Our analysis is divided into two parts. The first one is dedicated to the single-qubit architecture which shall permit us to investigate the dynamics of quantum coherence and its sensitivity to decay. The second part treats the two-qubit architecture for exploring to which extent the time of existence of quantum entanglement can be prolonged with respect to its natural disappearance time without the proposed engineered environment.

Single-qubit coherence preservation

The global system is made of a two-level atom (qubit) inside a lossy cavity C 1 which in turn interacts with another cavity C 2 , as depicted in Fig. 1. The Hamiltonian of the qubit and two cavities is given by (ħ = 1)

Figure 1 Scheme of the single-qubit architecture. A two-level atom (qubit) is embedded in a cavity C 1 which is in turn coupled to a second cavity C 2 by a coupling strength J. Both cavities are taken at zero temperature and can lose photons. Full size image

where is a Pauli operator for the qubit with transition frequency ω 0 , are the raising and lowering operators of the qubit, and the annihilation (creation) operators of cavities C 1 and C 2 which sustain modes with frequency ω 1 and ω 2 , respectively. The parameter κ denotes the coupling of the qubit with cavity C 1 and J the coupling between the two cavities. We take ω 1 = ω 2 = ω and, in order to consider both resonant and non-resonant qubit-C 1 interactions, ω 0 = ω+δ with δ being the qubit-cavity detuning. Taking the dissipations of the two cavities into account, the density operator ρ(t) of the atom plus the cavities obeys the following master equation65

where and Γ 1 (Γ 2 ) denotes the photon decay rate of cavity C 1 (C 2 ). The rate Γ n /2 physically represents the bandwidth of the Lorentzian frequency spectral density of the cavity C n , which is not a perfect single-mode cavity65. A cavity with a high quality factor will have a narrow bandwidth and therefore a small photon decay rate. Weak and strong coupling regimes for the qubit-C 1 interaction can be then individuated by the conditions and 41,65.

Let us suppose the qubit is initially in the excited state and both cavities in the vacuum states , so that the overall initial state is , where the first, second and third element correspond to the qubit, cavity C 1 and cavity C 2 , respectively. Since there exist at most one excitation in the total system at any time of evolution, we can make the ansatz for ρ(t) in the form

where with and with h(0) = 1 and . It is convenient to introduce the unnormalized state vector66,67

where represents the probability amplitude of the qubit and (n = 1, 2) that of the cavities being in their excited states. In terms of this unnormalized state vector we then get

The time-dependent amplitudes , , of Eq. (4) are determined by a set of differential equations as

The above differential equations can be solved by means of standard Laplace transformations combined with numerical simulations to obtain the reduced density operators of the atom as well as of each of the cavities. In particular, in the basis the density matrix evolution of the qubit can be cast as

where u t and z t are functions of the time t (see Methods).

An intuitive quantification of quantum coherence is based to the off-diagonal elements of the desired quantum state, being these related to the basic property of quantum interference. Indeed, it has been recently shown68 that the functional

where ρ ij (t) (i ≠ j) are the off-diagonal elements of the system density matrix, satisfies the physical requirements which make it a proper coherence measure68. In the following, we adopt C as quantifier of the qubit coherence and explore how to preserve and even trap it under various conditions. To this aim, we first consider the resonant atom-cavity interaction and then discuss the effects of detuning on the dynamics of coherence.

Suppose the qubit is initially prepared in the state (with ), namely, C , then at time t > 0 the coherence becomes C . Focusing on the weak coupling between the qubit and the cavity C 1 with κ = 0.24 Γ 1 , we plot the dynamics of coherence in Fig. 2(a). In this case, the qubit exhibits a Markovian dynamics with an asymptotical decay of the coherence in the absence of the cavity C 2 (with J = 0). However, by introducing the cavity C 2 with a sufficiently large coupling strength, quantum coherence undergoes non-Markovian dynamics with oscillations. Moreover, it is readily observed that the decay of coherence can be greatly inhibited by increasing the C 1 -C 2 coupling strength J. On the other hand, if the coupling between the atom and the cavity C 1 is initially in the strong regime with the occurrence of coherence collapses and revivals, the increasing of the C 1 -C 2 coupling strength J can drive the non-Markovian dynamics of the qubit to the Markovian one and then back to the non-Markovian one, as shown in Fig. 2(b). This behavior is individuated by the suppression and the successive reactivation of oscillations during the dynamics. It is worth noting that, although the qubit can experience non-Markovian dynamics again for large enough J, the non-Markovian dynamics curve is different from the original one for J = 0 in the sense that the oscillations arise before the coherence decays to zero. In general, the coupling of C 1 -C 2 can enhance the quantum coherence also in the strong coupling regime between the qubit and the cavity C 1 .

Figure 2 Coherence C of the qubit as a function of the scaled time Γ 1 t for different coupling strengths J between the two cavities for (a) κ = 0.24 Γ 1 , Γ 2 = 0.5 Γ 1 and (b) κ = 0.4 Γ 1 , Γ 2 = 0.5 Γ 1 . The qubit is initially prepared in the state with and resonant with the cavity (detuning δ = 0). The plots in panels (c,d) display the coherence trapping for a perfect cavity (Γ 2 = 0) with κ = 0.24 Γ 1 and κ = 0.4 Γ 1 , respectively. Full size image

The oscillations of coherence, in clear contrast to the monotonic smooth decay in the Markovian regime, constitute a sufficient condition to signify the presence of memory effects in the system dynamics, being due to information backflow from the environment to the quantum system69. The degree of a non-Markovian process, the so-called non-Markovianity, can be quantified by different suitable measures69,70,71,72. We adopt here the non-Markovianity measure which exploits the dynamics of the trace distance between two initially different states ρ 1 (0) and ρ 2 (0) of an open system to assess their distinguishability69. A Markovian evolution can never increase the trace distance, hence nonmonotonicity of the latter would imply a non-Markovian character of the system dynamics. Based on this concept, the non-Markovianity can be quantified by a measure defined as69

where is the rate of change of the trace distance, which is defined as , with . By virtue of , we plot in Fig. 3 the non-Markovianity of the qubit dynamics for the conditions considered in Fig. 2(a,b). We see that if the qubit is initially weakly coupled to the cavity C 1 (κ = 0.24 Γ 1 ) its dynamics can undergo a transition from Markovian to non-Markovian regimes by increasing the coupling strengths J between the two cavities. On the other hand, for strong qubit-cavity coupling (κ = 0.4 Γ 1 ), the non-Markovian dynamics occurring for J = 0 turns into Markovian and then back to non-Markovian by increasing J. We mention that such a behavior has been also observed in a different structured system where a qubit simultaneously interacts with two coupled lossy cavities73.

Figure 3 Non-Markovianity quantifier of Eq. (9) of the qubit dynamics as a function of J/Γ 1 for weak (κ = 0.24 Γ 1 ) and strong (κ = 0.4 Γ 1 ) coupling regimes to cavity C 1 and a fixed decay rate Γ 2 = 0.5 Γ 1 of the cavity C 2 . Full size image

Trapping qubit coherence in the long-time limit is a useful dynamical feature for itself that shall play a role for the preservation of quantum entanglement to be treated in the next section. We indeed find that the use of coupled cavities can achieve this result if the cavity C 2 is perfect, that is Γ 2 = 0 (no photon leakage). The plots in Fig. 2(c,d) demonstrate the coherence trapping in the long-time limit for both weak and strong coupling regimes between the qubit and the cavity C 1 for different coupling strengths J between the two cavities. This behavior can be explained by noticing that there exists a bound (decoherence-free) state of the qubit and the cavity C 2 of the form , with J and κ being the C 1 -C 2 and qubit-C 1 coupling strengths. Being this state free from decay, once the reduced initial state of the qubit and the cavity C 2 contains a nonzero component of this bound state , a long-living quantum coherence for the qubit can be obtained. For the initial state of the qubit and two cavities here considered and Γ 2 = 0, the coherence defined in Eq. (8) gets the asymptotic value C , which increases with J for a given κ. We further point out that the cavity C 1 acts as a catalyst of the entanglement for the hybrid qubit-C 2 system, in perfect analogy to the stationary entanglement exhibited by two qubits embedded in a common cavity24. In the latter case, in fact, the cavity mediates the interaction between the two qubits and performs as an entanglement catalyst for them.

We now discuss the effect of non-resonant qubit-C 1 interaction (δ ≠ 0) on the dynamics of coherence. In Fig. 4(a–d), we display the density plots of the coherence as functions of detuning δ = ω 0 − ω and rescaled time Γt for both weak and strong couplings. One observes that when δ departures from zero, the decay of coherence speeds up achieving the fastest decay around δ = J. It is interesting to highlight the role of the cavity-cavity coupling parameter J as a benchmark for having the fastest decay during the dynamics under the non-resonant condition. For larger detuning tending to the dispersive regime , the decay of coherence is instead strongly slowed down48. However, as shown in Fig. 5, stationary coherence is forbidden out of resonance when the cavity C 2 is perfect. Since our main aim is the long-time preservation of quantum coherence and thus of entanglement, in the following we only focus on the condition of resonance between qubit and cavity frequencies.

Figure 4 Density plots of coherence C of the qubit as functions of detuning δ and the scaled time Γ 1 t for (a) κ = 0.24 Γ 1 , Γ 2 = 0.2 Γ 1 , J = 0.5 Γ 1 ; (b) κ = 0.24 Γ 1 , Γ 2 = 0.2 Γ 1 , J = Γ 1 ; (c) κ = 0.4 Γ 1 , Γ 2 = 0.5 Γ 1 , J = 0.5 Γ 1 ; (d) κ = 0.4 Γ 1 , Γ 2 = 0.5 Γ 1 , J = Γ 1 . The initial state of the qubit is maximally coherent . The values of the coherence are within the range: [0, 1]. Full size image

Figure 5 Coherence C of the qubit as a function of the scaled time Γ 1 t for different values of the detuning δ in the case when the cavity C 2 is perfect, that is Γ 2 = 0. The qubit-C 1 and the C 1 -C 2 coupling strengths are, respectively, (a) κ = 0.24 Γ 1 , J = 0.3 Γ 1 ; (b) κ = 0.4 Γ 1 , J = 0.3 Γ 1 . Out of resonance (δ > 0) no coherence trapping is achievable. Full size image

Two-qubit entanglement preservation

So far, we have studied the manipulation of coherence dynamics of a qubit via an adjustment of coupling strength between two cavities. We now extend this architecture to explore the possibility to harness and preserve the entanglement of two independent qubits, labeled as A and B. We thus consider A (B) interacts locally with cavity C 1A (C 1B ) which is in turn coupled to cavity C 2A (C 2B ) with coupling strength J A (J B ), as illustrated in Fig. 6. That is, we have two independent dynamics with each one consisting of a qubit j (j = A, B) and two coupled cavities C 1j − C 2j . The total Hamiltonian is then given by the sum of the two independent Hamiltonians, namely, , where each H j is the single-qubit Hamiltonian of Eq. (1). Denoting with Γ 1j (Γ 2j ) the decay rate of cavity C 1j (C 2j ), we shall assume Γ 1A = Γ 1B = Γ as the unit of the other parameters.

Figure 6 Scheme of the two-qubit architecture. Two independent qubits A and B, initially entangled, are locally embedded in a cavity C 1j which is in turn coupled to a second cavity C 2j by a coupling strength J j (j = A, B). Full size image

As known for the case of independent subsystems, the complete dynamics of the two-qubit system can be obtained by knowing that of each qubit interacting with its own environment41,42. By means of the single-qubit evolution, we can construct the evolved density matrix of the two atoms, whose elements in the standard computational basis are

where ρ lm (0) are the density matrix elements of the two-qubit initial state and , are the time-dependent functions of Eq. (7).

We consider the qubits initially in an entangled state of the form . As is known, this type of entangled states with suffers from entanglement sudden death when each atom locally interacts with a dissipative environment7,8,9. As far as non-Markovian environments are concerned, partial revivals of entanglement can occur38,41,42,43,44,74,75,76,77,78,79,80,81,82,83,84 typically after asymptotically decaying to zero or after a finite dark period of complete disappearance. It would be useful in practical applications that the non-Markovian oscillations can occur when the entanglement still retain a relatively large value. With our cavity-based architecture, on the one hand we show that the Markovian dynamics of entanglement in the weak coupling regime between the atoms and the corresponding cavities (i.e., C 1A and C 1B ) can be turned into non-Markovian one by increasing the coupling strengths between the cavities C 1A -C 2A and (or) C 1B -C 2B ; on the other hand, we find that the appearance of entanglement revivals can be shifted to earlier times. We employ the concurrence85 to quantify the entanglement (see Methods), which for the two-qubit evolved state of Eq. (10) reads CAB . Notice that the concurrence of the Bell-like initial state is CAB . In Fig. 7(a) we plot the dynamics of concurrence CAB in the weak coupling regime between the two qubits with their corresponding cavities with ( has been assumed). For two-qubit initial states with , , the entanglement experiences sudden death without coupled cavities . By incorporating the additional cavities with relatively small coupling strength, e.g., J A = 0.5 Γ and J B = Γ, the concurrence still undergoes a Markovian decay but the time of entanglement disappearance is prolonged. Increasing the coupling strengths J A , J B of the relevant cavities drives the entanglement dynamics from Markovian regime to non-Markovian one. Moreover, the entanglement revivals after decay happen shortly after the evolution when the entanglement still has a large value. In general, the concurrences are enhanced pronouncedly with J A and J B . A comprehensive picture of the dynamics of concurrence as a function of coupling strength J is shown in Fig. 7(c) where we have assumed J A = J B = J. In Fig. 7(b) we plot the dynamics of in the strong coupling regime between qubit j and its cavity C 1j with for which the two-qubit dynamics is already non-Markovian in absence of cavity coupling, namely the entanglement can revive after dark periods. Remarkably, the figure shows that when the coupling J j between C 1j and C 2j is activated and gradually increased in each location, multiple transitions from non-Markovian to Markovian dynamics surface. We point out that the entanglement dynamics within the non-Markovian regime exhibit different qualitative behaviors with respect to the first time when entanglement oscillates. For instance, for , the non-Markovian entanglement oscillations (revivals) happen after its disappearance, while when and the entanglement oscillates before its sudden death. These dynamical features are clearly displayed in Fig. 7(d).

Figure 7 The dynamics of concurrence for different coupling strengths J A and J B in (a) weak qubit-cavity coupling regimes with and (b) strong qubit-cavity coupling regimes with . The initial state weights are chosen as (a) , and (b) , , while in both cases . The inset in (b) shows the long-time dynamics of concurrence for and . Panels (c,d) show the density plots of the two-qubit concurrence as a function of J (J A = J B = J is here assumed) and scaled time Γt, the others parameters being as in panels (a,b), respectively. The values of the concurrence in the density plots range within the interval: (c) [0, 0.6]; (d) [0, 1]. Full size image

As expected according to the results obtained before on the single-qubit coherence, a steady concurrence arises in the long-time limit if the secondary cavities C 2A , C 2B do not lose photons, i.e., . Figure 8(a) shows the dynamics of concurrence for qubits coupled to their cavities with strengths , . We can readily see that, in absence of coupling with the secondary cavities (J A = J B = 0), the entanglement disappear at a finite time without any revival. Contrarily, if the local couplings C 1j -C 2j are switched on and increased, the entanglement does not vanish at a finite time any more and reaches a steady value after undergoing non-Markovian oscillations. Furthermore, the steady value of concurrence is proportional to the local cavity coupling strengths J A , J B . In Fig. 8(b), the concurrence dynamics for is plotted under which the two-qubit entanglement experiences non-Markovian features, that is revivals after dark periods, already in absence of coupled cavities, as shown by the black solid curve for J A = J B = 0. Of course, in this case the entanglement eventually decays to zero. On the contrary, by adjusting suitable nonzero values of the local cavity couplings a considerable amount of entanglement can be trapped. As a peculiar qualitative dynamical feature, we highlight that the entanglement can revive and then be frozen after a finite dark period time of complete disappearance (e.g., see the inset of Fig. 8(b), for the short-time dynamics with , and also ). We finally point out that the the amount of preserved entanglement depends on the choice of the initial state (i.e., on the initial amount of entanglement) of the two qubits. As displayed in Fig. 9, the less initial entanglement, the less entanglement is in general maintained in the ideal case of . However, since there is not a direct proportionality between the evolved concurrence CAB and its initial value CAB , the maximal values of concurrence do not exactly appear at (corresponding to maximal initial entanglement) at any time in the evolution, as instead one could expect. It can be then observed that nonzero entanglement trapping is achieved for α > 0.2.

Figure 8 The dynamics of concurrence for different coupling strengths J- and J B in the presence of ideal coupled cavities C 2A and C 2B with for (a) , and (b) . The other parameters are chosen as , . The inset in (b) shows the short time dynamics of concurrence. Full size image

Figure 9 The concurrence as a function of the two-qubit initial state parameter α and the scaled time Γt for , , , and . The parameter α quantifies the initial entanglement according to the concurrence CAB . Full size image

Experimental paramaters

We conclude our study by discussing the experimental feasibility of the cavity-based architecture here proposed for the two-qubit assembly. Due to its cavity quantum electrodynamics characteristics, our engineered environment finds its natural realization in the well-established framework of circuit quantum electrodynamics (cQED) with transmon qubits and coplanar waveguide cavities64,86,87,88,89. The entangled qubits can be initialized by using the standard technique of a transmission-line resonator as a quantum bus64,90. Initial Bell-like states as the one we have considered here can be currently prepared with very high fidelity90. Considering up-to-date experimental parameters86,87,88,89 applied to our global system of Fig. 6, the average photon decay rate for the cavity C 1j (j = A, B) containing the qubit is , while the average photon lifetime for the high quality factor cavity C 2j is 87, which implies . The qubit-cavity interaction intensity κ j and the cavity-cavity coupling strength J j are usually of the same order of magnitude, with typical values . The typical cavity frequency is 64 while the qubit transition frequency can be arbitrarily adjusted in order to be resonant with the cavity frequency. The above experimental parameters put our system under the condition which guarantees the validity of the rotating wave approximation (RWA) for the qubit-cavity interaction here considered in the Hamiltonian of Eq. (1).

In order to assess the extent of entanglement preservation expected under these experimental conditions, we can analyze the concurrence evolution under the same parameters of Fig. 8(a) for κ j , J j , which are already within the experimental values, but with instead of being zero (ideal case), where . The natural estimated disappearance time of entanglement in absence of coupling between the cavities is , as seen from Fig. 8(a). When considering the experimental achievable decay rates for the cavities C 2j , we find that the entanglement is expected to be preserved until times t* orders of magnitude longer than , as shown in Table 1. In the case of higher quality factors for the cavities C 2j , such that the photon decay rate is of the order of , the entanglement can last even until the order of the seconds. These results provide a clear evidence of the practical powerful of our simple two-qubit architecture in significantly extending quantum entanglement lifetime for the implementation of given entanglement-based quantum tasks and algorithms14,90,91,92.

Table 1 Estimates of the experimental entanglement lifetimes t* for different values of the second cavities decay rates Γ 2 and the local cavity couplings J A , J B . Full size table

It is worth to mention that nowadays cQED technologies are also able to create a qubit-cavity coupling strength comparable to the cavity frequency, thus entering the so-called ultra-strong coupling regime93. In that case the RWA is to be relaxed and the counter-rotating terms in the qubit-cavity interaction have to be taken into account. According to known results for the single qubit evolution beyond the RWA94, it appears that the main effect of the counter-rotating terms in the Rabi Hamiltonian is the photon creation from vacuum under dephasing noise, which in turns induces a bit-flip error in the qubit evolution. This photon creation would be instead suppressed in the presence of dissipative (damping) mechanisms94. Since our cavity-based architecture is subject to amplitude damping noise, the qualitative long-time dynamics of quantum coherence and thus of entanglement are expected not to be significantly modified with respect to the case when RWA is retained. These argumentations stimulate a detailed study of the performance of our proposed architecture under the ultra-strong coupling regime out of RWA, to be addressed elsewhere.