Effect of control procedures on the evolution of entanglement in open quantum systems
The effect of a number of mechanisms designed to suppress decoherence in open quantum systems are studied with respect to their effectiveness at slowing down the loss of entanglement. The effect of photonic band-gap materials and frequency modulation of the system-bath coupling are along expected lines in this regard. However, other control schemes, like resonance fluorescence, achieve quite the contrary: increasing the strength of the control kills entanglement off faster. The effect of dynamic decoupling schemes on two qualitatively different system-bath interactions are studied in depth. Dynamic decoupling control has the expected effect of slowing down the decay of entanglement in a two-qubit system coupled to a harmonic oscillator bath under non-demolition interaction. However, non-trivial phenomena are observed when a Josephson charge qubit, strongly coupled to a random telegraph noise bath, is subject to decoupling pulses. The most striking of these reflects the resonance fluorescence scenario in that an increase in the pulse strength decreases decoherence but also speeds up the sudden death of entanglement. This demonstrates that the behaviour of decoherence and entanglement in time can be qualitatively different in the strong-coupling non-Markovian regime.
pacs:03.67.Pp, 03.65.Yz, 03.67.Bg
Entanglement is one of the basic features that distinguish quantum systems from their classical counterparts, and has its origins in the inherent non-locality of quantum mechanics (1). It is the most useful resource in quantum information theory (2), and is indispensible for diverse quantum information tasks such as quantum communication, teleportation, quantum error correction, superdense coding, one-way communication etc. In closed systems — that is, systems which are completely isolated from their surroundings — entanglement remains conserved under a local unitary evolution, and can change only under non-local evolution. This makes these systems ideal for quantum information tasks. Closed systems are, however, a rarity in the natural world. More often than not, quantum systems are open, that is, they are in contact with the surrounding environment — a thermodynamic reservoir, for example (3); (4); (5). Quantum systems are extremely fragile, and the dissipative effects of the environment gives rise to the phenomenon of quantum decoherence (6); (7). As a result, the system undergoes an asymptotic transition to classicality and hence loses all its entanglement, which is a purely quantum phenomenon. Nevertheless, this in itself is not a bad scenario, for if the decoherence rate is low, then entanglement takes a long time to completely disappear and such systems can function as useful quantum devices for sufficient periods of time. However, recent studies (8); (9) have uncovered systems where the rate of loss of entanglement is exponentially higher than the decoherence rate. This results in a finite time to classicality, and consequently, a finite time to the total loss of entanglement — a phenomenon given the name entanglement sudden death (ESD). Systems that suffer from ESD are rendered unusable for quantum tasks. Naturally then, ESD has dire implications for the success of quantum tasks, and has become one of the premier branches of quantum information study in recent times. Some of us have recently investigated this phenomenon for the case of -qubit states at finite temperature (10), as well as for spatially separated -mode Gaussian states coupled to local squeezed thermal baths (11).
Given the obvious importance of ESD regarding the success of quantum tasks, it is thus a worthwhile exercise to investigate ways and means of controlling the rate of loss of entanglement. Error-correcting codes (12); (13); (14); (15) and error-avoiding codes (16); (17); (18) ( which are also known as decoherence-free subspaces) are such attempts. Open loop decoherence control strategies (19); (20); (21); (22); (23); (24); (25); (26); (27); (28) are another class of widely used strategies used to this effect, where the system of interest is subjected to external, suitably designed, time-dependent drivings that are independent of the system dynamics. The aim is to cause an effective dynamic decoupling of the system from the ambient environment. A comparative analysis of some of these methods has been made in (29). An important generalization of the dynamic decoupling scheme, presented recently in (30), involves exploiting and merging the randomization and deterministic strategies such as symmetrization, concatenation and cyclic permutation to an qubit system. Another mechanism known to slow down the process of decoherence is through manipulation of the density of states. This has been put to use in photonic band-gap materials, which is used to address questions related to the phenomenon of localization of light (31); (32); (33); (34); (35).
In this paper, we analyze the evolution of entanglement in two-qubit systems connected to local baths (or reservoirs). A number of studies of entanglement in open quantum systems have been made (36); (37); (38). Here we address the need to have a control on the resulting nonunitary evolution, as motivated by the above discussion, and study several methods of doing so. These include manipulation of the density of states in photonic crystals, modulation of the frequency of the system-bath coupling and modulation of external driving on two-qubit systems. A significant part of the paper is devoted to the study of control methods in two-qubit systems undergoing non-Markovian evolution. The first of these is dynamic decoupling — which is an open-loop strategy — on a two-qubit system one qubit of which is in contact with a harmonic oscillator bath. This system undergoes a quantum non-demolition interaction, where dephasing occurs without the system getting damped. The second is a Josephson-junction charge qubit subject to random telegraph () noise due to charge impurities.
The surprising aspect of this study is that suppression of decoherence due to a control procedure need not necessarily mean preservation of entanglement. In fact, application of resonance fluorescence or dynamical decoupling on the Josephson junction charge qubit, undergoing non-Markovian evolution, results in earlier ESD even though decoherence gets suppressed.
The plan of the paper is as follows. In Section II, we introduce the basic techniques and formalism used in this paper, including the formal way of solving the Lindblad master equation. We also introduce the concept of channel-state duality and the factorization law of entanglement decay, both of which will be used subsequently. In Section III, we study the evolution of entanglement in photonic band gap materials and the effect of the special characteristics of such materials on ESD. The effect of frequency modulation of the system-bath coupling on ESD is studied in Section IV. This is followed by a study of ESD of a two two-level system, one of which is driven by an external resonant field which is in resonance with the transition frequency. Finally, in section VI, we study the effect of dynamic decoupling on the evolution of entanglement and ESD. We pay particular attention to the Bang-Bang strategy (19) with regard to the usual two-qubit system under a QND interaction in section VI (A) and also to a Josephson-junction charge qubit subjected to random telegraph noise, and make comparisons in section VI (B). We conclude our paper in section VII with further discussions. Appendices A, B and C deal with some of the explicit calculations.
An open quantum system, as defined in the introduction, is exposed to its environment, which is usually a thermal reservoir. The dynamics of such a system is naturally dictated by its interaction with its environment. If be the total Hamiltonian of an open system, then , where and are the system and reservoir Hamiltonians respectively and is the interaction Hamiltonian. Open systems undergo nonunitary evolution due to this interaction term, and, depending on the type of the system-reservoir () interaction, they can be broadly divided into two categories — dissipative and non-dissipative. In the former, the system Hamiltonian does not commute with the interaction Hamiltonian, , and dephasing occurs along with dissipation and decoherence. In the latter however, these two do commute — — and hence the interaction is characterized by a class of energy-preserving measurements where dephasing occurs without damping the system (39); (40). Such a non-dissipative system, as well as the corresponding interaction, is called a Quantum Non-Demolition (QND) system.
We are interested in the time evolution of open quantum system, i.e., of the system state . Let the initial state of the system-bath combination be , and let the state at time be , where is the time evolution operator. The state of the system alone is obtained from by simply tracing out the bath degrees of freedom: , where Tr implies a partial trace over the bath. The evolution of the system-bath combination is unitary, and
is the equation of motion. However, the evolution of the system itself is nonunitary, and thus requires a more general equation of motion which, after the application of the Born, Markov and rotating wave approximations, can be written as
This is a master equation in its Lindblad form. It can be written in super operator form as
where is the super operator acting on the system state and is effectively a time-derivative, and is the matrix representation of . In general, is time independent and the solution of the above equation can be written formally as
where is the matrix representation of the time evolution map . If the system is evolving under unitary evolution then the matrix is simply , where represents the complex conjugate of in a fixed basis.
Channel-State Duality: A quantum channel is a conduit for the transmission of quantum as well as classical information, and is essentially a completely positive map between spaces of operators. Any such physical quantum channel acting on a -dimensional quantum state can be mapped to a positive operator in dimensions, and, if the channel is trace-preserving, then the corresponding positive operator will have unit trace. Similarly, a valid density matrix in -dimensions can be mapped to a trace preserving physical channel acting on dimensional systems. Such a two-way mapping between a quantum state and quantum channel is called channel-state duality.
where is a maximally entangled state in Hilbert space. Here the channel is applied to one side of the maximally entangled state.
We shall use the symbols and for the matrix representation of the time evolution map and a valid density matrix corresponding to it, respectively, throughout the paper.
Factorization law of entanglement decay (43): This law says that the evolution of entanglement in a bipartite entangled state under a local one-sided channel can be fully characterized by its action on a maximally entangled state. The amount of entanglement at any time in a given initially entangled two-qubit pure state , under the action of a one-sided quantum channel, is equal to the product of the initial entanglement in the given state and the entanglement in the state which we get by applying the channel on one side of a two-qubit maximally entangled state. Mathematically this can be written as:
being the concurrence (44); (45). Therefore, it is enough to study the concurrence in the state obtained after the evolution of state, i.e, concurrence in the matrix . We will make use of this factorization law in our subsequent analysis. It can be easily extended to the case where a local quantum channel acts on both the qubits. It should be pointed out that through out the paper, we consider the action of thermal bath as well as the action of the controlling mechanisms on one of the two qubits (say qubit ) of a two-qubit system . Thus the corresponding action on any two-qubit initial state is of the form , where is the associated quantum channel. In case where both the qubits are exposed to individual thermal bath (with or without individual control mechanism for each qubit), the corresponding action will be of the form , where is the associated quantum channel acting on qubit and is the associated quantum channel acting on qubit . This is the legitimate quantum operation as the individual qubits are subject to local quantum actions under the associated quantum channels, which does not care whether has any entanglement. But one must guarantee that the individual qubits are subject to the action under quantum channels, which is the case in this paper for each of the control mechanisms described.
Iii Evolution of entanglement in the presence of photonic crystals
In this section we consider a system of two level atoms interacting with a periodic dielectric crystal, this particular structure of which gives rise to the photonic band gap (31); (46); (34). The effect of this on electromagnetic waves is analogous to the effect semiconductor crystals have on the propagation of electrons, and leads to interesting phenomena like strong localization of light (33), inhibition of spontaneous emission (34) and atom-photon bound states (35); (47); (48). The origin of such phenomena can ultimately be traced to the photon density of states changing at a rate comparable to the spontaneous emission rates. The photon density of states are of course estimated from the local photon mode density which constitutes the reservoir. It is this photonic band gap that suppresses decoherence (49).
Let us consider a two-qubit system, one qubit of which is locally coupled to a photonic crystal reservoir kept at zero temperature. In this case, entanglement dynamics can be obtained by studying the qubit in contact with the reservoir. We start with the following Hamiltonian:
where is the natural frequency of the two level atom, is the energy of the th mode and is the frequency dependent coupling between the qubit and the photonic crystal, the latter acting as the reservoir here. And also, and , are the Pauli matrices, with and being the annihilation and creation operators for the th mode. If we restrict the total atom-reservoir system to the case of a single excitation (50), the evolution of a given state of the qubit is then given by (49):
Here is the detuning of the atomic frequency and is the upper band-edge frequency. We have made use of the following photon-dispersion relation near the band edge: , where , is the atomic dipole moment and is the vacuum dielectric constant. in Eq. (10) is the equation of the system taking into account the influence of the reservoir. This invokes a prescription for the reservoir spectral density, which depends upon the frequency dependent system-reservoir coupling , and is codified in the form of the function , above.
The dynamics , given by Eq. (10), is guaranteed to be described by a quantum channel (say) whose matrix representation is
Channel-state duality, explained earlier in Section II, ensures that there exists a two-qubit density matrix for every single-qubit channel . This matrix can be written as
where is a two-qubit maximally entangled state. The concurrence of is , where is a complex-valued function of the detuning parameter and time . If we assume that , can then be written in the following simplified form:
where and . Therefore, we need to see the effect of on entanglement in . Since the entanglement in is , it is now a function of and . Invoking the factorization law of entanglement decay, it is sufficient to study entanglement in in order to understand the nature of evolution of entanglement in the two-qubit system.
We show the evolution of entanglement in for different values of in FIGS. a, b. The insets of the figures depict the evolution of entanglement – computed using concurrence (see appendix A) – for the usual case of zero band gap, while the main panels show the evolution of entanglement for increasing influence of the band gap. In FIG. a, the system is within a gap in the photonic spectrum, indicated by the negative value of and hence also , as a result of which coherence is preserved and the decay of entanglement is arrested. This feature is further highlighted in FIG. b, which is also for the case of negative of higher order of magnitude than that in FIG. a, and as a result there is a greater persistence of entanglement. Thus we find that with the increase of the influence of the photonic band gap on the evolution, entanglement is preserved longer. From Eq. (III), it can be seen that, following the arguments of the previous section, there is no ESD in this case, a feature corroborated by the FIGS. 1 provided we choose the initial two-qubit state as a pure entangled state. Since the evolution of the off-diagonal elements of a single-qubit density operator is governed by , the behaviour of the dynamics of coherence is similar to the entanglement dynamics.
Apart from these, Fig. 1 shows another interesting phenomenon – the temporally damped oscillations in the entanglement. This phenomenon is a signature of the emergence of non-Markovian characteristics in the evolution and implies that the action of detuning changes the character of the dynamics itself, turning it non-Markovian from a Markovian one.
Iv Frequency modulation
Agarwal and coworkers (25); (26); (27) introduced an open-loop control strategy which involved modulation of the system-bath coupling, with the proviso that the frequency modulation (to be introduced below) should be carried out at a time scale which is faster than the correlation time scale of the heat bath. The technique of frequency modulation has been used earlier to demonstrate the existence of population trapping states in a two-level system (51). Raghavan et al. (52) showed the connection between trapping in a two-level system under the action of frequency-modulated fields in quantum optics and dynamic localization of charges moving in a crystal under the action of a time-periodic electric field.
Consider the Hamiltonian given in Eq. (9). Frequency modulation essentially involves a modification of the coupling — the modulated coupling is , where is the amplitude and is the frequency of the modulation. The decay of the excited state population can be significantly arrested by choosing such that , where are the Bessel functions of order zero. The resulting master equation in the interaction picture, when applied to the evolution of a two-level system, is (25); (26); (27):
Here are the Pauli matrices. We have used the Bessel function expansion , where is the Bessel function of order one. Additionally, the modified bath correlation functions are assumed to have the forms and , where is the bath correlation frequency. Now, we have
where and is the matrix representation of . We obtain the matrices and using Eq. (13) and the dynamics turns out to be completely positive. Invoking channel-state duality and the factorization law of entanglement decay, the time to ESD () is
where , and . Detailed calculations are given in Appendix B. From FIG. 2, it can be seen that a higher frequency of modulation sustains entanglement longer. This result is not altogether surprising, for a higher degree of modulation is naturally expected to filter out the influence of the bath and increase the coherence which ultimately results in entanglement sustaining for a longer period of time. And hence the behaviour of the coherence dynamics and entanglement dynamics are qualitatively similar.
V Resonance fluorescence
In the previous section, we focused on the increase in the time to ESD by increasing the degree of frequency modulation of the system-bath coupling. In this section, we study a system where a two-level atomic transition is driven by an external coherent single-mode field which is in resonance with the transition itself. We shall show that, in this situation, an increase in the Rabi frequency — which plays the role of the modulator — produces the opposite effect by speeding up ESD. The behavior of such driven systems has been well studied in the literature and has found many applications. In contrast to the situation here, Lam and Savage (53) have investigated a two-level atom driven by polychromatic light. The phenomenon of tunneling in a symmetric double-well potential perturbed by a monochromatic driving force was analyzed by Grossmann et al., (54), while photon-assisted tunneling in a strongly driven double-barrier tunneling diode has been studied by Wagner (55).
The analysis of the said driven system begins with its Hamiltonian which, when written in the interaction picture, is . Here is the electric field strength of the driving mode (treated classically), is the atomic transition frequency and is the dipole moment operator in the interaction picture. The driven two-level system is coupled to a thermal reservoir of radiation modes. If be the spontaneous rate due to coupling with the thermal reservoir and be the Planck distribution at the atomic transition frequency , the evolution of this composite system is given by the following master equation (5):
where is the Rabi frequency and is the transition matrix element of the dipole operator. The term characterizes the interaction between the atom and the external driving field in the rotating wave approximation. As usual, are the atomic raising and lowering operators, respectively.
Let us consider two identical qubits and, as before, assume that one of them interacts locally with a thermal bath and is subject to monochromatic driving by an external coherent field. The master equation (Eq. V) yields the corresponding matrices and (where the subscript stands for resonance fluorescence) giving rise to a completely positive map:
Using these, we plot, in FIG. (3), concurrence vs the time to ESD for different values of the Rabi frequency and observe that decreases for an increase in . This is contrary to the result derived in the previous section, where an increase in the modulation frequency delayed the loss of entanglement. The decrease in does not however continue indefinitely, but rather saturates to a certain value for large values of the Rabi frequency. This is an interesting result, further analysis of which will be carried out in a future work. Figure (4) depicts an increase in the single-qubit coherence with an increase in the Rabi frequency , bringing out the fact that here coherence and entanglement behave in a different fashion. This puts into perspective the fact that coherence, a local property, need not be monotonic with entanglement, a non-local property of quantum correlations.
Let us now consider the situation where the system, consisting of the excited two-level atom, is at zero temperature. Let us also consider the evolution of entanglement for two cases demarcated by the relation between the Rabi frequency and the spontaneous rate of coupling with the thermal reservoir. For the underdamped case when , the quantity is real (since at temperature ) and hence both the upper level occupation and coherence exhibit exponentially damped oscillations. Conversely, in the overdamped case, is purely imaginary and both these quantities decay monotonically to their stationary values. The evolution of entanglement, however, works in an opposite way . Entanglement decays faster for the underdamped case than for overdamping, where the is higher. One possible reason for this could be the relative positions of the three Lorentzian peaks of the inelastic part of the resonance fluorescence spectrum. The central peak is at and the rest are at (5) for the underdamped case, whereas all three peaks are at for the overdamped case. This indicates that the decay of entanglement in the underdamped should be closely dependent on the quantity . This in turn depends on both the dissipation parameter and the Rabi frequency, the latter in itself a function of the driving strength of the external field and the dipole transition matrix elements. Thus, in the underdamped case, there exists greater avenues for the decay of quantum coherences as well as entanglement than the overdamped case. Phenomenologically, for the underdamped case (), the two-level atom interacts with the external monochromatic field multiple times before spontaneously radiating a photon (see, for example, chapter 10 of ref. (56)). Such numerous interactions allows quantum correlations to develop between the two atomic levels and the quantized levels of the field. The phenomenon of monogamy of entanglement (57) thus ensures that the amount of quantum correlation between the two qubits will decrease. Additionally, it can be seen that at a higher Rabi frequency, dominates the dissipation and thus causes a saturation of the time to ESD, as shown in FIG. 3.
Vi Dynamic decoupling and the effect on ESD
As discussed earlier, open-loop control strategies involve the application of suitably tailored control fields on the system of interest, with the aim of achieving dynamic decoupling of the system from the environment (19); (22); (24); (25); (26); (27). Bang-Bang control is a particular form of such decoupling where the decoupling interactions are switched on and off at a rate faster than the rate of interaction set by the environment. The application of suitable radio frequency (RF) pulses, applied fast enough, averages out unwanted effects of the environment and suppresses decoherence. In this section, we compare the effect of Bang-Bang decoupling on the evolution of entanglement, using channel-state duality and factorization law of entanglement decay, in systems connected to two different types of baths. One bath type is composed of infinitely many harmonic oscillators at a finite temperature and couples locally to a two-level atom acting as the qubit, while the other adds random telegraph noise to a Josephson-junction charge qubit. It has been shown for the former case that all two-qubit states shows ESD at non-zero (10).
vi.1 Bang-Bang decoupling when the bath consists of harmonic oscillators
Quantum Non-Demolition Interaction
Let us consider the interaction of a qubit with a bath of harmonic oscillators where the system Hamiltonian commutes with the interaction Hamiltonian so that there is no exchange of energy between the system and the bath — this is quantum non-demolition dynamics (39); (40). The only effect of the bath will be on the coherence elements of the qubit evolution, which will decay in time at the rate . The total Hamiltonian for the system plus bath is:
Here the system Hamiltonian commutes with the interaction Hamiltonian and the evolution of such a system is called pure dephasing. For simplicity we will work in the interaction picture where the density matrix of the system plus bath and the interaction Hamiltonian transform as:
From here we can write the total time evolution operator for the system plus bath as
where . We are interested in calculating
Assuming that the bath and the qubit were uncorrelated in the beginning and that the bath is in a thermal state, we have (19):
The matrix representation of the evolution operator (which corresponds to a completely positive map) can be written from here as:
The evolution of maximally entangled state provides sufficient information concerning the evolution of entanglement. The evolution of one subsystem in state gives rise to the density matrix:
The concurrence in the state is directly proportional to .
Dephasing under Bang-Bang dynamics
The function of Bang-Bang decoupling is to hit the system of interest with a sequence of fast radio-frequency pulses with the aim of slowing down decoherence (see FIG. 5). Adding the radio frequency term to the system-plus-bath Hamiltonian (Eq. (21)), we get
where , and
The term acts only on the system of interest which here is the qubit. It represents a sequence of identical pulses, each of duration , applied at instants . The separation between the pulses is . The decay rate for this pulsed sequence evolution is (19):
In (19) it has also been shown that which implies that decoherence is suppressed. Also, it is evident that a lower value of implies a lower value of . Consequently, we can conclude that Bang-Bang decoupling (which corresponds to a new completely positive map, i.e, a modification of the map corresponding to Eq. (28) due to the RF pulses) slows down entanglement decay.
vi.2 Josephson Junction qubit
Although solid state nanodevices satisfy the requirements of large scale integrability and flexibility in design, they are subject to various kinds of low-energy excitations in the environment and suffer from decoherence problems. There have been a number of proposals in this context about the implementation of quantum computers using superconducting nanocircuits (58); (59). Experiments highlighting the quantum properties of such devices have already been performed (60); (61). Here the concept of a Josephson-junction qubit comes into prominence. A charge-Josephson qubit is a superconducting island connected to a circuit via a Josephson junction and a capacitor. The computational states are associated with charge in the island and are mixed by Josephson tunneling. For temperatures much lower than the Josephson energy, (62); (63); (64); (65), we have the Hamiltonian
with the charging energy dominating the Josephson energy. Here, , is the capacitance of the capacitor connected to the island and is the external gate voltage (see Fig. 6).
Fluctuating background charges (BCs) (charge impurities) are an important source of decoherence in the operation of Josephson charge qubits. These are believed to originate in random traps for single electrons in dielectric materials surrounding the superconducting island. At low frequencies, these fluctuations cause the noise which is also known as random telegraph noise, and is directly observed in single electron tunneling devices (66); (67). This has also been studied in the context of fractional statistics in the Quantum Hall Effect (68). This noise, arising out of decoherence, is modeled (64); (65) by considering each of the BCs as a localized impurity level connected to a fermionic band, i.e., the quantum impurity is described by the Fano Anderson model. This is the quantum analogue of the classical model of independent, randomly activated bistable processes. For a single impurity, the total Hamiltonian is:
Here describes the BC Hamiltonian, represents the impurity charge in the localized level , the electron in the band with energy , and is as in Eq. (35). The impurity electron may tunnel to the band with amplitude . The BC produces an extra bias for the qubit via the coupling term . An important scale is the switching rate , where is the density of states of the band. It is assumed that we are working in the the relaxation regime of the BC where the tunneling rate to all fermionic bands are approximately same, hence gets replaced by , above. The fraction determines whether the operational regime of the qubit is weak () or strong (). Studying the single BC case is important, since it has been shown (64) that the effect of multiple BCs can be trivially extended from that of a single BC. For multiple strongly coupled BCs producing noise, the effect of a large number of slow fluctuators is minimal and pronounced features of discrete dynamics such as saturation and transient behavior are seen. There are two special operational points for the qubit related to Eq. (35): (a) , corresponding to charge degeneracy and (b) , for the case of pure dephasing (69); (70), where tunneling can be neglected. We will consider this case later in detail and make a comparison of ESD, for the case of pure dephasing, between the harmonic oscillator and baths.
The general procedure for studying the effect of the BC on the dynamics of the qubit is to calculate the unitary evolution of the entire system plus bath and then trace out the bath degree of freedom. Thus, , being the the full density matrix. In the weak coupling limit a master equation for can be written (71). The results in the standard weak coupling approach are obtained at lowest order in the coupling , but it has been pointed out that higher orders are important for a noise (62); (63); (64).
The failure of the standard weak coupling approach is due to the fact that the environment includes fluctuators which are very slow on the time scale of the reduced dynamics. To circumvent this problem one considers another approach in which a part of the bath is treated on the same footing as the system (65). We study the evolution of this new system and later trace out the extra part which belongs to the bath, i.e., . We then obtain from as , where the subscript stands for fermionic band. In that context we split the Hamiltonian (36) into a system Hamiltonian and environment Hamiltonian coupled by . The eigenstates of are product states of the form , e.g,
with corresponding energies
Here , are the two eigenstates of , respectively, the directions being specified by the polar angles with and with . The two level splittings are and , and . Here and .
The master equation for the reduced density matrix in the Schrödinger representation and in the basis of the eigenstates of reads:
where is the difference of the energies (, etc.) and are the elements of the Redfield tensor (71) where . These are given by
Here is the Fourier transform of and , therefore, . This problem has a very interesting symmetry: the diagonal and off diagonal elements do not mix if the initial state is a diagonal density matrix in the BC. Therefore, we can divide the Redfield tensor elements in two parts, one corresponding to population (diagonal elements) and other corresponding to coherence (off diagonal elements).
The elements which affect the population are:
Here and , and
Now we calculate the elements which are responsible for the coherence part. In the adiabatic regime we have and , i.e., where the BCs are not static and the mixing of and in Eq. (38), as well as their conjugates cannot be neglected. Hence the non-zero elements of tensor – which affect coherence – are the following: