Steady state entanglement beyond the thermodynamic limit F. Tacchino,1 A. Auff`eves,2 M. F. Santos,3 and D. Gerace4, ∗ 1
Dipartimento di Fisica, Universit` a di Pavia, via Bassi 6, I-27100, Pavia, Italy CNRS and Universit´e Grenoble Alpes, Institut N´eel, F-38042, Grenoble, France 3 Instituto de F´ısica, Universidade Federal do Rio de Janeiro, CP68528, Rio de Janeiro, RJ 21941-972, Brazil 4 Dipartimento di Fisica, Universit` a di Pavia, via Bassi 6, I-27100 Pavia, Italy
arXiv:1710.01377v1 [quant-ph] 3 Oct 2017
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We consider an elementary bipartite quantum system coupled to two independent baths, generating an out of equilibrium entangled steady state between its internal degrees of freedom. An upper boundary to the degree of entanglement exists if classical thermal reservoirs are assumed. Here we show that such a classical limit can be overcome in a driven-dissipative scenario, in which the pumping rate is larger than the dissipation rate of each coupled transition. It is shown that this device is equivalent to a quantum thermal machine working between incoherent reservoirs with a negative effective temperature. A practical implementation is discussed in a quantum optical model of a pair of incoherently driven non-interacting qubits resonantly coupled to a quantized and strongly dissipative cavity mode, and special attention is paid to realistic parameters in view of realizations and experimental tests in solid state cavity QED systems.
Introduction. Classical thermodynamics typically manifests in situations involving two heat baths as the basic ingredients for two main classes of steady state thermal machines: heat engines extracting work from a classical fluid owing to a temperature gradient, or refrigerators cooling down a cold bath (and heating up a hot bath) at the price of work performed onto the classical fluid, as schematically represented in Fig. 1a. In the quantum realm, the working medium may provide a non-classical inner structure whose properties are fundamentally different from the classical thermodynamic machines. In particular, new out-of-equilibrium scenarios can be envisioned, in which different quantized transitions can be coupled to different and independent heat baths in order to prepare the working “quantum fluid” in a given non-trivial target steady state, which in turn can be used, e.g., as a resource for quantum thermodynamic applications1,2 . Within this context, the generation of entangled steady states that are robust to decoherence caused by noisy environments has been explored in many different contexts3–26 . A few of these works addressed the problem of achieving steady state entanglement of qubits driven from purely incoherent resources11–13,15 , most of the attention being limited to equilibrium heat baths16,17,21–23,26 . However, the amount of entanglement that can be generated without any additional feedback or filtering operation is rather modest23,26 , with an upper theoretical limit that is asymptotically reached only under unrealistically large temperature gradients between the two reservoirs16,21 . Here we address this problem from the perspective of quantum optical models and cavity quantum electrodynamics (QED) devices, which can often be interpreted as quantum thermal machines out of equilibrium. The laser is a typical and well established example of an out-ofequilibrium heat engine. Remarkably, even the Carnot efficiency limit can eventually be overcome by exploiting coherent and/or entangled working fluids27–31 . Most im-
portantly, out-of-equilibrium quantum optical processes can often be described by defining negative effective temperatures (e.g., the laser itself). In this Letter we show that engineering driven-dissipative baths at negative effective temperatures and coupling them to elementary quantum fluids (see schematic picture in Fig. 1b) allows to increase the amount of steady state entanglement of
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FIG. 1: (a) Schemes of classical thermal machines: a classical working fluid (F) produces or receives work (W) while connected to two reservoirs in thermal equilibrium at hot (Th ) and cold (Tc ) temperatures, respectively. (b) Elementary model of a driven-dissipative quantum thermal machine with diamond-like internal level structure and degenerate symmetric (S) and antisymmetric (A) states; the collective eigenstates are assumed to be connected to two independent reservoirs, defined through their effective temperatures TS and TA , respectively (see Eq. 2). (c) Quantum optical scheme realizing the model represented in (b): a pair of two-level emitters is coupled to the same lossy cavity mode and incoherently pumped by an external drive.
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ˆ 0 ] + L(ρ) ∂t ρ = i[ρ, H
(1)
ˆ 0 = ω0 (ˆ where H c†1 cˆ1 + cˆ†2 cˆ2 ) is the Hamiltonian, and X Γ+ Γ− i i (2) L(ρ) = D ˆ† (ρ) + D ˆ (ρ) 2 Ji 2 Ji
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the bipartite system beyond the known limits imposed by classical heat baths at thermal equilibrium16,21 . Eventually, we show how this situation can be practically realized in an elementary driven-dissipative quantum optical model with realistic parameters: two independent and incoherently pumped qubits coupled to a leaky cavity mode32,33 , as schematically represented in Fig. 1c. Model and steady state entanglement. We assume the most elementary model of a bipartite quantum system, in which two independent qubits are described by destruction (creation) operators cˆi (ˆ c†i , i = 1, 2) that obey † anticommutation rules {ˆ ci , cˆj } = δij . In view of generically describing their mutual coupling, collective operators can be defined from the two independent degrees of freedom by taking the symmetric and antisymmetric linear combinations, JˆS = cˆ1 + cˆ2 and JˆA = cˆ1 − cˆ2 , respectively. The internal level structure of the composite system is then characterized by a diamond-like scheme, as represented in Fig. 1, with degenerate √ transitions energies ω0 = ωA − ω√G = ωS − ωG , where 2|Si = (|0i1 |1i2 + |1i1 |0i2 ) and 2|Ai = (|0i1 |1i2 − |1i1 |0i2 ) are the two maximally entangled Bell states, respectively. We assume the two qubits to be coupled to collective and incoherent dissipation baths, such that the dynamics is described by the master equation for the density matrix (~ = 1 and kB = 1 in the following)
bA FIG. 2: The value of steady state concurrence (Eq. 4) is plotted against βA = ω0 /TA and βS = ω0 /TS , respectively. White dashed lines explicitly mark the thermal region (βA,S ≥ 0); the red dashed curves show the contour line for the classical thermodynamic limit value, C = 1/3.
incoherently driven system, which is quantified by the degree of non-separability of the given steady state ρSS , i.e. ˆ = iL(ρ), through its concurrence35 . the solution of [ρ, H] For maximally entangled pure states such as, e.g., Bell states, the concurrence is bound to C[ρ] = 1. For the model above, the steady state concurrence can be analytically solved as (see details in the Appendix) C(ρ) = max {0, (N1 − N2 )/d}
(4)
N1 = |A (S/2 − 1)| 1/2 N2 = (S/2 + 1) 2S 2 + 2P(S − 2)
(5)
i=A,S
is the Liouvillian operator in Lindblad form, with Doˆ(ρ) = 2ˆ oρˆ o† −{ˆ o† oˆ, ρ}. Here we will assume a common + pumping rate for the two collective modes, i.e. Γ+ i =Γ for i = A, S, while we will allow for independent dissipa− tion rates, Γ− S and ΓA . Thanks to the Markovian nature of the dissipation baths assumed here, we can generically relate the corresponding pumping/dissipation rates to effective thermal reservoirs through the following relations34 Γ+ = e−βS ; Γ− S
Γ+ = e−βA Γ− A
(3)
which define two real and positive definite temperatures (in units of ω0 ), TS = ω0 /βS and TA = ω0 /βA only if Γ+ < Γ− A,S , respectively. However, in a generic drivendissipative scheme one can always assume Γ+ to be larger than one or both Γ− A,S . Thus, it is preferable to define effective thermal reservoirs at temperatures TS and TA , respectively, without restricting them to positive values. The scheme in Fig. 1b may thus be seen as a generalized quantum version of the classical thermal machines represented in the panels above. We are interested in the amount of steady state entanglement that can be generated out of equilibrium in this
with
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d = 1 + S + S(P + 3)/2 − P in which we defined S = exp(βA ) + exp(βS ), A = exp(βA ) − exp(βS ), and P = exp(βA + βS ). A plot of Eq. 4 is given in Fig. 2 as a function of the two effective thermal resources. As expected, the steady state is fully separable (C[ρSS ] = 0) for balanced reservoirs, i.e. when βS ' βA . In this case, most of the stationary population is either in |Gi or |Ei on average, while the rest is in an equal mixture of the two Bell states |Ai and |Si. On the other hand, an unbalance in the two thermal reservoirs allows for driving the system in a nonseparable steady state, with the population of either |Ai or |Si dominating over the other. The amount of entanglement is limited to the value C = 1/3 when classical thermal reservoirs at positive temperatures are assumed (see dashed lines superimposed to the color scale plot), as also inferred from the analytic expression above (see the Appendix for details). This limiting value is reached when the ∆T → ∞, i.e. when one of the two temperatures goes to zero and the other to infinite. Evidently,
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ˆT C = H
2 X i=1
ω0 cˆ†i cˆi + ωcav a ˆ† a ˆ+
2 X
g (ˆ c†i a ˆ + cˆi a ˆ† )
(6)
i=1
where a ˆ (ˆ a† ) is are the destruction (creation) operator of the single-mode cavity photons. The master equation describing the driven-dissipative system of Fig. 1c is thus ˆ T C ] + L(ρ), where the full Liouvillian explic∂t ρ = i[ρ, H itly reads p X γ X κ L(ρ) = Dcˆ† (ρ) + Dcˆ (ρ) + Daˆ (ρ) (7) 2 i=1,2 i 2 i=1,2 i 2
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in such a case the population of the Bell states is unbalanced such that 1/3 is in the one coupled to the hottest bath, while the other goes to zero. The rest of the population, 2/3, will be on the ground state |Gi. This result was also found in alternative models of bipartite quantum systems coupled to thermal reservoirs16,21 . Here we claim that such a classical limit can be overcome by relaxing the conditions of real thermal reservoirs with positive defined temperatures. In fact, in an out-ofequilibrium, driven-dissipative scenario (namely, allowing Γ+ > Γ− A,S ) one can always define negative effective temperatures from Eq. 3, which explain the maximal entanglement regions in Fig. 2: when one of the baths is at effective negative temperature, while the other is at a positive and small one, the system is pumped into the maximally entangled state with 1/2 stationary probability, giving the limiting value C[ρSS ] = 1/2. This is a key result of this work: a bipartite quantum system can be pumped into a maximally entangled steady state by exploiting purely incoherent resources, with the largest concurrence reaching the limiting value of 0.5 if one of the two collective states is at a negative effective temperature. In the absence of feedback or further purification of the steady state7,8,26 , this is the theoretical limiting value. Notice that the regions with the highest concurrence are all outside the thermal region, which could in principle be reached with classic thermal baths. Notice also that the lower left region, corresponding to both reservoirs being at negative effective temperature, gives C = 0 due to the largest occupancy of the fully separable |Ei state, i.e. corresponding to the population inversion of the diamond at large pumping. A cavity QED-based implementation. The natural question that arises is whether the theoretical model in Eq. 2 can be practically realized in a physical system that is amenable to experimental implementation. In fact, we show here that this is the case for a quite straightforward cavity QED situation where two independent and incoherently pumped qubits are coupled to a single radiation mode of an electromagnetic resonator, as schematically represented in Fig. 1c. Specifically, we consider a pair of point-like two level systems that are resonantly (ωcav = ω0 ) coupled to a single-mode resonator at the same rate g 1. In fact, under such conditions the cavity only acts as an additional dissipation channel in
4 the reduced two-qubits subspace,33 in which each qubit is further relaxed at a rate Γ = 4g 2 /κ in addition to the intrinsic spontaneous emission at rate γ. Hence Eq. 7 can be recast exactly as Eqs. 1 and 2, after straightforward algebra with the following relations − Γ+ = p/2 ; Γ− A = γ/2 ; ΓS = Γ + γ/2
(8)
The effective temperatures result now from combinations of the physical parameters: βA = log [γ/px ] and βS = log [(γ + 2Γ)/px ]. As it was already evidenced in Ref. 33, in the subradiant regime the system is optically pumped in the dark |Ai = |0; 0i state (i.e., the singlet in the |J; MJ i notation for eigenstates of the total angular momentum operator), thus creating an imbalanced population with respect to the |Si = |1; 0i (triplet) state, as schematically represented in the diamond-like level structure of Fig. 3a. Here we corroborate such hypothesis by explicitly showing in Fig. 3b the calculated steady state concurrence, which is evidently different from zero only when κ/g falls in the subradiant sector of this model. There exists an optical pumping range for which the system reaches its maximal concurrence, which depends on γ/g and κ/g (p/γ ' 5 in the Figure). In particular, here the maximal value is C ' 0.4, but it can be even larger and approaching the C = 0.5 limit for smaller values of γ/g (see, e.g., numerical results in the Appendix). Further instructive information is inferred from plotting the concurrence as a function of κ/g and for a few paradigmatic values of pumping, as shown in Figs. 3c,d. First, in Fig. 3c we show the ideal result for γ = 0, corresponding to the negative effective temperature reservoir coupled to the dark state, which gives the limiting value C[ρSS ] → 0.5 when p/Γ → 0 (in agreement with the results in Fig. 2); the full model only follows the effective model for κ/g > 1, i.e. until the adiabatic approximation holds. At difference with the general model of the previous section, here the A-S symmetry is broken since only the antisymmetric state is dark, since βS > βA from Eq. 8. Hence, with reference to Fig. 2, only the part above the βS = βA diagonal should be considered when dealing with the cavity QED implementation. In Fig. 3d we show the behavior of the steady state concurrence for γ = 10−3 g, which is usually the case in most practical realizations of this model (see discussion below). While it is evident that the regime of steady state entanglement narrows in κ as p increases, it should also be noted that for the proper values of κ the thermodynamic limit is overcome (i.e., C[ρSS ] > 1/3) as soon as p > γ. The latter condition corresponds to the onset of negative effective temperature for the dark state reservoir, as it is evident from Fig. 3b (see also the full plot in the Appendix). The theoretical cavity QED model presented here can be practically realized in a number of possible experimental platforms, where the steady state subradiant emission regime can be achieved. We hereby discuss two prominent examples where the physics of quantum thermal ma-
chines could be investigated in a controlled setting by using state-of-the art solid state cavity QED systems. A tolerance analysis against the main sources of decoherence and dephasing in realistic implementations, such as qubits pure dephasing and inhomogeneous broadening, as well as cavity incoherent pumping, is reported in SI. The first example relies on semiconductor quantum dots, behaving as artificial two level systems that can be coupled to a single mode of a photonic resonator. These systems allow for a controlled and fully deterministic coupling of the quantum emitters to the cavity mode36,37 . Spatial control now allows to simultaneously place more than a single artificial atom in deterministic optical coupling with the same cavity mode38 . Either optical or electrical control of these qubits has already been demonstrated. Typical parameters for quantum dots and semiconductor microcavities made of III-V materials are g ' 0.1 meV and κ ranging from 0.01 meV to a few meV, depending on the Q-factor of the corresponding resonator, easily allowing to access the region κ/g ≥ 1 in which the thermodynamic limit of 1/3 can be overcome, as shown in Fig. 3. As a further potential implementation of the proposed model we mention superconducting circuit quantum electrodynamic devices, in which artificial atoms are realized by Cooper pair boxes, while high-quality resonators are implemented by coplanar transmission lines39 . It should be noted that controlled coupling of a few qubits to a single resonator mode has already been experimentally tested20,25,40 . Given the high control capabilities reached for these state-of-art devices, from the tunability of the single qubits transition frequencies to their effective dissipation rates, this platform seems particularly suited to investigate the quantum thermodynamic aspects of elementary quantum thermal machines. We also notice that the coupling rates in the range of g ' 0.1 to 10 MHz, and the dissipation rates κ ' few kHz to hundreds MHz (e.g., by increasing the temperature above the superconducting critical temperature of the material constituting the transmission line), make these practical implementations of the model span almost the full available range of radiative emission properties.
Acknowledgements. The authors acknowledge several useful discussions with S. Carretta, E. Mascarenhas, M. Lostaglio, M. Richard, F. Troiani, H. E. T¨ ureci, J. P. Vasco Cano. This work was partly supported from COST Action MP1403 “Nanoscale Quantum Optics” through the Short Term Scientific Mission (STSM) program, the Italian Ministry of Education and Research (MIUR) through PRIN Project 2015 HYFSRT “Quantum Coherence in Nanostructures of Molecular Spin Qubits”, the Brazilian funding agency CNPq through project No. 305384/2015-5 and the PVE-Ciˆencia Sem Fronteiras Project No. 407167/2013-7, the CNRS French-Brazilian PICS program “Thermodynamics of Quantum Optics”.
5 Appendix A: Steady state concurrence of the bipartite quantum system
By using the definitions of effective temperatures (in units of ω0 ) as given in the paper Γ+ = e−βS ; Γ− S
From the master equation ˆ 0 ] + L(ρ) ∂t ρ = i[ρ, H
(A1)
ˆ 0 = ω0 (ˆ where H c†1 cˆ1 + cˆ†2 cˆ2 ) and X Γ+ Γ− i L(ρ) = DJˆ† (ρ) + i DJˆi (ρ) i 2 2
(A2)
i=A,S
(A11)
one gets, after some algebra N1 = |A (S/2 − 1)| 1/2 N2 = (S/2 + 1) 2S 2 + 2P(S − 2)
(A12)
d = 1 + S 2 + S(P + 3)/2 − P
one can find the steady state ρSS by imposing the condition ∂t ρ = 0
(A3)
The solution expressed in the computational basis {|00i, |01i, |10i, |11i} has the general form ρ00 0 0 0 0 ρ01 ρc 0 (A4) ρSS = 0 ρ∗c ρ10 0 0 0 0 ρ11 This can be easily interpreted as a consequence of the fact that the dissipative part of the Liouvillian involves coherent superpositions only in the {|Si, |Ai} subspace. According to the formal definition, the concurrence of a two-qubit density matrix ρ can be computed as follows: • define ρ˜ = (σy ⊗ σy )ρ∗ (σy ⊗ σy ) • find the spectral decomposition of ρ˜ ρ as ρ˜ ρ=
Γ+ = e−βA Γ− A
4 X
λ2i |ψi ihψi |
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• after ordering the eigenvalues as λ1 ≥ λ2 ≥ λ3 ≥ λ4
(A6)
the concurrence of ρ is C(ρ) = max{λ1 − λ2 − λ3 − λ4 , 0}
(A7)
For ρ = ρSS as given in (A4) this reduces to √ C(ρSS ) = 2 max {0, |ρc | − ρ00 ρ11 }
(A8)
and, using the explicit solution C(ρSS ) = max {0, (N1 − N2 )/d}
(A9)
with − − − + + + + N1 = (Γ− 1 /Γ − Γ2 /Γ ) Γ1 /Γ + Γ2 /Γ − 2 h − − + + + 2 + N2 = Γ − (Γ− 1 /Γ + Γ2 /Γ + 2 1 /Γ ) (Γ2 /Γ ) i1/2 − − − + 2 + 2 + 2 + + (Γ− /Γ ) + (Γ /Γ ) + (Γ /Γ ) (Γ /Γ ) 1 2 2 1 − + 2 + 2 d = 1 + (Γ− 1 /Γ ) + (Γ2 /Γ ) − − − + + + + + (Γ− 1 /Γ )(Γ2 /Γ ) + (3/2)(Γ1 /Γ + Γ2 /Γ ) − − − + + + + + (1/2)(Γ− 1 /Γ + Γ2 /Γ )(Γ1 /Γ )(Γ2 /Γ )
(A10)
in which we defined S = exp(βA ) + exp(βS ), A = exp(βA ) − exp(βS ), and P = exp(βA + βS ). It is easy to realize that both effective temperatures are treated symmetrically in the final expression, meaning that the system can in principle rely both on the symmetric or antisymmetric maximally entangled state to produce nonzero concurrence. Finally, it is also easy to show explicitly the behavior of the concurrence in some instructive cases. First, let us put βA = βS : this corresponds to equal effective temperatures that always produce a separable steady state. Indeed, we have A = 0 ⇒ N1 = 0 and N2 > 0, implying C(ρSS ) = 0. One can also show that under these conditions the collective dissipators appearing in the master equation decouple into local ones. On the other hand, when βA = 0 and βS → +∞ we have N1 ' exp(2βS )/2, N2 ' exp((3/2)βS ) and d ' (3/2) exp(2βS ), and we reach the thermal limit 2 1 1 −βS /2 C(ρSS ) ' −e → (A13) 3 2 3 Finally, we can consider the extreme case βA → −∞, for which S ' exp(βS ), A ' − exp(βS ) and P ' 0 for every finite value of βS . The factors appearing in the formula for the concurrence are now N1 ' exp(βS )(exp(βS )/2 − 1), N2 ' exp(βS )(exp(βS )+2)1/2 and d ' 1+exp(2βS )+ (3/2) exp(βS ), and if βS is positive and large enough (to be precise, we should still ask |βA | βS so that e.g. the P ' 0 limit holds) we get the maximum (1/2)e2βS − eβS eβS (eβS + 2)1/2 1 − → 2β β 1 + e S + (3/2)e S 1 + e2βS + (3/2)eβS 2 (A14) As it can be seen in Fig. 2 of the paper, this limit is already approached in a wide region for moderately negative values of βA and is well approximated even when |βA | < βS . Needless to say, given the symmetry of the problem all the calculations can be done in the same way for the case in which the roles of βA and βS are exchanged. C(ρSS ) '
Appendix B: Steady state concurrence of the full model
The master equation for the full model in the main text, describing two incoherently driven quantum emit-
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FIG. 4: Numerical results for the steady state concurrence in the real model, as a function of incoherent pumping rate px /g and of the cavity dissipation κ/g. Panel on the left is obtained for γ = 10−3 g, while panel on the right for γ = 10−4 g. The red dashed line shows the contour line for C = 1/3, while the white one shows the px = γ condition, i.e. the border between (infinite) positive and (infinite) negative effective hot temperature.
ters coupled to the same cavity mode, can be most efficiently solved in steady state by expressing the operators on a Fock basis of occupation numbers truncated to the most suitable photon number nmax priorly checked for convergence and then solving numerically the equation FρSS = 0 · ρSS , where F is the superoperator corresponding to the linear operator equation ˆ + L(ρ) = 0. This is obtained by the usual mapi[ρ, H] ping between the d-dimensional Hilbert space of the system and a d2 -dimensional Hilbert space via the relations X X ρ= rij |iihj| 7→ |ρii = rij |ii|ji (B1) ij
ij
and AρB 7→ |AρBii = (A ⊗ B T )|ρii
(B2)
In the present case, we have d = 2 · 2 · (nmax + 1) with nmax ≤ 15 in the simulations shown in this work, which are largely sufficient for convergence. The steady state concurrence is then calculated on the reduced density matrix of the two qubits (q1 ,q2 ) obtained after tracing out the cavity (C) degrees of freedom ρq1 ,q2 = TrC [ρq1 ,q2 ,C ]
(B3)
and following the procedure outlined in the previous Section: this is straightforwardly implemented in a routine ˜ = (σy ⊗ σy )(X)∗ (σy ⊗ σy ) that applies the spin-flip X operation and then finds and sorts the eigenvalues of ρq1 ,q2 ρ˜q1 ,q2 . Here we report the results of the scans over the model parameters p and κ (both in units of the qubit-cavity coupling), respectively, and for different values of the qubit relaxation rate, γ. As it can be seen in Fig. S4 of this Supplementary Information, for both cases there is an optimal region in the (px , κ) plane where the concurrence is at a maximum. Notice that this happens for px > γ and that the amount of steady state entanglement achievable
in the model can exceed the C = 1/3 thermal limit. In particular, for the cases that we report here, generated numerically for nmax = 15, we obtain Cmax ' 0.3869 for γ = 10−3 g and Cmax = 0.4479 for γ = 10−4 g. Appendix C: Pure dephasing and inhomogeneous two-level systems
In this last section we evaluate the robustness of the entangled steady state of the full cavity QED implementation of the model with respect to possibly detrimental processes and sources of noise in the system. In particular, we hereby analyze the performances of the modell for parameters corresponding to the maximal calculated concurrence of Fig. S4 for γ = 10−3 g, and introducing one of the following additional features: • individual pure dephasingh on the qubits, i i.e. a con† tribution L(γi,z , ρ) = γz σˆiz ρσˆiz − ρ in the master equation; • disorder in the form of non perfectly identical qubits, or inhomogeneous size distribution in the case of artificial atoms, which we detune from the cavity in a symmetric fashion as ω1 = ω + δ, ω2 = ω − δ; • finite non-zero temperature of the cavity bath, by introducing pump term L(pc , ρ) = † an incoherent (pc /2) 2ˆ a ρˆ a−a ˆa ˆ† ρ − ρˆ aa ˆ† . The results are presented in Fig. S5, where it can be seen that the orders of magnitude required for noise processes to destroy the quantum coherence in the steady state are not far from those obtained in comparable situations involving a coherent pumping of the system. In particular, we notice that our device retains a still significant amount of steady state entanglement even when the pure dephasing rate equals that of the individual relaxation
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FIG. 5: Steady state concurrence as a function of the main sources of decoherence and dephasing, such as (a) pure dephasing rate, (b) qubits detuning with respect to the cavity mode, and (c) finite temperature of the cavity bath.
and pump mechanisms on the qubits, and is even more robust, albeit with a sharper transition, with respect to incoherent driving of the cavity mode. For what concerns disorder, we note that fabrication inhomogeneities
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