Driven-dissipative preparation of entangled states in cascaded quantum-optical networks

We study the dissipative dynamics and the formation of entangled states in driven cascaded quantum networks, where multiple systems are coupled to a common unidirectional bath. Specifically, we identify the conditions under which emission and coherent reabsorption of radiation drives the whole network into a pure stationary state with non-trivial quantum correlations between the individual nodes. We illustrate this effect in more detail for the example of cascaded two-level systems, where we present an explicit preparation scheme that allows one to tune the whole network through"bright"and"dark"states associated with different multi-partite entanglement patterns. In a complementary setting consisting of cascaded non-linear cavities, we find that two cavity modes can be driven into a non-Gaussian entangled dark state. Potential realizations of such cascaded networks with optical and microwave photons are discussed.


Introduction
The coupling of a quantum system to an environment is often associated with decoherence. This is exemplified by the field of quantum computation, where the presence of additional reservoirs degrades the performance of quantum algorithms based on unitary operations executed on large many body systems [1]. On the other hand, in many quantum optical settings the environment is actually useful for realizing certain applications. Prominent examples are laser cooling or optical pumping, where quantum systems are prepared in highly pure states with the help of a reservoir [2], or continuous measurement schemes which enable the conditioned preparation of quantum states [3]. While quantum control schemes employing engineered unitary evolution are by now standard, the last decade has witnessed an increasing interest in alternative methods based on the concept of "quantum reservoir engineering", i.e., on controlling a quantum system by tailoring its coupling to an environment [4]. Efforts along these lines have lead, e.g., to a broad range of proposals for the dissipative preparation of entangled few body quantum states [5,6,7,8,9,10,11]. However, in recent years attention has in particular been devoted to the study of dissipative many body systems. In this context, it has been realized that quantum reservoir engineering allows one to dissipatively prepare interesting many body states [12,13,14,15,16], perform universal quantum computation [14], realize a dissipative quantum repeater [17], or a dissipatively protected quantum memory [18]. Meanwhile, first experiments demonstrating the dissipative preparation of GHZ states in systems of trapped ions [19] and EPR entangled states of two atomic ensembles [20,21] have been reported. In these experiments the underlying principle has been to carefully design and implement a many-particle master equation, where a fully dissipative dynamics drives the system into a unique pure steady state representing the entangled state of interest.
In this work, we study entanglement formation in driven few-and many-particle cascaded quantum networks as introduced by Gardiner and Carmichael [22,23], where the unconventional coupling of multiple systems to a common unidirectional bath offers remarkable new opportunities for dissipative preparation of highly correlated states. As illustrated in figure 1, a cascaded quantum network consists of N systems coupled to a 1D reservoir that has the unique feature that excitations can only propagate along a single direction, thereby driving successive systems in the network in a unidirectional way. While such a scenario is reminiscent of edge modes in quantum Hall systems [24,25,26,27,28], various artificial and more controlled realizations based on (integrated) non-reciprocal devices for optical [29,30] and microwave [31,32] photons are currently developed. Our goal below is to identify situations where cascaded quantum networks are driven by classical fields in such a way that they exhibit pure and entangled steady states. As shown in figure 1, such dark states emerge if the continuous stream of photons emitted by the first part ("A") of the network is coherently reabsorbed by the second part ("B"), such that no photons escape from the system and the output remains dark. The system hence acts as its own coherent quantum absorber while the constant stream of photons maintains entanglement between its two parts.
We will first discuss the construction of coherent quantum absorbers on general grounds. Given the first part A of the network, we show how to choose the second part B such that the whole system evolves into a dark state. In doing so, we make use of the unidirectionality of the reservoir, which allows us to solve the first part independently of the second. These formal developments are then illustrated by two A B Figure 1. A driven cascaded quantum network which is realized by a set of twolevel systems coupled to a unidirectional bath. The two-level systems are driven by classical fields ∼ Ω and the continuously emitted radiation propagates along the waveguide and excites successive nodes. Under specific conditions, all photons emitted in a subsystem A are coherently reabsorbed in subsystem B. In this case the system relaxes into a dark state where no radiation escapes from the network, but a constant stream of photons running from A to B establishes entanglement between the two subsystems.
settings in which the coherent quantum absorber scenario can be realized. The first is a many body cascaded network, where each of the N nodes consists of a driven two-level system (TLS, "spin"). We show that this system exhibits a whole class of multi-partite entangled dark states, whose entanglement structure can be adjusted by tuning local parameters. As a second, complementary system we consider a network consisting of two non-linear cavities described by bosonic mode operators. By choosing appropriate laser drives and Kerr-type non-linearities one can ensure that this system evolves into a non-Gaussian dark state in which the two modes are entangled. In both examples, the coherent quantum absorber scenario thus leads to a dissipative state preparation scheme for non-trivial entangled states. In a more general context, these networks realize a novel type of non-equilibrium (many body) quantum system, which by changing the system parameters can be tuned between "dark" or "passive" phases (with no scattered photons emerging) and "bright" or "active" phases (with light scattered), while the nodes are driven into pure entangled or mixed states, respectively. The remainder of this paper is structured as follows. In section 2 we present the general model for an N -node cascaded network and show how to construct the coherent quantum absorber subsystem B for some given subsystem A (cf. figure 1). Complementing these rather general developments, we discuss the two mentioned realizations of the coherent quantum absorber scenario in subsequent sections. Section 3 presents the driven cascaded spin-system for which we derive and discuss a multipartite entangled class of dark states and also comment on the influence of various imperfections. Subsequently, section 4 discusses the setup based on cascaded cavities with Kerr-type non-linearity. In the latter case, we also obtain a purification of the steady state density matrix of the well-known dispersive optical bi-stability problem [33] as an interesting by-product. Implementations of the proposed cascaded networks are discussed in section 5 and concluding remarks can be found in section 6.

Cascaded quantum networks
We consider the general setting of a cascaded quantum network as shown in figure 1. Here, N ≥ 2 subsystems located at positions x i are coupled to a 1D continuum of right-propagating bosonic modes b ω , which represent, for example, photons in an optical or microwave waveguide. The whole network can be modeled by a Hamiltonian ( = 1) where H bath = dω ω b † ω b ω is the free Hamiltonian of the bath modes and the integrals run over a broad bandwidth ∆ω around the characteristic system frequency ω 0 . In this frequency range the bath modes are assumed to exhibit a linear dispersion relation with speed of light v. In equation (1) the H i describe the dynamics of the individual systems and include classical driving fields (e.g. lasers or microwave fields). The system-bath coupling is determined by the "jump operators" c i and coupling constants g ω , which we assume to be approximately constant over the frequency range ∆ω. Implementations of an effective model of the form given in equation (1) can be achieved with atoms or solid state TLSs coupled to 1D optical or microwave waveguides and will be discussed in more detail in section 5 below.
The system-bath interaction in equation (1) breaks time reversal symmetry and while photons can be emitted to the right, drive successive subsystems and eventually leave the network, the reverse processes cannot occur. To study the effects of this unconventional coupling, we assume that ω 0 as well as the bandwidth ∆ω are large compared to the other relevant frequency scales and eliminate the bath modes in a Born-Markov approximation. This yields a generalized cascaded master equation (ME) for the reduced system density operator ρ [22,23,34], Here the first part describes the uncoupled evolution of each subsystem where the Lindblad terms D[x]ρ = xρx † − {x † x, ρ}/2 model dissipation due to emission of photons into the waveguide with a rate γ = 2πg 2 ω0 . The unidirectionality of the bath is reflected by the last term in equation (2), which accounts for the possibility to reabsorb photons emitted at system i by all successive nodes located at x j > x i . The explicit Lindblad form of the ME (2) readṡ where now includes the non-local coherent part of the environment-mediated coupling, while the only decay channel with collective jump operator c = i c i is associated with a photon leaving the system to the right. ‡ Note that equations (2) and (3) are understood in a rotating frame in order to account for the explicit time-dependence of the driving fields.

Coherent quantum absorbers
In the following we are interested in steady state situations where every photon emitted within the system is perfectly reabsorbed by successive nodes in the network, such that there is no spontaneous emission via the waveguide output and the system relaxes to a pure steady state ρ 0 = |ψ 0 ψ 0 |. To identify the general conditions for the existence of such states, we partition the network into two subsystems A and B as indicated in figure 1, with local Hamiltonians H A and H B , and jump operators c A and c B , respectively. Specifically, where the primed sums run over the nodes of part A, and corresponding expressions hold for B. Then, in equation (3), and c = c A + c B , and the conditions for the existence of a pure stationary state are (see Ref. [13] and further comments in Appendix A): The first condition implies that the waveguide output is dark, i.e., c † c = 0, and the second one ensures stationarity. Within a quantum trajectory picture, condition (I) means that there are no stochastic quantum jumps, which would lead to a mixed state. In situations where [H A,B , ρ 0 ] = 0 and c A,B |ψ 0 = 0 for each subsystem separately, the network can simply be devided into two smaller parts which are then treated independently. In the following, we thus focus on situations where such a division is not possible and where the steady state possesses non-trivial correlations between A and B, as characterized for example by a non-vanishing In view of (I) this third requirement can be expressed as and directly connects the correlations between A and B with the amount of radiation emitted from the first subsystem. Note that for a pure steady state, C = 0 also implies that A and B are entangled. Finally, we remark that under stationary conditions, a non-vanishing C implies a constant flow of energy from A to B, while the total scattered light vanishes.
In the examples discussed below this 'coherent' absorption of energy in B can be understood as a destructive interference of the signals scattered by the two subsystems. The conditions (I) − (III) will not be satisfied in general. However, given a system A described by a Hamiltonian H A and jump operator c A we can construct a perfect coherent absorber system B as follows. First, we point out that due to the unidirectional coupling the dynamics of A is unaffected by B, which can also be shown explicitly by tracing the ME (3) over system B. In particular, the steady state and assuming a unique solution we write its spectral decomposition as ρ 0 A = k p k |k k|. A pure state of the whole system is then given by we assumed A and B to have the same Hilbert space dimension and defined |k = V |k in terms of an arbitrary unitary V acting on |k . Now, we demonstrate in Appendix A that conditions (I) and (II) can be satisfied by the choice where A mn = m|H A,eff |n and H A,eff = H A − i γ 2 c † A c A is the effective non-hermitian Hamiltonian associated with L A , and we assumed a positive and non-degenerate spectrum {p k }. While equations (6) and (7) define a general absorber system B, we find that for many systems of interest the stationary state ρ 0 A satisfies √ p n k|c A |n = √ p k n|c A |k and √ p n k|H A,eff |n = √ p k n|H A,eff |k . In this case, the above relations such that up to a unitary basis transformation the absorber system is just the negative counterpart of A. In particular, this situation applies to the two examples of cascaded spin systems and cascaded nonlinear cavities, which we describe in more detail in the following sections.

Cascaded spin networks
Let us now be more specific and consider a set of N driven spins coupled to a unidirectional bosonic bath as shown in figure 1. In equation (3), the collective jump operator is now c = i σ i − and the cascaded Hamiltonian in the frame rotating at the frequency ω d of the external driving field reads Here the σ i µ are the usual Pauli operators on site i, the Ω i are local Rabi frequencies, and the δ i = ω i − ω d are the detunings of the individual spin transition frequencies ω i from the common classical driving frequency ω d . In the following, the basis states of the spins are denoted by |e , |g , such that σ − = |g e|. The classical fields which are used to drive the spins can either be applied via additional local channels (assuming that the associated decay rate is much smaller than γ) or via a coherent field which is sent through the common waveguide. Note that by omitting the cascaded interaction in equation (8) we recover the familiar Dicke model for multiple two level atoms decaying via a common field mode, where even for Ω = 0 a series of dark states can be identified from c|ψ 0 = 0. In contrast, in our cascaded setting these states are not stationary and non-trivial dark states can only emerge from an interplay between driving and cascaded coupling terms.

Construction of dark states
For N = 2 the dark state condition (I) restricts |ψ 0 to the subspace spanned by |gg and the singlet |S = (|eg − |ge )/ √ 2. Condition (II) can then be satisfied for Ω 1 = Ω 2 ≡ Ω and any δ 1 = −δ 2 ≡ δ, for which we obtain the unique and pure steady state |ψ 0 = |S δ , where The two spins thus realize a source and a matched absorber in the sense introduced in the previous section, and for the matrix representations of the various operators we For strong driving, |α| 1, the state |S δ approaches the singlet |S , where the mutual correlation |C| → 1 is maximized.
While for larger N a direct search for possible dark states is hindered by the exponential growth of the subspace defined by c|ψ 0 = 0, we can use the state (9) as a starting point and solve the cascaded system iteratively "from left to right": Suppose that for Ω i = Ω and δ 1 = −δ 2 the first two spins have evolved into the dark state |S δ1 such that no more photons are emitted into the waveguide. Then, the following two § We use this as a short-hand notation for an equivalent matrix representation of the two operators, n|c B |m = − n|V c A V † |m , etc. Here, the same set of states is used on both systems, which is permissible due to their equal Hilbert-space dimension.  spins effectively see an empty waveguide and evolve into the dark state |S δ3 , provided that δ 3 = −δ 4 . By iterating this argument we see that for any detuning profile with δ 2i−1 = −δ 2i (i = 1, 2, . . .) the steady state of the ME (3) is given by which is shown explicitly in Appendix B. In particular, for the homogenous case δ i 0, a strongly driven cascaded spin system relaxes into a chain of pairwise singlets. The dimer structure of the state |S 0 reflects the fact that radiation emitted from one node is immediately reabsorbed by the following one. We now consider situations where this reabsorption occurs by several of the following spins, leading to multipartite entangled states. To this end, note that by starting from the state in equation (10) we can construct another dark state |S = U |S 0 by any global unitary operation U with [U, c] = 0, while implementing the Hamiltonian H = U H casc U † would ensure stationarity. However, for arbitrary unitary transformations H will in general contain additional non-local terms, and we must hence restrict ourselves to unitaries U under which H casc is form invariant. As an example, we write H casc ≡ H casc (∆), where ∆ = (δ 1 , δ 2 , ...) is the detuning profile, and introduce the nearest-neighbor operations with a new detuning profile ∆ = P i,i+1 ∆, where P i,i+1 denotes the permutation of δ i and δ i+1 (this is demonstrated in Appendix B). Thus, by starting from a set ∆ 0 of alternating detunings as defined before equation (10), we can simply swap the detunings of nodes i and i+1 to implement a new cascaded spin network with a unique stationary state |S = U i (θ i )|S 0 . By repeating this argument, we obtain a different pure steady state for each permutation ∆ of ∆ 0 . This class of states is given by where U (∆ 0 → ∆ ) is a product of nearest-neighbor operations U i (θ i ), specified by the sequence of nearest-neighbor transpositions required for transforming ∆ 0 into ∆ . A graphical representation of U (∆ 0 → ∆ ) in terms of a circuit model is shown in figure 2(a) for a specific example.

Discussion
For large detuning differences |δ i − δ i+1 | γ the unitary transformations given in equation (11) are SWAP operations [1] between neighboring sites. In this limit the states |S remain approximately two-partite entangled, but with singlets shared between arbitrary nodes in the network. In contrast, for |δ i − δ i+1 | ≈ γ the U i correspond to highly entangling √ SWAP operations. Then, the entanglement structure can be much richer and in general the states |S contain multi-partite entanglement between several or even all nodes.
While in this case a full characterization is difficult, we point out that the U i conserve total angular momentum such that the |S approach multi-spin singlets in the strong driving limit. The amount of entanglement between subsystems now depends very much on the choice of the detuning profile, as can be seen from the two examples displayed in figure 2(b), showing oscillating and linearly growing block entropy, respectively. More generally, the cascaded network can be driven into different types of states by simply adjusting local detunings. This is illustrated in figure 3, where an adiabatic variation of the detunings in a six-node network is used to prepare pure steady states with 2-, 4and 6-partite entanglement, separated by "bright" (mixed state) phases where the conditions in equation (4) are violated.
For assessing the relaxation time that characterizes the approach of the system towards the presented steady states it is sufficient to examine the simple detuning profile leading to |S 0 , since the spectral properties of the Liouvillian are invariant Waveguide losses, where η is the fraction of photons that gets lost between the nodes (modeled by adding a factor √ 1 − η to the last term in equation (2)). (d) Deviation from the asymmetric detuning condition modeled by a symmetric detuning offset δ 1 = δ 2 = . In (a-c) we have chosen δ 1 = δ 2 = 0. (e) Influence of on-site decays on concurrences C i,i+1 of reduced two-spin density matrices ρ i,i+1 for N = 6 spins with δ i = 0. (f) Influence of on-site decays on the steady state four-partite entanglement in a four-spin network, quantified by the measure of Ref. [36], which is bounded by 0.5 in this case. The detuning profile is given by ∆ = (0, −γ/2, γ/2, 0). under the transformations (11). Numerical calculations for small systems suggest that the preparation time for these states scales efficiently with the number of nodes N . In particular, the uniqueness of |S 0 implies that effects related to non-unique steady states found in related systems [35] are absent in our case.

Imperfections
Under realistic conditions various imperfections like onsite decays or losses in the waveguide can violate the exact dark state condition and the system then evolves to a mixed ("bright") steady state. This is exemplified by the dashed lines in figure 3 for the case of onsite decays, modelled by adding an additional term L ρ = κ 0 i D[σ i − ]ρ to the ME (3). One clearly observes how the scattered intensity increases, while the purity drops as compared to the ideal case with κ 0 = 0. To study the entanglement properties in such non-ideal situations, we show in figure 4(a)-(d) the resulting steady state concurrence for various types of imperfections in a two-node network. In addition to onsite decays (panel (a)), we also consider intrinsic spin dephasing (panel (b)), waveguide losses (panel (c)), and small deviations from the ideal detuning profile (panel (d)). We see that different sources of imperfections lead to a qualitatively similar behavior and that in all cases the entanglement is quite robust and optimized for intermediate driving strengths Ω.
For larger networks, the scattering of photons from the first nodes due to in out (a) (b) Figure 5. (a) Two driven cascaded cavities with Kerr-type non-linearities. For simplicity, we assume the cavities to be driven through their back mirrors (red arrows), whereas the dominant decay channel γ is provided by the front mirrors. (b) Level scheme for the construction of the pure steady state in the occupation number basis |nm ≡ |n + |m − of the c ± modes (N = n + m is the total photon number). Straight arrows denote Hamiltonian matrix elements, while wavy ones correspond to the decay of the symmetric mode. Blue dots indicate population in the dark state (19).
imperfections also affects successive spins, as shown in figure 4(e) for a six-spin dimer chain |S 0 in the presence of onsite losses. One clearly observes that the bi-partite entanglement in the dimers decreases with increasing number of previous nodes. To study the robustness of genuine multi-partite entanglement in the presence of imperfections, we employ the entanglement measure proposed in Ref. [36]. It can be evaluated in a straight-forward way and figure 4(f) displays the results for a fourpartite entangled steady state in the presence of onsite decays. The behavior of this measure qualitatively agrees with the results for the concurrence in the two-spin case (cf. figure 4(a)). That is, we observe a tradeoff between the maximal achievable entanglement and the robustness of the state. Finally, figure 3 also shows that for a fixed N , different bi-and multi-partite entangled states are affected equally.

Cascaded non-linear cavities
The general concept of a coherent quantum absorber introduced in section 2 suggests that the formation of dark entangled states can exist also for systems other than spins. As a second non-trivial example where this can be shown explicitly, we now discuss a setting where the two cascaded systems A and B are represented by two Kerr non-linear cavities as depicted in figure 5(a). The resulting distribution scheme for continuous variable entanglement is an alternative to other cascaded settings considered in this context [37]. We denote the two bosonic cavity modes by a and b, and the dynamics of the system is governed by the ME (3) with collective jump operator c = a + b and cascaded Hamiltonian Here, the Hamiltonian of the first cavity (system A, frequency ω c,A ) is where we have already moved to a frame rotating at the frequency ω d of the external driving field, such that ∆ = ω c,A − ω d is the corresponding detuning of the cavity frequency and Ω the associated driving strength. Further, K denotes the strength of the Kerr non-linearity. Motivated by our analysis of the cascaded spin system above, we assume that the Hamiltonian for the second cavity is given by Here we have chosen all constants to be identical to those used in H A , such that the first two terms have opposite sign as compared to system A. As demonstrated below, this educated guess for H B ensures that the cascaded ME exhibits a dark state.

Steady state solution
In order to show that the ME (3) indeed exhibits a pure steady state for the local Hamiltonians chosen above we exploit once more the conditions given in equation (4). From condition (I), i.e. (a + b)|ψ 0 = 0, it is clear that the steady state should contain zero quanta in the symmetric mode. Therefore, it is convenient to change to symmetric and anti-symmetric modes c ± = (a ± b)/ √ 2 and to write the dark state ansatz as |ψ 0 = |0 + ⊗ |χ 0 − , with |χ 0 = n α n |n , where |n are Fock states in the occupation number basis. In order to exploit condition (II) we rewrite the cascaded Hamiltonian in terms of the symmetric and anti-symmetric modes, HereN = c † + c + + c † − c − is the total number of quanta, which is conserved by all terms except for those ∝ Ω. Condition (II) is equivalent to H casc |ψ 0 = λ|ψ 0 , and by projecting this equation onto |ψ 0 we see that it can only be fulfilled for λ = 0 and hence This recursion is readily solved, such that the unique solution of conditions (I) and (II) is given by |n . (19) Here N = 0 F 2 (x, x * ; 2| | 2 ) 1/2 and 0 F 2 denotes the generalized hyper-geometric function [38]. Due to the simple tensor-product structure of |ψ 0 its only non-vanishing normally ordered moments are those of the c − -mode, i.e.
and the moments of the original modes thus read Numerical studies suggest that there are no additional mixed steady states.

Entanglement properties of the steady state
Despite the simple structure of the steady state |ψ 0 in the c ± representation, it is generally entangled when written in terms of the original modes a and b according to where the α n can be read off from equation (19). We characterize the entanglement with respect to this bipartition by the mixedness of the reduced state ρ 0 A = tr B {|ψ 0 ψ 0 |} = BB † of the first cavity, as measured by its linear entropy The results for two different strengths of the non-linearity are displayed in figure 6. For a better orientation, we show in panels (a) and (b) the photon number of the first cavity, normalized to the resonant response in the linear case, i.e. ∆ = K = 0. Here, one clearly observes the well-known behavior of a single Kerr-non-linear cavity, which As shown in Ref. [13] a pure stationary state could also exist if |ψ 0 is an eigenstate of the jump operator, i.e. (a + b)|ψ 0 = λ |ψ 0 with λ ∈ C. However, in the present example this would imply that the symmetric mode is in a coherent state of amplitude λ . We do not expect this due to the non-linearity of the problem and it can indeed be shown that such an ansatz does not lead to a stationary solution unless λ = 0.
is characterized by a deformation of the response curve for increasing Ω. Above a certain critical driving strength Ω c the classical response becomes bi-stable, while the quantum mechanical response curve exhibits a sharp step [33]. From panels (c) and (d) we see that the regions of pronounced non-linear response are those where the entropy is large, signaling a high degree of entanglement between the two cavities. In general, the features of the response are more washed out for larger non-linearities, where also the regions of high entropy are more extended. Note that the state |χ 0 , and hence also |ψ 0 , is generally non-Gaussian, as can also be shown by evaluating appropriate measures [40]. Finally, we remark that for K → 0 the steady state approaches a product of coherent states, i.e. |ψ 0 → |β A ⊗ | − β B , with β = /x, where |β denotes a coherent state of amplitude β. We are thus able to recover the classical limit in which no entanglement persists between the cavities.

The inverse coherent absorber problem
Finally, let us take a different view on the analysis presented so far and emphasize once more that cascaded systems may be solved in a successive fashion "from left to right". This means that tracing the full cascaded ME (3) over the second subsystem B yields a closed equation for the first subsystem A, which in this case readṡ In section 2 we have assumed that the stationary mixed state solution of this equation including its spectral decomposition is known, which allowed us to explicitly construct a corresponding perfect absorber system B, such that the whole system is driven into a pure steady state. However, in many cases solving for the steady state ρ 0 A of equation (23) is a non-trivial problem by itself, which in the present case was first accomplished using phase space methods [33]. Note that by using an educated guess for the Hamiltonian H B of the absorber system and then solving for the pure steady state |ψ 0 of the whole network, we have also indirectly solved the mixed state problem of a single cavity by computing ρ 0 A = tr B {|ψ 0 ψ 0 |}. Its explicit matrix-elements in the Fock-state basis agree with the expressions found in the literature [41], as do its moments (a † ) n (a) m obtained as a special case of equation (20) [33]. The procedure presented above thus represents an elegant alternative way of obtaining the mixed steady state of the dispersive optical bi-stability problem.
Avoiding to work with mixed states by using pure states on larger Hilbert spaces has found wide-spread use in the quantum information community, one of the reasons being that pure states can be manipulated more easily [1]. From this point of view, the steady state |ψ 0 possesses additional relevance as a purification of the density matrix ρ 0 A . This is an interesting result in itself, since obtaining such a purification generally requires knowledge of the spectral decomposition of ρ 0 A -a strong requirement even if ρ 0 A is known explicitly. In the above example we were able to circumvent this difficulty by directly constructing the purification |ψ 0 as a steady state of the cascaded ME. Although so far this procedure is limited to systems where the corresponding absorber Hamiltonian H B can be obtained by an educated guess, it is intriguing to think about cascaded quantum systems also as an analytic tool for calculating stationary states of non-trivial open quantum systems.

Implementations
The two key ingredients for realizing cascaded networks as discussed in the previous sections are (i) the implementation of low-loss non-reciprocal devices for directional routing of photons, and (ii) achieving a coupling of single quantum systems to a 1D waveguide that exceeds local decoherence channels. In the following, we discuss several implementations that fulfill these requirements, where a focus lies on optical and microwave setups. Requirement (i) is crucial for realizing the unidirectional coupling between the nodes and can in principle be fulfilled by standard circulators based on the Faraday effect (see, e.g., Ref. [42] for an optical implementation), but also by exploiting unidirectional edge modes in media with broken time-reversal symmetry [25,26,27]. However, in recent years there has been increasing interest in designing non-reciprocal on-chip devices which are integrable with, e.g., microwave circuitry [31,32] or nanofabricated photonic components [29,30]. Such elements would allow one to build up cascaded networks in a very controlled way in both the optical and the microwave domain, and thus constitute a promising way of implementing the scenarios discussed in this work. Concerning the second requirement, we first discuss the case of the cascaded spin network presented in section 3. Realizing a coupling of TLSs to a 1D waveguide could, e.g., be achieved along the lines of the experiments reported in Refs. [45,46], where superconducting qubits or atoms are coupled to transmission lines or hollow core optical fibers, respectively. However, one can also do without a direct coupling of the TLS to the waveguide by using the generic and flexible approach depicted in figure 7(a). Here, the TLS is coupled to a cavity, whose output port is connected to a circulator or another non-reciprocal device. In the bad cavity limit κ g, where g is the TLS-cavity coupling, a series of these nodes results in the desired model (2), with an effective TLS-waveguide decay rate γ = 2g 2 /κ. A particular realization of this scheme in the context of circuit cavity quantum electrodynamics [47,31] is shown in figure 7(b). Finally note that other systems like optomechanical transducers (figure 7(c)) have been proposed to realize a similar unidirectional coupling [43,44]. In the case of cascaded cavities discussed in section 4, the most important requirement is the realization of strong Kerr-type non-linearities, which are of equal magnitude and opposite sign in the two cavities. Such a tunable non-linearity can be realized by a dispersive coupling of the cavity field to a suitable four-level system, which has been analyzed both for optical [48,49] as well as superconducting microwave cavities [50].
In summary, we find that the current experimental capabilities for realizing strong TLS-photon interactions in various systems, combined with the development of novel non-reciprocal devices in the optical and microwave domain, enable the implementation and design of various cascaded quantum networks. Here, the dissipative state preparation schemes described in this work could serve as an interesting application exploring the unconventional physical properties of such devices.

Conclusions
We have shown that photon emission and coherent reabsorption processes in cascaded quantum systems can lead to the formation of pure and highly entangled steady states. In the case of spin networks, this mechanism provides a tunable dissipative preparation scheme for a whole class of multi-partite entangled states, and instances of this scheme might serve as a basis for dissipative quantum communication protocols [17]. For the case of two cascaded non-linear cavities we have identified a dissipative preparation scheme for non-Gaussian entangled states, and we have illustrated how cascaded systems can serve as a new analytic tool to evaluate the stationary states of driven-dissipative systems. More generally, our findings show that such driven cascaded networks realize a novel type of non-equilibrium quantum many-body system, which can be implemented with currently developed integrated optical systems or superconducting devices. uniqueness, we introduce its spectral decomposition ρ 0 A = k p k |k k| , (A. 3) and assume that the eigenvalues p k are positive and non-degenerate. A potential pure steady state |ψ 0 of the whole system can then be written as a purification of ρ 0 A , i.e., |ψ 0 = k √ p k |k A ⊗ |k B [1]. Here, |k = V |k , and V denotes an arbitrary unitary and we assume subsystem B to have the same Hilbert space dimension as A.
In the following, we write out the tensor-products more explicitly, such that the dark-state condition c|ψ 0 = 0 reads To construct the Hamiltonian H B for system B we exploit the second condition [H casc , |ψ 0 ψ 0 |] = 0 needed for a pure steady state, which is equivalent to H casc |ψ 0 = λ|ψ 0 for λ ∈ R. Note that c|ψ 0 = 0 implies