Dephasing-assisted entanglement in a system of strongly coupled qubits

Creation of entangled states of quantum systems with low decoherence rates is a cornerstone in practical implementation of quantum computations. Processes of separate dephasing in each qubit in experimentally feasible systems is commonly accepted to destroy entanglement. In this work, we consider a system of two strongly coupled qubits that interact with dephasing reservoirs. We demonstrate that interaction with dephasing reservoirs can contribute to the formation of a long-lived mixed entangled state with nonzero concurrence. The weight of the subradiant state in this mixed state tends toward unity if the dephasing rate is much larger than the radiative rate and less than the coupling constant between qubits. The lifetime of this state is proportional to the exponent of the ratio of the coupling constant to environmental temperature and can be, by orders of magnitude, larger than the system's characteristic dephasing and dissipation times. Therefore, high dephasing, along with strong coupling, contributes to the creation of an entangled state with a long lifetime. This result paves the way for creation of long-lived entangled states.

above, revising the problem of two-qubit entanglement, in case of strong coupling, using recent progress in physics of open quantum systems is important [40].
In this paper, we consider the system of two strongly coupled qubits each of which interacts with its own dephasing reservoir. We show that dephasing not only results in the relaxation of non-diagonal terms of density matrix, but also leads to the transition between super-and sub-radiant states. If coupling constant between qubits is much larger than dephasing rate then there is exist long-lived mixed entangled state. Its lifetime is proportional to the exponent of ratio of coupling constant to environmental temperature. If coupling constant between qubits is much larger than the temperature of dephasing reservoir, the lifetime of the mentioned state is much larger than the characteristic dephasing and dissipation times of the system. It is demonstrated that the weight of entangled subradiant state in this mixed state tends to unity, if dephasing rate is much higher than the radiative rate, i.e., high dephasing contributes to entanglement. As a result, concurrence of this state is greater than zero which is a criterion of a state to be entangled.
The obtained results open the way to use dephasing as a source for the creation of long-lived entanglement.

II. FORMATION OF LONG-LIVED ENTANGLEMENT BY SEPARATE DEPHASING RESEVOIRS IN THE SYSTEM OF STRONGLY-COUPLED QUBITS
We consider the system of two strongly coupled TLSs with transition frequencies ω 1 ≈ ω 2 lying in the optical range and with the distance between TLSs r that is much smaller than the optical wavelength, i.e. r λ 1,2 = 2πc/ω 1,2 . We suppose that both TLSs interact with a common radiative reservoir, and each TLS interacts with its own dephasing reservoir. Each dephasing reservoir is associated with vibrational degrees of freedom (e.g. phonons [34]) of qubits. Because we are interested in the case of nearby qubit frequencies, |ω 1 − ω 2 | ω 1,2 we will use the rotating wave approximation [41]. Thus, the Hamiltonian of the system takes the form [33,34]: Here,σ 1,2 are lowering operators for the first and second TLSs, respectively. The last term in Eq. (1) describes the dipole-dipole interaction in the rotating wave approximation when dipoles occupy subwavelength volume [42], Ω = (d 1 d 2 − 3 (d 1 n) (d 2 n)) / r 3 is the constant of interaction, d 1,2 are dipole transition matrix elements of the first and second TLSs, and n is the unit vector from one TLS to another. First, we consider the case of zero detuning, ω 1 = ω 2 = ω, and Ω > 0. As such, the system eigenstates are the excited state of two TLSs, |ee , the superradiant symmetric state |s = (|eg + |ge ) / √ 2, the subradiant antisymmetric state |as = (|eg − |ge ) / √ 2, and the ground state |gg , with the eigenfrequencies 2ω, ω + Ω, ω − Ω, and 0, respectively.
The main consequence of usage of correct Lindblad operators (3) is that dephasing not only results in the relaxation of non-diagonal density matrix terms (in the eigenstate basis) but also leads to the transition between super-and subradiant states. Indeed, operatorsL dp,11 ,L dp,21 describe dephasing of both TLSs while operatorsL dp,12 ,L dp, 22 and L dp,13 ,L dp,23 describe forward and backward energy flows between the system and dephasing reservoirs, respectively. Notably, these operators have nonzero matrix elements between the system eigenstates |s and |as . This means that dephasing reservoirs result in transitions between super-and subradiant states. The reason for the energy flow between the system and the dephasing reservoir is interaction between the TLSs and expansion of the system reservoir interaction operators over eigenstates of interacting TLSs. These terms do not appear in the local approach, in which Lindblad superoperators are assumed as in the case of non-interacting TLSs [35,37]. Note that in [22] it has been mentioned that the dephasing reservoirs in the case of coupled qubits result in transition between super-and subradiant states. However, in [22] it has been considered stationary entanglement which is created by the external coherent drive. In contrast, here we are interested in the temporal dynamics of the entanglement between two qubits without any external coherent source.
OperatorL rad,1 defines the influence of the radiative reservoir and consist of two parts. The first part represents relaxation from the state |ee to the symmetric state |s and from the state |s to the ground state |gg . The second part is energy flow from one TLS to another. OperatorL rad,2 describes relaxation of nondiagonal density matrix terms.
Note that the matrix elements between the states |as and |gg of the Lindblad superoperators, arising from both dephasing and radiative reservoirs, are equal to zero. Thus, neither radiative nor dephasing reservoirs lead to transition from the subradiant state to the ground state. The only possibility of the system to leave the state |as is a temperature induced transition to the state |s which has a higher eigenfrequency. According to the KMS condition, the rate of this transition is smaller than the rate of transition from the state |s to the state |as by the factor exp (−2 Ω/kT ). From these qualitative arguments, it follows that if the system reaches the state |as , then the time during which the system remains in this state is larger than the characteristic relaxation times by the factor exp (2 Ω/kT ). Because this state is entangled, one can expect that entanglement between two TLSs will also be conserved during the time that is larger than the characteristic relaxation times by the factor exp (2 Ω/kT ).  Numerical simulations and analytical evaluations confirm these qualitative arguments. Results of the simulation of master equation (2), with the initial conditionρ S (0) = |ee ee|, are shown in the Fig. 1. Experimentally feasible parameters for molecules and quantum dots are γ dp 10 11 s −1 , γ rad 10 8 − 10 9 s −1 . The value of the Rabi constant depends on the dipole moment matrix element. We are interested in the case of strong coupling. Thus, we consider the case of quantum dots with a large dipole moment 50 D [7,8] and 10 nm distance between them, such that the corresponding Rabi constant is Ω 2 · 10 12 s −1 . In numerical simulations, we use the value 0.01 eV as a dimensionless unit of frequency such that the other parameters are γ dp1,2 (−2Ω) = γ dp = 2 · 10 −2 , γ rad (−ω ± Ω) = γ rad = 2 · 10 −4 , T dp = T rad = 2 · 10 −2 ( 5K), and Ω = 10 −1 .
In Fig. 1a, the dependencies of the non-zero matrix elements of the density matrix are shown, namely, the probabilities p ee , p as , p s , and p gg of the system to be in the states |ee , |as , |s , and |gg , respectively. There are two stages in the system dynamics. During the first stage (t ≤ 10 5 ), the system starting from the state |ee relaxes and primarily occupies the antisymmetric subradiant state. At the second stage, (t ≥ 10 5 ), the system leaves this state and relaxes to the ground state of the system, |gg . We calculated the entropy, S = −Tr (ρ S logρ S ), and concurrence, . The evolution of entropy, concurrence and occupancy of the antisymmetric state are presented in Fig. 1b.
At the first stage, entropy of the system grows ( Fig. 1b, blue dashed line) because the system transits from the state |ee to the subspace formed by states |eg and |ge . After the first stage finishes, the system primarily occupies the antisymmetric state (Fig. 1b, orange solid line) and stays in this state for a time which is much larger than all relaxation times in the system. As a consequence, the entropy at the second stage decreases while concurrence grows from zero to 0.95 (Fig. 1b, red dash-dotted line). This indicates that the system is in the entangled state. Finally, the system relaxes to the ground state which is also accompanied by the temporary grows of entropy.
As has been mentioned in the introduction, in general, separate reservoirs destruct entanglement. In our case, the main contribution to the formation of entanglement is made by separated dephasing reservoirs. We illustrate this fact with two examples II A and II B described below. Additionally, we examine the system with help of local approach in the third example II C to show that the local approach does not reveal the entanglement.
A. Long-lived entanglement in the case of one dephasing reservoir In the first example, we turn off the dephasing reservoirs associated with the first TLS, while the dephasing reservoir associated with the second TLS is not changed (i.e., γ 1,dp = 0 in Eq. (2)). In other words, we break the symmetry of interaction between the TLSs and dephasing reservoirs.
Numerical simulation shows (see Fig. 2) that no significant changes of entropy, concurrence, or occupancy of the antisymmetric state are observed, and time dynamics are similar to those from Fig. 1b. This is a consequence of the cross-relaxation processes described above. In the second example, we remove both dephasing reservoirs (γ 1,dp = γ 2,dp = 0), such that only the common radiative reservoir interacts with the TLSs. This case is analogous to the Dicke model of superradiance for two TLSs [44], such that only states |ee , |s , and |gg are occupied (see Fig. 3a). Relaxation of the system to the ground state with the rate 2γ rad occurs. Concurrence, in all moments of time, is zero (Fig. 3b, red dash-dotted line), which means that the system is in the disentangled state. Indeed, in the Dicke model of superradiance, entanglement is not generated [45,46].

C. Disappearance of entanglement in the local approach
In third example, we use local approach to describe the system. Under local approach, the Lindblad superoperators, arising from dephasing reservoirs, are assumed to coincide with the ones for noninteracting TLSs and have the form In the Fig. 4, the results of this model are presented. Both symmetric and antisymmetric states have equal populations at all moments of time. Concurrence is zero in all moments of time, thus, entanglement does not occur.

III. ANALYTICAL EVALUATION OF THE PROBABILITY OF SUBRADIANT STATE OCCUPATION AND ITS LIFETIME
To clarify the nature of the long-lived entangled state, from master equation (2), we find the approximate solution for p as in the limit Ω kT for time scale t ≥ 2γ −1 dp exp ( Ω/kT dp ). We rewrite the master equation in terms of occupancies p ee , p s , p as , p gg of the system eigenstates |ee , |s , |as , |gg and obtain the following equations: p ee = −2γ rad p ee ,ṗ s = 2γ rad p ee − (γ dp /2 + 2γ rad ) p s + γ dp 2 exp (−2Ω/T dp ) p as , p as = γ dp 2 p s − γ dp 2 exp (−2Ω/T dp ) p as ,ṗ gg = 2γ rad p s , with the initial condition p ee (0) = 1, p s (0) = p as (0) = p gg (0) = 0. Because the case of strong coupling is examined, Ω kT , at time scale γ −1 rad t γ −1 dp exp ( Ω/kT dp ), we can suppose that the rate of transition from the sub-to superradiant state is zero. In such an assumption, we obtain the following quasi-stationary solution: p qs ee = p qs s = 0, p qs as = γ dp /2 2γ rad + γ dp /2 , p qs gg = 2γ rad 2γ rad + γ dp /2 .
Analytical evaluation of antisymmetric state occupancy, p an as (t), is in a good agreement with exact solution, p as (t) at times t ≥ 2γ −1 dp exp ( Ω/kT dp ) (see Fig 5). Eq. (9) shows that the characteristic lifetime of the subradiant state, in the limit γ dp γ rad , is of the order t ent 0.5γ −1 rad exp (2 Ω/kT dp ) 2γ −1 dp , 0.5γ −1 rad , i.e., much greater than the characteristic time of dissipation and dephasing. Note that the long-lived state described above is a special case of metastable states proposed in [47].

IV. CONCURRENCE IN THE CASE OF NON-ZERO DETUNING BETWEEN QUBIT FREQUENCIES
The long-lived entangled state is also exists in the case of non-zero detuning between qubit frequencies, ω 1 = ω 2 . In the Appendix, we generalize the master equation for this case (see Eqs. (26) -(29)). Modeling this equation with initial conditionρ (0) = |ee ee|, one can get concurrence as a function of time and detuning (we fix ω 2 and change ω 1 ) (see Fig 6). Its seen that the lifetime of the entangled state decreases when detuning grows. However, there exists detuning range (|ω 1 − ω 2 | ∼ 0.1Ω), for which the system stays in the entangled state (concurrence is grater than 0.9) for the time which is much larger than the radiative and dephasing times. If detuning is out of this range, entangled state also is formed, but its lifetime is much shorter and its concurrence is smaller.

V. CONCLUSION
To conclude, in this work, we consider the system of two coupled two-level qubits with optical frequencies, which interact with the electromagnetic field of free space modes and with dephasing reservoirs. We show that when the coupling constant between qubits is much larger than the dephasing rate and dephasing rate is much larger than radiative rate, a long-lived mixed entangled state exists. The weight of the subradiant state in this mixed state, as well as concurrence, are equal to one up to the ratio γ rad /γ dp . The lifetime of this state is larger than the characteristic radiative time by the factor exp (2 Ω/kT dp ).
The obtained result, in principle, can be extended to the case of many qubits. If one creates such interaction between qubits that symmetry with respect to permutation is protected at least approximately, then super-and subradiant states may still be eigenstates of the system. For example, when qubits are placed in the ring and approximate interaction Hamiltonian is Ω . . e k . . . g , respectively (see [28]). There is dephasing-induced transition from the super-to subradiant state, analogously to the case of two qubits. Careful investigation we leave on further works. The obtained results pave the way for using dephasing as a resource for the creation of long-lived entanglement at experimentally realizable conditions. In the case of molecules, the characteristic dipole moment is 1 D. If one achieves the characteristic distance between them 10 nm, then the Rabi constant of interaction is Ω 10 9 s −1 . Thus, the factor exp (2 Ω/kT dp ) is much larger than unity for temperatures T dp ≤ 5 mK, and the lifetime of the entangled state can be ≥ γ −1 rad 10 −9 s. If one considers semiconductor quantum dots with the characteristic dipole moment 50 D [7,8] and distance between them 10 nm, one obtains Ω 2 · 10 12 s −1 and the factor Ω/kT dp much larger than unity for the temperatures T dp ≤ 10 K. For example, for temperature T dp ≤ 5 K the factor exp (2 Ω/kT dp ) 10 3 and the lifetime of the entangled state may reach 1 µs.
The predicted value of concurrence may be useful in contest of DLSZ protocol [48,49], where the value 0.9 ± 0.3 of concurrence is used. In this work, we obtain the value C ≈ 0.9. Also, this value is quite similar to maximal concurrence of two entangled photons achieved in recent experiments on optical quantum networks [50].
Note that the simple model developed in this work may shed light on the recently demonstrated robustness of the subradiant state decay time for an ensemble of cold atoms with the increasing of temperature [51]. In this regard, the toy-model that describes long-lived entanglement may be useful for qualitative analyses.

APPENDIX
Here we derive the master equation (2) and Lindblad superoperators (3) from the main text and consider the case of non-zero detuning.
To describe the interaction of two coupled qubits with the electromagnetic field of free space and separate dephasing reservoirs, we start with the Hamiltonian of the entire system. This Hamiltonian consists of three terms.Ĥ R represents the electromagnetic (EM) field of free space modes and degrees of freedom of dephasing reservoirs. The termĤ S Figure 6. The dependence of concurrence on time t and frequency detuning ω1 − ω2. The frequency of the first qubit is changed while other parameters of the system are the same as in the main text.
describes the coupled qubits, andĤ SR is responsible for the interaction between the EM field and qubit dipole moments, and for the interaction between dephasing reservoirs and qubits. The corresponding Hamiltonians have the form:Ĥ H SR =Ĥ S,1dp +Ĥ S,2dp +Ĥ S,rad ,Ĥ S,rad = k γ rad k σ † 1 +σ 1 +σ † 2 +σ 2 b † rad,k +b rad,k , Hereσ 1 andσ 2 are lowering operators for the first and second qubit, respectively; ω 1 and ω 2 are transition frequencies. The last term in Eq. (10) describes dipole-dipole interaction in the rotating wave approximation [41]. The coupling constant between qubits is Ω = (d 1 d 2 − 3 (d 1 n) (d 2 n)) / r 3 , d 1,2 are matrix elements of the first and second TLS dipole transitions, and n is the unit vector from one TLS to another. Operatorsb 1dp,k ,b 2dp,k denote lowering operators for the k-th degree of freedom of the first and second dephasing reservoirs, respectively. We suppose that they obey commutation relations b 1dp,k ,b † 1dp,k = δ kk , b 2dp,k ,b † 2dp,k = δ kk , and b 1dp,k ,b † 2dp,k = 0; this means that the reservoirs are independent. Operatorb rad,k denotes the lowering operator for the k-th EM field mode of free space with commutation relation b rad,k ,b † rad,k = δ kk The constants γ 1dp k , γ 2dp k , and γ rad k are interaction constants of the qubits with the k-th degree of freedom of dephasing and radiative reservoirs, respectively.