^{†}Corresponding author. Email: hszeng@hunnu.edu.cn
^{*}Project supported by the National Natural Science Foundation of China (Grant Nos. 11275064 and 11075050), the Specialized Research Fund for the Doctoral Program of Higher Education, China (Grant No. 20124306110003), the Program for Changjiang Scholars and Innovative Research Team in University, China (Grant No. IRT0964), and the Construct Program of the National Key Discipline, China.
From a quite general form of the Lindbladlike master equation of open twolevel systems (qubits), we study the effect of Lamb shift on the nonMarkovian dynamics. We find that the Lamb shift can induce a nonuniform rotation of the Bloch sphere, but that it does not affect the nonMarkovianity of the open system dynamics. We determine the optimal initialstate pairs that maximize the backflow of information for the considered master equation and find an interesting phenomenon–the sudden change of the nonMarkovianity. We relate the dynamics to the evolution of the Bloch sphere to help us comprehend the obtained results.
Quantum nonMarkovian dynamics, due to its wide range of existence such as in the quantum optical system, ^{[1]} quantum dot, ^{[2]} superconductor system, ^{[3]} quantum chemistry, ^{[4]} biological system^{[5]} and some possible applications such as in quantum metrology^{[6]} and quantum key distribution, ^{[7]} has received a great deal of attention in recent years. Several suggestions^{[8– 13]} for the measure of nonMarkovianity have been presented and various dynamical properties^{[14– 32]} of nonMarkovian processes have been investigated. Experimentally, the simulation of nonMarkovianity^{[33– 35]} under controlled environments has been realized.
In the aspect of measuring the nonMarkovianity of open quantum processes, Breuer, Laine, and Piilo (BLP) presented a physically intuitive method, ^{[8]} i.e., by the increase of the trace distance or distinguishability between pairs of evolving quantum states, which may be interpreted as the recovery of the lost information (the flow of the lost information from the environment back to the open system). Unfortunately, there is an optimization process in the definition, i.e., finding an optimal pair of initial states to maximize the backflow of information, which is a formidable task in practice. Though much effort has been devoted to this problem, ^{[36– 38]} it is still the main obstacle to studying nonMarkovianity. In this paper, we solve this problem for a quite general nonMarkovian master equation of open twolevel systems. Due to the typicality of the dynamical model under consideration, the result has good applicability.
In the theoretical study of open quantum dynamics, the effect of the electromagnetic field of the environment will cause shifts of atomic energies, the very well known Lamb shift.^{[39, 40]} For the timeindependent Markovian process, the Lamb shift is a constant and thus can be removed via the renormalization of energy, i.e., by taking the Lamb shift into the transition frequencies of the open system. For nonMarkovian process, however, the Lamb shift is timedependent whose effect needs to be reexamined. The second intention of the paper is designed for this point.
In order to make our research more concrete, we start from a specific interaction model which describes a twolevel system coupled to its environment via both amplitude and phasedamping ways. In the secular approximation and the limit of weakcoupling between the open system and its environment, we derive a master equation which has a very general Lindbladlike form. We then expand our research based on this master equation. Interestingly, the results presented in this paper can be understood intuitively by relating the dynamics to the evolution of a Bloch sphere.
The rest of this paper is organized as follows. In Section 2, we introduce the origin of the general Lindbladlike nonMarkovian master equation used in this paper. In Sections 3 and 4, we study respectively the effects of Lamb shift on the evolution of quantum states and on the nonMarkovianity of system dynamics. In Sections 5 and 6, we discuss respectively the issues of optimal initialstate pairs and the sudden change of nonMarkovianity. Finally, the conclusion is arranged in Section 7.
Consider a twolevel atom with Bohr frequency ω _{0} interacting with a zerotemperature bosonic reservoir modeled by an infinite chain of quantum harmonic oscillators. The total Hamiltonian for this system in the Schrö dinger picture is given by
where σ _{x} and σ _{z} are the Pauli operators of the atom, ω _{k}, b_{k}, and
The timeconvolution less (TCL) projection operator technique^{[1]} is most effective in dealing with the dynamics of open quantum systems. In the secular approximation and the limit of weak coupling between the system and its environment, by expanding the TCL generator to the second order with respect to the coupling strengths, the nonMarkovian master equation describing the evolution of the open system, in the interaction picture, can be written as
where
with σ _{± } the inversion operators of the atom, which is the Lamb shift Hamiltonian that describes the energy shifts of the eigenstates of the twolevel atom, and
describes the dissipation of the system. The Lamb shifts S_{± }(t) of the levels  0〉 and  1〉 , and the timedependent decay rate Γ (t) may be written respectively as
with ξ = {+ 1, − 1}. In the above derivation, we have used the continuum limit
The first, second, and third lines of the dissipator D[ρ (t)] describe respectively the dissipation, heating and purely dephasing of the atom to its environment. Equation (2), which may be viewed as the generalization of the Lindblad master equation, has a quite general form. In fact, many quantum dynamics of open twolevel systems in the secular approximations can be cast into this form. The timedependent decay rates and Lamb shifts may be different for different microscopic models. But our main results are in principle independent of the specific form of these expressions. It is the generality of the master equation that makes our results applicable.
In this section, we discuss the effect of Lamb shift on the evolution of quantum states. For this purpose, we first consider the case where the Lamb shift is removed temporarily, which leads to the very simple Bloch equations
where the components of the Bloch vector are defined by b_{j}(t) = Tr[ρ (t)σ _{j}] with j = x, y, z. The parameters γ (t) and γ _{z}(t) are defined by
The above Bloch equations can be solved very easily, which give the following solutions:
with
where λ is the width of the Lorentzian distribution, and we assume the Bohr frequency ω _{0} = 10λ , the detuning between the atom and the central frequency of the environment Δ = 0.5λ . In the figure, we take
When the Lamb shift is considered, the Bloch equation becomes
and the evolution of b_{z} is still given by Eq. (10). Now the evolutions of b_{x} and b_{y} are no longer independent. The simplest way to solve this set of equations is to employ the timedependent rotating transformation
with a timedependent rotating angle
The component b_{z}(t) is still given by Eq. (15).
From the above mathematical derivation, we see that the rotating transformation (19) on the one hand can tremendously simplify the process of solving the master equation (2), and on the other hand it shows the Lamb shift has a significant physical meaning: the Lamb shift induces a rotation of the Bloch sphere with respect to the zaxis. It is the use of this property that will be very convenient for future research of nonMarkovian dynamics.
In fact, the above rotating effect of the Lamb shift is quite understandable based on the Bloch theory of the evolution of a twolevel system.^{[43]} When the dissipator D[ρ (t)] in Eq. (2) is ignored, the system will undergo a unitary evolution induced by Lamb shift Hamiltonian H_{LS}(t), which up to a global phase is equivalent to the rotation transformation U(t) = exp[δ (t)σ _{z}/2]. The inclusion of the dissipator only gives rise to the deformation (expansion and contraction) of the Bloch sphere.
We now study the effect of the Lamb shift on the nonMarkovianity of the dynamical process. Among a few measures of nonMarkovianity so far, the BLP measure^{[8]} is a typical one which has physically intuitive interpretations. Note that Markovian processes always tend to continuously reduce the trace distance between quantum states, thus an increase of the trace distance during any time interval signifies the emergence of nonMarkovianity. In quantum information science, the trace distance is related to the distinguishability between quantum states and its change means the exchange of quantum information between an open system and its environment. NonMarkovian processes imply the flow of the lost information from the environment to the open system.
For a given pair of initial states ρ _{1, 2}(0) of the system, the change of the dynamical trace distance is described by its time derivative
where ρ _{1, 2}(t) are the dynamical states corresponding to the initial states ρ _{1, 2}(0), and the trace distance is defined as D(ρ _{1}, ρ _{2}) = tr ρ _{1} − ρ _{2} /2 with trace norm
as the measure of nonMarkovianity of a dynamical process. In order to reflect the degrees of nonMarkovianity of the whole dynamical process, the time integration is extended over all intervals in which σ is positive, and the maximum is taken over all initialstate pairs of the system. Obviously, 𝓝 = 0 for all Markovian processes. The larger the quantity 𝓝 is, the higher the nonMarkovianity of the process is.
For our considered dynamics and by use of the solution of Eqs. (20) and(21) as well as Eq. (15), we obtain
where G(t) = {e^{− 2Θ (t)}[(Δ b_{x})^{2} + (Δ b_{y})^{2}] + e^{− 2Λ (t)}(Δ b_{z})^{2}}^{− 1/2} and Δ b_{j} = b_{1j}(0) − b_{2j}(0) is the difference of the initial Bloch components. It is worthwhile noting that equation (24) is independent of the Lamb shift S_{± }(t); therefore, the Lamb shift does not affect the nonMarkovianity of the system dynamics. This result is interesting but not selfevident, because the Lamb shift can affect the evolution of quantum states.
Of course, we may understand this result more easily from the picture of a Bloch sphere. As verified in the previous section, a Lamb shift only induces a rotation of the Bloch sphere, which apparently does not change the Euclidean distance in the Bloch sphere picture. In the qubit case the Euclidean distance of the Bloch vectors is up to a constant the same as the trace distance between the corresponding quantum states.^{[44]} Therefore, the Lamb shift does not affect the nonMarkovianity of quantum processes. Another intuitive interpretation may come from the observation of the BLP measure for nonMarkovianity, which involves only the differences of quantum states. These differences obviously do not change under Lamb shifts, because the shifts are the same for all quantum states. So the nonMarkovianity is also unchangeable.
From Eq. (24), we can easily confirm that the sufficient and necessary condition for the backflow of information is
for some time intervals [Note that G(t) is positive]. Because if one of the inequalities is satisfied, then we can always find a pair of initial states so that σ > 0. For example, if γ (t) < 0 it suffices to choose the initial states satisfying Δ b_{z} = 0. Conversely, if σ > 0 at given time t, then at least one of the two inequalities must be satisfied.
In fact, the condition (25) is also intuitive. As mentioned in the previous section, the positivity of γ (t) and γ _{z}(t) is the sufficient and necessary condition for the monotonous contraction of a Bloch sphere, and the contraction leads to the decrease of the Euclidean (trace) distance between quantum states, thus the positivity of γ (t) and γ _{z}(t) is the sufficient and necessary condition of Markovian dynamics, or equivalently, equation (25) is the sufficient and necessary condition of nonMarkovian dynamics.
In the calculation of a BLP measure of nonMarkovianity, a key step is to find the optimal initialstate pair so that equation (23) is maximized. It is a formidable problem in practice. Here we solve this problem for the quite general nonMarkovian master equation (2). Let us first intuitively analyze the problem from the picture of a Bloch sphere. It was already proved that^{[38]} for a qubit system, the optimal initialstate pair must be located at the antipodal points of the Bloch sphere. As the increase of the trace distance corresponds to the inflation of the Bloch sphere, we then infer that the optimal initialstate pair should be such antipodal points for which the Bloch sphere oscillates most strongly in that direction, i.e., the sum of the inflations in that direction in the whole dynamical process is the largest. As the Bloch sphere governed by the master Eq. (2) is rotation symmetrical with respect to the z axis, the most strongest oscillating direction is probably either the pole direction or any direction in the equatorial plane. In other words, the optimal initialstate pair is either { 0〉 ,  1〉 } or { + 〉 ,  − 〉 } with
Similarly, for the initialstate pair { 0〉 ,  1〉 }, we have
By comparing the values of 𝓝 _{1} and 𝓝 _{2}, we can finally determine the optimal initialstate pair and the corresponding nonMarkovianity 𝓝 = max{𝓝 _{1}, 𝓝 _{2}}.
The correctness of the above conjecture may be examined through concrete calculations. As the optimal initialstate pair must fit (Δ b_{x})^{2} + (Δ b_{y})^{2} + (Δ b_{z})^{2} = 4, thus equation (24) may be viewed as a function of (Δ b_{z})^{2}. For any fixed time t yielding σ > 0, equation (24) is maximized either at the endpoints (Δ b_{z})^{2} = 0, 4, or at (Δ b_{z})^{2} = α _{0}, where α _{0} is a point for which the derivative of σ with respect to (Δ b_{z})^{2} vanishes. Straightforward calculation shows that there is only one extremum point
which is the minimum point of σ . Thus the maximum of σ takes place at the endpoints of (Δ b_{z})^{2} irrespective of the time t, which indicates { 0〉 ,  1〉 } and { + 〉 ,  − 〉 } are the possible optimal initialstate pairs.
In the previous section, we provided two alternative pairs of initial states, for which the real choice for optimal initial states dependends on the structure parameters of the environment. For some parameter areas, the optimal initialstate pair may be { + 〉 ,  − 〉 } and the nonMarkovianity is calculated by 𝓝 _{1}. While for other parameter areas, the optimal initialstate pair may be { 0〉 ,  1〉 } and the nonMarkovianity is calculated by 𝓝 _{2}. On the boundary of the two kinds of parameter areas, a sudden change of the nonMarkovianity 𝓝 may take place. In order to demonstrate this point, we still take the Lorentzian density of Eq. (16) as an example and plot the nonMarkovianity as a function of the dimensionless detuning as in Fig. 2, where we take λ = 0.1ω _{0},
It is worthwhile stressing that the sudden change of nonMarkovianity in this model is related closely to the rotation symmetry of the Bloch sphere with respect to the z axis. It is the rotation symmetry that makes the optimal initialstate pairs either along the z axis or in the equatorial plane. Thus if you change the parameters of the dynamics, eventually you will find ones for which the optimal initialstate pair jumps from the z axis to the equatorial plane or vice versa unless the optimal initialstate pair is unique.
In conclusion, by considering a specific interaction model that describes a twolevel system coupling to a zerotemperature structured environment via amplitudephase dampings, in the limit of weak coupling between the system and its reservoir and by use of the secular approximation, we derived a nonMarkovian master equation for the reduced state of the open system which has a quite general Lindbladlike form. Based on this, we found on the one hand that the timedependent Lamb shift can induce a nonuniform rotation of the Bloch sphere and thus alter the evolution of quantum states. Therefore, in quantum information processing, the Lamb shift should be taken into consideration. On the other hand however, it does not affect the nonMarkovianity of the open system dynamics and thus can be removed from the master equation.
We also, for the general nonMarkovian master equation, obtained the sufficient and necessary conditions for the backflow of information, and found the optimal initialstate pairs that maximize the backflow of information. There are two alternative initialstate pairs depending on the environmental structure parameters, the polar or the equatorial state pair, for the calculation of nonMarkovianity. The change of the optimal initialstate pairs can lead to the sudden change of nonMarkovianity.
One key factor in our study is the use of the rotating transformation which not only tremendously simplifies the solving process of the master equation, but also reveals the significant physical meaning of the Lamb shift, i.e., the Lamb shift induces a rotation of the Bloch sphere with respect to the z axis, which establishes the foundation of our research work. Almost all of the results listed in this paper can be explained intuitively from the rotation and the symmetry under the rotation.
Though the discussion was started from a specific interaction model that describes amplitudephase dampings of a twolevel open system, the results have good applicability. Because on the one hand the derived Lindbladlike master equation has a quite general form, and on the other hand the results in principle do not depend on specific environmental structures. We believe our results are very helpful for the study of nonMarkovian dynamics.
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