generalised hydrodynamic picture of this heating process in integrable systems has been developed in Ref. [606]; while Ref. [607] has discussed the dissipative dynamics in the framework of a non-Hermitian linear response theory. Furthermore there have been studies focusing on the spreading of correlations in interacting and non-interacting models under dephasing [608, 609] as well as the growth of operator entanglement, which determines the efficiency of the Matrix-Product Operator representation for the density matrix [610].

Finally, we note that even if the stationary state is featureless—as occurs in the examples above where the stationary state is the infinite temperature state—the dynamical correlations developed in the approach to the steady state contain rich information about the thermalisation process. These features have been investigated in the case of a free fermionic chain under dephasing [302, 304]. For strongly correlated systems with charge and spin degrees of freedom, such as the Fermi–Hubbard model, selective dephasing of the spin sector has been shown to lead to a non-trivial charge order in the stationary state; this is related to the so-called η–pairing state [466]. Slow dynamics and anomalous diffusion in open quantum many-body systems can also emerge in presence of long-range hopping [611] or kinetic constraints [138, 140–142] (see also discussion in Sec. 6.2.4).

### 7.1.1 Quantum spin chains under dephasing

We begin our discussion by considering a spin chain subject to a dephasing noise. One such example is the dissipative XX chain studied in Ref. [301, 590], as described by the Hamiltonian and jump operators

$$ \hat{H} = -J \sum_{i} (\hat{\sigma}_{i}^{+} \hat{\sigma}_{i+1}^{-} + \text{H.c.}) + h \sum_{i} \hat{\sigma}_{i}^{z}, \quad \hat{L}_{i} = \sqrt{\gamma} \hat{\sigma}_{i}^{z}. \qquad (192) $$

We note that such dephasing processes can be physically realised by considering the interaction of spins with a magnetic field pointing in the z direction with strength fluctuating in time. The Lindblad master equation follows from averaging such a model over noise realisations, assuming the field correlations are those of white noise. When $J = 0$ the system is composed of decoupled sites and the single-site matrix element of the density matrix, quantified by the expectation value of $\hat{\sigma}_i^+$, decays in time as $e^{-2\gamma t}$.

If we consider $J \neq 0$ a few simple considerations help understanding the main aspects of the physics of the many-body problem. We begin by focusing on the total magnetisation of the chain $\hat{S}^z = \sum_i \hat{\sigma}_i^z$; the adjoint Lindblad equation introduced in Sec. 3.4.2, that for Hermitian Lindblad operators reads: $\frac{d}{dt}\hat{S}^z = +i[\hat{H}, \hat{S}^z] - \frac{1}{2}\sum_i[\hat{L}_i, [\hat{L}_i, \hat{S}^z]]$, allows us to conclude that the total magnetisation of the chain is conserved during the dynamics because $[\hat{H}, \hat{S}^z] = 0$ and $[\hat{L}_i, \hat{S}^z] = 0$. On the other hand, the local magnetisation is not and spreads through the spin chain, its dynamics being constrained by the continuity equation $\frac{d}{dt}\hat{\sigma}_j^z = +i[\hat{H}, \hat{\sigma}_j^z]$. For closed systems, the local magnetisation evolves by spreading in a ballistic way. For open systems, dephasing enters through the equation of motion for the spin current: the end result is the emergence of a diffusive transport at long times [300, 302, 612].

As such one can expect a power law decay of the magnetisation at long times. This has been discussed extensively in the fermionic language. Indeed the Hamiltonian in Eq. (192) can be reduced via a Jordan-Wigner transformation to a quadratic fermionic Hamiltonian, which is non-interacting; however, because of the dephasing Lindblad operators, the Lindbladian is not quadratic. As discussed in Sec. 4.1, two-point fermionic observables can nonetheless be solved exactly; thanks to this property one can study for instance the dynamics of the local magnetisation without further approximations. We refer to Refs. [293, 295] for a detailed study in the more general case of the XY model and of several aspects of its dynamics, among which the ADR. The evolution of this model towards its steady state can be understood qualitatively since
<figure><img src="image_38.png" /><figcaption>Figure 35: Numerical study of the dissipative XXZ model in Eq. (193). (a) Algebraic decay of the inter-site coherences for several values of the couplings parameters $J_{xy}$ and $J_z$ obtained with MPS-based techniques; the authors fit a late-time decay as $t^{-1.58}$ independent of the parameters of the system. (b) Lindbladian gap obtained with exact-diagonalization techniques; in all parameter regimes considered, the authors fit a decay as $1/N_s^2$. From Ref. [590] [Copyright (2013) by the American Physical Society].</figcaption></figure>

the dynamics of diagonal matrix elements is expected to rapidly become effectively classical and diffusive [297–299]. This result can be also understood from the spectral properties of the associated Lindbladian. For a tight-binding chain the Lindbladian spectrum can be exactly diagonalised using Bethe-Ansatz techniques [303].

A different dynamical behaviour is expected for quantities which do not coincide with the conserved quantity, nor are a trivial function thereof, such as for example the coherences. If we consider the operator $\hat{A} = \sum_i (\hat{\sigma}_i^+ \hat{\sigma}_{i+1}^- + H.c.)$, it commutes with the Hamiltonian. Interestingly, one can prove by direct calculation that $\sum_i [\hat{L}_i, [\hat{L}_i, \hat{A}]] \propto \hat{A}$, which implies an exponential decay of its expectation value with a decay rate that does not depend on the chain length, $\langle \hat{A} \rangle \propto e^{-4\gamma t}$. If translational invariance of the initial state is assumed, a similar result holds for $\hat{\sigma}_i^+ \hat{\sigma}_{i+1}^-$. Several works have focused on the role of inter-site couplings in changing the dynamical behaviour of the system and giving rise to power-law decays in the coherences $\langle \hat{\sigma}_i^+ \hat{\sigma}_{i+1}^- \rangle$ [590, 613]; for this to occur, a minimum requirement is that there must be a vanishing gap of the Lindbladian in the thermodynamic limit, represented by a set of many Lindbladian eigenvalues corresponding to slow decay. This is well illustrated by the case of a XXZ spin chain with dephasing:

$$ \hat{H}_{XXZ} = \sum_i \left( \frac{J_{xy}}{2} (\hat{\sigma}_i^+ \hat{\sigma}_{i+1}^- + \hat{\sigma}_i^- \hat{\sigma}_{i+1}^+) + J_z \hat{\sigma}_i^z \hat{\sigma}_{i+1}^z \right), \quad \hat{L}_i = \sqrt{\gamma} \hat{\sigma}_i^z. \quad (193) $$

This model was studied via DMRG and exact-diagonalization techniques in Ref. [590], and
was found to display a power-law decay of the inter-site coherences as a function of time, and a Lindbladian gap vanishing as $\Delta \sim 1/N_s^2$, as reported in Fig. 35.

This result can be understood by writing down an effective Lindbladian in the strongly dissipative regime, $\gamma \gg J_{xy}, J_z$ and by splitting the Lindbladian into two parts:

$$ \hat{\mathcal{L}}_0[\hat{\rho}] = -i[J_z \sum_i \hat{\sigma}_i^z \hat{\sigma}_{i+1}^z, \hat{\rho}] + \sum_i \hat{D}[\hat{L}_i, \hat{\rho}], \quad (194) $$

$$ \hat{\mathcal{L}}_1[\hat{\rho}] = -i[(J_{xy}/2) \sum_i (\hat{\sigma}_i^+ \hat{\sigma}_{i+1}^- + \hat{\sigma}_i^- \hat{\sigma}_{i+1}^+), \hat{\rho}], \quad (195) $$

the idea is to treat $\hat{\mathcal{L}}_1$ as a perturbation. The steady state of $\hat{\mathcal{L}}_0$ is highly degenerate, with the degenerate steady states taking the form $\rho_{ss} = |\{\sigma\}\rangle\langle\{\sigma\}|$, with $|\{\sigma\}\rangle = \prod_i |\sigma_i\rangle$ and $|\sigma_i\rangle$ is an eigenstate of $\hat{\sigma}_i^z$. The perturbation $\hat{\mathcal{L}}_1$ has non-vanishing matrix elements between these terms. Treating $\hat{\mathcal{L}}_1$ at second order leads to an effective Lindbladian that takes the form [590]

$$ \hat{\mathcal{L}}_{\text{eff}}[|\{\sigma\}\rangle\langle\{\sigma\}|] = -\sum_{\{\tau\}} \langle\{\tau\}|\hat{\mathcal{K}}|\{\sigma\}\rangle \times |\{\tau\}\rangle\langle\{\tau\}|, \quad (196) $$

and lifts the degeneracy. The operator $\hat{\mathcal{K}}$ here is an effective Hamiltonian that describes a ferromagnetic Heisenberg model

$$ \hat{\mathcal{K}} = -\frac{J_{xy}^2}{\gamma} \sum_i \left( \frac{1}{2} (\hat{\sigma}_i^+ \hat{\sigma}_{i+1}^- + \hat{\sigma}_i^- \hat{\sigma}_{i+1}^+) + \hat{\sigma}_i^z \hat{\sigma}_{i+1}^z - \frac{1}{4} \right), \quad (197) $$

whose low-energy spectrum is gapless, with a gap closing as $1/N_s^2$ due to the fact that it is composed of magnons with wavevectors $k$, energy proportional to $k^2$, and minimal wavevector $k \propto N_s^{-1}$. This provides a simple theoretical framework to explain the numerical results reported in the bottom panel of Fig. 35. To the best of our knowledge, the superoperator $\hat{\mathcal{L}}_{\text{eff}}$ in Eq. (196) has never been used to theoretically compute the late-time decay in the top panel of Fig. 35 although a calculation of such decay exponent, equal to 3/2, is presented in Ref. [303] for the case $J_z = 0$.

Finally, we mention the study in Ref. [590] of the dissipative quantum Ising model in a transverse field coupled to dephasing noise; here the total magnetisation is not conserved by the dynamics and several of the previous considerations need to be modified accordingly. Here the on-site spin-flip coherence of the state, $\langle\hat{\sigma}_i^+\rangle$, decays exponentially in time, although with a decay rate that depends on the parameters of the model. We refer to the original article for further details.

### 7.1.2 Bose-Hubbard model under dephasing

Another well-studied example of interaction-induced anomalous dissipative dynamics is provided by the Bose-Hubbard model with dephasing that was introduced in Sec. 3.1.5. In one dimension, we consider the Hamiltonian introduced in Eq. (36) and the jump operator proportional to the local density operator:

$$ \hat{H} = -t_H \sum_{\langle i,j \rangle} (\hat{a}_i^\dagger \hat{a}_j + \text{H.c.}) + \frac{U}{2} \sum_i \hat{n}_i(\hat{n}_i - 1), \quad \hat{L}_{i\phi} = \sqrt{\gamma_{\phi}} \hat{n}_i. \quad (198) $$

Here we focus on the strongly-interacting regime and follow the discussion of Refs. [614,615]. In the absence of hopping, $t_H = 0$, any density matrix that is diagonal in Fock space is a steady state of the dynamics. A small hopping removes this degeneracy and when the initial state
<figure><img src="image_39.png" /><figcaption>Figure 36: Left: Dephasing dynamics and anomalous diffusion in the two-site Bose-Hubbard model: we see the dynamics of the diagonal matrix-elements $\rho_{n,n}$ of the density matrix showing long tails as time grows. The simulation is performed for $N = 80$ and the red line corresponds to $n = 40$, the situation where both sites are equally populated. The thin blue lines correspond to larger values of $n$, from 41 to 80. The dashed green lines correspond to the anomalous diffusion in Fock space of the equation (201). The inset shows a three-dimensional plot of the evolution. From Ref. [614] [Copyright (2012) by the American Physical Society.].</figcaption></figure>

has a well-defined number of bosons it drives the system towards a unique steady state: the completely-mixed state of all Fock states with total number of particles given by the initial condition. The long-time dynamics can be captured by treating the small off-diagonal matrix elements of the density matrix at second order in perturbation theory and writing down a rate equation for the diagonal ones in this basis.

It is instructive to follow Ref. [614] and to discuss the case $N_s = 2$ of the dissipative Bose-Hubbard model in Eq. (198): two coupled sites with dephasing and interaction. Since the total number of bosons, $N$, is fixed, we can define a basis of two-site states $\{|n\rangle\}$ labelled by the number $n = 0, 1, \dots, N$ of bosons in the left site (the number of bosons in the right site being $N-n$). We can then parametrise the density matrix as $\hat{\rho} = \sum_{n,m} \rho_{n,m} |n\rangle\langle m|$. Writing the equation of motion for the diagonal $\rho_{n,n}$ and off-diagonal $\rho_{n,n+1}$ matrix elements of the density matrix, and then eliminating the latter, one thus obtains:

$$ \begin{aligned} \frac{d}{dt} \rho_{n,n} &= \left( \frac{t_H^2 \gamma_\phi}{2N^2 U^2} \right) 2N^2 \left( W_{n+1} \rho_{n+1,n+1} - [W_n + W_{n+1}] \rho_{n,n} + W_n \rho_{n-1,n-1} \right) \\ &= \left( \frac{t_H^2 \gamma_\phi}{2N^2 U^2} \right) (2W_{n+1} \Delta \rho_n - 2W_n \Delta \rho_{n-1}), \end{aligned} \quad (199) $$

where $W_{n+1} = (n+1)(N-n)/(n-N/2+1/2)^2$ and $\Delta\rho_n = N^2(\rho_{n+1,n+1} - \rho_{n,n})$. Interestingly, all the dependence on the physical parameters enters only in the rescaled time $\tau = t/t^*$, with $t^* = 2N^2U^2/t_H^2\gamma_\phi$.

In the limit of large bosonic occupation, $N \gg 1$, one can map Eq. (199) into a diffusion equation for the probability distribution $p(s, \tau) = N\rho_{n,n}(\tau)$ with $s = n/N - 1/2 \in [-1/2, 1/2]$ that takes the form:

$$ \frac{\partial}{\partial \tau} p(s, \tau) = \frac{\partial}{\partial s} \left( D(s) \frac{\partial}{\partial s} p(s, \tau) \right). \quad (200) $$

The diffusion function reads $D(s) = \frac{1}{4s^2} - 1$ and thus features a strong spatial dependence
<figure><img src="image_40.png" /><figcaption>Figure 37: Spatial coherence of a one-dimensional many-body bosonic gas in an optical lattice and subject to dephasing due to induced spontaneous-emission processes. (Panels a,b,c) Snapshots of the momentum distribution function $\pi(k_x)$ as a function of time; the initial spatial coherence, reflected by a peaked distribution, is lost as times goes by. (Panels d,e) Dynamics of spatial coherence of the gas; $C_1(t)$ is well fitted by the solid blue line that crosses over from an initial exponential decay to a long-time algebraic decay. From Ref. [100].</figcaption></figure>

which diverges at $s = 0$, corresponding to the symmetric occupation of the two sites, and vanishes at the boundaries $s = \pm 1/2$. The slow diffusion at the boundaries corresponds to the slow rate of populating energetically-costly configurations. While continuing our discussion, it is important to keep in mind that the variable $s$ is not a true spatial variable, as it represents a sort of density of bosons in the left site. Importantly, one can find a scaling solution for this non-Brownian motion which, at short-times ($\tau \ll 1$), reads:

$$p(s, \tau) = \frac{\sqrt{2}}{\Gamma(1/4)} \frac{1}{\tau^{1/4}} \exp\left(-\frac{s^4}{4\tau}\right). \qquad (201)$$

This analytical solution shows that the system displays anomalous diffusion in Fock space and matches the numerics well, as shown in Fig. 36. Given this form of the local occupation probabilities, several physical consequences follow. In particular, one can show that the coherence between two neighbouring sites $C(t) = \langle \hat{b}_1^\dagger \hat{b}_2 + \text{H.c.} \rangle_t$ can be written as $C(t) \approx \frac{t_H}{U} C(t/t^*)$ with

$$C(\tau) = \int ds \frac{\partial}{\partial s} p(s, \tau) \frac{s^2 - 1/4}{s}. \qquad (202)$$

In the power-law regime, when $\tau \ll 1$ one obtains $C(\tau) \propto 1/\sqrt{\tau}$ for the coherence.

The anomalous decay of the coherence was shown to survive beyond the two-site case in the strong-interaction limit of the Bose-Hubbard model, using a time-dependent Gutzwiller Ansatz [615], similar to that discussed in Section 4.4. Such anomalous decay has been also observed in experiments with ultracold atoms under controlled dissipative processes [100,
616]. We follow Ref. [100], where the authors report the preparation of a nearly-ideal Bose-Einstein condensate of Yb atoms with broad spatial coherence loaded in a two-dimensional array of one-dimensional tubes with a superimposed optical lattice.

The gas is exposed to dissipation by shining a near-resonant laser, leading to a spontaneous emission from the excited state. The effect of such a drive can be modelled as a dephasing process and the loss of spatial coherence is associated to the fact that after spontaneous emission the atom recoils in a random direction. By measuring the dynamics of momentum distribution, and in particular the decay in time of the central peak and its broadening, one could track the decay in time of the coherence (see Fig. 37, top panels, for three snapshots at different times). The result shows a behaviour that is qualitatively different from the exponential decay expected for non-interacting atoms, and which is consistent with a power-law decay. With a procedure that we do not want to detail here, the authors extract from these data the coherence function $C_s(t) = \langle \sum_j \hat{b}_j^\dagger \hat{b}_{j+s} + \text{H.c.} \rangle_t$; the bottom panels of Fig. 37 shows the experimental data of $C_1(t)$ and $C_2(t)$. If we simply focus on the function $C_1(t)$, plotted with blue circles, the experimental data are well fitted by a function that crosses over from an initial exponential decay to a long-time algebraic decay. By adapting the reasoning proposed at the very beginning of this section on the heating of the XX model, for $U = 0$ the value of $C_1(t)$ is expected to decay exponentially in time. A long-time decay that is not exponential is the hallmark of correlations among the bosons created during the dissipative dynamics, and hence is a genuine many-body effect.

## 7.2 Depletion dynamics of bosonic gases under many-body losses

We now address the specific kind of open-system dynamics induced by atom losses. In contrast with what was discussed in Section 5.3.3, we will not focus here on the existence of intriguing dark states: our goal is to characterise dynamically how losses decrease the population of the gas, and the kind of nonequilibrium states through which the gas is driven during the process. This task is required in order to produce a quantitative description of the experiments which are always plagued by these effects (see Ref. [76] for an early study). Usually, these kinds of theoretical studies require the ability to fully treat the non-trivial interplay of the classical fluctuations induced by the loss events and the quantum fluctuations induced by the Hamiltonian dynamics; as such, understanding this physics has a broader relevance. For simplicity, we will focus on situations where the only stationary state is the vacuum and thus we discuss only the lossy dynamics of bosonic gases.

In general, the population of a bosonic gas subject to m-body losses evolves according to an equation that links the density to the local quantum correlations of the gas [617]. We consider for simplicity a gas on a lattice described by the Bose-Hubbard Hamiltonian $\hat{H}_{\text{BH}}$ in Eq. (36) and by the jump operators describing m-body losses,

$$ \hat{L}_{j,(m)} = \sqrt{\kappa^{(m)}} \hat{a}_j^m. \qquad (203) $$

One may compute the time evolution of the expectation value of the number operator $\hat{N} = \sum_j \hat{a}_j^\dagger \hat{a}_j$ using the adjoint equation as described in Sec. 3.4.2. This takes the form:

$$ \begin{aligned} \frac{d}{dt} \langle \hat{N} \rangle &= i \langle [\hat{H}_{\text{BH}}, \hat{N}] \rangle + \sum_j \left\langle \hat{L}_{j,(m)}^\dagger \hat{N} \hat{L}_{j,(m)} - \frac{1}{2} \{ \hat{N}, \hat{L}_{j,(m)}^\dagger \hat{L}_{j,(m)} \} \right\rangle \\ &= \frac{1}{2} \left\langle \sum_j \hat{L}_{j-(m)}^\dagger [\hat{N}, \hat{L}_{j-(m)}] + \text{H.c.} \right\rangle, \end{aligned} \qquad (204) $$

where we used the fact that the Hamiltonian conserves the number of particles. If we now consider a basis of the Hilbert space composed of states that have a definite number of bosons,
it is not difficult to show that $[\hat{N}, \hat{L}_{j,(m)}] = -m\hat{L}_{j,(m)}$ and thus:

$$ \frac{d}{dt} \langle \hat{N} \rangle = -\kappa^{(m)} m \sum_{j} \langle \hat{a}_{j}^{\dagger m} \hat{a}_{j}^{m} \rangle. \qquad (205) $$

If we assume spatial homogeneity and introduce the density $n(t) = \langle \hat{N} \rangle / N_s$ and the normalised zero-distance m-body correlation function $g_m(0, t)$, the equation reads:

$$ \frac{d}{dt} n(t) = -\kappa^{(m)} m g_m(0, t) n(t)^m, \quad \text{with} \quad g_m(0, t) = \frac{1}{N_s} \frac{\langle \sum_j \hat{a}_j^{\dagger m} \hat{a}_j^m \rangle}{n(t)^m} = \frac{\langle \hat{a}_0^{\dagger m} \hat{a}_0^m \rangle}{n(t)^m}. \qquad (206) $$

This equation clearly shows that the dynamics of the population contains important information on the correlations among the particles of the gas. For this reason, already in the early days of Bose-Einstein condensates, three-body loss rates have been employed as a sensitive probe of the statistical correlations between atoms [75]. More recently similar techniques have been employed in the first experiments on Efimov physics, where the appearance of the Borromean three-body bound state was signalled by an abrupt modification of the three-body loss rate [618].

In the remainder of this section, we will concentrate on two-body losses ($m = 2$), which arguably induce the simplest non-trivial loss dynamics. Before we focus on the $m = 2$ case, we make a few comments about two other heavily studied case, $m = 1$ and $m = 3$.

The case of one-body losses, $m = 1$, can be treated exactly as discussed in Sec. 4.1, as they correspond to a simple non-interacting limit. Furthermore, since for $m = 1$ it is always true that $g_1(0, t) = 1$, Eq. (206) does not depend on particle correlations, and instead shows a simple exponential decay (although, as discussed in Refs. [617,619], it can have non-trivial effects on the rapidity distribution of the integrable and one-dimensional Bose gas). We note however that there can be non-trivial behaviour with one-body losses. For example, several recent studies have considered the effect of one-body losses that do not act on the entire sample, but only on a part of it, e.g. a single site of an optical lattice, finding interesting non-trivial results [88, 620–639]; moreover, one-body losses have applications in the detection of topological invariants [640,641]. More recently, the study of non-local one-body loss processes in correlated bosonic gases has also proved to be the source of intriguing effects, such as critical behaviour [642]. There has also been significant interest in the study of three-body losses in quantum simulation and in characterising many-body states [75–77, 483, 643–649].

Two-body losses have been significantly studied both theoretically and experimentally. One point that contributes to their appeal is the fact that several aspects of this dynamics can be understood using a non-Hermitian Hamiltonian with complex interaction constant, describing at the same time elastic and inelastic interactions. Here below, we want to briefly describe the theoretical ideas and the experimental results that are employed in the two limiting regimes of weak and strong two-body losses. We conclude with a very concise list of other notable results presented in the literature.

### 7.2.1 Ideal Bose gas and weak two-body losses

We start by characterising the dynamics of a one-dimensional non-interacting Bose gas under weak two-body losses in a lattice of $N_s$ sites with periodic boundary conditions; our discussion here is based on Ref. [617] (see also the review in Ref. [650]). The Hamiltonian for this system can be written in momentum space as: $\hat{H} = -2t_H \sum_k \cos(k) \hat{a}_k^\dagger \hat{a}_k$ using standard Fourier-transform techniques. If we look at the theoretical approach that we are going to introduce from the viewpoint of the methods detailed in Sec. 4, the solution of the dynamics is based on an Ansatz that displays a mean-field factorisation in momentum space, as discussed in
Sec. 4.4.2; moreover, we will also see that this state is a bosonic Gaussian state and thus that it is particularly simple to treat since it satisfies the Wick's theorem. Rather than following this abstract approach, however, we are going to introduce the Ansatz for the density matrix from a more physical viewpoint.

It is well known that the properties of a non-interacting Bose gas can be defined by its momentum occupation function, $n_k = \langle \hat{a}_k^\dagger \hat{a}_k \rangle$. Under the Hamiltonian dynamics, the operators $\hat{a}_k^\dagger \hat{a}_k$ are constants of motion because they all commute with the Hamiltonian (and in fact, they all mutually commute); one may use them to construct a generalised Gibbs ensemble (GGE) state characterised by a set of Lagrange multipliers $\lambda_k$:

$$ \hat{\rho}_{\text{GGE}} = \prod_k \frac{e^{-\lambda_k} \hat{a}_k^\dagger \hat{a}_k}{Z_k}, \quad \text{with} \quad Z_k = \text{Tr}[e^{-\lambda_k} \hat{a}_k^\dagger \hat{a}_k]. \qquad (207) $$

The interest of this quantum state is that it can reproduce the local (in real-space) properties of a pure state that has evolved under a long unitary time-evolution with the non-interacting bosonic Hamiltonian; in order to do that it is simply necessary to tune the $\{\lambda_k\}$ so that $\text{Tr}[\hat{\rho}_{\text{GGE}} \hat{a}_k^\dagger \hat{a}_k]$ matches the value of the pure state under consideration [651, 652].

In the presence of losses, the operators $\hat{a}_k^\dagger \hat{a}_k$ are no longer conserved quantities of the dynamics. Nonetheless, when losses are weak (sometimes referred to in literature as *adiabatic losses*), one can introduce a *time-dependent* GGE that accounts for the time evolution of $\langle \hat{a}_k^\dagger \hat{a}_k \rangle$ by making the Lagrange multipliers depend on time [333, 653]:

$$ \hat{\rho}_{\text{tGGE}}(t) = \prod_k \frac{e^{-\lambda_k(t)} \hat{a}_k^\dagger \hat{a}_k}{Z_k(t)}, \quad \text{with} \quad Z_k(t) = \text{Tr}[e^{-\lambda_k(t)} \hat{a}_k^\dagger \hat{a}_k]. \qquad (208) $$

We will discuss the evolution of this Ansatz under the Lindblad master equation with the jump operators as defined in Eq. (203) for $m=2$.

The state $\hat{\rho}_{\text{tGGE}}(t)$ is Gaussian and once it is used as an Ansatz in the Lindblad master equation, it leads to simple mean-field dynamical equations for the momentum occupation functions and for the total density $n(t) = \sum_k n_k(t)$:

$$ \frac{d}{dt} n_k(t) = -2\kappa_2 n(t) n_k(t) \quad \Rightarrow \quad \frac{d}{dt} n(t) = -2\kappa_2 n(t)^2. \qquad (209) $$

By comparing the second equation of Eq. (209) to the expression in Eq. (206), one sees that this implies the gas maintains a zero-distance 2-body correlation function equal to one during the dynamics, $g_2(0, t) = 1$. The solution of the dynamics of the density of the gas is equally simple:

$$ n(t) = \frac{n(0)}{1 + 2\kappa_2 n(0)t}, \quad n_k(t) = \frac{n_k(0)}{1 + 2\kappa_2 n(0)t}. \qquad (210) $$

This shows a decay at late times as $1/t$. For a non-interacting gas, thus, weak losses do not appear to induce any peculiar quantum properties, although it is important to note that even when $n_k(0)$ corresponds to a Bose-Einstein distribution, $n_k(t)$ does not remain an equilibrium distribution.

### 7.2.2 Fermionised regime and quantum-Zeno effect

We now focus on the limit of strong two-body losses, following the theoretical discussion in Refs. [345, 528, 617, 619, 654]. We start from a simple observation that is valid when one neglects hopping between lattice sites: lattice sites with double or higher occupancies undergo some lossy dynamics, whereas lattice sites with at most one particle are stable. The Fock space
<figure><img src="image_41.png" /><figcaption>Figure 38: Strong two-body losses in a one-dimensional molecular bosonic gas. (Left) Dynamics of the molecular population; losses begin at $t = 0$. The dashed line shows the expectation of an uncorrelated system. The observed loss is much slower than the dashed line because of strong correlations. (Right) Loss dynamics in presence of an optical lattice. The strength of the optical lattice, and thus the strength of the bare loss rate increases from black to red to green. We observe a clear signature of the quantum Zeno effect: for strong rate effects, the life time of the gas increases instead of decreasing. From Ref. [86] [N. Syassen et al., Science 320, 1329 (2008)]. Reprinted with permission from AAAS.</figcaption></figure>

is thus the direct sum of two orthogonal subspaces, $\mathcal{H}_{\text{stable}}$ and $\mathcal{H}_{\text{loss}}$, whose different dynamical properties are best highlighted by the different action of the non-Hermitian Hamiltonian associated to the process:

$$ \hat{H}_{\text{nH}} = -t_H \sum_j [\hat{a}_j^\dagger \hat{a}_{j+1} + \text{H.c.}] + (U - i\frac{\gamma}{2}) \sum_j \hat{n}_j(\hat{n}_j - 1). \quad (211) $$

In the limit $t_H = 0$ of no hopping processes, $\mathcal{H}_{\text{stable}}$ is the kernel of the Hamiltonian, whereas $\mathcal{H}_{\text{loss}}$ is the complement.

We consider the effect on $\mathcal{H}_{\text{stable}}$ of a non-zero but small hopping rate, $0 < t_H \ll \gamma$. The hopping process can connect states with two singly-occupied neighbouring lattice sites with an unstable state where two bosons sit on the same site, a state that belongs to the dissipative space $\mathcal{H}_{\text{loss}}$. We thus conclude that also the states belonging to $\mathcal{H}_{\text{stable}}$ become unstable and a second-order perturbative calculation allows us to determine the strength of the effective complex interaction:

$$ \frac{t_H^2}{U + i \frac{\gamma}{2}} = \frac{t_H^2}{U^2 + \left(\frac{\gamma}{2}\right)^2} (U - i \frac{\gamma}{2}). \quad (212) $$

This simple analysis contains the whole essence of the typical timescales of strongly lossy gases: the effective loss rate $\Gamma = \frac{\gamma}{2} \frac{t_H^2}{U^2 + \left(\frac{\gamma}{2}\right)^2}$ scales as $1/\gamma$ and thus the lifetime of the gas increases as the loss-rate of the gas is increased. This counterintuitive result has been the object of several experimental analyses and Fig. 38 displays some of the main results obtained in the first reported experiment [86]. This perturbative analysis can also be performed also for the Lindbladian superoperator and not just for non-Hermitian Hamiltonians, we refer the reader to the original literature for more details [654].

Before continuing, it is interesting to briefly discuss the name “quantum Zeno effect” that is commonly employed to describe this phenomenology. The quantum Zeno effect was originally introduced to describe the fact that a frequently monitored unstable quantum system experiences a longer decay time than an unmonitored one [68] (see also Sec. 2.1). This is
<figure><img src="image_42.png" /><figcaption>Figure 39: (Left) Time evolution of the number of atoms in the strongly lossy regime according to the rate equations (214) for the initial state with one boson per site (dashed red line). The result is benchmarked with simulations based on quantum trajectories for $N_s = 10, 12$ and $14$ (each point is averaged over $10^4$ trajectories). The dot-dashed black line represents the solution $N(t)/N_s$ in Eq. (210) using $\Gamma$ instead of $\kappa_2$. The inset highlights the different long-time decay as $t^{-1}$ for the spatially-uncorrelated solution and as $t^{-1/2}$ for the rate equation. (Centre) Fermionic rapidity distribution and (right) bosonic quasi-momentum distributions. Dashed lines are the predictions using the rate equation. Data from quantum-trajectory simulations for $N_s = 14$ (symbols) are presented for two times. From Ref. [345] [Copyright (2021) by the American Physical Society].</figcaption></figure>

exactly what happens in the system described above, as long as the action of the environment is considered as an unread measurement. Strong two-body losses can be interpreted as an action of the environment that repeatedly checks whether in the system any lattice site is doubly occupied. By frequently performing this check, the environment effectively prevents two particles from occupying the same site and thus increases the lifetime of the state where all lattice sites have at most one particle [655].

We now show that the method of the GGE Ansatz—as employed above for the discussion of weak losses—can also be employed to solve the full dynamics of the system in the regime of strong losses. The crucial point is the observation that the system is characterised by an effective separation of time scales. In this section we have already assumed that $\gamma \gg t_H$. Inspection of Eq. (212) allows us to also write that $t_H \gg \Gamma$. It is important in the following to keep in mind these two inequalities, which state that after the fast loss of all doubly occupied sites, with a rate $\gamma$, the system is a weakly lossy gas of hard-core bosons, with a rate $\Gamma$.

Based on the above, the study of strong two-body losses is performed via an effective perturbative master equation that only acts on $\mathcal{H}_{stable}$. It is customary to say that a gas described by a state in $\mathcal{H}_{stable}$ is fermionised because it is composed of hard-core bosons and can only have at most a particle per site. Note however that the operators creating such excitations on different sites commute, so these are not true fermions. To avoid confusion, we use the Pauli matrices to represent the hard-core bosons. The Hamiltonian and the quantum jump operators of the effective master equation read:

$$ \hat{H} = -J \sum_j [\hat{\sigma}_j^+ \hat{\sigma}_{j+1}^- + \text{H.c.}], \quad \hat{L}_j = \sqrt{\Gamma} \hat{\sigma}_j^- (\hat{\sigma}_{j-1}^- + \hat{\sigma}_{j+1}^-). \quad (213) $$

In Fig. 39 we compare Eq. (210) with a numerical solution of the hardcore boson model, Eq. (213) obtained with state-of-the-art techniques based on quantum trajectories [269] for
sizes up to $L = 14$. These numerical simulations do not rely on physical approximations and serve here as a benchmark. We propose a first comparison with the solution of the simple equation that appears in the right side of Eq. (209), that is routinely employed to describe two-body losses: $\frac{d}{dt}n(t) = -2\Gamma n(t)^2$. This equation does not assume any spatial correlation of the gas and unsurprisingly the solution, dubbed “uncorrelated” in the figure, agrees with the numerics only at short times because the initial state is uncorrelated. Increasingly strong deviations appear at long times, indicating the build-up of correlations that the mean-field model fails to capture.

An alternate analytic approximation for the hard-core bosons can be found by using a Jordan-Wigner transformation [254] to map the hard-core bosons into fermions. The corresponding Hamiltonian can be diagonalised in momentum space and reads $\hat{H} = \sum_k -2J \cos(k) \hat{c}_k^\dagger \hat{c}_k$, where $\hat{c}_k^{(\dagger)}$ are fermionic operators obeying canonical anticommutation relations. Since the operators $\hat{c}_k^\dagger \hat{c}_k$ mutually commute and are constants of the motion under the Hamiltonian dynamics, we may again characterise the local properties of the lossy gas using a time-dependent GGE that is a fermionic Gaussian state which factorises in momentum space, $\hat{\rho}_{tGGE}(t) = \prod_k \frac{1}{Z_k(t)} e^{-\lambda_k(t) \hat{c}_k^\dagger \hat{c}_k}$, with $Z_k(t) = \text{Tr}[e^{-\lambda_k(t) \hat{c}_k^\dagger \hat{c}_k}]$. Writing the Lindblad master equation in the fermionic formulation we obtain, after some algebra, the following rate equations for the expectation values $n_k^{(c)}(t) = \text{Tr}[\rho_{tGGE}(t) \hat{c}_k^\dagger \hat{c}_k]$. Note that this quantity is the fermionic occupation number, and is also referred to as the rapidity distribution of the gas:

$$ \frac{d}{dt} n_k^{(c)}(t) = -\frac{4\Gamma}{L} \sum_q [\sin(k) - \sin(q)]^2 n_q^{(c)}(t) n_k^{(c)}(t). \quad (214) $$

These equations are easily solved numerically and the solution, dubbed “rate equation”, is plotted in Fig. 39, where we observe an excellent agreement between the predictions of the rate equation and the numerical simulations for all times shown. Unlike the spatially-uncorrelated solution, which predicts the scaling $\sim t^{-1}$ at long times, the rate equations (214) predict that $n(t)$ decays to zero as $t^{-1/2}$. This result is highlighted in the inset of Fig. 39 and can be analytically proven. This algebraic decay is the hallmark of the spatial correlations that build up after dissipation is enabled.

In addition to considering the total population, Fig. 39(centre) also shows the time evolution of the fermionic population (rapidity distribution) $n_k^{(c)}(t)$. In the long-time limit $t > \Gamma^{-1}$ this is well-approximated by

$$ n_k^{(c)}(t) \approx \frac{1}{(8\pi\Gamma t)^{1/4}} e^{-\sin^2(k)\left(\frac{\Gamma t}{\pi}\right)^{1/2}}. \quad (215) $$

Although at initial times the population is uniformly spread among the different momenta, a double-peaked distribution emerges for long times, with maxima at $k = 0, \pi$. The interplay between two-body losses and coherent free-fermion dynamics has thus created a nonequilibrium fermionic gas where the notion of Fermi sea is completely lost. Standard time-of-flight measurements give instead access to the bosonic momentum distribution function $n_k(t) = \langle \hat{a}_k^\dagger \hat{a}_k \rangle_t$, discussed above. The link between $n_k^{(c)}(t)$ and $n_k(t)$ is known explicitly but is complicated so not written here. The result of the calculation is shown in Fig. 39(right).

At this point instead it is interesting to observe the fact that these results are obtained using a density matrix that is of mean-field form in the sense that it displays momentum-space factorisation. However, differently from what done in the discussion of the non-interacting Bose gas, the density matrix is not a bosonic Gaussian state; it is a fermionic Gaussian state, but this property does not trivially transfer to the bosonic language. As an example, this means that the bosonic correlation function $g_2(0, t)$ is different from zero and thus a non-trivial dynamics can take place.
### 7.2.3 Related results

The study of the dynamics of lossy gases has been the object of several other publications that have addressed aspects of the problem that we did not discuss here. For example, the effect of an harmonic confinement [619, 656, 657], the nonequilibrium properties of the gas [658] or the effect of spin in the fermionic case [460, 659, 660]. Recently, the same problem has been addressed with Keldysh field theory, obtaining similar results [661, 662], which have raised interesting questions on the universal properties of the gas decay discussed also in Refs. [319]. For superfluid fermionic gases with two-body losses the dynamics of particle density depletion, together with the destruction of the superfluid order parameter, have been studied in Ref. [245].

The theoretical results presented above are based on the notion of time-dependent GGE, that is possible when an open- or closed-system dynamical effect perturbs a closed system with conserved quantities. A general discussion of this theory and of some applications beyond lossy gases can be found in Refs. [333, 653, 663–666]. As a notable example, we mention here the recent development of the quantum theory of reaction-diffusion processes, that employs exactly the same formalism [667, 668]. There have also been works studying random Lindbladians, and the observation of phase transitions in the relaxation dynamics between slow and fast relaxation [669].

## 8 Monitored quantum systems

In this last section we return to the description of open quantum systems introduced in Sec. 3.3, based on the *unravelling* the Lindblad master equation into stochastic quantum trajectories [255–258]. As discussed in that section, the physical interpretation of the resulting stochastic dynamics is that of a system evolving under continuous monitoring of certain observables. The quantum trajectory produced this way represents the evolution of the state conditioned on a given measurement record. Although such ideas are well established in the field of few body quantum optics [256, 258, 259, 268, 271], these tools have recently attracted much attention in the context of open many-body systems [269] as well as in hybrid quantum circuits [670] (that is, circuits with both unitary gates and measurement processes). In the context of these Lectures Notes, we will focus only on monitored quantum many-body systems which are described by a continuous-time Lindblad master equation, or the corresponding unravelling. The particular setting we will consider in this section is a system, described by a given Hamiltonian $H$, which is coupled on each lattice site to a monitored environment, as illustrated in Fig. 40. As such, we will not consider the role of projective measurements and of random unitary circuits, for which other reviews exist [670–672].

### 8.1 Trajectory-resolved averaging

A key point in the field of monitored quantum systems is the idea that there may be new physics contained in the quantum trajectories, i.e. in the conditional state, that is not captured by the average state. To appreciate this point it is helpful to consider a specific choice of unravelling from among those introduced in Sec. 3.3, such as Quantum State Diffusion. (Similar considerations can be drawn for other unravellings). As we have mentioned, given the evolution of
<figure><img src="image_43.png" /><figcaption>Figure 40: Cartoon of the monitored Quantum Ising Chain. (a) Quantum state diffusion protocol: on each lattice site a two-level system (Ising spin) interacts with its neighbours and is subject to a weak continuous measurement of the up-polarised state. (b) Quantum jump protocol: the spins in the chain interact through a non-Hermitian Hamiltonian. Occasionally a quantum jump take place, projecting the measured degree of freedom in the up-polarised state. The no-click limit corresponds to post-selecting only trajectories without jumps. Adapted from Ref. [673].</figcaption></figure>

the conditional state:

$$d|\psi(\xi_t, t)\rangle = \left[ -i\hat{H} - \frac{1}{2} \sum_{\mu} (\hat{L}_{\mu}^{\dagger} \hat{L}_{\mu} + \langle \hat{L}_{\mu}^{\dagger} \rangle_{\xi_t, t} \langle \hat{L}_{\mu} \rangle_{\xi_t, t} - 2 \langle \hat{L}_{\mu}^{\dagger} \rangle_{\xi_t, t} \hat{L}_{\mu}) \right] dt |\psi(\xi_t, t)\rangle + \sum_{\mu} (\hat{L}_{\mu} - \langle \hat{L}_{\mu} \rangle_{\xi_t, t}) d\xi_t^{\mu} |\psi(\xi_t, t)\rangle, \quad (216)$$

one can show that by averaging over the noise, the conditional density matrix $\hat{\rho}(\xi_t, t) = |\psi(\xi_t)\rangle\langle\psi(\xi_t)|$ gives back an average state evolving with a Lindblad master equation, with jump operators given by the monitored operator $\hat{L}_{\mu}$.

As a consequence of the linearity of the average over the noise it is sufficient to know the mean density matrix $\hat{\rho}(\xi_t, t)$ to compute all observables which are linear in the state $O[\hat{\rho}] = \hat{O}\hat{\rho}$. Specifically, for any function $f(\hat{O})$, we have

$$\overline{\mathrm{Tr}(\hat{\rho}(\xi_t, t)f(\hat{O}))} \equiv \mathrm{Tr}(f(\hat{O})\hat{\rho}(t)). \quad (217)$$

This is no longer the case if one is interested in the noise-average of quantities which depend non-linearly on the conditional density matrix [674]. A simple example is the purity $P(\hat{\rho}) = \mathrm{Tr}\hat{\rho}^2$: for the conditional state $P(\hat{\rho}(\xi_t, t)) = 1$ (and hence its average over trajectories $P(\hat{\rho}(\xi_t, t)) = 1$), while for the mean state we have $P(\hat{\rho}(t)) < 1$. Another particularly relevant example is provided by the entanglement entropy. Significant work and attention has been devoted to understanding the dynamics of this quantity under continuous monitoring. Given a partition of the system into two parts, $A \cup B$, the (conditional) reduced density matrix $\hat{\rho}_A(\xi_t) = \mathrm{Tr}_B \hat{\rho}(\xi_t)$ encodes the bipartite entanglement, via the entanglement entropy:

$$S(\xi_t) = -\mathrm{Tr}_A[\hat{\rho}_A(\xi_t) \ln(\hat{\rho}_A(\xi_t))]. \quad (218)$$

If the overall state is pure this is a well defined measure of entanglement, which yields the amount of Bell pairs that can be distilled from the quantum state. On the other hand for a mixed state, such as the averaged state $\hat{\rho}(\xi_t, t)$, entanglement with the environment also enters into the entropy of the subsystem density matrix. Since the unravelled system density matrix corresponds to a pure state, there is however no ambiguity about bipartite entanglement vs entanglement with the environment. After evolving the state under Eq. (216), one
<figure><img src="image_44.png" /><figcaption>Figure 41: Sketch of different entanglement entropy scalings with system size in monitored quantum systems undergoing a Measurement-Induced Phase Transition.</figcaption></figure>

can compute the average entropy given by

$$ \bar{s} = \int \mathcal{D}\xi_t P(\xi_t)S(\xi_t), \qquad (219) $$

which is clearly a non-linear functional of the conditional state and therefore can be sensitive to physics not captured by the average state.

In general, similar issues of trajectory-resolved averaging arise when one is interested in more subtle statistical correlation functions of the stochastic process, for example in the so-called overlaps [119] $\langle O_\mu(t) \rangle \langle O_\nu(t) \rangle = \text{Tr}(\hat{O}_\mu \hat{\rho}(\xi_t, t))\text{Tr}(\hat{O}_\nu \hat{\rho}(\xi_t, t))$. We note that these questions are closely related to the difference between annealed and quenched averages in disordered systems [675].

## 8.2 Measurement-induced phase transitions

Based on the previous discussion we can expect that, as a function of the monitoring rate, the entanglement entropy of a monitored system might display non trivial behaviour which one would not see in the averaged dynamics (see Fig. 41). This statement can be understood intuitively: one can expect that while the unitary dynamics creates entanglement the effect of monitoring or measurements tends to project the system into an eigenstate of the (site-local) measurement operator, thus reducing the entanglement. Therefore one can quite generically expect a crossover or phase transition in the entanglement content. The prediction of a phase transition was put forward in two theoretical works that focused on random unitary circuits with projective measurements [676,677] leading to the concept of Measurement-Induced Phase Transitions (MIPT), separating a volume-law entangling phase from an area-law disentangling phase. The latter can be interpreted as arising from a many-body Zeno effect, where quantum measurements prevent the spreading of quantum information. Such MIPT have been discussed in the context of purification transitions for mixed states [678,679]: for weak monitoring an initial mixed state purifies in a time that diverges with system size, typically exponentially for chaotic dynamics, while for large monitoring the environment succeeds in locally purifying the system at a constant rate, independent of system size. This robust mixed phase, which corresponds to the volume-law entanglement phase for the pure state protocol discussed above, can be also interpreted from a quantum information perspective as an error correcting phase, where unitary dynamics protects quantum information from external measurements and errors.
<figure><img src="image_45.png" /><figcaption>Figure 42: Interplay between measurement-induced phase transitions and symmetry-breaking dissipative phase transitions.</figcaption></figure>

It was later realised that MIPT are not restricted to random circuits with projective measurements but can generically occur in some classes of open quantum many-body systems under suitable unravelling. One such example is non-interacting systems under continuous monitoring of the particle density, which have been extensively studied both numerically [673, 680–688] —taking advantage of the Gaussianity of the state—and analytically—by developing a replica field theory approach [689–691]. Despite the non-interacting nature of the problem the phenomenology is extremely rich and still under active investigation. Monitored one-dimensional free fermions with conserved particle number for example are believed to display a crossover from a logarithmic entanglement phase to an area law [680, 690, 692]. When the symmetry of the model is reduced to a discrete symmetry, such as for example in the monitored Ising chain [673] or for random Majorana fermions, the transition is believed to remain sharp [691]. Sharp entanglement transitions are also known to occur in the fully post-selected case of no-click evolution [693–696] and the role of quantum jumps and deviations from full post-selection on these transitions has been recently discussed [697, 698]. Finally, the dependence of the entanglement dynamics and the associated MIPT on the specific unravelling scheme has been also discussed [405, 673, 680, 685, 699], as well as the role of the monitored operator, in particular in the case of particle losses [700–703].

While the vast majority of theoretical and numerical works have focused on non-interacting, Gaussian, monitored systems there are few works that have discussed the interacting case [704–709] with projective or weak measurements. Here of course, due to the genuine many-body nature of the problem, numerical simulations are challenging and so only much smaller systems can be accessed. As such, understanding of such models is still at its infancy.

Finally, although MIPT are examples of entanglement phase transitions, they are not however limited to the entanglement. Indeed as discussed in the previous section there are other quantities that can be sensitive to the conditional state. Among these we mention connected correlation functions [680], as well as histograms of local observables and full counting statistics [682, 710, 711].

## 8.3 Post-selection problem

As we have discussed so far, monitored quantum systems allow one to explore the effect of quantum measurements on the system dynamics. There is however a fundamental challenge associated to achieving this goal in a genuine many-body context, which is rather fundamental in nature and referred to as the “post-selection problem”. This problem arises as follows: in order to perform averages of observables along a given trajectory experimentally one should be able to reproduce the same sequence of measurements outcomes. However, the number of possible measurement records grow exponentially with both system size and time. As such the probability of acquiring the same measurement record multiple times is very low. For this
reason, experimental results have so far been limited to a few sites, and considerable efforts
were required to increase the lattice length.

From a theoretical point of view it is important to find robust mitigation techniques for the post-selection problem. Different strategies have been suggested and explored experimentally, including using space-time duality [712, 713] as has been experimentally achieved in [714] with the Google processor. A different strategy has proposed to use adaptive quantum circuits with measurements and feedback [715,716], i. e., conditioning the evolution on the measurement record to make the MIPT visible at the average (density matrix) level. However feedback has been shown to alter crucially the dynamics of the system and to lead to a different type of phase transition as compared to the original MIPT [717–720].

Finally, a recent theoretical proposal suggested an implementation of MIPT using atomic ensembles of laser-driven atoms and collective dissipation [721]. Here the slow disentangling dynamics of collective jump operators leads to a saturation time for the entanglement entropy growing only as $\tau \sim \log(N)$, with N the number of atoms, leading to a polynomial overhead for post-selection (quantum state diffusion for this model has been previously studied in [722]).

The post-selection problem remains a formidable hurdle to overcome, and the search for cases where it can be mitigated is necessary for experimental progress in monitoring quantum many-body systems.

## 8.4 Experimental evidence for MIPT

Despite the challenges associated to the post-selection problem, there has been recent progress on the experimental side to detect signatures of MIPT on small quantum devices.

The first quantum simulation experiment with trapped ions [69] used the idea of probing the transition via a reference qubit, in the spirit of a purification transition [679, 723], as discussed earlier. Specifically the experimental set-up comprised of a chain of trapped ion qubits, used as the system, reference qubit and measurement ancilla qubits. After initialisation, the reference qubit is entangled with a randomly selected system qubit and then undergoes unitary evolution interspersed with projective measurement. Measurements were performed at the end of the circuit, using some ancilla qubits. Postselection on the reference qubit is performed in order to obtain the value of the reference qubit Pauli operators, conditioned on the measurement outcomes. Finally, the reference qubit density matrix is reconstructed and from this the entropy can be obtained, revealing the purification transition. Subsequent works followed by the IBM and Google teams, exploring MIPT using superconducting qubits. In a first experiment explicit post-selection was performed on a small superconducting quantum processor characterised by random unitary evolution and measurements. Differently from the trapped ion experiment, here the measurements were performed mid-circuit, on a fast time-scale and with high-fidelity. The density matrix was reconstructed via tomography and the Renyi entropy estimated, showing a transition from volume to area law scaling [724]. Finally, the experimental realisation from the Google group took advantage of space-time dualities [714] to avoid mid-circuit measurements and thus probe the MIPT for a chain of 70 qubits. A key step of this implementation was the use of classical post-processing of the measurement record to characterise the structure of the phases. These pioneering experiments represent a first step to explore MIPT but it is clear that in the future more experimental evidence is needed, particularly in the regime of large number of qubits.

# 9 Summary and outlook

The aim of these Lectures Notes was to give, to the best of our abilities, a self-contained introduction to the properties of open many-body synthetic systems as realised in the experimental
platforms where the coupling to an environment can be tuned at will, and that could be also
viewed as open-system quantum simulators. As briefly summarised in Section 2, such plat-
forms range from cold atoms to trapped ions, from cavity arrays in superconducting networks
to BECs in cavities. A common feature of all these systems is that all of them are well described
by the assumption that the coupling to the environment is Markovian, i.e. memoryless: in Sec-
tion 3 we described the main theoretical and conceptual frameworks in which such dynamics
can be formulated. After introducing the main analytical and numerical theoretical tools to
study the many-body Lindblad master equation in Section 4, we dealt with several conse-
quences of the above-mentioned interplay between Hamiltonian and dissipative contributions
on the evolution of the systems.

In general, both the steady state and the approach to it are the result of the interplay
between the unitary evolution—dictated by the system Hamiltonian—and the sources of dis-
sipation and decoherence, such as photon and atom losses, or local dephasing processes. Con-
centrating first on the steady state, in Section 5 we showed that appropriately designed baths
and Hamiltonians can represent powerful ways to engineer many-body states: this is a well-
known technique in quantum optics that was more recently extended to many-body systems.
The realisation of many-body dark states may however require some fine tuning of coupling
constants. By tuning the relative ratios of the parameters of the setup, the steady-state changes
and the system can enter different phases separated by transition points or lines: these are the
dissipative phase transitions that we analysed in Section 6. In contrast to thermal or quantum
phase transitions, dissipative phase transition are of nonequilibrium nature, so that the sta-
bilisation of different type of ordering is not based on (free-)energy arguments: new phases,
not allowed at equilibrium, are possible, and the critical behaviour may be governed by new
critical exponents. A rich, interesting phenomenology appears also in the approach to the
steady-state where the relaxation dynamics can be dictated by cooperative effects, as we dis-
cussed in Section 7; this is the case, for example, of the depletion dynamics of interacting
Fermi/Bose particles subject to loss events. Finally, in Section 8 we focused on the striking
phenomenology that can appear when it is possible to follow the stochastic dynamics along
quantum trajectories: measurement-induced phase transitions can then occur that are not
detectable via any averaged physical observable.

There are several very important aspects of the physics of open quantum many-body sys-
tems that we did not cover in these lecture notes. The general perimeter of this work has
been presented in the introduction and in each specific section we have made reference to
subjects and works that were not explicitly discussed, either for pedagogical reasons or simply
for keeping the discussion limited in space. Instead of re-listing them here, we prefer to men-
tion some topics that have not been discussed and that are likely to flourish in the next years;
the following list reflects our personal perspective and is not meant to be a thorough review
of these areas.

i) Non-Markovian dynamics. While there is a huge literature on this theory and its applica-
tion to a few qubits or few-level systems [11,725], less is known in the many-body case;
the development of experimental platforms that we are witnessing will surely motivates
further studies in this direction.

ii) Interplay between boundary and bulk dissipation. In these lecture notes we considered the case of dissipative terms in the bulk, whereas an extensive discussion of the role of boundary dissipative baths can be found in [726]. The simultaneous presence of boundary terms in driven-dissipative systems can be a very interesting avenue for exploring how quantum transport could take place in a non-equilibrium phase, or is affected by a dissipative phase transition.
iii) Adiabatic dynamics and the Kibble-Zurek mechanism in open systems. An early intro-
duction to the topic in the context of closed quantum systems can be found in [727].
Whereas the discussion of adiabatic dynamics in open systems has significantly devel-
oped in the context of few-body physics, the extension to Markovian many-body systems
looks like a natural one.

iv) Time-dependent Lindbladians. This topic embraces several different, very interesting, situations. These include the case of dissipative Floquet dynamics (see for example Ref. [728]), the whole topic of quantum control [729, 730] and related work on shortcuts to adiabaticity [731] in many-body open systems. A recent example of this topic, based on the variational approach of [732] can be found in Ref. [733].

v) Integrability and chaos in open quantum systems. Although the topics have already developed and the community has produced some first significant results, the number of studies devoted to this subject is small with respect to those addressing closed systems. The question of what it means to have an integrable Lindbladian, and what are the consequences for physical properties, is still beginning to be understood. Additional references can be found, for example in Ref. [317, 322] and the references cited in the bibliography. Similarly, the question of how to define and characterise quantum chaos in open quantum systems has attracted attention from a broad community in recent years [734–738], with many questions still open to be explored.

vi) Topological physics. The subject of topology has been discussed in these lecture notes in only a very tangential way; yet, this is a very important and rapidly evolving research topic. Excellent reviews exist, see for example [27, 522] where topological classification of non-Hermitian systems is discussed in depth. The topological properties of systems governed by a Lindblad dynamics are less explored and additional work is required. More information on this rapidly growing area of activities can be found in [499, 506, 511, 739] and the references therein.

vii) Adaptive protocols, feedback and active quantum matter. In Sec. 8 we discussed moni-
tored quantum systems whose evolution is conditioned to the measurement outcomes.
A novel frontier which has started to be explored is to actively use the measurement
outcomes to modify or correct the subsequent evolution, thus performing adaptive dy-
namics and feedback [740–743]. On the one hand this could allow to characterise
measurement-induced transitions at the level of the average state, a task which remains
non-trivial and requires appropriate entanglement measures/witnesses [744]. In addi-
tion, this could open the way to completely new dynamical behaviour, which generalise
to the quantum setting the physics explored in classical active and non-reciprocal mat-
ter [642, 745–748].

viii) Frustration and disorder. Very few works address the possible formation of glassy behaviour in driven-dissipative systems. Recent contributions in this direction include [118-120, 749, 750].

To conclude, we would like to mention two aspects of the entire field that may be of interest for future studies. The choice, of course, is biased by our own interests and does not represent by any means an objective priority list.

In the last section our discussion on monitored systems focusing on measurement induced phase transitions reflects the current status of the interests on this questions. The number of questions that may arise is nevertheless much wider and concerns the whole area of cor- related/collective dynamics of quantum jumps. For example, there are connections between
properties that are only visible through individual trajectories of a monitored quantum system, and properties revealed by the distribution of replicas in spin glasses [119]. Further exploration of correlated dynamics in monitored quantum systems is an open research area that may lead to unexpected results but that however requires to find strategies to monitor many-body systems and to find efficient experimental techniques for revealing entanglement phase transitions without suffering from the burden of performing a postselection, a problem that still waits for a solution.

As a second and final point, we would like to mention the interest of exploring how open many-body systems can be of relevance for quantum technologies, for example to realise quantum memories [751]. Quantum simulators are at the heart of the current discussion on the subject, promising the possibility of addressing specific problems that are out of reach for classical computers. The same may be true for open-quantum system simulators, which could be employed to study different problems.

However, this should not limit the field of possible applications to quantum technologies: other applications such as quantum thermodynamics could be envisioned [25, 752] or collective effects in sensing and metrology are also a promising direction.

## Acknowledgments

We would like to thank all the people with whom we interacted and collaborated on the topics covered in these lecture notes. We had many interesting and enlightening discussions with K. B. Arnardóttir, A. Biella, I. Bouchoule, M. Capone, I. Carusotto, G. Chiriacò, J. I. Cirac, C. Ciuti, A. A. Clerk, A. J. Daley, M. Dalmonte, A. Delmonte, A. De Luca, J. De Nardis, S. Diehl, J. Dubail, P. Fowler-Wright, G. Fux, D. Gerace, F. Gerbier, V. Giovannetti, S. Gopalakrishnan, L. Gotta, M. Hartman, F. Iemini, A. Imamoglu, A. Insall, J. Jin, C. Joshi, H. Katsura, P. Kirton, H. Landa, Z. Lenarcic, A. Lerose, I. Lesanovsky, B. L. Lev, Y. Le Gal, Z. Li, P. B. Littlewood, B. W. Lovett, P. Lucignano, M. Maghrebi, A. Marché, J. Marino, B. Marsh, F. Minganti, G. Misguich, A. Nardin, F. B. F. Nissen, G. Pagano, G. Passarelli, D. Poletti, D. Rossini, L. Rosso, A. Russomanno, O. Scarlatella, M. Seclì, K. Seetharam, S. Sharma, P. Sierant, R. D. Soares, F. Surace, M. H. Szymańska, A. Tomadin, H. E. Türeci, X. Turkeshi, M. Vanhoecke, O. Viyuela, B. Xing, H. Yoshida and P. Zoller. We are particularly grateful for helpful comments on earlier versions of these lecture notes from A. Biella, I. Carusotto, F. Gerbier, F. Minganti, G. Pagano, T. Tomita and O. Scarlatella.

**Funding information.** R.F. is supported by the European Research Council via the grant agreements RAVE, 101053159 and the Italian PNRR MUR project PE0000023-NQSTI (R.F.). L.M. acknowledges support from the ANR project LOQUST ANR-23-CE47-0006-02; this work is also part of HQI (www.hqi.fr) initiative and is supported by France 2030 under the French National Research Agency grant number ANR-22-PNCQ-0002. M.S. acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant agreement No. 101002955 - CONQUER). Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. This research was supported in part by (i) the International Centre for Theoretical Sciences (ICTS) for R.F. and J.K. participating in the program "Periodically and quasi-periodically driven complex systems" (code: ICTS/pdcs2023/6); and by (ii) the Institut Pascal of Université Paris-Saclay in the framework of the program "Open quantum many-body physics 2023" to which R.F., L.M. and M.S. participated.
# A Derivation of Lindblad master equation

In this Appendix we briefly recapitulate some basic notions related to deriving the Lindblad equation. The discussion contained here is not intended to offer a complete and rigorous derivation of the Lindblad equation or a comprehensive discussion of its properties. Our main goal is to give a quick introduction to few concepts that underlie the discussion throughout the lecture notes. For a complete discussion, the reader is referred to the references [190, 191, 753, 754] where the topic is treated in greater detail. We will first consider a formal derivation, based on the properties of the density matrix. In the second part we briefly outline how this equation is recovered in a system and environment dynamics assuming a weak coupling approximation.

## A.1 From quantum channels to the Lindblad equation

A remarkable fact of the Lindblad equation is that its form can be derived by only requiring that during the dynamical evolution the density matrix retains all its properties (i.e. remains Hermitian, unit trace, semi-positive definite) and that the dynamics is Markovian. Note that this universal character is lost in the case of non-Markovian dynamics for which no general dynamical equation is known (or believed to exist).

Any transformation that maps density matrices into density matrices can be expressed in the Kraus form [190, 191]

$$ \hat{\rho}' = \sum_{\mu} \hat{K}_{\mu} \hat{\rho} \hat{K}_{\mu}^{\dagger}, \quad \text{with} \quad \sum_{\mu} \hat{K}_{\mu}^{\dagger} \hat{K}_{\mu} = \mathbb{1}. \qquad (A.1) $$

It is easy to verify that if $\hat{\rho}$ obeys all the properties of a density matrix then so will $\hat{\rho}'$. To derive the Lindblad equation, one can consider Eq. (A.1) in the case of an infinitesimal transformation, linking the state of the system at time $t$ to that at time $t+dt$. For such a transformation one may easily see that one of the Kraus operators, say $\hat{K}_0$, must be the identity up to corrections of the order of $dt$ while all the other Kraus operators must scale as $\sqrt{dt}$:

$$ \begin{aligned} \hat{K}_0 &= \mathbb{1} + (-i\hat{H} + \hat{K})dt, \\ \hat{K}_\mu &= \hat{L}_\mu \sqrt{dt}, \quad \mu \neq 0. \end{aligned} \qquad (A.2) $$

The decomposition in $\hat{K}_0$ of the operator proportional to $dt$ into its real and imaginary parts immediately suggests that $\hat{H}$ can be identified with the (effective) Hamiltonian of the system and $\hat{K}$ should satisfy $\hat{K} = -(1/2) \sum_{\mu \neq 0} \hat{L}_{\mu}^{\dagger} \hat{L}_{\mu}$. Substituting the expressions of Eq. (A.2) in Eq. (A.1) the Lindblad equation follows

$$ \frac{d\hat{\rho}}{dt} = -i[\hat{H}, \hat{\rho}] + \sum_{\mu \neq 0} \left( \hat{L}_{\mu} \hat{\rho} \hat{L}_{\mu}^{\dagger} - \frac{1}{2} \{ \hat{L}_{\mu}^{\dagger} \hat{L}_{\mu}, \hat{\rho} \} \right). \qquad (A.3) $$

The number of allowed operators $\hat{L}_{\mu}$ extends up to the square of the dimension of the Hilbert space. For a many body system, this will thus scale exponentially with the system size and so is potentially very large; however, in many cases of relevance for these lecture notes, the jump operators are local and so the number of terms in the sum scales linearly with the system size rather than exponentially.

The operators $\hat{H}$ and $\hat{L}_{\mu}$ have a fundamental physical meaning: the first is the Hamiltonian and the second accounts for the action of the environment on the system. As described extensively in the lecture notes, the choice of the jump operators usually follows from the derivation of the reduced dynamics from global models of system and environment; however, in many situations this can be done on phenomenological grounds by considering relevant processes
associated to the action of the environment on the systems or by approximate methods. In the next subsection we briefly recap a very popular and effective approach based on a weak-coupling expansion in the system-environment interaction. One should always bear in mind that the derivation from microscopic principles of appropriate dynamical equations is a highly non-trivial tasks and may require sophisticated methods.

Before concluding, let us remark that depending on the choice of the jump operators, the steady state may or may not describe an equilibrium state. If one wants a Lindblad equation that reaches the thermal equilibrium state at long times, this will generally require non-local jump operators. In this case the steady state is diagonal in the eigenstates of the Hamiltonian and the jump operators describe transitions between the eigenstates with rates obeying detailed balance. Since these eigenstates are often delocalised, the corresponding operators must also be.

## A.2 Born–Markov master equation

The derivation of the Lindblad dynamics for Markovian open quantum systems that we presented in the previous section leaves open important questions on how to determine the form of the jump operators from first principles; several approaches are discussed in Refs. [191, 753, 754]. In this Appendix we summarise a simple method—often referred to as the Bloch–Redfield approach—valid in the weak-coupling regime, that is however illuminating in understanding several features of quantum open system dynamics.

The starting point is to assume that the system and its environment evolve globally in a unitary fashion governed by the Hamiltonian

$$ \hat{H} = \hat{H}_S + \hat{H}_E + \hat{H}_{S-E}, \qquad (A.4) $$

where $\hat{H}_S$ is the Hamiltonian of the system, $\hat{H}_E$ that of the environment and $\hat{H}_{S-E}$ their interaction. The interaction can be chosen, for simplicity, as given by the product of an operator of the system $\hat{Y}_S$ with one of the environment $\hat{X}_E$, and so $\hat{H}_{S-E} = \hat{Y}_S \hat{X}_E^\dagger + \text{H.c.}$ The evolution of the state of the “universe”, $\hat{\rho}_U$, is unitary and follows $\frac{d}{dt}\hat{\rho}_U = -i[\hat{H}, \hat{\rho}_U]$. The goal is to find a dynamical equation for $\hat{\rho} = \text{Tr}_E[\hat{\rho}_U]$ by tracing over the environmental degrees of freedom. The question boils down to finding (approximate) methods for the computation of the partial trace $\text{Tr}_E[\hat{H}, \hat{\rho}_U]$. This trace can be easily evaluated by expanding $\hat{\rho}_U$ in powers of the interaction. By going to the interaction picture and substituting the formal solution for the density matrix back into the Liouville equation (we drop for simplicity the subscript indicating the interaction picture) one finds:

$$ \frac{d}{dt} \hat{\rho} = - \int_{t_0 \to -\infty}^{t} dt' \text{Tr}_E[\hat{H}_{S-E}(t), [\hat{H}_{S-E}(t'), \hat{\rho}_U(t')]]. \qquad (A.5) $$

In this expression we dropped an irrelevant term depending on the density matrix at $t_0$. The system Hamiltonian can be always renormalized in order to make this term to disappear.

Equation (A.5) has the right form to discuss the key approximations:

1. Born approximation. In general, $\hat{\rho}_U$ can be written as the factorised form $\hat{\rho}_{E,\beta} \otimes \hat{\rho}(t)$, where $\rho_\beta$ is the (assumed thermal) state of the environment, plus terms that arise due to the interaction of the system and the environment. Since the right hand side of Eq. (A.5) is already of second order in the interaction, parts of $\hat{\rho}_U$ that are of higher order in the coupling can be dropped. This means that one can substitute it with the product of the system and bath states, $\hat{\rho}_U(t) \approx \hat{\rho}_{E,\beta} \otimes \hat{\rho}(t)$. Note that this Born approximation further implies that the state of the environment does not change due to the interaction with the system.
2. Markov approximation. The trace over the environment amounts to computing correlators of the form $\text{Tr}_E[\hat{X}_E(t_1)\hat{X}_E(t_2)\hat{\rho}_{E,\beta}]$. If these correlations are sufficiently short-ranged in time then it is legitimate to substitute $\hat{\rho}_U(t') \to \hat{\rho}_U(t)$ in Eq. (A.5).

With these approximations, the equation of motion for the density matrix becomes:

$$ \frac{d}{dt}\hat{\rho} = -\int_{-\infty}^{t} dt' \text{Tr}_E[\hat{H}_{S-E}(t), [\hat{H}_{S-E}(t'), \hat{\rho}_{E,\beta} \otimes \hat{\rho}(t)]] \equiv \hat{\mathcal{L}}_{BM}[\hat{\rho}(t)], \quad (A.6) $$

where the subscript in the superoperator indicate the Born-Markov approximation.

There are several features that emerge already at this stage, that are worth pointing out. In the right hand side of Eq. (A.6) the degrees of freedom of the system and the environment “factorise”. Let us examine the two of them separately, starting from the system. The double commutator appearing in Eq. (A.6) leads naturally to a Lindblad-like sequence of terms. If we consider the simplest case, in which $\hat{Y}_S$ is an eigenoperator of the system Hamiltonian (where $\hat{Y}_S$ evolves monocrhomatically under $\hat{H}_S$, i.e. $e^{i\hat{H}_S t}\hat{Y}_S e^{-i\hat{H}_S t} = \hat{Y}_S e^{-i\epsilon t}$, or equivalently, $[\hat{Y}_S, \hat{H}_S] = \epsilon \hat{Y}_S$), then the terms that arise in the density matrix equation of motion are $\hat{Y}_S \hat{\rho} \hat{Y}_S^\dagger$, $\hat{Y}_S^\dagger \hat{Y}_S \hat{\rho}$, $\hat{\rho} \hat{Y}_S^\dagger \hat{Y}_S$ and their Hermitian conjugates. The jump operators appearing in the Lindblad equations in this special case are proportional to the operators $\hat{Y}_S$, i.e. the microscopic coupling between the system and the environment also determine how the dynamics of the reduced density matrix is affected. For example, if $\hat{Y}_S$ is a spin-1/2 raising/lowering operator $\sigma_i^\pm$ coupled, in each site, to a bath of bosons (photons for instance), the process accounted in Eq. (A.6) is the absorption/emission of a quantum of energy from/to the bath.

The perturbative character of the Born-Markov approximations appears in the fact that the only operators that appear are first order in the system-bath coupling. For example, in the spin raising case, the jump operators $\hat{\sigma}_i^\pm$ are indicating that the only allowed process is the emission or absorption of a single photon. Two- or higher-order processes can be found going beyond the second order (for example, $\hat{\sigma}_i^+ \hat{\sigma}_j^-$). The perturbation expansion in Eq. (A.6) should be interpreted as an expansion of the self-energy in the Dyson expansion of the many-body Green's function [755]. It includes only selected terms (in this example the single photon absorption/emission) resummed to infinite orders.

In the general case, where $\hat{Y}_S$ is not an eigenoperator of $\hat{H}_S$, there is still some extra work to do to bring Eq. (A.6) into a Lindblad form. In this case the time evolution can be considered as having the effect of decomposing $\hat{Y}_S$ into eigenoperators which individually evolve monochromatically with $\hat{H}_S$, i.e. $e^{i\hat{H}_S t}\hat{Y}_S e^{-i\hat{H}_S t} = \sum_{\mu} \hat{Y}_{S,\mu} e^{-i\epsilon_{\mu} t}$. All the details can be found, for example, in [754]. In general Eq. (A.6) will then take the form:

$$ \frac{d}{dt}\hat{\rho} = \sum_{\mu\nu} L_{\mu\nu} \left( \hat{Y}_{S,\nu} \hat{\rho} \hat{Y}_{S,\mu}^{\dagger} - \frac{1}{2} \{ \hat{Y}_{S,\mu}^{\dagger} \hat{Y}_{S,\nu} \hat{\rho} \} \right). \quad (A.7) $$

This expression is not yet in Lindblad form, as the matrix $L_{\mu\nu}$ is not diagonal, and crucially may have both positive and negative eigenvalues. The Lindblad form corresponds to working in the diagonal basis, but also requiring the matrix $L$ has only positive eigenvalues, corresponding to rates of each process. There are various procedures that can be applied to ensure a positive form, such as “secularisation”, which consists of neglecting all terms that are time-dependent in the interaction picture [191]. However, in a number of cases secularisation leads to a form that is of Lindblad form but generates incorrect dynamics [756, 757].

The other very important element of Eq. (A.6) is the “environmental part”, i.e. the correlators $\text{Tr}_E[\hat{X}_E(t_1)\hat{X}_E(t_2)\hat{\rho}_{E,\beta}]$ integrated over time. This term will give the magnitude of the coupling, entering in the jump operator. Assuming a thermal bath, one may find that the different correlators (i.e., the different patterns of orders of times) are related to the familiar
rates of emission and absorption obtained by Fermi's golden rule. The important aspect to highlight here is that with this method it is possible to compute how all the terms forming the density matrix evolve under the action of the external environment. With the Fermi golden rule argument alone, only the transition rates are obtained.

## References

[1] J. von Neumann, *Mathematical foundations of quantum mechanics*, Princeton University Press, Princeton, USA, ISBN 9780691028934 (1955).

[2] W. H. Zurek, *Decoherence, einselection, and the quantum origins of the classical*, Rev. Mod. Phys. **75**, 715 (2003), doi:10.1103/RevModPhys.75.715 [preprint doi:10.48550/arXiv.quant-ph/0105127].

[3] W. H. Zurek, *Decoherence and the transition from quantum to classical – Revisited*, in *Quantum decoherence*, Birkhäuser, Basel, Switzerland, ISBN 9783764378073 (2006), doi:10.1007/978-3-7643-7808-0_1.

[4] W. P. Schleich et al., *Quantum technology: From research to application*, Appl. Phys. B **122**, 130 (2016), doi:10.1007/s00340-016-6353-8.

[5] L. Chirolli and G. Burkard, *Decoherence in solid-state qubits*, Adv. Phys. **57**, 225 (2008), doi:10.1080/00018730802218067 [preprint doi:10.48550/arXiv:0809.4716].

[6] D. Suter and G. A. Álvarez, *Colloquium: Protecting quantum information against environmental noise*, Rev. Mod. Phys. **88**, 041001 (2016), doi:10.1103/RevModPhys.88.041001.

[7] Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean and T. E. O'Brien, *Quantum error mitigation*, Rev. Mod. Phys. **95**, 045005 (2023), doi:10.1103/RevModPhys.95.045005 [preprint doi:10.48550/arXiv.2210.00921].

[8] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler and P. Zoller, *Quantum states and phases in driven open quantum systems with cold atoms*, Nat. Phys. **4**, 878 (2008), doi:10.1038/nphys1073 [preprint doi:10.48550/arXiv.0803.1482].

[9] F. Verstraete, M. M. Wolf and J. I. Cirac, *Quantum computation and quantum-state engineering driven by dissipation*, Nat. Phys. **5**, 633 (2009), doi:10.1038/nphys1342.

[10] R. P. Feynman, *Simulating physics with computers*, Int. J. Theor. Phys. **21**, 467 (1982), doi:10.1007/BF02650179.

[11] I. de Vega and D. Alonso, *Dynamics of non-Markovian open quantum systems*, Rev. Mod. Phys. **89**, 015001 (2017), doi:10.1103/RevModPhys.89.015001.

[12] I. M. Georgescu, S. Ashhab and F. Nori, *Quantum simulation*, Rev. Mod. Phys. **86**, 153 (2014), doi:10.1103/RevModPhys.86.153 [preprint doi:10.48550/arXiv.1308.6253].

[13] E. Altman et al., *Quantum simulators: Architectures and opportunities*, PRX Quantum **2**, 017003 (2021), doi:10.1103/PRXQuantum.2.017003 [preprint doi:10.48550/arXiv.1912.06938].

[14] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and W. Zwerger, *Dynamics of the dissipative two-state system*, Rev. Mod. Phys. **59**, 1 (1987), doi:10.1103/RevModPhys.59.1.
[15] K. Le Hur, *Entanglement entropy, decoherence, and quantum phase transitions of a dissipative two-level system*, Ann. Phys. **323**, 2208 (2008), doi:10.1016/j.aop.2007.12.003 [preprint doi:10.48550/arXiv:0711.2301].

[16] H. Deng, H. Haug and Y. Yamamoto, *Exciton-polariton Bose-Einstein condensation*, Rev. Mod. Phys. **82**, 1489 (2010), doi:10.1103/RevModPhys.82.1489.

[17] I. Carusotto and C. Ciuti, *Quantum fluids of light*, Rev. Mod. Phys. **85**, 299 (2013), doi:10.1103/RevModPhys.85.299 [preprint doi:10.48550/arXiv:1205.6500].

[18] A. Kavokin, J. Baumberg, G. Malpuech and F. Laussy, *Microcavities*, Oxford University Press, Oxford, UK, ISBN 9780198782995 (2017), doi:10.1093/oso/9780198782995.001.0001.

[19] N. Proukakis, D. Snoke and P. Littlewood, *Universal themes of Bose-Einstein condensation*, Cambridge University Press, Cambridge, UK, ISBN 9781107085695 (2017), doi:10.1017/9781316084366.

[20] R. Kosloff and A. Levy, *Quantum heat engines and refrigerators: Continuous devices*, Annu. Rev. Phys. Chem. **65**, 365 (2014), doi:10.1146/annurev-physchem-040513-103724.

[21] G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens and N. Y. Halpern, *The resource theory of informational nonequilibrium in thermodynamics*, Phys. Rep. **583**, 1 (2015), doi:10.1016/j.physrep.2015.04.003 [preprint doi:10.48550/arXiv:1309.6586].

[22] S. V. Aradhya and L. Venkataraman, *Single-molecule junctions beyond electronic transport*, Nat. Nanotechnol. **8**, 399 (2013), doi:10.1038/nnano.2013.91.

[23] P. Gehring, J. M. Thijssen and H. S. J. van der Zant, *Single-molecule quantum-transport phenomena in break junctions*, Nat. Rev. Phys. **1**, 381 (2019), doi:10.1038/s42254-019-0055-1.

[24] R. Kosloff, *Quantum thermodynamics: A dynamical viewpoint*, Entropy **15**, 2100 (2013), doi:10.3390/e15062100 [preprint doi:10.48550/arXiv:1305.2268].

[25] S. Vinjanampathy and J. Anders, *Quantum thermodynamics*, Contemp. Phys. **57**, 545 (2016), doi:10.1080/00107514.2016.1201896 [preprint doi:10.48550/arXiv:1508.06099].

[26] H. Weimer, A. Kshetrimayum and R. Orús, *Simulation methods for open quantum many-body systems*, Rev. Mod. Phys. **93**, 015008 (2021), doi:10.1103/RevModPhys.93.015008 [preprint doi:10.48550/arXiv:1907.07079].

[27] Y. Ashida, Z. Gong and M. Ueda, *Non-Hermitian physics*, Adv. Phys. **69**, 249 (2020), doi:10.1080/00018732.2021.1876991 [preprint doi:10.48550/arXiv:2006.01837].

[28] A. A. Houck, H. E. Türeci and J. Koch, *On-chip quantum simulation with superconducting circuits*, Nat. Phys. **8**, 292 (2012), doi:10.1038/nphys2251.

[29] C. Noh and D. G. Angelakis, *Quantum simulations and many-body physics with light*, Rep. Prog. Phys. **80**, 016401 (2016), doi:10.1088/0034-4885/80/1/016401.

[30] M. J. Hartmann, *Quantum simulation with interacting photons*, J. Opt. **18**, 104005 (2016), doi:10.1088/2040-8978/18/10/104005.
[31] L. M. Sieberer, M. Buchhold and S. Diehl, *Keldysh field theory for driven open quantum systems*, Rep. Prog. Phys. **79**, 096001 (2016), doi:10.1088/0034-4885/79/9/096001.

[32] L. M. Sieberer, M. Buchhold, J. Marino and S. Diehl, *Universality in driven open quantum matter*, Rev. Mod. Phys. **97**, 025004 (2025), doi:10.1103/RevModPhys.97.025004 [preprint doi:10.48550/arXiv:2312.03073].

[33] F. Minganti and A. Biella, *Open quantum systems – A brief introduction*, (arXiv preprint) doi:10.48550/arXiv:2407.16855.

[34] R. Fazio, *Quantum phase transitions and vortex dynamics in superconducting networks*, Phys. Rep. **355**, 235 (2001), doi:10.1016/S0370-1573(01)00022-9.

[35] R. Graham and H. Haken, *Laserlight – First example of a second-order phase transition far away from thermal equilibrium*, Z. Phys. **237**, 31 (1970), doi:10.1007/BF01400474.

[36] H. Haken, *The semiclassical and quantum theory of the laser*, in *Quantum optics: Proceedings of the tenth session of the Scottish universities summer school in physics*, 1969, Academic Press, Cambridge, USA, ISBN 9780124037502 (1970).

[37] H. Haken, *Cooperative phenomena in systems far from thermal equilibrium and in non-physical systems*, Rev. Mod. Phys. **47**, 67 (1975), doi:10.1103/RevModPhys.47.67.

[38] U. Weiss, *Quantum dissipative systems*, 4th edition, World Scientific, Singapore, ISBN 9789814374910 (2012), doi:10.1142/8334.

[39] P. W. Anderson, *Special effects in superconductivity*, in *Lectures on the many-body problems*, Academic Press, New York, USA, ISBN 9780123956163 (1964), doi:10.1016/B978-0-12-395616-3.50010-3.

[40] A. J. Leggett, *Macroscopic quantum systems and the quantum theory of measurement*, Prog. Theor. Phys. Suppl. **69**, 80 (1980), doi:10.1143/PTP.69.80.

[41] R. Newrock, C. J. Lobb, U. Geingemueller and M. Octavio, *The two-dimensional physics of Josephson-junction arrays*, in *Solid state physics*, Academic Press, New York, USA, ISBN 9780126077544 (2000), doi:10.1016/S0081-1947(08)60250-7.

[42] D. J. Resnick, J. C. Garland, J. T. Boyd, S. Shoemaker and R. S. Newrock, *Kosterlitz-Thouless transition in proximity-coupled superconducting arrays*, Phys. Rev. Lett. **47**, 1542 (1981), doi:10.1103/PhysRevLett.47.1542.

[43] P. Martinoli, P. Lerch, C. Leemann and H. Beck, *Arrays of Josephson junctions: Model systems for two-dimensional physics*, Jpn. J. Appl. Phys. **26**, 1999 (1987), doi:10.7567/JJAPS.26S3.1999.

[44] L. J. Geerligs, M. Peters, L. E. M. de Groot, A. Verbruggen and J. E. Mooij, *Charging effects and quantum coherence in regular Josephson junction arrays*, Phys. Rev. Lett. **63**, 326 (1989), doi:10.1103/PhysRevLett.63.326.

[45] S. Chakravarty, G.-L. Ingold, S. Kivelson and A. Luther, *Onset of global phase coherence in Josephson-junction arrays: A dissipative phase transition*, Phys. Rev. Lett. **56**, 2303 (1986), doi:10.1103/PhysRevLett.56.2303.

[46] A. J. Rimberg, T. R. Ho, Ç. Kurdak, J. Clarke, K. L. Campman and A. C. Gossard, *Dissipation-driven superconductor-insulator transition in a two-dimensional Josephson-junction array*, Phys. Rev. Lett. **78**, 2632 (1997), doi:10.1103/PhysRevLett.78.2632.
[47] Y. Takahide, R. Yagi, A. Kanda, Y. Ootuka and S.-i. Kobayashi, Superconductor-insulator transition in a two-dimensional array of resistively shunted small Josephson junctions, Phys. Rev. Lett. 85, 1974 (2000), doi:10.1103/PhysRevLett.85.1974.

[48] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977), doi:10.1103/RevModPhys.49.435.

[49] H. Callen, Thermodynamics and an introduction to thermostatistics, Wiley, Hoboken, USA, ISBN 9780471862567 (1991).

[50] K. Huang, Introduction to statistical physics, Taylor & Francis, New York, USA, ISBN 9781420055764 (2001).

[51] P. Hänggi, P. Talkner and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62, 251 (1990), doi:10.1103/RevModPhys.62.251.

[52] P. Talkner and P. Hänggi, Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Rev. Mod. Phys. 92, 041002 (2020), doi:10.1103/RevModPhys.92.041002 [preprint doi:10.48550/arXiv.1911.11660].

[53] J. D. Cresser and J. Anders, Weak and ultrastrong coupling limits of the quantum mean force Gibbs state, Phys. Rev. Lett. 127, 250601 (2021), doi:10.1103/PhysRevLett.127.250601 [preprint doi:10.48550/arXiv.2104.12606].

[54] A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press, Cambridge, UK, ISBN 9781108769266 (2023), doi:10.1017/9781108769266.

[55] R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966), doi:10.1088/0034-4885/29/1/306.

[56] R. Kubo, M. Toda and N. Hashitsume, Statistical physics II: Nonequilibrium statistical mechanics, Springer, Berlin, Heidelberg, Germany, ISBN 9783540538332 (1991), doi:10.1007/978-3-642-58244-8.

[57] I. Carusotto, A. A. Houck, A. J. Kollár, P. Roushan, D. I. Schuster and J. Simon, Photonic materials in circuit quantum electrodynamics, Nat. Phys. 16, 268 (2020), doi:10.1038/s41567-020-0815-y.

[58] I. Bloch, J. Dalibard and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012), doi:10.1038/nphys2259.

[59] H. Ritsch, P. Domokos, F. Brennecke and T. Esslinger, Cold atoms in cavity-generated dynamical optical potentials, Rev. Mod. Phys. 85, 553 (2013), doi:10.1103/RevModPhys.85.553 [preprint doi:10.48550/arXiv.1210.0013].

[60] F. Mivehvar, F. Piazza, T. Donner and H. Ritsch, Cavity QED with quantum gases: New paradigms in many-body physics, Adv. Phys. 70, 1 (2021), doi:10.1080/00018732.2021.1969727 [preprint doi:10.48550/arXiv.2102.04473].

[61] R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012), doi:10.1038/nphys2252.

[62] C. D. Bruzewicz, J. Chiaverini, R. McConnell and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Appl. Phys. Rev. 6, 021314 (2019), doi:10.1063/1.5088164 [preprint doi:10.48550/arXiv.1904.04178].
[63] I. Buluta and F. Nori, *Quantum simulators*, Science **326**, 108 (2009), doi:10.1126/science.1177838.

[64] D. Leibfried, R. Blatt, C. Monroe and D. Wineland, *Quantum dynamics of single trapped ions*, Rev. Mod. Phys. **75**, 281 (2003), doi:10.1103/RevModPhys.75.281.

[65] W. M. Itano, D. J. Heinzen, J. J. Bollinger and D. J. Wineland, *Quantum Zeno effect*, Phys. Rev. A **41**, 2295 (1990), doi:10.1103/PhysRevA.41.2295.

[66] P. Schindler et al., *Quantum simulation of dynamical maps with trapped ions*, Nat. Phys. **9**, 361 (2013), doi:10.1038/nphys2630.

[67] J. T. Barreiro et al., *An open-system quantum simulator with trapped ions*, Nature **470**, 486 (2011), doi:10.1038/nature09801 [preprint doi:10.48550/arXiv.1104.1146].

[68] B. Misra and E. G. Sudarshan, *The Zeno's paradox in quantum theory*, J. Math. Phys. **18**, 756 (1977), doi:10.1063/1.523304.

[69] C. Noel et al., *Measurement-induced quantum phases realized in a trapped-ion quantum computer*, Nat. Phys. **18**, 760 (2022), doi:10.1038/s41567-022-01619-7.

[70] T. Behrle et al., *Phonon laser in the quantum regime*, Phys. Rev. Lett. **131**, 043605 (2023), doi:10.1103/PhysRevLett.131.043605 [preprint doi:10.48550/arXiv.2301.08156].

[71] K. P. Horn, F. Reiter, Y. Lin, D. Leibfried and C. P. Koch, *Quantum optimal control of the dissipative production of a maximally entangled state*, New J. Phys. **20**, 123010 (2018), doi:10.1088/1367-2630/aaf360.

[72] D. C. Cole et al., *Resource-efficient dissipative entanglement of two trapped-ion qubits*, Phys. Rev. Lett. **128**, 080502 (2022), doi:10.1103/PhysRevLett.128.080502 [preprint doi:10.48550/arXiv.2108.03136].

[73] M. Foss-Feig et al., *Entanglement from tensor networks on a trapped-ion quantum computer*, Phys. Rev. Lett. **128**, 150504 (2022), doi:10.1103/PhysRevLett.128.150504 [preprint doi:10.48550/arXiv.2104.11235].

[74] V. So, M. D. Suganthi, A. Menon, M. Zhu, R. Zhuravel, H. Pu, P. G. Wolynes, J. N. Onuchic and G. Pagano, *Trapped-ion quantum simulation of electron transfer models with tunable dissipation*, Sci. Adv. **10**, eads8011 (2024), doi:10.1126/sciadv.ads8011 [preprint doi:10.48550/arXiv.2405.10368].

[75] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell and C. E. Wieman, *Coherence, correlations, and collisions: What one learns about Bose-Einstein condensates from their decay*, Phys. Rev. Lett. **79**, 337 (1997), doi:10.1103/PhysRevLett.79.337.

[76] J. Söding, D. Guéry-Odelin, P. Desbiolles, F. Chevy, H. Inamori and J. Dalibard, *Three-body decay of a rubidium Bose-Einstein condensate*, Appl. Phys. B **69**, 257 (1999), doi:10.1007/s003400050805 [preprint doi:10.48550/arXiv.cond-mat/9811339].

[77] B. Tolra, K. O'Hara, J. Huckans, W. Phillips, S. Rolston and J. Porto, *Observation of reduced three-body recombination in a correlated 1D degenerate Bose gas*, Phys. Rev. Lett. **92**, 190401 (2004), doi:10.1103/PhysRevLett.92.190401.

[78] S. Knoop, J. S. Borbely, R. van Rooij and W. Vassen, *Nonexponential one-body loss in a Bose-Einstein condensate*, Phys. Rev. A **85**, 025602 (2012), doi:10.1103/PhysRevA.85.025602 [preprint doi:10.48550/arXiv.1112.4754].
[79] A. Browaeys et al., *Two body loss rate in a magneto-optical trap of metastable He*, Eur. Phys. J. D - At. Mol. Opt. Phys. **8**, 199 (2000), doi:10.1007/s100530050027.

[80] A. Yamaguchi, S. Uetake, D. Hashimoto, J. M. Doyle and Y. Takahashi, *Inelastic collisions in optically trapped ultracold metastable ytterbium*, Phys. Rev. Lett. **101**, 233002 (2008), doi:10.1103/PhysRevLett.101.233002 [preprint doi:10.48550/arXiv.0802.0461].

[81] R. Bouganne, M. B. Aguilera, A. Dareau, E. Soave, J. Beugnon and F. Gerbier, *Clock spectroscopy of interacting bosons in deep optical lattices*, New J. Phys. **19**, 113006 (2017), doi:10.1088/1367-2630/aa8c45.

[82] L. Franchi, L. F. Livi, G. Cappellini, G. Binella, M. Inguscio, J. Catani and L. Fallani, *State-dependent interactions in ultracold <sup>174</sup>Yb probed by optical clock spectroscopy*, New J. Phys. **19**, 103037 (2017), doi:10.1088/1367-2630/aa8fb4.

[83] T. Weber, J. Herbig, M. Mark, H.-C. Nägerl and R. Grimm, *Three-body recombination at large scattering lengths in an ultracold atomic gas*, Phys. Rev. Lett. **91**, 123201 (2003), doi:10.1103/PhysRevLett.91.123201 [preprint doi:10.48550/arXiv.physics/0304052].

[84] P. Grišins, B. Rauer, T. Langen, J. Schmiedmayer and I. E. Mazets, *Degenerate Bose gases with uniform loss*, Phys. Rev. A **93**, 033634 (2016), doi:10.1103/PhysRevA.93.033634 [preprint doi:10.48550/arXiv.1411.4946].

[85] B. Rauer, P. Grišins, I. E. Mazets, T. Schweigler, W. Rohringer, R. Geiger, T. Langen and J. Schmiedmayer, *Cooling of a one-dimensional Bose gas*, Phys. Rev. Lett. **116**, 030402 (2016), doi:10.1103/PhysRevLett.116.030402 [preprint doi:10.48550/arXiv.1505.04747].

[86] N. Syassen, D. M. Bauer, M. Lettner, T. Volz, D. Dietze, J. J. Garciía-Ripoll, J. I. Cirac, G. Rempe and S. Duürr, *Strong dissipation inhibits losses and induces correlations in cold molecular gases*, Science **320**, 1329 (2008), doi:10.1126/science.1155309 [preprint doi:10.48550/arXiv.0806.4310].

[87] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin and J. Ye, *Observation of dipolar spin-exchange interactions with lattice-confined polar molecules*, Nature **501**, 521 (2013), doi:10.1038/nature12483 [preprint doi:10.48550/arXiv.1305.5598].

[88] G. Barontini, R. Labouvie, F. Stubenrauch, A. Vogler, V. Guarrera and H. Ott, *Controlling the dynamics of an open many-body quantum system with localized dissipation*, Phys. Rev. Lett. **110**, 035302 (2013), doi:10.1103/PhysRevLett.110.035302 [preprint doi:10.48550/arXiv.1212.4824].

[89] T. Tomita, S. Nakajima, I. Danshita, Y. Takasu and Y. Takahashi, *Observation of the Mott insulator to superfluid crossover of a driven-dissipative Bose-Hubbard system*, Sci. Adv. **3**, e1701513 (2017), doi:10.1126/sciadv.1701513 [preprint doi:10.48550/arXiv.1705.09942].

[90] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch and I. Bloch, *Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms*, Nature **415**, 39 (2002), doi:10.1038/415039a [preprint doi:10.48550/arXiv.2506.21303].

[91] T. Tomita, S. Nakajima, Y. Takasu and Y. Takahashi, *Dissipative Bose-Hubbard system with intrinsic two-body loss*, Phys. Rev. A **99**, 031601 (2019), doi:10.1103/PhysRevA.99.031601 [preprint doi:10.48550/arXiv.1809.09989].
[92] K. Nagao, I. Danshita and S. Yunoki, Discrete-phase-space method for driven-dissipative dynamics of strongly interacting bosons in optical lattices, Phys. Rev. A **110**, 063310 (2024), doi:10.1103/PhysRevA.110.063310 [preprint doi:10.48550/arXiv:2307.16170].

[93] J. P. Gordon and A. Ashkin, Motion of atoms in a radiation trap, Phys. Rev. A **21**, 1606 (1980), doi:10.1103/PhysRevA.21.1606.

[94] J. Dalibard and C. Cohen-Tannoudji, Atomic motion in laser light: Connection between semiclassical and quantum descriptions, J. Phys. B: At. Mol. Phys. **18**, 1661 (1985), doi:10.1088/0022-3700/18/8/019.

[95] D. Witthaut, F. Trimborn and S. Wimberger, Dissipation induced coherence of a two-mode Bose-Einstein condensate, Phys. Rev. Lett. **101**, 200402 (2008), doi:10.1103/PhysRevLett.101.200402 [preprint doi:10.48550/arXiv:0809.1776].

[96] F. Gerbier and Y. Castin, Heating rates for an atom in a far-detuned optical lattice, Phys. Rev. A **82**, 013615 (2010), doi:10.1103/PhysRevA.82.013615 [preprint doi:10.48550/arXiv:1002.5018].

[97] H. Pichler, A. J. Daley and P. Zoller, Nonequilibrium dynamics of bosonic atoms in optical lattices: Decoherence of many-body states due to spontaneous emission, Phys. Rev. A **82**, 063605 (2010), doi:10.1103/PhysRevA.82.063605 [preprint doi:10.48550/arXiv:1009.0194].

[98] Y. S. Patil, S. Chakram and M. Vengalattore, Measurement-induced localization of an ultracold lattice gas, Phys. Rev. Lett. **115**, 140402 (2015), doi:10.1103/PhysRevLett.115.140402 [preprint doi:10.48550/arXiv:1411.2678].

[99] H. P. Lüschen et al., Signatures of many-body localization in a controlled open quantum system, Phys. Rev. X **7**, 011034 (2017), doi:10.1103/PhysRevX.7.011034 [preprint doi:10.48550/arXiv:1610.01613].

[100] R. Vatré, R. Bougarne, M. B. Aguilera, A. Ghermaoui, J. Beugnon, R. Lopes and F. Gerbier, Dynamics of spatial phase coherence in a dissipative Bose-Hubbard atomic system, C. R. Phys. **24**, 263 (2024), doi:10.5802/crphys.166 [preprint doi:10.48550/arXiv:2405.03226].

[101] R. Bougarne, M. B. Aguilera, A. Ghermaoui, J. Beugnon and F. Gerbier, Anomalous decay of coherence in a dissipative many-body system, Nat. Phys. **16**, 21 (2019), doi:10.1038/s41567-019-0678-2.

[102] F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl and T. Esslinger, Cavity QED with a Bose-Einstein condensate, Nature **450**, 268 (2007), doi:10.1038/nature06120 [preprint doi:10.48550/arXiv:0706.3411].

[103] I. D. Leroux, M. H. Schleier-Smith and V. Vuletić, Implementation of cavity squeezing of a collective atomic spin, Phys. Rev. Lett. **104**, 073602 (2010), doi:10.1103/PhysRevLett.104.073602 [preprint doi:10.48550/arXiv:0911.4065].

[104] K. C. Cox, G. P. Greve, J. M. Weiner and J. K. Thompson, Deterministic squeezed states with collective measurements and feedback, Phys. Rev. Lett. **116**, 093602 (2016), doi:10.1103/PhysRevLett.116.093602 [preprint doi:10.48550/arXiv:1512.02150].

[105] K. Baumann, Experimental realization of the Dicke quantum phase transition, PhD thesis, ETH Zürich, Zurich, Switzerland (2011), doi:10.3929/ethz-a-006961248.
[106] F. Dimer, B. Estienne, A. S. Parkins and H. J. Carmichael, Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system, Phys. Rev. A 75, 013804 (2007), doi:10.1103/PhysRevA.75.013804 [preprint doi:10.48550/arXiv.quant-ph/0607115].

[107] K. Rzażewski, K. Wódkiewicz and W. Żakowicz, Phase transitions, two-level atoms, and the $A^2$ term, Phys. Rev. Lett. 35, 432 (1975), doi:10.1103/PhysRevLett.35.432.

[108] K. Hepp and E. H. Lieb, Equilibrium statistical mechanics of matter interacting with the quantized radiation field, Phys. Rev. A 8, 2517 (1973), doi:10.1103/PhysRevA.8.2517.

[109] K. Hepp and E. H. Lieb, On the superradiant phase transition for molecules in a quantized radiation field: The Dicke maser model, Ann. Phys. 76, 360 (1973), doi:10.1016/0003-4916(73)90039-0.

[110] K. Baumann, C. Guerlin, F. Brennecke and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, Nature 464, 1301 (2010), doi:10.1038/nature09009 [preprint doi:10.48550/arXiv:0912.3261].

[111] A. J. Kollár, A. T. Papageorge, K. Baumann, M. A. Armen and B. L. Lev, An adjustable-length cavity and Bose-Einstein condensate apparatus for multimode cavity QED, New J. Phys. 17, 043012 (2015), doi:10.1088/1367-2630/17/4/043012.

[112] J. Léonard, A. Morales, P. Zupancic, T. Esslinger and T. Donner, Supersolid formation in a quantum gas breaking a continuous translational symmetry, Nature 543, 87 (2017), doi:10.1038/nature21067 [preprint doi:10.48550/arXiv:1609.09053].

[113] J. Léonard, A. Morales, P. Zupancic, T. Donner and T. Esslinger, Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas, Science 358, 1415 (2017), doi:10.1126/science.aan2608 [preprint doi:10.48550/arXiv:1704.05803].

[114] V. D. Vaidya, Y. Guo, R. M. Kroeze, K. E. Ballantine, A. J. Kollár, J. Keeling and B. L. Lev, Tunable-range, photon-mediated atomic interactions in multimode cavity QED, Phys. Rev. X 8, 011002 (2018), doi:10.1103/PhysRevX.8.011002 [preprint doi:10.48550/arXiv.1708.08933].

[115] Y. Guo, R. M. Kroeze, B. P. Marsh, S. Gopalakrishnan, J. Keeling and B. L. Lev, An optical lattice with sound, Nature 599, 211 (2021), doi:10.1038/s41586-021-03945-x.

[116] A. Periwal, E. S. Cooper, P. Kunkel, J. F. Wienand, E. J. Davis and M. Schleier-Smith, Programmable interactions and emergent geometry in an array of atom clouds, Nature 600, 630 (2021), doi:10.1038/s41586-021-04156-0.

[117] G. Bentsen, I.-D. Potirniche, V. B. Bulchandani, T. Scaffidi, X. Cao, X.-L. Qi, M. Schleier-Smith and E. Altman, Integrable and chaotic dynamics of spins coupled to an optical cavity, Phys. Rev. X 9, 041011 (2019), doi:10.1103/PhysRevX.9.041011 [preprint doi:10.48550/arXiv.1904.10966].

[118] B. P. Marsh, Y. Guo, R. M. Kroeze, S. Gopalakrishnan, S. Ganguli, J. Keeling and B. L. Lev, Enhancing associative memory recall and storage capacity using confocal cavity QED, Phys. Rev. X 11, 021048 (2021), doi:10.1103/PhysRevX.11.021048 [preprint doi:10.48550/arXiv.2009.01227].
[119] B. P. Marsh, R. M. Kroeze, S. Ganguli, S. Gopalakrishnan, J. Keeling and B. L. Lev, *Entanglement and replica symmetry breaking in a driven-dissipative quantum spin glass*, Phys. Rev. X **14**, 011026 (2024), doi:10.1103/PhysRevX.14.011026 [preprint doi:10.48550/arXiv:2307.10176].

[120] R. M. Kroeze, B. P. Marsh, D. A. Schuller, H. S. Hunt, A. N. Bourzutschky, M. Winer, S. Gopalakrishnan, J. Keeling and B. L. Lev, *Directly observing replica symmetry breaking in a vector quantum-optical spin glass*, (arXiv preprint) doi:10.48550/arXiv.2311.04216.

[121] Y. Guo, V. D. Vaidya, R. M. Kroeze, R. A. Lunney, B. L. Lev and J. Keeling, *Emergent and broken symmetries of atomic self-organization arising from Gouy phase shifts in multimode cavity QED*, Phys. Rev. A **99**, 053818 (2019), doi:10.1103/PhysRevA.99.053818 [preprint doi:10.48550/arXiv:1810.11085].

[122] J. Klinder, H. Keßler, M. R. Bakhtiari, M. Thorwart and A. Hemmerich, *Observation of a superradiant Mott insulator in the Dicke-Hubbard model*, Phys. Rev. Lett. **115**, 230403 (2015), doi:10.1103/PhysRevLett.115.230403 [preprint doi:10.48550/arXiv:1511.00850].

[123] V. Helson, T. Zwettler, F. Mivehvar, E. Colella, K. Roux, H. Konishi, H. Ritsch and J.-P. Brantut, *Density-wave ordering in a unitary Fermi gas with photon-mediated interactions*, Nature **618**, 716 (2023), doi:10.1038/s41586-023-06018-3.

[124] H. Keßler, J. G. Cosme, M. Hemmerling, L. Mathey and A. Hemmerich, *Emergent limit cycles and time crystal dynamics in an atom-cavity system*, Phys. Rev. A **99**, 053605 (2019), doi:10.1103/PhysRevA.99.053605 [preprint doi:10.48550/arXiv:1905.02751].

[125] N. Dogra, M. Landini, K. Kroeger, L. Hruby, T. Donner and T. Esslinger, *Dissipation-induced structural instability and chiral dynamics in a quantum gas*, Science **366**, 1496 (2019), doi:10.1126/science.aaw4465 [preprint doi:10.48550/arXiv:1901.05974].

[126] C.-M. Halati, A. Sheikhan, H. Ritsch and C. Kollath, *Numerically exact treatment of many-body self-organization in a cavity*, Phys. Rev. Lett. **125**, 093604 (2020), doi:10.1103/PhysRevLett.125.093604 [preprint doi:10.48550/arXiv:1909.07335].

[127] R. Lin, C. Georges, J. Klinder, P. Molignini, M. Büttner, A. U. J. Lode, R. Chitra, A. Hemmerich and H. Keßler, *Mott transition in a cavity-boson system: A quantitative comparison between theory and experiment*, SciPost Phys. **11**, 030 (2021), doi:10.21468/SciPostPhys.11.2.030 [preprint doi:10.48550/arXiv:2104.11253].

[128] S. Gopalakrishnan, B. L. Lev and P. M. Goldbart, *Emergent crystallinity and frustration with Bose-Einstein condensates in multimode cavities*, Nat. Phys. **5**, 845 (2009), doi:10.1038/nphys1403 [preprint doi:10.48550/arXiv:0903.2254].

[129] K. Seetharam, A. Lerose, R. Fazio and J. Marino, *Correlation engineering via nonlocal dissipation*, Phys. Rev. Res. **4**, 013089 (2022), doi:10.1103/PhysRevResearch.4.013089 [preprint doi:10.48550/arXiv:2101.06445].

[130] K. Seetharam, A. Lerose, R. Fazio and J. Marino, *Dynamical scaling of correlations generated by short- and long-range dissipation*, Phys. Rev. B **105**, 184305 (2022), doi:10.1103/PhysRevB.105.184305 [preprint doi:10.48550/arXiv:2110.09547].

[131] M. Saffman, T. G. Walker and K. Mølmer, *Quantum information with Rydberg atoms*, Rev. Mod. Phys. **82**, 2313 (2010), doi:10.1103/RevModPhys.82.2313 [preprint doi:10.48550/arXiv:0909.4777].
[132] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker and M. Saffman, *Observation of Rydberg blockade between two atoms*, Nat. Phys. **5**, 110 (2009), doi:10.1038/nphys1178 [preprint doi:10.48550/arXiv:0805.0758].

[133] N. Malossi, M. M. Valado, S. Scotto, P. Huillery, P. Pillet, D. Ciampini, E. Arimondo and O. Morsch, *Full counting statistics and phase diagram of a dissipative Rydberg gas*, Phys. Rev. Lett. **113**, 023006 (2014), doi:10.1103/PhysRevLett.113.023006 [preprint doi:10.48550/arXiv:1308.1854].

[134] H. Weimer, M. Müller, I. Lesanovsky, P. Zoller and H. P. Büchler, *A Rydberg quantum simulator*, Nat. Phys. **6**, 382 (2010), doi:10.1038/nphys1614 [preprint doi:10.48550/arXiv:0907.1657].

[135] G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky and P. Zoller, *Strongly correlated gases of Rydberg-dressed atoms: Quantum and classical dynamics*, Phys. Rev. Lett. **104**, 223002 (2010), doi:10.1103/PhysRevLett.104.223002 [preprint doi:10.48550/arXiv:1001.0519].

[136] J. E. Johnson and S. L. Rolston, *Interactions between Rydberg-dressed atoms*, Phys. Rev. A **82**, 033412 (2010), doi:10.1103/PhysRevA.82.033412 [preprint doi:10.48550/arXiv:1006.3212].

[137] J. Honer, H. Weimer, T. Pfau and H. P. Büchler, *Collective many-body interaction in Rydberg dressed atoms*, Phys. Rev. Lett. **105**, 160404 (2010), doi:10.1103/PhysRevLett.105.160404 [preprint doi:10.48550/arXiv:1004.2499].

[138] I. Lesanovsky and J. P. Garrahan, *Kinetic constraints, hierarchical relaxation, and onset of glassiness in strongly interacting and dissipative Rydberg gases*, Phys. Rev. Lett. **111**, 215305 (2013), doi:10.1103/PhysRevLett.111.215305 [preprint doi:10.48550/arXiv:1307.8078].

[139] C. Pérez-Espigares, M. Marcuzzi, R. Gutiérrez and I. Lesanovsky, *Epidemic dynamics in open quantum spin systems*, Phys. Rev. Lett. **119**, 140401 (2017), doi:10.1103/PhysRevLett.119.140401 [preprint doi:10.48550/arXiv:1705.06994].

[140] F. Ritort and P. Sollich, *Glassy dynamics of kinetically constrained models*, Adv. Phys. **52**, 219 (2003), doi:10.1080/0001873031000093582 [preprint doi:10.48550/arXiv:cond-mat/0210382].

[141] B. Olmos, I. Lesanovsky and J. P. Garrahan, *Facilitated spin models of dissipative quantum glasses*, Phys. Rev. Lett. **109**, 020403 (2012), doi:10.1103/PhysRevLett.109.020403 [preprint doi:10.48550/arXiv:1203.6585].

[142] F. Carollo, E. Gillman, H. Weimer and I. Lesanovsky, *Critical behavior of the quantum contact process in one dimension*, Phys. Rev. Lett. **123**, 100604 (2019), doi:10.1103/PhysRevLett.123.100604 [preprint doi:10.48550/arXiv:1902.04515].

[143] H. Schempp et al., *Full counting statistics of laser excited Rydberg aggregates in a one-dimensional geometry*, Phys. Rev. Lett. **112**, 013002 (2014), doi:10.1103/PhysRevLett.112.013002 [preprint doi:10.48550/arXiv:1308.0264].

[144] F. Letscher, O. Thomas, T. Niederprüm, M. Fleischhauer and H. Ott, *Bistability versus metastability in driven dissipative Rydberg gases*, Phys. Rev. X **7**, 021020 (2017), doi:10.1103/PhysRevX.7.021020 [preprint doi:10.48550/arXiv:1611.00627].
[145] S. Helmrich, A. Arias, G. Lochhead, T. M. Wintermantel, M. Buchhold, S. Diehl and S. Whitlock, *Signatures of self-organized criticality in an ultracold atomic gas*, Nature **577**, 481 (2020), doi:10.1038/s41586-019-1908-6.

[146] L. W. Clark, N. Schine, C. Baum, N. Jia and J. Simon, *Observation of Laughlin states made of light*, Nature **582**, 41 (2020), doi:10.1038/s41586-020-2318-5.

[147] N. Schine, A. Ryou, A. Gromov, A. Sommer and J. Simon, *Synthetic Landau levels for photons*, Nature **534**, 671 (2016), doi:10.1038/nature17943 [preprint doi:10.48550/arXiv:1511.07381].

[148] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin and R. J. Schoelkopf, *Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation*, Phys. Rev. A **69**, 062320 (2004), doi:10.1103/PhysRevA.69.062320 [preprint doi:10.48550/arXiv:cond-mat/0402216].

[149] D. L. Underwood, W. E. Shanks, J. Koch and A. A. Houck, *Low-disorder microwave cavity lattices for quantum simulation with photons*, Phys. Rev. A **86**, 023837 (2012), doi:10.1103/PhysRevA.86.023837 [preprint doi:10.48550/arXiv:1203.5363].

[150] Y. Makhlin, G. Schön and A. Shnirman, *Quantum-state engineering with Josephson-junction devices*, Rev. Mod. Phys. **73**, 357 (2001), doi:10.1103/RevModPhys.73.357 [preprint doi:10.48550/arXiv:cond-mat/0011269].

[151] A. Blais, A. L. Grimsmo, S. M. Girvin and A. Wallraff, *Circuit quantum electrodynamics*, Rev. Mod. Phys. **93**, 025005 (2021), doi:10.1103/RevModPhys.93.025005 [preprint doi:10.48550/arXiv:2005.12667].

[152] D. Sanvitto and S. Kéna-Cohen, *The road towards polaritonic devices*, Nat. Mater. **15**, 1061 (2016), doi:10.1038/nmat4668.

[153] J. Keeling, F. Marchetti, M. Szymańska and P. Littlewood, *Collective coherence in planar semiconductor microcavities*, Semicond. Sci. Technol. **22**, R1 (2007), doi:10.1088/0268-1242/22/5/R01.

[154] J. Keeling and S. Kéna-Cohen, *Bose-Einstein condensation of exciton-polaritons in organic microcavities*, Annu. Rev. Phys. Chem. **71**, 435 (2020), doi:10.1146/annurev-physchem-010920-102509.

[155] T. Low et al., *Polaritons in layered two-dimensional materials*, Nat. Mater. **16**, 182 (2017), doi:10.1038/nmat4792 [preprint doi:10.48550/arXiv:1610.04548].

[156] R. Su, A. Fieramosca, Q. Zhang, H. S. Nguyen, E. Deleporte, Z. Chen, D. Sanvitto, T. C. H. Liew and Q. Xiong, *Perovskite semiconductors for room-temperature exciton-polaritonics*, Nat. Mater. **20**, 1315 (2021), doi:10.1038/s41563-021-01035-x.

[157] D. N. Basov, A. Asenjo-Garcia, P. J. Schuck, X. Zhu and A. Rubio, *Polariton panorama*, Nanophotonics **10**, 549 (2020), doi:10.1515/nanoph-2020-0449.

[158] M. H. Szymańska, J. Keeling and P. B. Littlewood, *Nonequilibrium quantum condensation in an incoherently pumped dissipative system*, Phys. Rev. Lett. **96**, 230602 (2006), doi:10.1103/PhysRevLett.96.230602 [preprint doi:10.48550/arXiv:cond-mat/0603447].
[159] M. Wouters and I. Carusotto, *Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons*, Phys. Rev. Lett. **99**, 140402 (2007), doi:10.1103/PhysRevLett.99.140402 [preprint doi:10.48550/arXiv:cond-mat/0702431].

[160] J. Keeling and N. G. Berloff, *Spontaneous rotating vortex lattices in a pumped decaying condensate*, Phys. Rev. Lett. **100**, 250401 (2008), doi:10.1103/PhysRevLett.100.250401 [preprint doi:10.48550/arXiv:0706.3686].

[161] J. Kasprzak et al., *Bose-Einstein condensation of exciton polaritons*, Nature **443**, 409 (2006), doi:10.1038/nature05131.

[162] Y. Sun, P. Wen, Y. Yoon, G. Liu, M. Steger, L. N. Pfeiffer, K. West, D. W. Snoke and K. A. Nelson, *Bose-Einstein condensation of long-lifetime polaritons in thermal equilibrium*, Phys. Rev. Lett. **118**, 016602 (2017), doi:10.1103/PhysRevLett.118.016602 [preprint doi:10.48550/arXiv:1601.02581].

[163] J. Keeling, L. M. Sieberer, E. Altman, L. Chen, S. Diehl and J. Toner, *Superfluidity and phase correlations of driven dissipative condensates*, in *Universal themes of Bose-Einstein condensation*, Cambridge University Press, Cambridge, UK, ISBN 9781316084366 (2017), doi:10.1017/9781316084366.013.

[164] J. J. Baumberg, P. G. Savvidis, R. M. Stevenson, A. I. Tartakovskii, M. S. Skolnick, D. M. Whittaker and J. S. Roberts, *Parametric oscillation in a vertical microcavity: A polariton condensate or micro-optical parametric oscillation*, Phys. Rev. B **62**, R16247 (2000), doi:10.1103/PhysRevB.62.R16247.

[165] C. Ciuti, P. Schwendimann and A. Quattropani, *Theory of polariton parametric interactions in semiconductor microcavities*, Semicond. Sci. Technol. **18**, S279 (2003), doi:10.1088/0268-1242/18/10/301.

[166] G. Muñoz-Matutano et al., *Emergence of quantum correlations from interacting fibre-cavity polaritons*, Nat. Mater. **18**, 213 (2019), doi:10.1038/s41563-019-0281-z.

[167] A. Delteil, T. Fink, A. Schade, S. Höfling, C. Schneider and A. İmamoğlu, *Towards polariton blockade of confined exciton-polaritons*, Nat. Mater. **18**, 219 (2019), doi:10.1038/s41563-019-0282-y.

[168] D. Bajoni, P. Senellart, E. Wertz, I. Sagnes, A. Miard, A. Lemaître and J. Bloch, *Polariton laser using single micropillar GaAs-GaAlAs semiconductor cavities*, Phys. Rev. Lett. **100**, 047401 (2008), doi:10.1103/PhysRevLett.100.047401 [preprint doi:10.48550/arXiv:0710.2060].

[169] T. Jacqmin et al., *Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons*, Phys. Rev. Lett. **112**, 116402 (2014), doi:10.1103/PhysRevLett.112.116402 [preprint doi:10.48550/arXiv:1310.8105].

[170] P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch and A. Amo, *Lasing in topological edge states of a one-dimensional lattice*, Nat. Photonics **11**, 651 (2017), doi:10.1038/s41566-017-0006-2.

[171] S. Klembt et al., *Exciton-polariton topological insulator*, Nature **562**, 552 (2018), doi:10.1038/s41586-018-0601-5.

[172] A. Amo and J. Bloch, *Exciton-polaritons in lattices: A non-linear photonic simulator*, C. R. Phys. **17**, 934 (2016), doi:10.1016/j.crhy.2016.08.007.
[173] C. Schneider, K. Winkler, M. D. Fraser, M. Kamp, Y. Yamamoto, E. A. Ostrovskaya and S. Höfling, *Exciton-polariton trapping and potential landscape engineering*, Rep. Prog. Phys. **80**, 016503 (2016), doi:10.1088/0034-4885/80/1/016503.

[174] P. Cristofolini, G. Christmann, S. I. Tsintzos, G. Deligeorgis, G. Konstantinidis, Z. Hatzopoulos, P. G. Savvidis and J. J. Baumberg, *Coupling quantum tunneling with cavity photons*, Science **336**, 704 (2012), doi:10.1126/science.1219010.

[175] I. Rosenberg, D. Liran, Y. Mazuz-Harpaz, K. West, L. Pfeiffer and R. Rapaport, *Strongly interacting dipolar-polaritons*, Sci. Adv. **4**, eaat8880 (2018), doi:10.1126/sciadv.aat8880.

[176] J. Gu et al., *Enhanced nonlinear interaction of polaritons via excitonic Rydberg states in monolayer $WSe_2$*, Nat. Commun. **12**, 2269 (2021), doi:10.1038/s41467-021-22537-x.

[177] K. Orfanakis, S. K. Rajendran, V. Walther, T. Volz, T. Pohl and H. Ohadi, *Rydberg exciton-polaritons in a $Cu_2O$ microcavity*, Nat. Mater. **21**, 767 (2022), doi:10.1038/s41563-022-01230-4.

[178] R. P. A. Emmanuele et al., *Highly nonlinear trion-polaritons in a monolayer semiconductor*, Nat. Commun. **11**, 3589 (2020), doi:10.1038/s41467-020-17340-z.

[179] A. Badolato, K. Hennessy, M. Atatuüre, J. Dreiser, E. Hu, P. M. Petroff and A. Imamoğlu, *Deterministic coupling of single quantum dots to single nanocavity modes*, Science **308**, 1158 (2005), doi:10.1126/science.1109815.

[180] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatuüre, S. Gulde, S. Fält, E. L. Hu and A. Imamoğlu, *Quantum nature of a strongly coupled single quantum dot-cavity system*, Nature **445**, 896 (2007), doi:10.1038/nature05586 [preprint doi:10.48550/arXiv.quant-ph/0610034].

[181] G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch and A. Scherer, *Vacuum Rabi splitting in semiconductors*, Nat. Phys. **2**, 81 (2006), doi:10.1038/nphys227.

[182] R. Chikkaraddy et al., *Single-molecule strong coupling at room temperature in plasmonic nanocavities*, Nature **535**, 127 (2016), doi:10.1038/nature17974.

[183] C. Denz, M. Schwab and C. Weilnau, *Transverse-pattern formation in photorefractive optics*, Springer, Berlin, Heidelberg, Germany, ISBN 9783540021094 (2003), doi:10.1007/b13583.

[184] M. Scully and M. Zubairy, *Quantum optics*, Cambridge University Press, Cambridge, UK, ISBN 9780521435956 (1997).

[185] P. Kirton and J. Keeling, *Superradiant and lasing states in driven-dissipative Dicke models*, New J. Phys. **2**, 015009 (2018), doi:10.1088/1367-2630/aaa11d.

[186] Y. Shchadilova, M. M. Roses, E. G. D. Torre, M. D. Lukin and E. Demler, *Fermionic formalism for driven-dissipative multilevel systems*, Phys. Rev. A **101**, 013817 (2020), doi:10.1103/PhysRevA.101.013817 [preprint doi:10.48550/arXiv.1804.03543].

[187] K. C. Stitely, A. Giraldo, B. Krauskopf and S. Parkins, *Lasing and counter-lasing phase transitions in a cavity-QED system*, Phys. Rev. Res. **4**, 023101 (2022), doi:10.1103/PhysRevResearch.4.023101 [preprint doi:10.48550/arXiv.2201.12775].
[188] J. Klaers, J. Schmitt, F. Vewinger and M. Weitz, *Bose-Einstein condensation of photons in an optical microcavity*, Nature **468**, 545 (2010), doi:10.1038/nature09567 [preprint doi:10.48550/arXiv:1007.4088].

[189] F. P. Schäfer, *Dye lasers*, Springer, Berlin, Heidelberg, Germany, ISBN 9783540515586 (1990), doi:10.1007/3-540-51558-5.

[190] M. A. Nielsen and I. L. Chuang, *Quantum computation and quantum information*, Cambridge University Press, Cambridge, UK, ISBN 9781107002173 (2010), doi:10.1017/CBO9780511976667.

[191] H.-P. Breuer and F. Petruccione, *The theory of open quantum systems*, Oxford University Press, Oxford, UK, ISBN 9780191706349 (2007), doi:10.1093/acprof:oso/9780199213900.001.0001.

[192] B. Buča and T. Prosen, *A note on symmetry reductions of the Lindblad equation: Transport in constrained open spin chains*, New J. Phys. **14**, 073007 (2012), doi:10.1088/1367-2630/14/7/073007.

[193] V. V. Albert and L. Jiang, *Symmetries and conserved quantities in Lindblad master equations*, Phys. Rev. A **89**, 022118 (2014), doi:10.1103/PhysRevA.89.022118 [preprint doi:10.48550/arXiv:1310.1523].

[194] H. Spohn, *Approach to equilibrium for completely positive dynamical semigroups of N-level systems*, Rep. Math. Phys. **10**, 189 (1976), doi:10.1016/0034-4877(76)90040-9.

[195] H. Spohn, *An algebraic condition for the approach to equilibrium of an open N-level system*, Lett. Math. Phys. **2**, 33 (1977), doi:10.1007/BF00420668.

[196] B. Baumgartner, H. Narnhofer and W. Thirring, *Analysis of quantum semigroups with GKS-Lindblad generators: I. Simple generators*, J. Phys. A: Math. Theor. **41**, 065201 (2008), doi:10.1088/1751-8113/41/6/065201.

[197] B. Baumgartner and H. Narnhofer, *Analysis of quantum semigroups with GKS-Lindblad generators: II. General*, J. Phys. A: Math. Theor. **41**, 395303 (2008), doi:10.1088/1751-8113/41/39/395303.

[198] D. Nigro, *On the uniqueness of the steady-state solution of the Lindblad-Gorini-Kossakowski-Sudarshan equation*, J. Stat. Mech.: Theory Exp. 043202 (2019), doi:10.1088/1742-5468/ab0c1c.

[199] H. Yoshida, *Uniqueness of steady states of Gorini-Kossakowski-Sudarshan-Lindblad equations: A simple proof*, Phys. Rev. A **109**, 022218 (2024), doi:10.1103/PhysRevA.109.022218 [preprint doi:10.48550/arXiv:2309.00335].

[200] Z. Zhang, J. Tindall, J. Mur-Petit, D. Jaksch and B. Buča, *Stationary state degeneracy of open quantum systems with non-Abelian symmetries*, J. Phys. A: Math. Theor. **53**, 215304 (2020), doi:10.1088/1751-8121/ab88e3.

[201] Y. Li, P. Sala and F. Pollmann, *Hilbert space fragmentation in open quantum systems*, Phys. Rev. Res. **5**, 043239 (2023), doi:10.1103/PhysRevResearch.5.043239 [preprint doi:10.48550/arXiv:2305.06918].

[202] Y. Li, F. Pollmann, N. Read and P. Sala, *Highly entangled stationary states from strong symmetries*, Phys. Rev. X **15**, 011068 (2025), doi:10.1103/PhysRevX.15.011068 [preprint doi:10.48550/arXiv:2406.08567].
[203] F. Minganti, A. Biella, N. Bartolo and C. Ciuti, *Spectral theory of Liouvillians for dissipative phase transitions*, Phys. Rev. A **98**, 042118 (2018), doi:**10.1103/PhysRevA.98.042118** [preprint doi:**10.48550/arXiv:1804.11293**].

[204] E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin and J. I. Cirac, *Dissipative phase transition in a central spin system*, Phys. Rev. A **86**, 012116 (2012), doi:**10.1103/PhysRevA.86.012116** [preprint doi:**10.48550/arXiv:1205.3341**].

[205] T. Prosen, *Third quantization: A general method to solve master equations for quadratic open Fermi systems*, New J. Phys. **10**, 043026 (2008), doi:**10.1088/1367-2630/10/4/043026**.

[206] A. A. Dzhioev and D. S. Kosov, *Super-fermion representation of quantum kinetic equations for the electron transport problem*, J. Chem. Phys. **134**, 044121 (2011), doi:**10.1063/1.3548065** [preprint doi:**10.48550/arXiv:1007.4643**].

[207] U. Harbola and S. Mukamel, *Superoperator nonequilibrium Green's function theory of many-body systems; applications to charge transfer and transport in open junctions*, Phys. Rep. **465**, 191 (2008), doi:**10.1016/j.physrep.2008.05.003** [preprint doi:**10.48550/arXiv:0805.4260**].

[208] A. Dorda, M. Nuss, W. von der Linden and E. Arrigoni, *Auxiliary master equation approach to nonequilibrium correlated impurities*, Phys. Rev. B **89**, 165105 (2014), doi:**10.1103/PhysRevB.89.165105** [preprint doi:**10.48550/arXiv:1312.4586**].

[209] E. Arrigoni and A. Dorda, *Master equations versus Keldysh Green's functions for correlated quantum systems out of equilibrium*, in *Out-of-equilibrium physics of correlated electron systems*, Springer, Cham, Switzerland, ISBN 9783319949550 (2018), doi:**10.1007/978-3-319-94956-7_4**.

[210] D. Werner, J. Lotze and E. Arrigoni, *Configuration interaction based nonequilibrium steady state impurity solver*, Phys. Rev. B **107**, 075119 (2023), doi:**10.1103/PhysRevB.107.075119** [preprint doi:**10.48550/arXiv:2210.09623**].

[211] Y. Takahashi and H. Umezawa, *Thermo field dynamics*, Int. J. Mod. Phys. B **10**, 1755 (1996), doi:**10.1142/S0217979296000817**.

[212] I. Ojima, *Gauge fields at finite temperatures – “Thermo field dynamics” and the KMS condition and their extension to gauge theories*, Ann. Phys. **137**, 1 (1981), doi:**10.1016/0003-4916(81)90058-0**.

[213] V. V. Albert, *Lindbladians with multiple steady states: Theory and applications*, (arXiv preprint) doi:**10.48550/arXiv:1802.00010**.

[214] T. Mori and T. Shirai, *Resolving a discrepancy between Liouvillian gap and relaxation time in boundary-dissipated quantum many-body systems*, Phys. Rev. Lett. **125**, 230604 (2020), doi:**10.1103/PhysRevLett.125.230604** [preprint doi:**10.48550/arXiv:2006.10953**].

[215] Y. Nakanishi and T. Sasamoto, *PT phase transition in open quantum systems with Lindblad dynamics*, Phys. Rev. A **105**, 022219 (2022), doi:**10.1103/PhysRevA.105.022219** [preprint doi:**10.48550/arXiv:2104.07349**].

[216] M. Lax, *Formal theory of quantum fluctuations from a driven state*, Phys. Rev. **129**, 2342 (1963), doi:**10.1103/PhysRev.129.2342**.
[217] Á. Rivas, A. Douglas K. Plato, S. F. Huelga and M. B. Plenio, *Markovian master equations: A critical study*, New J. Phys. **12**, 113032 (2010), doi:10.1088/1367-2630/12/11/113032.

[218] P. P. Hofer, M. Perarnau-Llobet, L. D. M. Miranda, G. Haack, R. Silva, J. B. Brask and N. Brunner, *Markovian master equations for quantum thermal machines: Local versus global approach*, New J. Phys. **19**, 123037 (2017), doi:10.1088/1367-2630/aa964f.

[219] J. O. González, L. A. Correa, G. Nocerino, J. P. Palao, D. Alonso and G. Adesso, *Testing the validity of the 'local' and 'global' GKLS master equations on an exactly solvable model*, Open Syst. Inf. Dyn. **24**, 1740010 (2017), doi:10.1142/S1230161217400108 [preprint doi:10.48550/arXiv.1707.09228].

[220] G. De Chiara, G. Landi, A. Hewgill, B. Reid, A. Ferraro, A. J. Roncaglia and M. Antezza, *Reconciliation of quantum local master equations with thermodynamics*, New J. Phys. **20**, 113024 (2018), doi:10.1088/1367-2630/aaecee.

[221] M. Cattaneo, G. L. Giorgi, S. Maniscalco and R. Zambrini, *Local versus global master equation with common and separate baths: Superiority of the global approach in partial secular approximation*, New J. Phys. **21**, 113045 (2019), doi:10.1088/1367-2630/ab54ac.

[222] A. Hewgill, G. De Chiara and A. Imparato, *Quantum thermodynamically consistent local master equations*, Phys. Rev. Res. **3**, 013165 (2021), doi:10.1103/PhysRevResearch.3.013165 [preprint doi:10.48550/arXiv.2008.04742].

[223] P. P. Potts, A. A. S. Kalaee and A. Wacker, *A thermodynamically consistent Markovian master equation beyond the secular approximation*, New J. Phys. **23**, 123013 (2021), doi:10.1088/1367-2630/ac3b2f.

[224] A. S. Trushechkin, M. Merkli, J. D. Cresser and J. Anders, *Open quantum system dynamics and the mean force Gibbs state*, AVS Quantum Sci. **4**, 012301 (2022), doi:10.1116/5.0073853 [preprint doi:10.48550/arXiv.2110.01671].

[225] G. W. Ford and R. F. O'Connell, *There is no quantum regression theorem*, Phys. Rev. Lett. **77**, 798 (1996), doi:10.1103/PhysRevLett.77.798.

[226] G. Guarnieri, A. Smirne and B. Vacchini, *Quantum regression theorem and non-Markovianity of quantum dynamics*, Phys. Rev. A **90**, 022110 (2014), doi:10.1103/PhysRevA.90.022110 [preprint doi:10.48550/arXiv.1406.5340].

[227] D. Kilda and J. Keeling, *Fluorescence spectrum and thermalization in a driven coupled cavity array*, Phys. Rev. Lett. **122**, 043602 (2019), doi:10.1103/PhysRevLett.122.043602 [preprint doi:10.48550/arXiv.1709.06361].

[228] G. E. Fux, D. Kilda, B. W. Lovett and J. Keeling, *Tensor network simulation of chains of non-Markovian open quantum systems*, Phys. Rev. Res. **5**, 033078 (2023), doi:10.1103/PhysRevResearch.5.033078 [preprint doi:10.48550/arXiv.2201.05529].

[229] I. Reichental, A. Klempner, Y. Kafri and D. Podolsky, *Thermalization in open quantum systems*, Phys. Rev. B **97**, 134301 (2018), doi:10.1103/PhysRevB.97.134301 [preprint doi:10.48550/arXiv.1711.02699].

[230] T. Shirai and T. Mori, *Thermalization in open many-body systems based on eigenstate thermalization hypothesis*, Phys. Rev. E **101**, 042116 (2020), doi:10.1103/PhysRevE.101.042116 [preprint doi:10.48550/arXiv.1812.09713].
[231] P. Kirton, M. M. Roses, J. Keeling and E. G. D. Torre, *Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa*, Adv. Quantum Technol. **2**, 1800043 (2019), doi:10.1002/qute.201800043 [preprint doi:10.48550/arXiv.1805.09828].

[232] E. G. D. Torre, Y. Shchadilova, E. Y. Wilner, M. D. Lukin and E. Demler, *Dicke phase transition without total spin conservation*, Phys. Rev. A **94**, 061802 (2016), doi:10.1103/PhysRevA.94.061802 [preprint doi:10.48550/arXiv.1608.06293].

[233] P. Kirton and J. Keeling, *Suppressing and restoring the Dicke superradiance transition by dephasing and decay*, Phys. Rev. Lett. **118**, 123602 (2017), doi:10.1103/PhysRevLett.118.123602 [preprint doi:10.48550/arXiv.1611.03342].

[234] T. E. Lee, S. Gopalakrishnan and M. D. Lukin, *Unconventional magnetism via optical pumping of interacting spin systems*, Phys. Rev. Lett. **110**, 257204 (2013), doi:10.1103/PhysRevLett.110.257204 [preprint doi:10.48550/arXiv.1304.4959].

[235] M. Foss-Feig, K. R. A. Hazzard, J. J. Bollinger and A. M. Rey, *Nonequilibrium dynamics of arbitrary-range Ising models with decoherence: An exact analytic solution*, Phys. Rev. A **87**, 042101 (2013), doi:10.1103/PhysRevA.87.042101 [preprint doi:10.48550/arXiv.1209.5795].

[236] S. G. Schirmer and X. Wang, *Stabilizing open quantum systems by Markovian reservoir engineering*, Phys. Rev. A **81**, 062306 (2010), doi:10.1103/PhysRevA.81.062306 [preprint doi:10.48550/arXiv.0909.1596].

[237] T. E. Lee, H. Häffner and M. C. Cross, *Antiferromagnetic phase transition in a nonequilibrium lattice of Rydberg atoms*, Phys. Rev. A **84**, 031402 (2011), doi:10.1103/PhysRevA.84.031402 [preprint doi:10.48550/arXiv.1104.0908].

[238] T. E. Lee, H. Häffner and M. C. Cross, *Collective quantum jumps of Rydberg atoms*, Phys. Rev. Lett. **108**, 023602 (2012), doi:10.1103/PhysRevLett.108.023602 [preprint doi:10.48550/arXiv.1109.6506].

[239] C. Joshi, F. Nissen and J. Keeling, *Quantum correlations in the one-dimensional driven dissipative XY model*, Phys. Rev. A **88**, 063835 (2013), doi:10.1103/PhysRevA.88.063835 [preprint doi:10.48550/arXiv.1310.0964].

[240] M. P. A. Fisher, P. B. Weichman, G. Grinstein and D. S. Fisher, *Boson localization and the superfluid-insulator transition*, Phys. Rev. B **40**, 546 (1989), doi:10.1103/PhysRevB.40.546.

[241] L. Pitaevskii and S. Stringari, *Bose-Einstein condensation*, Clarendon Press, Oxford, UK, ISBN 9780198507192 (2003).

[242] L. M. Sieberer, S. D. Huber, E. Altman and S. Diehl, *Dynamical critical phenomena in driven-dissipative systems*, Phys. Rev. Lett. **110**, 195301 (2013), doi:10.1103/PhysRevLett.110.195301 [preprint doi:10.48550/arXiv.1301.5854].

[243] J. R. Schrieffer, *Theory of superconductivity*, CRC Press, Boca Raton, USA, ISBN 9780429495700 (2018), doi:10.1201/9780429495700.

[244] K. Yamamoto, M. Nakagawa, N. Tsuji, M. Ueda and N. Kawakami, *Collective excitations and nonequilibrium phase transition in dissipative fermionic superfluids*, Phys. Rev. Lett. **127**, 055301 (2021), doi:10.1103/PhysRevLett.127.055301 [preprint doi:10.48550/arXiv.2006.06169].
[245] G. Mazza and M. Schirò, Dissipative dynamics of a fermionic superfluid with two-body losses, Phys. Rev. A **107**, L051301 (2023), doi:10.1103/PhysRevA.107.L051301 [preprint doi:10.48550/arXiv:2208.02739].

[246] M. Yamaguchi, K. Kamide, T. Ogawa and Y. Yamamoto, BEC-BCS-laser crossover in Coulomb-correlated electron-hole-photon systems, New J. Phys. **14**, 065001 (2012), doi:10.1088/1367-2630/14/6/065001.

[247] M. Yamaguchi, K. Kamide, R. Nii, T. Ogawa and Y. Yamamoto, Second thresholds in BEC-BCS-laser crossover of exciton-polariton systems, Phys. Rev. Lett. **111**, 026404 (2013), doi:10.1103/PhysRevLett.111.026404 [preprint doi:10.48550/arXiv:1301.4838].

[248] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo and R. Walser, Resonance superfluidity in a quantum degenerate Fermi gas, Phys. Rev. Lett. **87**, 120406 (2001), doi:10.1103/PhysRevLett.87.120406 [preprint doi:10.48550/arXiv:cond-mat/0103479].

[249] E. Timmermans, K. Furuya, P. W. Milonni and A. K. Kerman, Prospect of creating a composite Fermi-Bose superfluid, Phys. Lett. A **285**, 228 (2001), doi:10.1016/S0375-9601(01)00346-2 [preprint doi:10.48550/arXiv:cond-mat/0103327].

[250] Y. Ohashi and A. Griffin, BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance, Phys. Rev. Lett. **89**, 130402 (2002), doi:10.1103/PhysRevLett.89.130402 [preprint doi:10.48550/arXiv:cond-mat/0201262].

[251] M. H. Szymańska, J. Keeling and P. B. Littlewood, Mean-field theory and fluctuation spectrum of a pumped decaying Bose-Fermi system across the quantum condensation transition, Phys. Rev. B **75**, 195331 (2007), doi:10.1103/PhysRevB.75.195331 [preprint doi:10.48550/arXiv:cond-mat/0611456].

[252] R. Hanai, A. Edelman, Y. Ohashi and P. B. Littlewood, Non-Hermitian phase transition from a polariton Bose-Einstein condensate to a photon laser, Phys. Rev. Lett. **122**, 185301 (2019), doi:10.1103/PhysRevLett.122.185301 [preprint doi:10.48550/arXiv:1809.01273].

[253] F. Thompson and A. Kamenev, Field theory of many-body Lindbladian dynamics, Ann. Phys. **455**, 169385 (2023), doi:10.1016/j.aop.2023.169385 [preprint doi:10.48550/arXiv:2301.02953].

[254] A. Altland and B. D. Simons, Condensed matter field theory, Cambridge University Press, Cambridge, UK, ISBN 9780511789984 (2010), doi:10.1017/CBO9780511789984.

[255] N. Gisin and I. C. Percival, The quantum-state diffusion model applied to open systems, J. Phys. A: Math. Gen. **25**, 5677 (1992), doi:10.1088/0305-4470/25/21/023.

[256] J. Dalibard, Y. Castin and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. **68**, 580 (1992), doi:10.1103/PhysRevLett.68.580.

[257] R. Dum, P. Zoller and H. Ritsch, Monte Carlo simulation of the atomic master equation for spontaneous emission, Phys. Rev. A **45**, 4879 (1992), doi:10.1103/PhysRevA.45.4879.

[258] H. J. Carmichael, Quantum trajectory theory for cascaded open systems, Phys. Rev. Lett. **70**, 2273 (1993), doi:10.1103/PhysRevLett.70.2273.
[259] H. M. Wiseman and G. J. Milburn, *Quantum measurement and control*, Cambridge University Press, Cambridge, UK, ISBN 9780521804424 (2009), doi:10.1017/CBO9780511813948.

[260] H. Nha and H. J. Carmichael, *Entanglement within the quantum trajectory description of open quantum systems*, Phys. Rev. Lett. **93**, 120408 (2004), doi:10.1103/PhysRevLett.93.120408 [preprint doi:10.48550/arXiv.quant-ph/0408060].

[261] S. Vogelsberger and D. Spehner, *Average entanglement for Markovian quantum trajectories*, Phys. Rev. A **82**, 052327 (2010), doi:10.1103/PhysRevA.82.052327 [preprint doi:10.48550/arXiv.1006.1317].

[262] T. Vovk and H. Pichler, *Entanglement-optimal trajectories of many-body quantum Markov processes*, Phys. Rev. Lett. **128**, 243601 (2022), doi:10.1103/PhysRevLett.128.243601 [preprint doi:10.48550/arXiv.2111.12048].

[263] T. Vovk and H. Pichler, *Quantum trajectory entanglement in various unravelings of Markovian dynamics*, Phys. Rev. A **110**, 012207 (2024), doi:10.1103/PhysRevA.110.012207.

[264] R. L. Hudson and K. R. Parthasarathy, *Quantum Ito's formula and stochastic evolutions*, Commun. Math. Phys. **93**, 301 (1984), doi:10.1007/BF01258530.

[265] A. Chenu, M. Beau, J. Cao and A. del Campo, *Quantum simulation of generic many-body open system dynamics using classical noise*, Phys. Rev. Lett. **118**, 140403 (2017), doi:10.1103/PhysRevLett.118.140403 [preprint doi:10.48550/arXiv.1608.01317].

[266] J. Nobakht, M. Carlesso, S. Donadi, M. Paternostro and A. Bassi, *Unitary unraveling for the dissipative continuous spontaneous localization model: Application to optomechanical experiments*, Phys. Rev. A **98**, 042109 (2018), doi:10.1103/PhysRevA.98.042109 [preprint doi:10.48550/arXiv.1808.01143].

[267] M. Bauer, D. Bernard and T. Jin, *Equilibrium fluctuations in maximally noisy extended quantum systems*, SciPost Phys. **6**, 045 (2019), doi:10.21468/SciPostPhys.6.4.045 [preprint doi:10.48550/arXiv.1811.09427].

[268] M. B. Plenio and P. L. Knight, *The quantum-jump approach to dissipative dynamics in quantum optics*, Rev. Mod. Phys. **70**, 101 (1998), doi:10.1103/RevModPhys.70.101 [preprint doi:10.48550/arXiv.quant-ph/9702007].

[269] A. J. Daley, *Quantum trajectories and open many-body quantum systems*, Adv. Phys. **63**, 77 (2014), doi:10.1080/00018732.2014.933502 [preprint doi:10.48550/arXiv.1405.6694].

[270] G. T. Landi, M. J. Kewming, M. T. Mitchison and P. P. Potts, *Current fluctuations in open quantum systems: Bridging the gap between quantum continuous measurements and full counting statistics*, PRX Quantum **5**, 020201 (2024), doi:10.1103/PRXQuantum.5.020201 [preprint doi:10.48550/arXiv.2303.04270].

[271] C. Gardiner and P. Zoller, *Quantum noise: A handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics*, Springer, Berlin, Heidelberg, Germany, ISBN 9783540223016 (2004).

[272] B. Zhu, A. M. Rey and J. Schachenmayer, *A generalized phase space approach for solving quantum spin dynamics*, New J. Phys. **21**, 082001 (2019), doi:10.1088/1367-2630/ab354d.
[273] J. Huber, P. Kirton and P. Rabl, *Phase-space methods for simulating the dissipative many-body dynamics of collective spin systems*, SciPost Phys. **10**, 045 (2021), doi:10.21468/SciPostPhys.10.2.045 [preprint doi:10.48550/arXiv.2021.10049].

[274] D. Walls and G. Milburn, *Quantum optics*, Springer, Berlin, Heidelberg, Germany, ISBN 9783642795046 (2012), doi:10.1007/978-3-540-28574-8.

[275] P. D. Drummond and C. W. Gardiner, *Generalised $P$-representations in quantum optics*, J. Phys. A: Math. Gen. **13**, 2353 (1980), doi:10.1088/0305-4470/13/7/018.

[276] A. Gilchrist, C. W. Gardiner and P. D. Drummond, *Positive $P$ representation: Application and validity*, Phys. Rev. A **55**, 3014 (1997), doi:10.1103/PhysRevA.55.3014.

[277] P. Deuar, A. Ferrier, M. Matuszewski, G. Orso and M. H. Szymańska, *Fully quantum scalable description of driven-dissipative lattice models*, PRX Quantum **2**, 010319 (2021), doi:10.1103/PRXQuantum.2.010319 [preprint doi:10.48550/arXiv.2012.02014].

[278] D. Roberts, A. Lingenfelter and A. A. Clerk, *Hidden time-reversal symmetry, quantum detailed balance and exact solutions of driven-dissipative quantum systems*, PRX Quantum **2**, 020336 (2021), doi:10.1103/PRXQuantum.2.020336 [preprint doi:10.48550/arXiv.2011.02148].

[279] T. Prosen and T. H. Seligman, *Quantization over boson operator spaces*, J. Phys. A: Math. Theor. **43**, 392004 (2010), doi:10.1088/1751-8113/43/39/392004.

[280] A. McDonald and A. A. Clerk, *Third quantization of open quantum systems: Dissipative symmetries and connections to phase-space and Keldysh field-theory formulations*, Phys. Rev. Res. **5**, 033107 (2023), doi:10.1103/PhysRevResearch.5.033107 [preprint doi:10.48550/arXiv.2302.14047].

[281] M. Zwolak and G. Vidal, *Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm*, Phys. Rev. Lett. **93**, 207205 (2004), doi:10.1103/PhysRevLett.93.207205 [preprint doi:10.48550/arXiv.cond-mat/0406440].

[282] S. Finazzi, A. Le Boité, F. Storme, A. Baksic and C. Ciuti, *Corner-space renormalization method for driven-dissipative two-dimensional correlated systems*, Phys. Rev. Lett. **115**, 080604 (2015), doi:10.1103/PhysRevLett.115.080604 [preprint doi:10.48550/arXiv.1502.05651].

[283] J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio and D. Rossini, *Cluster mean-field approach to the steady-state phase diagram of dissipative spin systems*, Phys. Rev. X **6**, 031011 (2016), doi:10.1103/PhysRevX.6.031011 [preprint doi:10.48550/arXiv.1602.06553].

[284] O. Scarlatella, A. A. Clerk, R. Fazio and M. Schiró, *Dynamical mean-field theory for Markovian open quantum many-body systems*, Phys. Rev. X **11**, 031018 (2021), doi:10.1103/PhysRevX.11.031018 [preprint doi:10.48550/arXiv.2008.02563].

[285] A. Nagy and V. Savona, *Variational quantum Monte Carlo method with a neural-network Ansatz for open quantum systems*, Phys. Rev. Lett. **122**, 250501 (2019), doi:10.1103/PhysRevLett.122.250501 [preprint doi:10.48550/arXiv.1902.09483].

[286] M. J. Hartmann and G. Carleo, *Neural-network approach to dissipative quantum many-body dynamics*, Phys. Rev. Lett. **122**, 250502 (2019), doi:10.1103/PhysRevLett.122.250502 [preprint doi:10.48550/arXiv.1902.05131].
[287] F. Vicentini, A. Biella, N. Regnault and C. Ciuti, *Variational neural-network Ansatz for steady states in open quantum systems*, Phys. Rev. Lett. **122**, 250503 (2019), doi:10.1103/PhysRevLett.122.250503 [preprint doi:10.48550/arXiv.1902.10104].

[288] J. R. Johansson, P. D. Nation and F. Nori, *QuTiP: An open-source Python framework for the dynamics of open quantum systems*, Comput. Phys. Commun. **183**, 1760 (2012), doi:10.1016/j.cpc.2012.02.021 [preprint doi:10.48550/arXiv.1110.0573].

[289] J. R. Johansson, P. D. Nation and F. Nori, *QuTiP 2: A Python framework for the dynamics of open quantum systems*, Comput. Phys. Commun. **184**, 1234 (2013), doi:10.1016/j.cpc.2012.11.019 [preprint doi:10.48550/arXiv.1211.6518].

[290] A. C. Y. Li, F. Petruccione and J. Koch, *Perturbative approach to Markovian open quantum systems*, Sci. Rep. **4**, 4887 (2014), doi:10.1038/srep04887 [preprint doi:10.48550/arXiv.1311.3227].

[291] A. C. Y. Li, F. Petruccione and J. Koch, *Resummation for nonequilibrium perturbation theory and application to open quantum lattices*, Phys. Rev. X **6**, 021037 (2016), doi:10.1103/PhysRevX.6.021037 [preprint doi:10.48550/arXiv.1511.01551].

[292] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro and S. Lloyd, *Gaussian quantum information*, Rev. Mod. Phys. **84**, 621 (2012), doi:10.1103/RevModPhys.84.621 [preprint doi:10.48550/arXiv.1110.3234].

[293] B. Horstmann, *Quantum simulations of out-of-equilibrium phenomena*, PhD thesis, Technische Universität München, Munich, Germany (2012).

[294] L. Mazza, *Quantum simulation of topological states of matter*, PhD thesis, Technische Universität München, Munich, Germany (2012).

[295] B. Horstmann, J. I. Cirac and G. Giedke, *Noise-driven dynamics and phase transitions in fermionic systems*, Phys. Rev. A **87**, 012108 (2013), doi:10.1103/PhysRevA.87.012108 [preprint doi:10.48550/arXiv.1207.1653].

[296] Y. Zhang and T. Barthel, *Criticality and phase classification for quadratic open quantum many-body systems*, Phys. Rev. Lett. **129**, 120401 (2022), doi:10.1103/PhysRevLett.129.120401 [preprint doi:10.48550/arXiv.2204.05346].

[297] M. Esposito and P. Gaspard, *Exactly solvable model of quantum diffusion*, J. Stat. Phys. **121**, 463 (2005), doi:10.1007/s10955-005-7577-x.

[298] M. Esposito and P. Gaspard, *Emergence of diffusion in finite quantum systems*, Phys. Rev. B **71**, 214302 (2005), doi:10.1103/PhysRevB.71.214302.

[299] V. Eisler, *Crossover between ballistic and diffusive transport: The quantum exclusion process*, J. Stat. Mech.: Theory Exp. P06007 (2011), doi:10.1088/1742-5468/2011/06/p06007.

[300] M. Žnidarič, *A matrix product solution for a nonequilibrium steady state of an XX chain*, J. Phys. A: Math. Theor. **43**, 415004 (2010), doi:10.1088/1751-8113/43/41/415004.

[301] M. Žnidarič, *Exact solution for a diffusive nonequilibrium steady state of an open quantum chain*, J. Stat. Mech.: Theory Exp. L05002 (2010), doi:10.1088/1742-5468/2010/05/L05002.
[302] X. Turkeshi and M. Schiró, *Diffusion and thermalization in a boundary-driven dephasing model*, Phys. Rev. B **104**, 144301 (2021), doi:10.1103/PhysRevB.104.144301 [preprint doi:10.48550/arXiv:2106.13180].

[303] M. V. Medvedyeva, F. H. L. Essler and T. Prosen, *Exact Bethe Ansatz spectrum of a tight-binding chain with dephasing noise*, Phys. Rev. Lett. **117**, 137202 (2016), doi:10.1103/PhysRevLett.117.137202 [preprint doi:10.48550/arXiv:1606.09122].

[304] T. Jin, J. S. Ferreira, M. Filippone and T. Giamarchi, *Exact description of quantum stochastic models as quantum resistors*, Phys. Rev. Res. **4**, 013109 (2022), doi:10.1103/PhysRevResearch.4.013109 [preprint doi:10.48550/arXiv:2106.14765].

[305] M. Žnidarič, B. Žunkovič and T. Prosen, *Transport properties of a boundary-driven one-dimensional gas of spinless fermions*, Phys. Rev. E **84**, 051115 (2011), doi:10.1103/PhysRevE.84.051115 [preprint doi:10.48550/arXiv:1107.2889].

[306] M. Dykman, *Heating and cooling of local and quasilocal vibrations by a nonresonance field*, Sov. Phys. Solid State **20**, 1306 (1978).

[307] P. D. Drummond and D. F. Walls, *Quantum theory of optical bistability. I. Nonlinear polarisability model*, J. Phys. A: Math. Gen. **13**, 725 (1980), doi:10.1088/0305-4470/13/2/034.

[308] L. A. Lugiato, *II Theory of optical bistability*, in *Progress in optics*, Elsevier, Amsterdam, Netherlands, ISBN 9780444867612 (1984), doi:10.1016/S0079-6638(08)70122-7.

[309] S. Chaturvedi and V. Srinivasan, *Class of exactly solvable master equations describing coupled nonlinear oscillators*, Phys. Rev. A **43**, 4054 (1991), doi:10.1103/PhysRevA.43.4054.

[310] N. Bartolo, F. Minganti, W. Casteels and C. Ciuti, *Exact steady state of a Kerr resonator with one- and two-photon driving and dissipation: Controllable Wigner-function multimodality and dissipative phase transitions*, Phys. Rev. A **94**, 033841 (2016), doi:10.1103/PhysRevA.94.033841 [preprint doi:10.48550/arXiv:1607.06739].

[311] D. Roberts and A. A. Clerk, *Driven-dissipative quantum Kerr resonators: New exact solutions*, photon blockade and quantum bistability, Phys. Rev. X **10**, 021022 (2020), doi:10.1103/PhysRevX.10.021022 [preprint doi:10.48550/arXiv:1910.00574].

[312] S. Lieu, R. Belyansky, J. T. Young, R. Lundgren, V. V. Albert and A. V. Gorshkov, *Symmetry breaking and error correction in open quantum systems*, Phys. Rev. Lett. **125**, 240405 (2020), doi:10.1103/PhysRevLett.125.240405 [preprint doi:10.48550/arXiv:2008.02816].

[313] X. H. H. Zhang and H. U. Baranger, *Driven-dissipative phase transition in a Kerr oscillator: From semiclassical $PT$ symmetry to quantum fluctuations*, Phys. Rev. A **103**, 033711 (2021), doi:10.1103/PhysRevA.103.033711 [preprint doi:10.48550/arXiv:2007.01422].

[314] K. Stannigel, P. Rabl and P. Zoller, *Driven-dissipative preparation of entangled states in cased quantum-optical networks*, New J. Phys. **14**, 063014 (2012), doi:10.1088/1367-2630/14/6/063014.

[315] A. McDonald and A. A. Clerk, *Exact solutions of interacting dissipative systems via weak symmetries*, Phys. Rev. Lett. **128**, 033602 (2022), doi:10.1103/PhysRevLett.128.033602 [preprint doi:10.48550/arXiv:2109.13221].
[316] D. Roberts and A. A. Clerk, *Exact solution of the infinite-range dissipative transverse-field Ising model*, Phys. Rev. Lett. **131**, 190403 (2023), doi:10.1103/PhysRevLett.131.190403 [preprint doi:10.48550/arXiv:2307.06946].

[317] M. de Leeuw, C. Paletta and B. Pozsgay, *Constructing integrable Lindblad superoperators*, Phys. Rev. Lett. **126**, 240403 (2021), doi:10.1103/PhysRevLett.126.240403 [preprint doi:10.48550/arXiv:2101.08279].

[318] D. Karevski, V. Popkov and G. M. Schütz, *Exact matrix product solution for the boundary-driven Lindblad XXZ chain*, Phys. Rev. Lett. **110**, 047201 (2013), doi:10.1103/PhysRevLett.110.047201 [preprint doi:10.48550/arXiv:1211.7010].

[319] A. Marché, H. Yoshida, A. Nardin, H. Katsura and L. Mazza, *Universality and two-body losses: Lessons from the effective non-Hermitian dynamics of two particles*, Phys. Rev. A **110**, 033321 (2024), doi:10.1103/PhysRevA.110.033321 [preprint doi:10.48550/arXiv:2405.04789].

[320] D. A. Rowlands and A. Lamacraft, *Noisy spins and the Richardson-Gaudin model*, Phys. Rev. Lett. **120**, 090401 (2018), doi:10.1103/PhysRevLett.120.090401 [preprint doi:10.48550/arXiv:1711.00828].

[321] P. W. Claeys and A. Lamacraft, *Dissipative dynamics in open XXZ Richardson-Gaudin models*, Phys. Rev. Res. **4**, 013033 (2022), doi:10.1103/PhysRevResearch.4.013033 [preprint doi:10.48550/arXiv:2108.01677].

[322] F. H. L. Essler and L. Piroli, *Integrability of one-dimensional Lindbladians from operator-space fragmentation*, Phys. Rev. E **102**, 062210 (2020), doi:10.1103/PhysRevE.102.062210 [preprint doi:10.48550/arXiv:2009.11745].

[323] B. Buča, C. Booker, M. Medenjak and D. Jaksch, *Bethe Ansatz approach for dissipation: Exact solutions of quantum many-body dynamics under loss*, New J. Phys. **22**, 123040 (2020), doi:10.1088/1367-2630/abd124.

[324] J. Oitmaa, C. Hamer, and W. Zheng, *Series expansion methods for strongly interacting lattice models*, Cambridge University Press, Cambridge, UK, ISBN 9780511584398 (2006), doi:10.1017/CBO9780511584398.

[325] C. Domb and M. S. Green, *Phase transitions and critical phenomena, volume 3*, Academic Press, New York, USA, ISBN 9780122203039 (1964).

[326] M. P. Gelfand, R. R. P. Singh and D. A. Huse, *Perturbation expansions for quantum many-body systems*, J. Stat. Phys. **59**, 1093 (1990), doi:10.1007/BF01334744.

[327] A. Biella, J. Jin, O. Viyuela, C. Ciuti, R. Fazio and D. Rossini, *Linked cluster expansions for open quantum systems on a lattice*, Phys. Rev. B **97**, 035103 (2018), doi:10.1103/PhysRevB.97.035103 [preprint doi:10.48550/arXiv:1708.08666].

[328] B. Tang, E. Khatami and M. Rigol, *A short introduction to numerical linked-cluster expansions*, Comput. Phys. Commun. **184**, 557 (2013), doi:10.1016/j.cpc.2012.10.008 [preprint doi:10.48550/arXiv:1207.3366].

[329] J. Qiao, W. Chang, X. Li and J. Jin, *Numerical linked-cluster expansion for the dissipative XYZ model on a triangular lattice*, J. Phys. Commun. **4**, 015020 (2020), doi:10.1088/2399-6528/ab6e13.
[330] R. Rota, F. Storme, N. Bartolo, R. Fazio and C. Ciuti, *Critical behavior of dissipative two-dimensional spin lattices*, Phys. Rev. B **95**, 134431 (2017), doi:10.1103/PhysRevB.95.134431 [preprint doi:10.48550/arXiv.1609.02848].

[331] A. Kshetrimayum, H. Weimer and R. Orús, *A simple tensor network algorithm for two-dimensional steady states*, Nat. Commun. **8**, 1291 (2017), doi:10.1038/s41467-017-01511-6.

[332] F. Benatti, A. Nagy and H. Narnhofer, *Asymptotic entanglement and Lindblad dynamics: A perturbative approach*, J. Phys. A: Math. Theor. **44**, 155303 (2011), doi:10.1088/1751-8113/44/15/155303.

[333] Z. Lenarčič, F. Lange and A. Rosch, *Perturbative approach to weakly driven many-particle systems in the presence of approximate conservation laws*, Phys. Rev. B **97**, 024302 (2018), doi:10.1103/PhysRevB.97.024302 [preprint doi:10.48550/arXiv.1706.05700].

[334] Z. Lenarčič, E. Altman and A. Rosch, *Activating many-body localization in solids by driving with light*, Phys. Rev. Lett. **121**, 267603 (2018), doi:10.1103/PhysRevLett.121.267603 [preprint doi:10.48550/arXiv.1806.04772].

[335] P. A. M. Dirac, *Note on exchange phenomena in the Thomas atom*, Math. Proc. Camb. Philos. Soc. **26**, 376 (1930), doi:10.1017/S0305004100016108.

[336] J. Frenkel, *Wave mechanics, advanced general theory*, Clarendon Press, Oxford, UK (1934).

[337] J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde and F. Versraete, *Time-dependent variational principle for quantum lattices*, Phys. Rev. Lett. **107**, 070601 (2011), doi:10.1103/PhysRevLett.107.070601 [preprint doi:10.48550/arXiv.1103.0936].

[338] M. C. Gutzwiller, *Effect of correlation on the ferromagnetism of transition metals*, Phys. Rev. Lett. **10**, 159 (1963), doi:10.1103/PhysRevLett.10.159.

[339] W. Krauth, M. Caffarel and J.-P. Bouchaud, *Gutzwiller wave function for a model of strongly interacting bosons*, Phys. Rev. B **45**, 3137 (1992), doi:10.1103/PhysRevB.45.3137.

[340] M. Schiró, C. Joshi, M. Bordyuh, R. Fazio, J. Keeling and H. E. Türeci, *Exotic attractors of the nonequilibrium Rabi-Hubbard model*, Phys. Rev. Lett. **116**, 143603 (2016), doi:10.1103/PhysRevLett.116.143603 [preprint doi:10.48550/arXiv.1503.04456].

[341] J. Huber, P. Kirton and P. Rabl, *Nonequilibrium magnetic phases in spin lattices with gain and loss*, Phys. Rev. A **102**, 012219 (2020), doi:10.1103/PhysRevA.102.012219 [preprint doi:10.48550/arXiv.1908.02290].

[342] J. Jin, D. Rossini, R. Fazio, M. Leib and M. J. Hartmann, *Photon solid phases in driven arrays of nonlinearly coupled cavities*, Phys. Rev. Lett. **110**, 163605 (2013), doi:10.1103/PhysRevLett.110.163605 [preprint doi:10.48550/arXiv.1302.2242].

[343] R. A. Barankov, L. S. Levitov and B. Z. Spivak, *Collective Rabi oscillations and solitons in a time-dependent BCS pairing problem*, Phys. Rev. Lett. **93**, 160401 (2004), doi:10.1103/PhysRevLett.93.160401 [preprint doi:10.48550/arXivcond-mat/0312053].
[344] R. A. Barankov and L. S. Levitov, *Synchronization in the BCS pairing dynamics as a critical phenomenon*, Phys. Rev. Lett. **96**, 230403 (2006), doi:10.1103/PhysRevLett.96.230403 [preprint doi:10.48550/arXiv:cond-mat/0603317].

[345] D. Rossini, A. Ghermaoui, M. B. Aguilera, R. Vatré, R. Bouganne, J. Beugnon, F. Gerbier and L. Mazza, *Strong correlations in lossy one-dimensional quantum gases: From the quantum Zeno effect to the generalized Gibbs ensemble*, Phys. Rev. A **103**, L060201 (2021), doi:10.1103/PhysRevA.103.L060201 [preprint doi:10.48550/arXiv:cond-mat/040318].

[346] M. Capello, F. Becca, M. Fabrizio, S. Sorella and E. Tosatti, *Variational description of Mott insulators*, Phys. Rev. Lett. **94**, 026406 (2005), doi:10.1103/PhysRevLett.94.026406 [preprint doi:10.48550/arXiv:cond-mat/0403430].

[347] M. Capello, F. Becca, M. Fabrizio and S. Sorella, *Mott transition in bosonic systems: Insights from the variational approach*, Phys. Rev. B **77**, 144517 (2008), doi:10.1103/PhysRevB.77.144517 [preprint doi:10.48550/arXiv:0801.0952].

[348] G. Mahan, *Many-particle physics*, Springer, New York, USA, ISBN 9780306463389 (2000), doi:10.1007/978-1-4757-5714-9.

[349] G. Parisi, *Statistical field theory*, Addison-Wesley, Redwood City, USA, ISBN 9780201059854 (1988).

[350] A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, *Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions*, Rev. Mod. Phys. **68**, 13 (1996), doi:10.1103/RevModPhys.68.13.

[351] H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka and P. Werner, *Nonequilibrium dynamical mean-field theory and its applications*, Rev. Mod. Phys. **86**, 779 (2014), doi:10.1103/RevModPhys.86.779.

[352] L. F. Cugliandolo, *Recent applications of dynamical mean-field methods*, Annu. Rev. Condens. Matter Phys. **15**, 177 (2024), doi:10.1146/annurev-conmatphys-040721-022848.

[353] K. Byczuk and D. Vollhardt, *Correlated bosons on a lattice: Dynamical mean-field theory for Bose-Einstein condensed and normal phases*, Phys. Rev. B **77**, 235106 (2008), doi:10.1103/PhysRevB.77.235106 [preprint doi:10.48550/arXiv:0706.0839].

[354] P. Anders, E. Gull, L. Pollet, M. Troyer and P. Werner, *Dynamical mean field solution of the Bose-Hubbard model*, Phys. Rev. Lett. **105**, 096402 (2010), doi:10.1103/PhysRevLett.105.096402 [preprint doi:10.48550/arXiv:1004.0510].

[355] Y. Murakami, D. Golež, M. Eckstein and P. Werner, *Photoinduced nonequilibrium states in Mott insulators*, Rev. Mod. Phys. **97**, 035001 (2025), doi:10.1103/tkjh-lr83 [preprint doi:10.48550/arXiv:2310.05201].

[356] J. Panas, M. Pasek, A. Dhar, T. Qin, A. Geißler, M. Hafez-Torbati, M. E. Sorantin, I. Titvinidze and W. Hofstetter, *Density-wave steady-state phase of dissipative ultra-cold fermions with nearest-neighbor interactions*, Phys. Rev. B **99**, 115125 (2019), doi:10.1103/PhysRevB.99.115125 [preprint doi:10.48550/arXiv:1811.07369].

[357] N. Makri, *The linear response approximation and its lowest order corrections: An influence functional approach*, J. Phys. Chem. B **103**, 2823 (1999), doi:10.1021/jp9847540.
[358] T. Mori, *Exactness of the mean-field dynamics in optical cavity systems*, J. Stat. Mech.: Theory Exp. P06005 (2013), doi:10.1088/1742-5468/2013/06/P06005.

[359] F. Carollo and I. Lesanovsky, *Exactness of mean-field equations for open Dicke models with an application to pattern retrieval dynamics*, Phys. Rev. Lett. **126**, 230601 (2021), doi:10.1103/PhysRevLett.126.230601 [preprint doi:10.48550/arXiv.2009.13932].

[360] E. Fiorelli, M. Müller, I. Lesanovsky and F. Carollo, *Mean-field dynamics of open quantum systems with collective operator-valued rates: Validity and application*, New J. Phys. **25**, 083010 (2023), doi:10.1088/1367-2630/ace470.

[361] P. Fowler-Wright, K. B. Arnardóttir, P. Kirton, B. W. Lovett and J. Keeling, *Determining the validity of cumulant expansions for central spin models*, Phys. Rev. Res. **5**, 033148 (2023), doi:10.1103/PhysRevResearch.5.033148 [preprint doi:10.48550/arXiv.2303.04410].

[362] F. Carollo, *Non-Gaussian dynamics of quantum fluctuations and mean-field limit in open quantum central spin systems*, Phys. Rev. Lett. **131**, 227102 (2023), doi:10.1103/PhysRevLett.131.227102 [preprint doi:10.48550/arXiv.2305.15547].

[363] L. Sá, P. Ribeiro and T. Prosen, *Lindbladian dissipation of strongly-correlated quantum matter*, Phys. Rev. Res. **4**, L022068 (2022), doi:10.1103/PhysRevResearch.4.L022068 [preprint doi:10.48550/arXiv.2112.12109].

[364] K. Kawabata, A. Kulkarni, J. Li, T. Numasawa and S. Ryu, *Dynamical quantum phase transitions in Sachdev-Ye-Kitaev Lindbladians*, Phys. Rev. B **108**, 075110 (2023), doi:10.1103/PhysRevB.108.075110 [preprint doi:10.48550/arXiv.2210.04093].

[365] P. Anders, E. Gull, L. Pollet, M. Troyer and P. Werner, *Dynamical mean-field theory for bosons*, New J. Phys. **13**, 075013 (2011), doi:10.1088/1367-2630/13/7/075013.

[366] H. U. R. Strand, M. Eckstein and P. Werner, *Nonequilibrium dynamical mean-field theory for bosonic lattice models*, Phys. Rev. X **5**, 011038 (2015), doi:10.1103/PhysRevX.5.011038 [preprint doi:10.48550/arXiv.1405.6941].

[367] H. U. R. Strand, M. Eckstein and P. Werner, *Nonequilibrium dynamical mean-field theory for bosonic lattice models*, Phys. Rev. X **5**, 011038 (2015), doi:10.1103/PhysRevX.5.011038 [preprint doi:10.48550/arXiv.1405.6941].

[368] O. Scarlatella, A. A. Clerk and M. Schirò, *Stability of dissipatively-prepared Mott insulators of photons*, Phys. Rev. Res. **6**, 013033 (2024), doi:10.1103/PhysRevResearch.6.013033 [preprint doi:10.48550/arXiv.2303.09673].

[369] E. Arrigoni, M. Knap and W. von der Linden, *Nonequilibrium dynamical mean-field theory: An auxiliary quantum master equation approach*, Phys. Rev. Lett. **110**, 086403 (2013), doi:10.1103/PhysRevLett.110.086403 [preprint doi:10.48550/arXiv.1210.4167].

[370] M. Seclì, M. Capone and M. Schirò, *Steady-state quantum Zeno effect of driven-dissipative bosons with dynamical mean-field theory*, Phys. Rev. A **106**, 013707 (2022), doi:10.1103/PhysRevA.106.013707 [preprint doi:10.48550/arXiv.2201.03191].

[371] M. Schiro and O. Scarlatella, *Quantum impurity models coupled to Markovian and non-Markovian baths*, J. Chem. Phys. **151**, 044102 (2019), doi:10.1063/1.5100157 [preprint doi:10.48550/arXiv.1904.07679].
[372] O. Scarlatella and M. Schirò, *Self-consistent dynamical maps for open quantum systems*, SciPost Phys. **16**, 026 (2024), doi:10.21468/SciPostPhys.16.1.026 [preprint sci-post.org/submissions/2107.05553v5/].

[373] P. Degenfeld-Schonburg and M. J. Hartmann, *Self-consistent projection operator theory for quantum many-body systems*, Phys. Rev. B **89**, 245108 (2014), doi:10.1103/PhysRevB.89.245108 [preprint doi:10.48550/arXiv.1307.7027].

[374] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer and P. Werner, *Continuous-time Monte Carlo methods for quantum impurity models*, Rev. Mod. Phys. **83**, 349 (2011), doi:10.1103/RevModPhys.83.349 [preprint doi:10.48550/arXiv.1012.4474].

[375] M. Schiró and M. Fabrizio, *Real-time diagrammatic Monte Carlo for nonequilibrium quantum transport*, Phys. Rev. B **79**, 153302 (2009), doi:10.1103/PhysRevB.79.153302 [preprint doi:10.48550/arXiv.0808.0589].

[376] P. Werner, T. Oka and A. J. Millis, *Diagrammatic Monte Carlo simulation of nonequilibrium systems*, Phys. Rev. B **79**, 035320 (2009), doi:10.1103/PhysRevB.79.035320 [preprint doi:10.48550/arXiv.0810.2345].

[377] G. Cohen, E. Gull, D. R. Reichman and A. J. Millis, *Taming the dynamical sign problem in real-time evolution of quantum many-body problems*, Phys. Rev. Lett. **115**, 266802 (2015), doi:10.1103/PhysRevLett.115.266802 [preprint doi:10.48550/arXiv.1510.03534].

[378] M. Vanhoecke and M. Schirò, *Diagrammatic Monte Carlo for dissipative quantum impurity models*, Phys. Rev. B **109**, 125125 (2024), doi:10.1103/PhysRevB.109.125125 [preprint doi:10.48550/arXiv.2311.17839].

[379] J. Thoenniss, M. Sonner, A. Lerose and D. A. Abanin, *Efficient method for quantum impurity problems out of equilibrium*, Phys. Rev. B **107**, L201115 (2023), doi:10.1103/PhysRevB.107.L201115 [preprint doi:10.48550/arXiv.2211.10272].

[380] N. Ng, G. Park, A. J. Millis, G. K.-L. Chan and D. R. Reichman, *Real-time evolution of Anderson impurity models via tensor network influence functionals*, Phys. Rev. B **107**, 125103 (2023), doi:10.1103/PhysRevB.107.125103 [preprint doi:10.48550/arXiv.2211.10430].

[381] M. Cygorek, M. Cosacchi, A. Vagov, V. M. Axt, B. W. Lovett, J. Keeling and E. M. Gauger, *Simulation of open quantum systems by automated compression of arbitrary environments*, Nat. Phys. **18**, 662 (2022), doi:10.1038/s41567-022-01544-9.

[382] M. Eckstein and P. Werner, *Nonequilibrium dynamical mean-field calculations based on the noncrossing approximation and its generalizations*, Phys. Rev. B **82**, 115115 (2010), doi:10.1103/PhysRevB.82.115115 [preprint doi:10.48550/arXiv.1005.1872].

[383] M. Vanhoecke and M. Schirò, *Kondo-Zeno crossover in the dynamics of a monitored quantum dot*, Nat. Commun. **16**, 6155 (2025), doi:10.1038/s41467-025-61287-y [preprint doi:10.48550/arXiv.2405.17348].

[384] N. Tsuji, T. Oka and H. Aoki, *Nonequilibrium steady state of photoexcited correlated electrons in the presence of dissipation*, Phys. Rev. Lett. **103**, 047403 (2009), doi:10.1103/PhysRevLett.103.047403 [preprint doi:10.48550/arXiv.0903.2332].
[385] J. Li, C. Aron, G. Kotliar and J. E. Han, *Electric-field-driven resistive switching in the dissipative Hubbard model*, Phys. Rev. Lett. **114**, 226403 (2015), doi:10.1103/PhysRevLett.114.226403 [preprint doi:10.48550/arXiv:1410.0626].

[386] Y. Murakami and P. Werner, *Nonequilibrium steady states of electric field driven Mott insulators*, Phys. Rev. B **98**, 075102 (2018), doi:10.1103/PhysRevB.98.075102 [preprint doi:10.48550/arXiv:1804.08257].

[387] J. Li, D. Golez, P. Werner and M. Eckstein, *η-paired superconducting hidden phase in photodoped Mott insulators*, Phys. Rev. B **102**, 165136 (2020), doi:10.1103/PhysRevB.102.165136 [preprint doi:10.48550/arXiv:1908.08693].

[388] M. Eckstein and P. Werner, *Photoinduced states in a Mott insulator*, Phys. Rev. Lett. **110**, 126401 (2013), doi:10.1103/PhysRevLett.110.126401 [preprint doi:10.48550/arXiv:1207.0402].

[389] F. Peronaci, O. Parcollet and M. Schiró, *Enhancement of local pairing correlations in periodically driven Mott insulators*, Phys. Rev. B **101**, 161101 (2020), doi:10.1103/PhysRevB.101.161101 [preprint doi:10.48550/arXiv:1904.00857].

[390] T. M. Mazzocchi, P. Gazzaneo, J. Lotze and E. Arrigoni, *Correlated Mott insulators in strong electric fields: Role of phonons in heat dissipation*, Phys. Rev. B **106**, 125123 (2022), doi:10.1103/PhysRevB.106.125123 [preprint doi:10.48550/arXiv:2207.01921].

[391] F. Verstraete, V. Murg and J. I. Cirac, *Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems*, Adv. Phys. **57**, 143 (2008), doi:10.1080/14789940801912366 [preprint doi:10.48550/arXiv:0907.2796].

[392] U. Schollwöck, *The density-matrix renormalization group in the age of matrix product states*, Ann. Phys. **326**, 96 (2011), doi:10.1016/j.aop.2010.09.012 [preprint doi:10.48550/arXiv:1008.3477].

[393] R. Orús, *A practical introduction to tensor networks: Matrix product states and projected entangled pair states*, Ann. Phys. **349**, 117 (2014), doi:10.1016/j.aop.2014.06.013 [preprint doi:10.48550/arXiv:1306.2164].

[394] J. I. Cirac, D. Pérez-García, N. Schuch and F. Verstraete, *Matrix product states and projected entangled pair states: Concepts, symmetries, theorems*, Rev. Mod. Phys. **93**, 045003 (2021), doi:10.1103/RevModPhys.93.045003 [preprint doi:10.48550/arXiv:2011.12127].

[395] F. Verstraete, J. J. García-Ripoll and J. I. Cirac, *Matrix product density operators: Simulation of finite-temperature and dissipative systems*, Phys. Rev. Lett. **93**, 207204 (2004), doi:10.1103/PhysRevLett.93.207204 [preprint doi:10.48550/arXiv:cond-mat/0406426].

[396] U. Schollwöck, *The density-matrix renormalization group*, Rev. Mod. Phys. **77**, 259 (2005), doi:10.1103/RevModPhys.77.259 [preprint doi:10.48550/arXiv:cond-mat/0409292].

[397] S. R. White and R. M. Noack, *Real-space quantum renormalization groups*, Phys. Rev. Lett. **68**, 3487 (1992), doi:10.1103/PhysRevLett.68.3487.
[398] A. H. Werner, D. Jaschke, P. Silvi, M. Kliesch, T. Calarco, J. Eisert and S. Montangero, *Positive tensor network approach for simulating open quantum many-body systems*, Phys. Rev. Lett. **116**, 237201 (2016), doi:10.1103/PhysRevLett.116.237201 [preprint doi:10.48550/arXiv:1412.5746].

[399] R. Orús and G. Vidal, *Infinite time-evolving block decimation algorithm beyond unitary evolution*, Phys. Rev. B **78**, 155117 (2008), doi:10.1103/PhysRevB.78.155117 [preprint doi:10.48550/arXiv:0711.3960].

[400] E. Mascarenhas, H. Flayac and V. Savona, *Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays*, Phys. Rev. A **92**, 022116 (2015), doi:10.1103/PhysRevA.92.022116 [preprint doi:10.48550/arXiv:1504.06127].

[401] J. Cui, J. I. Cirac and M. C. Bañuls, *Variational matrix product operators for the steady state of dissipative quantum systems*, Phys. Rev. Lett. **114**, 220601 (2015), doi:10.1103/PhysRevLett.114.220601 [preprint doi:10.48550/arXiv:1501.06786].

[402] Y. Zhou, E. M. Stoudenmire and X. Waintal, *What limits the simulation of quantum computers?*, Phys. Rev. X **10**, 041038 (2020), doi:10.1103/PhysRevX.10.041038 [preprint doi:10.48550/arXiv:2002.07730].

[403] K. Noh, L. Jiang and B. Fefferman, *Efficient classical simulation of noisy random quantum circuits in one dimension*, Quantum **4**, 318 (2020), doi:10.22331/q-2020-09-11-318.

[404] K. Modi, A. Brodutch, H. Cable, T. Paterek and V. Vedral, *The classical-quantum boundary for correlations: Discord and related measures*, Rev. Mod. Phys. **84**, 1655 (2012), doi:10.1103/RevModPhys.84.1655 [preprint doi:10.48550/arXiv:1112.6238].

[405] M. Kolodrubetz, *Optimality of Lindblad unfolding in measurement phase transitions*, Phys. Rev. B **107**, L140301 (2023), doi:10.1103/PhysRevB.107.L140301 [preprint doi:10.48550/arXiv:2111.11391].

[406] G. Preisser, D. Wellnitz, T. Botzung and J. Schachenmayer, *Comparing bipartite entropy growth in open-system matrix-product simulation methods*, Phys. Rev. A **108**, 012616 (2023), doi:10.1103/PhysRevA.108.012616 [preprint doi:10.48550/arXiv:2303.09426].

[407] D. Kilda, A. Biella, M. Schirò, R. Fazio and J. Keeling, *On the stability of the infinite projected entangled pair operator Ansatz for driven-dissipative 2D lattices*, SciPost Phys. Core **4**, 005 (2021), doi:10.21468/SciPostPhysCore.4.1.005 [preprint doi:10.48550/arXiv:2012.03095].

[408] C. Mc Keever and M. H. Szymańska, *Stable iPEPO tensor-network algorithm for dynamics of two-dimensional open quantum lattice models*, Phys. Rev. X **11**, 021035 (2021), doi:10.1103/PhysRevX.11.021035 [preprint doi:10.48550/arXiv:2012.12233].

[409] J. Prior, A. W. Chin, S. F. Huelga and M. B. Plenio, *Efficient simulation of strong system-environment interactions*, Phys. Rev. Lett. **105**, 050404 (2010), doi:10.1103/PhysRevLett.105.050404 [preprint doi:10.48550/arXiv:1003.5503].

[410] A. W. Chin, Á. Rivas, S. F. Huelga and M. B. Plenio, *Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials*, J. Math. Phys. **51**, 092109 (2010), doi:10.1063/1.3490188 [preprint doi:10.48550/arXiv:1006.4507].