Notes

Time-invariant Discord: High Temperature Limit And Initial Environmental Correlations
We present a thorough investigation of the phenomena of frozen and time-invariant quantum discord for two-qubit systems independently interacting with local reservoirs. Our work takes into account several significant effects present in decoherence models, which have not been yet explored in the context of time-invariant quantum discord, but which in fact must be typically considered in almost all realistic models. Firstly, we study the combined influence of dephasing, dissipation and heating reservoirs at finite temperature. Contrarily to previous claims in the literature, we show the existence of time-invariant discord at high temperature limit in the weak coupling regime, and also examine the effect of thermal photons on the dynamical behaviour of frozen discord. Secondly, we explore the consequences of having initial correlations between the dephasing reservoirs. We demonstrate in detail how the time-invariant discord is modified depending on the relevant system parameters such as the strength of the initial amount of entanglement between the reservoirs.

pacs:
03.65.Yz, 42.50.Lc, 03.65.Ud, 05.30.Rt

Quantum theory is undoubtedly a cornerstone of modern physics and one of the pillars of all natural sciences. Even though it is a century old, some of its counter intuitive features have been started to be exploited on a fundamental level, only for the last two decades, with the emergence of the quantum information science and quantum computation theory nielsenbook . Quantum algorithms and quantum communication protocols have the potential to offer tremendous advantage over their corresponding classical counter parts. Genuine quantum correlations among the constituents of quantum systems are considered as the main resource for quantum technologies. Although the concept of quantum entanglement has been the only resource for almost all known quantum information tasks for many years entreview , several other quantifiers of genuine quantum correlations have been introduced in recent years. Among them, quantum discord has stood out and been extensively studied in the recent literature for its role as a significant alternative resource for quantum technologies discordreview . However, quantum systems are extremely fragile in real world conditions as they tend to rapidly lose their characteristic quantum features, such as quantum correlations, and become classical by uncontrollably interacting with their environment openbook .

Therefore, one of the major challenges for the practical implementation of quantum technologies is the development of reliable methods to retaliate or avoid the destructive effects of this unavoidable system-environment interaction. One way of protecting the precious quantum correlations in the system is to actively modify the properties of quantum processes, through the use of various methods such as dynamical decoupling techniques, to counter the effects of decoherence dd1 ; dd2 ; dd3 . On the other hand, an alternative strategy is to initiate our system in an appropriate state depending on the properties of the system-environment interaction such that quantum correlations in the system become frozen for a certain time interval despite the detrimental effects of the environment. It has been demonstrated both theoretically frozthry1 and experimentally frozexp1 ; frozexp2 that, under a suitable setting, quantum correlations quantified by quantum discord might become frozen while classical correlations keep decaying until a critical time is reached. At this point, a sudden transition takes place, classical correlations freeze and quantum discord begins to diminish, giving rise to a curious phenomenon known as the sudden transition between classical and quantum decoherence.

Frozen quantum discord has been first shown to occur in a Markovian pure dephasing model, where a bipartite system interacts with two independent environments, for a family of Bell-diagonal initial states frozthry1 . Later on, it has been found out that multiple intervals of frozen discord can emerge during the dynamics under a non-Markovian random telegraph noise setting frozthry2 . Even more remarkably, it has been revealed for a dephasing model with an ohmic type spectral density that quantum discord can in fact become forever frozen throughout the time evolution of the system, that is, no sudden transition point exists and discord becomes time-invariant, remaining constant at its initial value at all times invdisc . The existence of time-invariant discord is known to be inherently related with the emergence of non-Markovian memory effects.

Even though a lot of effort has put into the exploration of frozen and time-invariant discord phenomena, possible effects of a general non-Markovian noise setting, including the influences of dephasing, dissipation and heating channels, has not been investigated in the literature. Another unexplored problem is related to the consequences of having initial correlations between the environments for the occurrence of frozen and time-invariant discord. In other words, the main questions that we aim to answer in this work are the following: How are the properties of frozen discord affected by the presence of thermal reservoirs and what is the influence of thermal photons on this phenomenon? How is the frozen behaviour of quantum discord modified in case we initially have correlations between the environments? Specifically, here we will consider bipartite quantum systems initially prepared in Bell-diagonal states. Studying two different decoherence models, i.e., firstly a quite general combination of independent non-Markovian two-qubit dephasing, dissipation and heating channels at the high temperature limit, and secondly a two-qubit dephasing channel with initial environmental correlations, we will extensively explore the characteristics of the frozen and time-invariant discord phenomena. Our results indeed reveal the actual behaviour of time-invariant and frozen discord in decoherence models having realistic effects.

This paper is structured as follows. The concept of quantum discord is introduced in Sec. II. The time-invariant discord is investigated in the high temperature limit in Sec. III and the effects of various parameters of the environment on its existence is discussed. Sev. IV deals with the possible influences of initial environmental correlations on the behaviour of time-invariant discord. Sec. V includes the summary of our outcomes and our conclusion.

Before investigating some of the remarkable dynamical properties of quantum discord under decoherence channels, let us first briefly introduce its definition. The sum of classical and quantum correlations present in a bipartite quantum state can be measured with the help of quantum mutual information

I(ρAB)=S(ρA)+S(ρB)-S(ρAB),𝐼superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐴𝐵I(rho^AB)=S(rho^A)+S(rho^B)-S(rho^AB),italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) , (1)
where ρksuperscript𝜌𝑘rho^kitalic_ρ start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT (k=A,B)𝑘𝐴𝐵(k=A,B)( italic_k = italic_A , italic_B ) and ρABsuperscript𝜌𝐴𝐵rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT are respectively the reduced density matrices of the subsystems and the total system, with S(ρ)=-Tr(ρlog2ρ)𝑆𝜌Tr𝜌subscriptlog2𝜌S(rho)=-textmdTr(rhotextmdlog_2rho)italic_S ( italic_ρ ) = - Tr ( italic_ρ log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ρ ) being the von-Neumann entropy. Classical correlations in a bipartite quantum system, on the other hand, can be quantified as disc1

C(ρAB)=supΠk(S(ρA)-S(ρ|Πk)),Csuperscript𝜌𝐴𝐵subscriptsupremumsubscriptΠ𝑘𝑆superscript𝜌𝐴𝑆conditional𝜌subscriptΠ𝑘emphC(rho^AB)=sup_\Pi_k(S(rho^A)-S(rho|\Pi_k)),( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = roman_sup start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ | roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) , (2)
where the optimization is evaluated over all projective measurements ΠksubscriptΠ𝑘\Pi_k roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , performed locally on the subsystem B𝐵Bitalic_B. The quantum conditional entropy with respect to this measurement is then given by S(ρ|Πk)=∑kPkS(ρk)𝑆conditional𝜌subscriptΠ𝑘subscript𝑘subscript𝑃𝑘𝑆subscript𝜌𝑘S(rho|\Pi_k)=sum_kP_kS(rho_k)italic_S ( italic_ρ | roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), where the conditional density operator ρksubscript𝜌𝑘rho_kitalic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT associated with the measurement result k𝑘kitalic_k can be written as

ρk=(I⊗Πk)ρ(I⊗Πk)PKsubscript𝜌𝑘tensor-product𝐼subscriptΠ𝑘𝜌tensor-product𝐼subscriptΠ𝑘subscript𝑃𝐾rho_k=frac(IotimesPi_k)rho(IotimesPi_k)P_Kitalic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG ( italic_I ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_ρ ( italic_I ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG start_ARG italic_P start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_ARG (3)
with the probability Pk=Tr[(I⊗Πk)ρ(I⊗Πk)]subscript𝑃𝑘𝑇𝑟delimited-[]tensor-product𝐼subscriptΠ𝑘𝜌tensor-product𝐼subscriptΠ𝑘P_k=Tr[(IotimesPi_k)rho(IotimesPi_k)]italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_T italic_r [ ( italic_I ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_ρ ( italic_I ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ]. The amount of genuine quantum correlations quantified by the quantum discord can then be expressed as disc2

D(ρAB)=I(ρAB)-C(ρAB).𝐷superscript𝜌𝐴𝐵𝐼superscript𝜌𝐴𝐵𝐶superscript𝜌𝐴𝐵D(rho^AB)=I(rho^AB)-C(rho^AB).italic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_C ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) . (4)

Although it can be in general a very difficult task to analytically calculate quantum discord, for the bipartite quantum states that we will consider in this work, analytical expressions are available. In particular, assuming we have a bipartite density matrix in the following X-shaped form

ρ=(a00d0bw00wb0d00d),𝜌matrix𝑎00𝑑0𝑏𝑤00𝑤𝑏0𝑑00𝑑rho=beginpmatrixa&0&0&d\ 0&b&w&0\ 0&w&b&0\ d&0&0&d\ endpmatrix,italic_ρ = ( start_ARG start_ROW start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_d end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_b end_CELL start_CELL italic_w end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_w end_CELL start_CELL italic_b end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_d end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_d end_CELL end_ROW end_ARG ) , (5)
where the off-diagonal terms are real numbers, quantum discord can be evaluated analytically discformula . It is given by

D(ρ)=minD1,D2,𝐷𝜌subscript𝐷1subscript𝐷2D(rho)=minD_1,D_2,italic_D ( italic_ρ ) = roman_min italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (6)
where the minimum is simply calculated for the terms

D1=subscript𝐷1absentdisplaystyle D_1=italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = S(ρA)-S(ρAB)-alog2(aa+b)-blog2(ba+b)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐴𝐵𝑎subscript2𝑎𝑎𝑏𝑏subscript2𝑏𝑎𝑏displaystyle S(rho^A)-S(rho^AB)-alog_2(fracaa+b)-blog_2(% fracba+b)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_a roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG italic_a end_ARG start_ARG italic_a + italic_b end_ARG ) - italic_b roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG italic_b end_ARG start_ARG italic_a + italic_b end_ARG )

-displaystyle-- blog2(bd+b)-dlog2(dd+b),𝑏subscript2𝑏𝑑𝑏𝑑subscript2𝑑𝑑𝑏displaystyle blog_2(fracbd+b)-dlog_2(fracdd+b),italic_b roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG italic_b end_ARG start_ARG italic_d + italic_b end_ARG ) - italic_d roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG italic_d end_ARG start_ARG italic_d + italic_b end_ARG ) ,

D2=subscript𝐷2absentdisplaystyle D_2=italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = S(ρA)-S(ρAB)-Δ+log2Δ+-Δ-log2Δ-,𝑆superscript𝜌𝐴𝑆superscript𝜌𝐴𝐵subscriptΔ𝑙𝑜subscript𝑔2subscriptΔsubscriptΔ𝑙𝑜subscript𝑔2subscriptΔdisplaystyle S(rho^A)-S(rho^AB)-Delta_+log_2Delta_+-Delta_-% log_2Delta_-,italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - roman_Δ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_Δ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT - roman_Δ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_Δ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ,
where Δ±=12(1±M)subscriptΔplus-or-minus12plus-or-minus1𝑀Delta_pm=frac12(1pm M)roman_Δ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 ± italic_M ) and M2=(a-d)2+4(|z|+|w|)2superscript𝑀2superscript𝑎𝑑24superscript𝑧𝑤2M^2=(a-d)^2+4(|z|+|w|)^2italic_M start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( italic_a - italic_d ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 ( | italic_z | + | italic_w | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

III Time-invariant Discord at the High Temperature Limit

We commence this section by introducing a heuristic model where a qubit is coupled to a composite reservoir including the effects of dephasing, dissipation and heating model1 ; model11 . Let us consider the following time-local master equation

dρdt=-i2(ω+h(t))[σz,ρ]+γz(t)2[σzρσz-ρ]𝑑𝜌𝑑𝑡𝑖2𝜔ℎ𝑡subscript𝜎𝑧𝜌subscript𝛾𝑧𝑡2delimited-[]subscript𝜎𝑧𝜌subscript𝜎𝑧𝜌displaystylefracdrhodt=frac-i2(omega+h(t))[sigma_z,rho]+frac% gamma_z(t)2[sigma_zrhosigma_z-rho]divide start_ARG italic_d italic_ρ end_ARG start_ARG italic_d italic_t end_ARG = divide start_ARG - italic_i end_ARG start_ARG 2 end_ARG ( italic_ω + italic_h ( italic_t ) ) [ italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , italic_ρ ] + divide start_ARG italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG 2 end_ARG [ italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT italic_ρ italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT - italic_ρ ]

+γ1(t)2(σ+ρσ--12σ-σ+,ρ)subscript𝛾1𝑡2subscript𝜎𝜌subscript𝜎12subscript𝜎subscript𝜎𝜌displaystylehskip 22.76219pt+fracgamma_1(t)2(sigma_+rhosigma_-% -frac12\sigma_-sigma_+,rho)+ divide start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG 2 end_ARG ( italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT italic_ρ italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_ρ )

+γ2(t)2(σ-ρσ+-12σ+σ-,ρ),subscript𝛾2𝑡2subscript𝜎𝜌subscript𝜎12subscript𝜎subscript𝜎𝜌displaystylehskip 22.76219pt+fracgamma_2(t)2(sigma_-rhosigma_+% -frac12\sigma_+sigma_-,rho),+ divide start_ARG italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG 2 end_ARG ( italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT italic_ρ italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT , italic_ρ ) , (7)
with σ±subscript𝜎plus-or-minussigma_pmitalic_σ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT being the raising and lowering operators of the qubit, σzsubscript𝜎𝑧sigma_zitalic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT the Pauli spin operator in the z-direction, ω𝜔omegaitalic_ω the transition frequency of the qubit, h(t)ℎ𝑡h(t)italic_h ( italic_t ) a time-dependent frequency shift, and γ1,2,zsubscript𝛾12𝑧gamma_1,2,zitalic_γ start_POSTSUBSCRIPT 1 , 2 , italic_z end_POSTSUBSCRIPT time-dependent decay rates. While the first term describes the Lamb shift corrections to the free Hamiltonian, the second, third and the fourth terms describe dephasing, heating, and dissipation, respectively. The state of the qubit at time t𝑡titalic_t can then be written as ρ(t)=Λω(t)ρ(0)𝜌𝑡subscriptΛ𝜔𝑡𝜌0rho(t)=Lambda_omega(t)rho(0)italic_ρ ( italic_t ) = roman_Λ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_t ) italic_ρ ( 0 ), where ρ(0)𝜌0rho(0)italic_ρ ( 0 ) is the initial state and Λω(t)subscriptΛ𝜔𝑡Lambda_omega(t)roman_Λ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_t ) is given by

Λω(t)=(10000η⟂(t)cos(ϕ(t))-η⟂(t)sin(ϕ(t))00η⟂(t)sin(ϕ(t))η⟂(t)cos(ϕ(t))0κ(t)00η∥(t)),subscriptΛ𝜔𝑡matrix10000subscript𝜂perpendicular-to𝑡italic-ϕ𝑡subscript𝜂perpendicular-to𝑡italic-ϕ𝑡00subscript𝜂perpendicular-to𝑡italic-ϕ𝑡subscript𝜂perpendicular-to𝑡italic-ϕ𝑡0𝜅𝑡00subscript𝜂parallel-to𝑡Lambda_omega(t)=beginpmatrix1&0&0&0\ 0&eta_perp(t)cos(phi(t))&-eta_perp(t)sin(phi(t))&0\ 0&eta_perp(t)sin(phi(t))&eta_perp(t)cos(phi(t))&0\ kappa(t)&0&0&eta_parallel(t)\ endpmatrix,roman_Λ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_t ) = ( start_ARG start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_η start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ( italic_t ) roman_cos ( italic_ϕ ( italic_t ) ) end_CELL start_CELL - italic_η start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ( italic_t ) roman_sin ( italic_ϕ ( italic_t ) ) end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_η start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ( italic_t ) roman_sin ( italic_ϕ ( italic_t ) ) end_CELL start_CELL italic_η start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ( italic_t ) roman_cos ( italic_ϕ ( italic_t ) ) end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_κ ( italic_t ) end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_η start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT ( italic_t ) end_CELL end_ROW end_ARG ) , (8)
where the elements of the above matrix Λω(t)subscriptΛ𝜔𝑡Lambda_omega(t)roman_Λ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_t ) read

ϕ(t)=ωt+θ(t),italic-ϕ𝑡𝜔𝑡𝜃𝑡displaystylephi(t)=omega t+theta(t),italic_ϕ ( italic_t ) = italic_ω italic_t + italic_θ ( italic_t ) ,

η∥=e-Γ(t),subscript𝜂parallel-tosuperscript𝑒Γ𝑡displaystyleeta_parallel=e^-Gamma(t),italic_η start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - roman_Γ ( italic_t ) end_POSTSUPERSCRIPT ,

η⟂=e-Γ(t)/2-Γz(t),subscript𝜂perpendicular-tosuperscript𝑒Γ𝑡2subscriptΓ𝑧𝑡displaystyleeta_perp=e^-Gamma(t)/2-Gamma_z(t),italic_η start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - roman_Γ ( italic_t ) / 2 - roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ,

κ(t)=-e-Γ(t)(1+2G(t))+1,𝜅𝑡superscript𝑒Γ𝑡12𝐺𝑡1displaystylekappa(t)=-e^-Gamma(t)(1+2G(t))+1,italic_κ ( italic_t ) = - italic_e start_POSTSUPERSCRIPT - roman_Γ ( italic_t ) end_POSTSUPERSCRIPT ( 1 + 2 italic_G ( italic_t ) ) + 1 , (9)
and the four terms appearing in Eq. (III) are expressed as

θ(t)=∫0t𝑑t′h(t′),𝜃𝑡subscriptsuperscript𝑡0differential-dsuperscript𝑡′ℎsuperscript𝑡′displaystyletheta(t)=int^t_0dt^primeh(t^prime),italic_θ ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_d italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_h ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ,

Γ(t)=∫0t𝑑t′(γ1(t′)+γ2(t′))/2,Γ𝑡subscriptsuperscript𝑡0differential-dsuperscript𝑡′subscript𝛾1superscript𝑡′subscript𝛾2superscript𝑡′2displaystyleGamma(t)=int^t_0dt^primehskip 2.84526pt(gamma_1(t^% prime)+gamma_2(t^prime))/2,roman_Γ ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_d italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) + italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) / 2 ,

Γz(t)=∫0t𝑑t′γz(t′),subscriptΓ𝑧𝑡subscriptsuperscript𝑡0differential-dsuperscript𝑡′subscript𝛾𝑧superscript𝑡′displaystyleGamma_z(t)=int^t_0dt^primehskip 2.84526ptgamma_z(t% ^prime),roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_d italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ,

G(t)=∫0t𝑑t′eΓ(t′)γ2(t′)/2.𝐺𝑡subscriptsuperscript𝑡0differential-dsuperscript𝑡′superscript𝑒Γsuperscript𝑡′subscript𝛾2superscript𝑡′2displaystyle G(t)=int^t_0dt^primehskip 2.84526pte^Gamma(t^prime% )gamma_2(t^prime)/2.italic_G ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_d italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Γ ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 . (10)

In this work, we ignore the effect of the first term in Eq. (III) and focus on a thermal reservoir at temperature T𝑇Titalic_T. We assume that the heating and the dissipation decay rates are respectively given by γ1(t)/2=(N)f(t)subscript𝛾1𝑡2𝑁𝑓𝑡gamma_1(t)/2=(N)f(t)italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) / 2 = ( italic_N ) italic_f ( italic_t ) and γ2(t)/2=(N+1)f(t)subscript𝛾2𝑡2𝑁1𝑓𝑡gamma_2(t)/2=(N+1)f(t)italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) / 2 = ( italic_N + 1 ) italic_f ( italic_t ), where N𝑁Nitalic_N represents the mean number of photons in the modes of the thermal reservoir at temperature T𝑇Titalic_T. We note that, in case of a zero temperature reservoir, the heating rate vanishes, that is, γ1(t)=0subscript𝛾1𝑡0gamma_1(t)=0italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = 0, and the dissipation rate is simply given by γ2(t)/2=f(t)subscript𝛾2𝑡2𝑓𝑡gamma_2(t)/2=f(t)italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) / 2 = italic_f ( italic_t ). For the model considered in our study, the spectral density is taken as J(ω)=γ0λ2/2π[(ω0-Δ-ω)2+λ2]𝐽𝜔subscript𝛾0superscript𝜆22𝜋delimited-[]superscriptsubscript𝜔0Δ𝜔2superscript𝜆2J(omega)=gamma_0lambda^2/2pi[(omega_0-Delta-omega)^2+lambda^2]italic_J ( italic_ω ) = italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_π [ ( italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - roman_Δ - italic_ω ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ], where γ0subscript𝛾0gamma_0italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is an effective coupling constant which is related to the relaxation time of the system τR≈1/γ0subscript𝜏𝑅1subscript𝛾0tau_Rapprox 1/gamma_0italic_τ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ≈ 1 / italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and λ𝜆lambdaitalic_λ is the width of the Lorentzian spectrum that is connected to the reservoir correlation time τB≈1/λsubscript𝜏𝐵1𝜆tau_Bapprox 1/lambdaitalic_τ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ≈ 1 / italic_λ. Additionally, Δ=ω-νcΔ𝜔subscript𝜈𝑐Delta=omega-
u_croman_Δ = italic_ω - italic_ν start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is the detuning of ω𝜔omegaitalic_ω and νcsubscript𝜈𝑐
u_citalic_ν start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is the centeral frequency of the thermal reservoir. It is worth noting that the effective coupling between the qubit and its environment decreases when the value of the detuning ΔΔDeltaroman_Δ increases coupdetun . Taking into account these considerations, the function f(t)𝑓𝑡f(t)italic_f ( italic_t ) can be written in the following form model11

f(t)=-2ℜC˙(t)C(t),𝑓𝑡2˙𝐶𝑡𝐶𝑡displaystyle f(t)=-2Re\fracdotC(t)C(t),italic_f ( italic_t ) = - 2 roman_ℜ divide start_ARG over˙ start_ARG italic_C end_ARG ( italic_t ) end_ARG start_ARG italic_C ( italic_t ) end_ARG ,

C(t)=e-(λ-iΔ)t/2(cosh(dt2)+λ-iΔdsinh(dt2))C(0),𝐶𝑡superscript𝑒𝜆𝑖Δ𝑡2𝑑𝑡2𝜆𝑖Δ𝑑𝑑𝑡2𝐶0displaystyle C(t)=e^-(lambda-iDelta)t/2(cosh(fracdt2)+fraclambda% -iDeltadsinh(fracdt2))C(0),italic_C ( italic_t ) = italic_e start_POSTSUPERSCRIPT - ( italic_λ - italic_i roman_Δ ) italic_t / 2 end_POSTSUPERSCRIPT ( roman_cosh ( divide start_ARG italic_d italic_t end_ARG start_ARG 2 end_ARG ) + divide start_ARG italic_λ - italic_i roman_Δ end_ARG start_ARG italic_d end_ARG roman_sinh ( divide start_ARG italic_d italic_t end_ARG start_ARG 2 end_ARG ) ) italic_C ( 0 ) ,
with d=(λ-iΔ)2-2γ0λ𝑑superscript𝜆𝑖Δ22subscript𝛾0𝜆d=sqrt(lambda-iDelta)^2-2gamma_0lambdaitalic_d = square-root start_ARG ( italic_λ - italic_i roman_Δ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_λ end_ARG. We can also define R=γ0/λ𝑅subscript𝛾0𝜆R=gamma_0/lambdaitalic_R = italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_λ in order to distinguish the strong coupling regime from the weak coupling regime. It has been demonstrated that in the weak coupling regime, R≪1much-less-than𝑅1Rll 1italic_R ≪ 1, for sufficiently large detunings, the function f(t)𝑓𝑡f(t)italic_f ( italic_t ) might take on negative values within certain time intervals, hence the dynamics of the qubit becomes nondivisible and non-Markovian nonmarkov .

Supposing that the dephasing reservoir is at temperature T𝑇Titalic_T, then the time-dependent dephasing rate takes the form

γz(t)=∫𝑑ωJ(ω)coth(ℏω/2kBT)sin(ωt)ω.subscript𝛾𝑧𝑡differential-d𝜔𝐽𝜔hyperbolic-cotangentPlanck-constant-over-2-pi𝜔2subscript𝑘𝐵𝑇𝜔𝑡𝜔gamma_z(t)=int domega J(omega)coth(hbaromega/2k_BT)fracsin(% omega t)omega.italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) = ∫ italic_d italic_ω italic_J ( italic_ω ) roman_coth ( roman_ℏ italic_ω / 2 italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T ) divide start_ARG roman_sin ( italic_ω italic_t ) end_ARG start_ARG italic_ω end_ARG . (12)
In the high temperature limit, the above equation simply reads

γz(t)=2kBTℏ∫𝑑ωJ(ω)sin(ωt)ω2,subscript𝛾𝑧𝑡2subscript𝑘𝐵𝑇Planck-constant-over-2-pidifferential-d𝜔𝐽𝜔𝜔𝑡superscript𝜔2gamma_z(t)=frac2k_BThbarint domega J(omega)fracsin(omega t)% omega^2,italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG 2 italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_ARG start_ARG roman_ℏ end_ARG ∫ italic_d italic_ω italic_J ( italic_ω ) divide start_ARG roman_sin ( italic_ω italic_t ) end_ARG start_ARG italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , (13)
where ωT=kBT/ℏsubscript𝜔𝑇subscript𝑘𝐵𝑇Planck-constant-over-2-piomega_T=k_BT/hbaritalic_ω start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T / roman_ℏ is the thermal frequency and we assume that the spectral density is of the ohmic type, i.e., J(ω)=α(ωs/ωcs-1)e-ω/ωc𝐽𝜔𝛼superscript𝜔𝑠subscriptsuperscript𝜔𝑠1𝑐superscript𝑒𝜔subscript𝜔𝑐J(omega)=alpha(omega^s/omega^s-1_c)e^-omega/omega_citalic_J ( italic_ω ) = italic_α ( italic_ω start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT / italic_ω start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_ω / italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, with ωcsubscript𝜔𝑐omega_citalic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT being the cutoff frequency, s𝑠sitalic_s the Ohmicity parameter, and α𝛼alphaitalic_α the coupling constant. In addition, γz(t)subscript𝛾𝑧𝑡gamma_z(t)italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) takes temporarily negative values provided s>scrit=3𝑠subscript𝑠𝑐𝑟𝑖𝑡3s>s_crit=3italic_s >italic_s start_POSTSUBSCRIPT italic_c italic_r italic_i italic_t end_POSTSUBSCRIPT = 3. Therefore, if we have a super-Ohmic spectral density with s>3𝑠3s>3italic_s >3, information can flow back form the environment to the system giving rise to memory effects invdisc .

Consequently, with the help of Eqs. (III-13), one can obtain the following expressions to fully describe the dynamics of a qubit coupled to the considered reservoirs,

Γ(t)=-ℜ[ln(x(t)2N+1)],Γ𝑡𝑥superscript𝑡2𝑁1displaystyleGamma(t)=-Re[ln(x(t)^2N+1)],roman_Γ ( italic_t ) = - roman_ℜ [ roman_ln ( italic_x ( italic_t ) start_POSTSUPERSCRIPT 2 italic_N + 1 end_POSTSUPERSCRIPT ) ] ,

Γz(t)=α2kBTℏωcΓ~(-2+s)(1-(1+ωc2t2)(-s+2)/2fragmentssubscriptΓ𝑧fragments(t)α2subscript𝑘𝐵𝑇Planck-constant-over-2-pisubscript𝜔𝑐~Γfragments(2s)fragments(1superscriptfragments(1superscriptsubscript𝜔𝑐2superscript𝑡2)𝑠22displaystyleGamma_z(t)=alphahskip 2.84526ptfrac2k_BThbarhskip 2.% 84526ptomega_ctildeGamma(-2+s)(1-(1+omega_c^2hskip 2.84526ptt^2% )^(-s+2)/2roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) = italic_α divide start_ARG 2 italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_ARG start_ARG roman_ℏ italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG over~ start_ARG roman_Γ end_ARG ( - 2 + italic_s ) ( 1 - ( 1 + italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ( - italic_s + 2 ) / 2 end_POSTSUPERSCRIPT

×cos((-2+s)arctan(ωct))),fragmentsfragments(fragments(2s)fragments(subscript𝜔𝑐t))),displaystylehskip 28.45274pttimescos((-2+s)arctan(omega_chskip 2.8452% 6ptt))),× roman_cos ( ( - 2 + italic_s ) roman_arctan ( italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_t ) ) ) ,

κ(t)=-12N+1(1-exp[-ℜ[ln(x(t)2N+1)]]),𝜅𝑡12𝑁11𝑥superscript𝑡2𝑁1displaystylekappa(t)=frac-12N+1(1-exp[-Re[ln(x(t)^2N+1)]]),italic_κ ( italic_t ) = divide start_ARG - 1 end_ARG start_ARG 2 italic_N + 1 end_ARG ( 1 - roman_exp [ - roman_ℜ [ roman_ln ( italic_x ( italic_t ) start_POSTSUPERSCRIPT 2 italic_N + 1 end_POSTSUPERSCRIPT ) ] ] ) , (14)
where x(t)=C(t)/C(0)2𝑥𝑡superscript𝐶𝑡𝐶02x(t)=C(t)/C(0)^2italic_x ( italic_t ) = italic_C ( italic_t ) / italic_C ( 0 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, Γ~(s)~Γ𝑠tildeGamma(s)over~ start_ARG roman_Γ end_ARG ( italic_s ) is the Euler gamma function. The memory time of the dephasing environment can be defined by ωc-1subscriptsuperscript𝜔1𝑐omega^-1_citalic_ω start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. As a consequence, we can define β=ωc/λ𝛽subscript𝜔𝑐𝜆beta=omega_c/lambdaitalic_β = italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_λ to characterize the relation between the cut-off frequency of the dephasing environment and the width of the Lorentzian spectrum of the thermal reservoir.

We have dealt with the description of the dynamics of a single qubit up to this point in our paper. Let us now suppose that we have a bipartite quantum system composed of two identical qubits, labelled as A𝐴Aitalic_A and B𝐵Bitalic_B, that are locally coupled to their own environments. We also assume that the individual environments are identical and not correlated with each other. Hence, it is possible to obtain the time evolution of the two-qubit system from the single qubit dynamics in a straightforward fashion as ρAB(t)=(ΛωA(t)⊗ΛωB(t))ρAB(0)superscript𝜌𝐴𝐵𝑡tensor-productsubscriptsuperscriptΛ𝐴𝜔𝑡subscriptsuperscriptΛ𝐵𝜔𝑡superscript𝜌𝐴𝐵0rho^AB(t)=(Lambda^A_omega(t)otimesLambda^B_omega(t))rho^AB(0)italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ( italic_t ) = ( roman_Λ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_t ) ⊗ roman_Λ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_t ) ) italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ( 0 ). In the course of our work, we choose the initial state of the two-qubit open system in the form of Bell-diagonal states

ρS(0)=14(I⊗I+∑i=03miσi⊗σi),subscript𝜌𝑆014tensor-product𝐼𝐼subscriptsuperscript3𝑖0tensor-productsubscript𝑚𝑖subscript𝜎𝑖subscript𝜎𝑖rho_S(0)=frac14(Iotimes I+sum^3_i=0m_isigma_iotimessigma_% i),italic_ρ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( 0 ) = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( italic_I ⊗ italic_I + ∑ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , (15)
where misubscript𝑚𝑖m_iitalic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are three real number such that -1≤mi≤11subscript𝑚𝑖1-1leq m_ileq 1- 1 ≤ italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 1, and σisubscript𝜎𝑖sigma_iitalic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are the Pauli spin operators in the x,y and z directions. Therefore, the time-evolution of our system is given by

ρS(t)=(ρ1100ρ140ρ22ρ2300ρ23ρ330ρ1400ρ44),subscript𝜌𝑆𝑡matrixsubscript𝜌1100subscript𝜌140subscript𝜌22subscript𝜌2300subscript𝜌23subscript𝜌330subscript𝜌1400subscript𝜌44rho_S(t)=beginpmatrixrho_11&0&0&rho_14\ 0&rho_22&rho_23&0\ 0&rho_23&rho_33&0\ rho_14&0&0&rho_44\ endpmatrix,italic_ρ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_t ) = ( start_ARG start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , (16)
where the density matrix elements can be evaluated as

ρ11=14((1+κ(t))2+η∥2(t)m3),subscript𝜌1114superscript1𝜅𝑡2subscriptsuperscript𝜂2parallel-to𝑡subscript𝑚3displaystylerho_11=frac14((1+kappa(t))^2+eta^2_parallel(t)m_% 3),italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( ( 1 + italic_κ ( italic_t ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT ( italic_t ) italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,

ρ22=ρ33=14(1-κ(t)2-η∥2(t)m3),subscript𝜌22subscript𝜌33141𝜅superscript𝑡2subscriptsuperscript𝜂2parallel-to𝑡subscript𝑚3displaystylerho_22=rho_33=frac14(1-kappa(t)^2-eta^2_% parallel(t)m_3),italic_ρ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 - italic_κ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT ( italic_t ) italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,

ρ44=14((1-κ(t))2+η∥2(t)m3),subscript𝜌4414superscript1𝜅𝑡2subscriptsuperscript𝜂2parallel-to𝑡subscript𝑚3displaystylerho_44=frac14((1-kappa(t))^2+eta^2_parallel(t)m_% 3),italic_ρ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( ( 1 - italic_κ ( italic_t ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT ( italic_t ) italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,

ρ23=m1+m24η⟂2(t),subscript𝜌23subscript𝑚1subscript𝑚24subscriptsuperscript𝜂2perpendicular-to𝑡displaystylerho_23=fracm_1+m_24eta^2_perp(t),italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT = divide start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ( italic_t ) ,

ρ14=m1-m24η⟂2(t).subscript𝜌14subscript𝑚1subscript𝑚24subscriptsuperscript𝜂2perpendicular-to𝑡displaystylerho_14=fracm_1-m_24eta^2_perp(t).italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT = divide start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ( italic_t ) . (17)

III.1 Pure Dephasing

In this subsection, we will only consider the pure dephasing case at the high temperature limit in the weak coupling regime, when the two qubits interact with independent reservoirs. We choose the initial state of our two-qubit system from a family of Bell-diagonal states, namely, from the states given by Eq. (15) having the three real parameters m1=1subscript𝑚11m_1=1italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 and m2=-m3=msubscript𝑚2subscript𝑚3𝑚m_2=-m_3=mitalic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_m, with |m|<1𝑚1|m|<1| italic_m | <1. For such initial states and in the presence of pure dephasing dynamics, classical correlations and mutual information can be written in a compact form in the following way

C(ρAB)𝐶superscript𝜌𝐴𝐵displaystyle C(rho^AB)italic_C ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) =∑j=121+(-1)jχ(t)2log2[1+(-1)jχ(t)],absentsuperscriptsubscript𝑗121superscript1𝑗𝜒𝑡2subscriptlog2delimited-[]1superscript1𝑗𝜒𝑡displaystyle=sum_j=1^2frac1+(-1)^jchi(t)2textlog_2[1+(-1)^% jchi(t)],= ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_χ ( italic_t ) end_ARG start_ARG 2 end_ARG log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_χ ( italic_t ) ] , (18)

I(ρAB(t))𝐼superscript𝜌𝐴𝐵𝑡displaystyle I(rho^AB(t))italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ( italic_t ) ) =∑j=121+(-1)jm2log2[1+(-1)jm]absentsuperscriptsubscript𝑗121superscript1𝑗𝑚2subscriptlog2delimited-[]1superscript1𝑗𝑚displaystyle=sum_j=1^2frac1+(-1)^jm2textlog_2[1+(-1)^jm]= ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_m end_ARG start_ARG 2 end_ARG log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_m ]

+∑j=121+(-1)je-2Γz(t)2log2[1+(-1)je-2Γz(t)],superscriptsubscript𝑗121superscript1𝑗superscript𝑒2subscriptΓ𝑧𝑡2subscriptlog2delimited-[]1superscript1𝑗superscript𝑒2subscriptΓ𝑧𝑡displaystyle+sum_j=1^2frac1+(-1)^je^-2Gamma_z(t)2textlog_% 2[1+(-1)^je^-2Gamma_z(t)],+ ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - 2 roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - 2 roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ] , (19)
with χ(t)=maxe-2Γz(t),m𝜒𝑡superscript𝑒2subscriptΓ𝑧𝑡𝑚chi(t)=maxe^-2Gamma_z(t),mitalic_χ ( italic_t ) = roman_max italic_e start_POSTSUPERSCRIPT - 2 roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , italic_m , and thus quantum discord D(ρAB)𝐷superscript𝜌𝐴𝐵D(rho^AB)italic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) can be simply evaluated from their difference. Using these equations, one can define a transition time t~~𝑡tildetover~ start_ARG italic_t end_ARG as

e-2Γz(t~)=m,superscript𝑒2subscriptΓ𝑧~𝑡𝑚e^-2Gamma_z(tildet)=m,italic_e start_POSTSUPERSCRIPT - 2 roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( over~ start_ARG italic_t end_ARG ) end_POSTSUPERSCRIPT = italic_m , (20)
below which (t t~𝑡~𝑡t>tildetitalic_t >over~ start_ARG italic_t end_ARG), classical correlations freeze and discord starts to decrease.

Let us now assume that α=ℏωc/2kBT𝛼Planck-constant-over-2-pisubscript𝜔𝑐2subscript𝑘𝐵𝑇alpha=hbaromega_c/2k_BTitalic_α = roman_ℏ italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / 2 italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T in the second line of Eq. (III) and hence Γz(t)subscriptΓ𝑧𝑡Gamma_z(t)roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) is independent of the temperature. With gaming news in mind, as we are working at the high temperature limit, i.e., 2kBT≫ℏωcmuch-greater-than2subscript𝑘𝐵𝑇Planck-constant-over-2-pisubscript𝜔𝑐2k_BTgghbaromega_c2 italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T ≫ roman_ℏ italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, the coupling between the open system and its environment will be weak.

Dynamical behaviours of the quantum discord (dashed red line) and the classical correlations (dotted-dashed green line) are plotted in Figs. 1(a) and 1(b) as a function of ωctsubscript𝜔𝑐𝑡omega_ctitalic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_t for the ohmicity parameters s=2.5𝑠2.5s=2.5italic_s = 2.5 and s=3.5𝑠3.5s=3.5italic_s = 3.5, respectively, where the initial state is chosen as m=0.1𝑚0.1m=0.1italic_m = 0.1. Since we work at the high temperature limit in the weak coupling regime, we consider that α=0.01𝛼0.01alpha=0.01italic_α = 0.01 and 2kBT/ℏωc=1002subscript𝑘𝐵𝑇Planck-constant-over-2-pisubscript𝜔𝑐1002k_BT/hbaromega_c=1002 italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T / roman_ℏ italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 100. Note that for the non-Markovian memory effects to emerge at the high temperature limit, the ohmicity parameter should satisfy the condition s≥3𝑠3sgeq 3italic_s ≥ 3. Looking at Fig. 1, we clearly see that whereas we have frozen discord for a finite time interval in case of Markovian dynamics, time-invariant discord can be observed for non-Markovian evolution. In other words, while Eq. (20) has a solution for s=2.5𝑠2.5s=2.5italic_s = 2.5, there exists no solution for it when s=3.5𝑠3.5s=3.5italic_s = 3.5, giving rise to time-invariant discord. Therefore, we find out that the inherent connection between the non-Markovianity and the occurrence of time-invariant discord, as first explained in Ref. invdisc , still holds at the high temperature limit. However, contrarily to what has been claimed in Ref. invdisc , there indeed exists s𝑠sitalic_s and m𝑚mitalic_m even at the high temperature limit, such that the condition given in Eq. (20) is never satisfied and the phenomenon of time-invariant discord can still be observed. We emphasize that the key point here leading the emergence of time-invariant discord at the high temperature limit is the fact that we are working in the weak coupling regime as no such phenomenon would be present if the coupling was strong.

In Fig. 2(a), we display the outlook of correlation dynamics in the presence of independent pure dephasing reservoirs in the s-ωct𝑠subscript𝜔𝑐𝑡s-omega_ctitalic_s - italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_t plane. From this plot, one can see the range of values for s𝑠sitalic_s and ωctsubscript𝜔𝑐𝑡omega_ctitalic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_t for which Eq. (20) has a solution (frozen discord and sudden transition), and for which it does not have a solution (time-invariant discord). The intersection between the quantum and classical decoherence regions, as respectively shown by pink (dark gray) and yellow (gray) areas in the figure, correspond to the sudden transition point, after which point quantum discord begins to decay. Fig. 2(b) displays the asymptotic long-time limit for the transition condition given in Eq. (20). Yellow shaded (gray) region in the figure demonstrates the area defined by the values of the ohmicity parameter s𝑠sitalic_s and the initial state parameter m𝑚mitalic_m for which the phenomenon of time-invariant discord exists at the high temperature limit. Outside the yellow shaded region one will always see a sudden transition from classical to quantum decoherence and thus quantum discord can be frozen only for a finite time interval.

III.2 Dissipation and Heating

This section deals with the dynamics of quantum discord for two qubits independently interacting with dissipation/heating reservoirs. Using the analytical expression given in Eq. (6), we present a plot of the time-evolution of quantum discord as a function of λt𝜆𝑡lambda titalic_λ italic_t in Fig. 3. We consider the weak coupling regime with the parameter R=0.01𝑅0.01R=0.01italic_R = 0.01 and the initial Bell-diagonal state with m=0.1𝑚0.1m=0.1italic_m = 0.1. We also recall that in the weak coupling regime, Markovian dynamics emerge in case of sufficiently small detuning parameter, e.g., Δ=0Δ0Delta=0roman_Δ = 0. In the figure, the dotted-dashed red line displays the case where the dynamics is Markovian and there are no thermal photons. As can be clearly seen, although discord is initially amplified, it tends to monotonically decay and vanish in the long time limit. In fact, this behaviour can be observed for the class of initial states such that |m|<0.2𝑚0.2|m|<0.2| italic_m | <0.2. Moving to the non-Markovian regime with Δ=50λΔ50𝜆Delta=50lambdaroman_Δ = 50 italic_λ, still in the absence of thermal photons, dotted blue line shows that, even though discord is not actually time-invariant, it decays very slowly due to energy exchange between the system and the environment, and for all practical purposes, it can be considered almost invariant. Lastly, the effect of the thermal photons is demonstrated by the dashed green line. We observe that in the short time limit quantum discord is still almost time-invariant, despite the fact that thermal photons hasten its decay in the long time limit. It is important to notice that the sudden transition does not exists under this type of noise.

III.3 Dephasing, Dissipation and Heating

In this section, we take into account the combined effect of dephasing, dissipation/heating reservoirs which are once again interacting with individual qubits independently. We assume that the memory time of the dephasing and dissipation/heating environments is the same, i.e., we choose the parameter β=ωc/λ𝛽subscript𝜔𝑐𝜆beta=omega_c/lambdaitalic_β = italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_λ to be unity. Let us also work in the weak coupling limit of both models and therefore take α=0.01𝛼0.01alpha=0.01italic_α = 0.01 and R=0.01𝑅0.01R=0.01italic_R = 0.01. At the high temperature limit of the dephasing model with these conditions, and supposing detuning parameter is taken as Δ=50λΔ50𝜆Delta=50lambdaroman_Δ = 50 italic_λ, the number of the thermal photons in dissipation/heating reservoir become approximately N≈10𝑁10Napprox 10italic_N ≈ 10.

Time evolution of the classical and quantum correlations are shown in Fig. 4(a) and 4(b) when s=2.5𝑠2.5s=2.5italic_s = 2.5 (Markovian) and s=3.5𝑠3.5s=3.5italic_s = 3.5 (non-Markovian) for the dephasing reservoir, respectively. We note that the effects of the dissipation/heating reservoir is also present in these plots and the dissipation/heating dynamics is non-Markovian due to the detuning parameter acquiring the value Δ=50λΔ50𝜆Delta=50lambdaroman_Δ = 50 italic_λ. Thus, in Fig. 4(a), while the dephasing dynamics is Markovian, the dissipation/heating dynamics exhibit non-Markovian behaviour. We can observe that the sudden transition still exists even in the presence of non-Markovian dissipative environments and consequently almost time-invariant discord cannot be present. To put it differently, comparing Fig. 3 to 4(a), one can see that the presence of even Markovian dephasing can dominate the dynamics over non-Markovian dissipative reservoirs in terms of time-invariant discord, causing a sudden transition and subsequent rapid decay of quantum correlations. We should stress that the quantum (classical) correlations before (after) the sudden transition point here is not actually constant but rather decay very slowly with time, which is due to the effect of dissipative reservoirs. In Fig. 4(b), both the dephasing and dissipation/heating dynamics are non-Markovian. This implies that the occurrence of almost time-invariant discord is exclusively related to the non-Markovian memory effects in the purely dephasing dynamics. As can be seen in inset, it is expected that quantum discord once again degrades very slowly for long times and is not actually time-invariant. Finally, in Fig. 4(c), we take R=0.001𝑅0.001R=0.001italic_R = 0.001, Δ=0Δ0Delta=0roman_Δ = 0, s=2.5𝑠2.5s=2.5italic_s = 2.5, m=0.5𝑚0.5m=0.5italic_m = 0.5, and N=10𝑁10N=10italic_N = 10. In other words, here both the dephasing and dissipation/heating dynamics are Markovian and there exists no memory effects. We see that the sudden transition takes place as a result of the Markovian dephasing dynamics.

IV Time-invariant Discord in the Presence of Correlated Reservoirs

In this section, we turn our attention to a different decoherence model that takes into account the correlations between the environments. In particular, we consider an open system S𝑆Sitalic_S that contains two qubits interacting with a composite environment E𝐸Eitalic_E, which itself is composed of two subsystems. We suppose that each qubit interacts locally with one of the environments, and S𝑆Sitalic_S and E𝐸Eitalic_E are initially not correlated, but the two environments are initially in a correlated composite state. The total Hamiltonian is given as model2

H=𝐻absentdisplaystyle H=italic_H = H0+Hint(t),H0=∑i=12(HSi+HEi),subscript𝐻0subscript𝐻𝑖𝑛𝑡𝑡subscript𝐻0subscriptsuperscript2𝑖1subscriptsuperscript𝐻𝑖𝑆subscriptsuperscript𝐻𝑖𝐸displaystyle H_0+H_int(t),hskip 22.76219ptH_0=sum^2_i=1(H^i_S% +H^i_E),italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT ( italic_t ) , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ∑ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( italic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ) ,

HSisubscriptsuperscript𝐻𝑖𝑆displaystyle H^i_Sitalic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT =ϵiσzi,HEi=∑kωkibki†bki,formulae-sequenceabsentsubscriptitalic-ϵ𝑖subscriptsuperscript𝜎𝑖𝑧subscriptsuperscript𝐻𝑖𝐸subscript𝑘subscriptsuperscript𝜔𝑖𝑘subscriptsuperscript𝑏𝑖†𝑘subscriptsuperscript𝑏𝑖𝑘displaystyle=epsilon_isigma^i_z,hskip 42.67912ptH^i_E=sum_k% omega^i_kb^idagger_kb^i_k,= italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , italic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_ω start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (21)
with bki†(bki)subscriptsuperscript𝑏𝑖†𝑘subscriptsuperscript𝑏𝑖𝑘b^idagger_k(b^i_k)italic_b start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_b start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) being the creation (annihilation) operator of the k𝑘kitalic_kth mode of environment i=1,2𝑖12i=1,2italic_i = 1 , 2, and σzisubscriptsuperscript𝜎𝑖𝑧sigma^i_zitalic_σ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT the Pauli matrix and ϵisubscriptitalic-ϵ𝑖epsilon_iitalic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT the energy gap of the i𝑖iitalic_ith qubit. The interaction Hamiltonian is given by Hint(t)=∑iHinti(t)subscript𝐻𝑖𝑛𝑡𝑡subscript𝑖subscriptsuperscript𝐻𝑖𝑖𝑛𝑡𝑡H_int(t)=sum_iH^i_int(t)italic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT ( italic_t ), in which local interactions are specified by

Hinti(t)=χi(t)∑kσzi⊗(gkibki†+gki∗bki),subscriptsuperscript𝐻𝑖𝑖𝑛𝑡𝑡subscript𝜒𝑖𝑡subscript𝑘tensor-productsubscriptsuperscript𝜎𝑖𝑧subscriptsuperscript𝑔𝑖𝑘subscriptsuperscript𝑏𝑖†𝑘subscriptsuperscript𝑔𝑖∗𝑘subscriptsuperscript𝑏𝑖𝑘displaystyle H^i_int(t)=chi_i(t)sum_ksigma^i_zotimes(g^i_k% b^idagger_k+g^iast_kb^i_k),italic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT ( italic_t ) = italic_χ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⊗ ( italic_g start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_g start_POSTSUPERSCRIPT italic_i ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , (22)
where gkisubscriptsuperscript𝑔𝑖𝑘g^i_kitalic_g start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the coupling constant between qubit i𝑖iitalic_i and k𝑘kitalic_kth mode of its environment, gki∈ℜsubscriptsuperscript𝑔𝑖𝑘g^i_kinReitalic_g start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ roman_ℜ for i=1,2𝑖12i=1,2italic_i = 1 , 2 and all k𝑘kitalic_k. The step function χi(t)subscript𝜒𝑖𝑡chi_i(t)italic_χ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) is also given by

χi(t)=1,t∈[tis,tif]0,otherwisesubscript𝜒𝑖𝑡cases1𝑡subscriptsuperscript𝑡𝑠𝑖subscriptsuperscript𝑡𝑓𝑖0otherwisedisplaystylechi_i(t)=begincases1,&tin[t^s_i,t^f_i]\ 0,&textrmotherwiseendcasesitalic_χ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) = c

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