Biexponential decay and ultralong coherence of a qubit

A quantum two-state system, weakly coupled to a heat bath, is traditionally studied in the Born-Markov regime under the secular approximation with completely positive linear master equations. Despite its success, this microscopic approach exclusively predicts exponential decays and Lorentzian susceptibility profiles, in disagreement with a number of experimental findings. On the contrary, in the absence of the secular approximation they can be explained but with the risk of jeopardizing the positivity of the density matrix. To avoid these drawbacks, we use a physically motivated nonlinear master equation being both thermodynamically and statistically consistent. We find that, beyond a temperature-dependent threshold, a bifurcation in the decoherence time T2 takes place; it gives rise to a biexponential decay and a susceptibility profile being neither Gaussian nor Lorentzian. This implies that, for suitable initial states, a major prolongation of the coherence can be obtained in agreement with recent experiments. Moreover, T2 is no longer limited by the energy relaxation time T1 offering novel perspectives to elaborate devices for quantum information processing.

In this letter we describe the emergence of two T 2 decoherence times, namely a short and a long one, for a qubit undergoing a physically sound nonlinear Markovian dynamics beyond the WCL 1 . Thereby, the polarizations follow a biexponential decay coming along with a non-Lorentzian susceptibility profile. In this context, we explain how an appropriate choice of the initial state can slow down the decay of the polarizations which in turn can be exploited to overcome the well-known T 2 ≤ 2T 1 bound.
Traditional paradigm. -Let us consider a qubit connected to a heat bath at inverse temperature β described by a 2 × 2 density matrix ρ (with the natural units k B = 1 and = 1). In a conventional approach, the dynamics of the system is modelled with the LDME [10,12] The first contribution, associated to the system Hamiltonian H S , produces a reversible time-evolution whereas the second one, expressed in terms of the so-called Lindblad eigenoperators A ω and a spectral function h(ω), induces relaxation and decoherence processes. The connection to an underlying Hamiltonian dynamics is established by linking the A ω and h(ω) to the total Hamiltonian the bath Hamiltonian while Q and Φ are, respectively, the system and bath self-adjoint coupling operators. More concretely, the A ω satisfy [A ω , H S ] = ωA ω and are provided in terms of Q through the Kronecker delta relations using the matrix elements' notation (·) ij = E i | · |E j in the system energy eigenbasis {|E i }. On the other hand, the spectral func- where π B is the bath equilibrium state, respects the KMS condition h(ω) = e βω h(−ω) [11] implying that the system converges towards the Gibbs state π = e −βHS /tr(e −βHS ) assuming Q and H S have no common eigenspace. Prior to the SA, another microscopically derived equation is often used to study open quantum systems, namely the BRME [10,73] 1 Small but non-vanishing coupling strength contrary to the WCL.
Below, all relevant quantities are expressed in terms of the absorption rate a(ω) = h(−ω) for ω > 0. For the sake of simplicity, we use for the qubit a parametrization in terms of the Pauli matrices 2 σ x , σ y and σ z yielding H S = −(Δ/2)σ z with Δ = E 2 − E 1 the energy gap. The operator Q, dimensionless so that the units of energy are fully assigned to h(ω), carries the real elements Q 11 and Q 22 on the diagonal while the off-diagonal elements read Q 12 = e iθ = Q * 21 with θ ∈ [0, 2π). Thermodynamic formulation. -To go beyond the standard linear master equation (1), exclusively producing exponential decays, we make use of the nonlinear thermodynamic master equation (NTME) [74] where S(ρ) = − ln ρ is the von Neumann entropy operator. This equation, inspired by a derivation [64] as well as by thermodynamical [70,71] and statistical [75] arguments, generates a modular dynamical semigroup ensuring the preservation of the hermiticity, the trace and the positivity of ρ as expected from a physical master equation 3 . Moreover, it converges to the Gibbs state and gives rise to a positive entropy production. Besides, the NTME (3) gives back the LDME (1) in the WCL, asymptotically describing the exact Hamiltonian dynamics in the long time limit [76], by applying the time-averaging procedure mentioned in [75]. The NTME (3) carries a sum over two sets of Bohr frequencies like the BRME (2), due to the absence of the SA, but without jeopardizing the positivity of ρ. Pechukas [51] and Romero [69] noted that beyond the WCL nothing forbids to have a nonlinear equation with respect to the state for the reduced system. For example, the nonlinearity of a reduced system can arise by eliminating the irrelevant degrees of freedom of the total system's density matrix using generalized Nakajima-Zwanzig methods in the absence and in the presence of systemenvironment correlations [61,62,64,68]. Of course, the full system evolves under a linear von Neumann equation. In this respect, the present nonlinear Markovian reduced dynamics is pertinent from a physical perspective.
Linear response regime. -Given the nonlinear nature of the NTME, the various lifetimes can only be accessed through the first-order susceptibility. As shown in [77] the latter is obtained by means of the fluctuationdissipation theorem (FDT) in the time-domain through the expression χ AB (t) = −β ∂ t tr(Ae Lt K π B) relative to the self-adjoint observables A and B. It requires the lineariza-tionρ = Lρ of (3) near equilibrium obtained from 1 0 dλ π λ Aπ 1−λ known as the equilibrium Kubo-Mori superoperator [78]. Note that, contrary to the thermodynamically robust NTME, its linearized version does not preserve positivity far away from equilibrium and one should limit its use to the linear response regime where the lifetimes/decay rates are defined 4 .
To perform the calculations, it is advantageous to switch to the Liouville space (see, e.g., Chapt. 3 of [9]) highlighted hereafter with a bold notation. Choosing the vector representation ρ = (ρ 11 , ρ 12 , ρ 21 , ρ 22 ) T for the density matrix coefficients in the Hamiltonian basis, the linearized NTME (3), (4) readsρ = Lρ. Defining the dimensionless temperatureβ = βΔ, we compute the 4 × 4 matrix L for zero/large pure dephasing; the latter is associated to the decay rate Γ Thus, the populations ρ 11 and ρ 22 decouple from the coherence ρ 12 leading to the Liouville generator 5 depending on three real positive variables for the NTME. For the LDME we get L with z = 0 whereas for the BRME we obtain it with z = (1 + eβ) a(Δ)/2.

Spectral analysis. -
The four eigenvalues of the generator (5) are 0, −Γ 1 and −Λ ± given by Γ 1 = (1 + eβ) x and Λ ± = y ± Ω with Ω = √ z 2 − Δ 2 being real or imaginary. From L and its eigenvalues it naturally follows that Γ 1 = 1/T 1 is the energy relaxation rate. On the other hand, the real part of Λ ± is associated to either one decoherence rate Γ 2 = 1/T 2 = y, for z < Δ, or two decoherence rates Γ 2± = 1/T 2± = Λ ± for z > Δ. It is noteworthy that all decay rates are relative either to energy relaxation or decoherence, i.e. no Γ 3 coupling populations and coherence appears in the model. The transition from one to 4 Decay times have an experimental significance for long times near equilibrium [42]. The decay rates obtained by fitting the polarizations far away from equilibrium are not equal to the one defined in the linear response regime; this is even more true with the NTME which is nonlinear (i.e. distinct initial conditions can yield different decay rates; match is only recovered close to equilibrium). 5 For large pure dephasing, |Q11 − Q22| is negligible in regard of |Q11 − Q22| 2 producing the zero entries in L. two decay rates happens for z > Δ, or equivalently, past the absorption rate a(Δ) thresholds a NTME thr = Δ e −β/2 tanh (β/2)/(β/2), a BRME thr = 2Δ/(1 + eβ), obtained by computing z = Δ for each equation. Before the threshold, we observe for the NTME and BRME an analogy with the LDME equally predicting a single decoherence time. The "branching" of the decoherence times beyond the threshold is always reached for low enough temperatures (a NTME thr , a BRME thr → 0 forβ 1). The microscopic origin of the threshold can be tracked back to the non-secular terms of the BRME (2), i.e. with ω =ω, yielding the element L 23 = z e 2iθ = L * 32 . Thus, the Liouville matrix (5) for z = 0 offers a straightforward modeling tool to describe the bifurcation in the decoherence time. It is remarkable that such a temperature-dependent transition from one to two decoherence times, which are associated to a biexponential decay of the coherence, was reported in low-temperature experiments of nuclear spins' impurities in silicon crystals [19,20] or InGaAs QDs [27].
Dynamical analysis. -To fully capture the implications of the two decoherence times, we solve the system of equations generated by the Liouville generator (5) for ρ 12 with the initial condition ρ 12 (0) = r 0 e iφ0 , whose modulus must respect r 0 2 ≤ ρ 11 (0) ρ 22 (0), yielding Increasing a(Δ) beyond the thresholds (7), (8) induces a transition from a single oscillating exponential decay (i.e. Λ ± is complex) towards a non-oscillating biexponential one (i.e. Λ ± is real) associated to a short T 2+ and a long T 2− decoherence time. Both cases are presented in fig. 1. The dependence on the orientation φ 0 of the initial Bloch vector, displayed by the NTME and BRME but not by the LDME, has been qualitatively measured in the presence of a spin bath [84,85]. Moreover, the crossover from an oscillatory towards a biexponential damping has been observed for the spin dynamics in a 2D electron gas [86]. More generally, a biexponential decay was found in spins [16,[18][19][20], NV centers [21,22], light-harvesting complexes [23,24] or in QDs [17,[25][26][27][28][29][30][31]82]. To understand which physical mechanisms produce these unorthodox decays, one should investigate each setup thoroughly. This is done, for example, for GaAs QDs in Chapt. 4 of [81] where the origins of the non-exponential decay and non-Lorentzian profiles are discussed (spin-orbit coupling or electron-phonon/hyperfine interactions and so on) 8 . With these insights, one could get a reasonable model by specifying the parameters of the master equations (2) or (3). 7 The NTME has been designed to be Markovian and we checked this fact with a non-Markovianity measure [83]. 8 Note that the full decay pattern is much richer than what the NTME predicts in the linear response regime, e.g., see fig. 14 on In general, one would like to obtain a time-evolution solely driven by the longest decay time T 2-to preserve the coherence as long as possible. To this end, we set the first term in eq. (10) equal to zero yielding To satisfy this condition for a given setup, one should start from an initial state with φ 0 equal to the critical Bloch angle φ c = θ − arccos(−Ω/z)/2. Thus, it is possible to generate a prolonged time-evolution of the coherence by taking an initial Bloch vector being oriented along the critical angle φ c , i.e. producing an optimal initial state. This could drastically increase the coherence time being the cardinal resource to perform quantum gate operations. Note that there exist many strategies to prepare such an initial state [87,88] and that an iterative algorithm to track optimal Bloch vectors has been developed for materials doped with rare-earth ions [89].
T 1 T 2 ratio. -Beyond the thresholds (7), (8) the NTME and the BRME predict that the decoherence time T 2− is no longer restricted by the energy relaxation time T 1 , even in the weak-coupling regime, as illustrated in fig. 2. Moreover, their respective thresholds behave very similarly. The T 2 ≤ 2T 1 bound violation was already obtained by Laird and co-workers [38][39][40]. However, the bifurcation was not recognized 9 and the positivity of ρ is not ensured [42,45,46]. On the contrary, the NTME (but not the BRME) provides a thermodynamically and statistically safe way to discuss this phenomenon. Furthermore, a merely formal nonlinear master equation obtained in [47], being positive but not CP, also does not meet the inequality. Indeed, there exists a one-by-one link between this inequality and CP [53]. Moreover, the standard notion of CP only makes sense for linear dynamics [49,52,90] and, as mentioned by Pechukas [50] as well as by Shaji and Sudarshan [54], it does not need to be a physical requirement in spite of its mathematical attractiveness.
Conclusion and perspective. -The secular LDME is over-restricted by CP and does not allow to interpret p. 93 of [81] for an overview of decays at short, intermediate and long times. 9 Look, for example, at Laird's eqs. (A1)-(A4) in the appendix of [38] where they overlook the case of Ω = √ z 2 − Δ 2 being real.
numerous experimental results, for example the problem of Cooper-pair pumping [91]. On the contrary, the nonsecular terms lead to a bifurcation phenomenon beyond a temperature-dependent threshold, associated with a biexponential decay of the polarizations and non-Lorentzian susceptibility profiles. Unfortunately, the BRME does not in general preserve the positivity of ρ and leads to a negative entropy production [56]. The phenomenological NTME predicts the same unorthodox effects as the BRME but it is free of its drawbacks. The biexponential decay can be used to drastically prolong the coherence for optimal orientation of the initial Bloch vectors. Moreover, according to our analysis nothing forbids to overcome the bound T 2 ≤ 2T 1 although its experimental realization would clearly be challenging. Such conclusions could be made directly from the Liouville matrix (5) and its associated Bloch equation which can be obtained from various other models supporting the idea that the presented bifurcation phenomenon is of ubiquitous nature. For example, it can be observed for Laird and co-workers' equation [38] or the dynamically time coarse-grained master equations proposed in [74].
Remarkably, ultralong coherence is not limited to a single qubit and could be "scaled-up" since biexponential decays were measured in epitaxial QD arrays [92]. It could allow to delay entanglement sudden-death and enhance revival [93][94][95]. Away from the linear response regime, the decay pattern becomes richer. One then finds initial optimal states either by screening strategies or applying search/reinforcement learning algorithms. Moreover, at intermediate pure dephasing, where the populations and coherences are interwoven, the NTME predicts population beating and coherence revival which have been observed in photosynthetic complexes [5,6], quantum kicked rotors [7,8] as well as anions solvated in water [96]. Taking all this into account, we hope that the present findings will provide some leads to develop ground-breaking nanoscale devices in the near future. * * * We would like to thank R. Alicki and G. M. Graf for the stimulating discussion concerning the nonlinear nature of the thermodynamic master equation.