Equivalence between Redfield and master equation approaches for a time-dependent quantum system and coherence control
We present a derivation of the Redfield formalism for treating the dissipative dynamics of a time-dependent quantum system coupled to a classical environment. We compare such a formalism with the master equation approach where the environments are treated quantum mechanically. Focusing on a time-dependent spin- system we demonstrate the equivalence between both approaches by showing that they lead to the same Bloch equations and, as a consequence, to the same characteristic times and (associated with the longitudinal and transverse relaxations, respectively). These characteristic times are shown to be related to the operator-sum representation and the equivalent phenomenological-operator approach. Finally, we present a protocol to circumvent the decoherence processes due to the loss of energy (and thus, associated with ). To this end, we simply associate the time-dependence of the quantum system to an easily achieved modulated frequency. A possible implementation of the protocol is also proposed in the context of nuclear magnetic resonance.
pacs:03.65.Yz; 03.67.Pp; 03.67.-a
The rapid development of quantum information science has brought together several areas of theoretical and experimental physics (1). Much effort has been concentrated in the search for solutions to sensitive problems that prevent the efficient realization of quantum information processing (2). We first mention the system-environment coupling which induces the decoherence of quantum states (3), apart from other barriers such as scalability (4) and optimal control of individual systems (5). These challenges motivate both fundamental physical phenomena and outstanding technological issues such as individually addressing quantum systems, separated by only few m, with small errors (6).
Potential platforms for the implementation of quantum logic operations appeared in many fields such as condensed matter, quantum optics, and atomic physics (1). However, the problems mentioned above are faced by all the different communities when employing their particular techniques. In the particular case of the dissipation and decoherence phenomena —in which we focus in the present work— the Redfield formalism (7); (8) and the master equation (9) have been the most applied approaches to address the environment effects on the proposed protocols for quantum information processing. Whereas the semiclassical Redfield formalism relies on a classical noise source, a quantum environment is assumed in the master equation approach. In this article, considering the general case of a time-dependent system, we discuss general similarities and differences between both approaches and show that they are equivalent, in the sense that they lead to the same phenomenological Bloch equations (10). Consequently, both of them result in the same characteristic relaxation times and associated with the longitudinal and transverse relaxations, respectively (7); (8). From this identification we show how these characteristic times are related to the operator-sum representation (2) and the phenomenological-operator approach (11).
The Redfield formalism was intended to offer a microscopic description of the relaxation phenomenon, thus providing a deeper understanding of the parameters and . Whereas the classical noise source employed in the Redfield theory suffices to derive both relaxation times, two distinct quantum environments must be adopted to derive these time scales from the master equation formalism. On this regard, an amplitude and a phase damping environment are assumed to define the longitudinal and the transverse relaxation times, respectively. These quantum environments represent an energy-draining and a phase-shuffle channel by which the system loses excitations and phase relations.
After presenting a detailed derivation of the Redfield theory and comparing the derived characteristics times with those obtained from the master equation, we finally apply these equivalent formulations to the problem of state protection. We note that several distinct techniques have been proposed to control the effects of decoherence on quantum states, aiming to enlarge the fidelity of quantum information protocols. Among others, we mention the quantum-error correction codes (12), environments engineering (13), decoherence-free subspaces (14), and dynamical decoupling (15). We finally mention that in a previous work (16), addressing the energy draining and decoherence of a harmonic oscillator, it was demonstrated that the inevitable action of the environment can be substantially weakened when considering appropriate non-stationary quantum systems. Reasoning by analogy with the technique presented in Ref. (16), we show how to enlarge the longitudinal relaxation time associated with the amplitude-damping channel focusing a spin- system. The ideas presented here for decoherence control can be easily implemented in the nuclear magnetic resonance (NMR) context.
This article is organized as follows: In Sec. II we present the derivation of the Redfield equation for the general scenario of a time-dependent system. In Sec. III we apply the master equation approach to the same case. In Sec. IV we show the equivalence between the Redfield and the master equation formalisms by deriving the Bloch equations from both approaches. Focusing on a time-dependent spin- system, in Secs. V and VI we present the operator-sum representation and the phenomenological operator approaches and their relation with the previous techniques. As an application of the theory, in Sec. VII we address the state protection of a non-stationary spin- system. Finally, Sec. VIII is dedicated to our final remarks where we discuss the generalization of the methods presented in this article for larger systems. In general, we will adopt the language of the NMR quantum information processing (17), although the theory presented here is valid for several other platforms, as quantum dots (18), superconducting artificial atoms (19), etc. Throughout the article we will use natural units such that .
Ii Redfield formalism for a time-dependent spin system
Considering the interaction of a time-dependent spin system, described by the Hamiltonian , with a spin lattice modelling the environment and represented by the Hamiltonian , the total density operator in the Schrödinger picture, evolves as
being the time-dependent spin-lattice interaction. In the interaction picture, defined by the unitary transformation , where , we simplify the above evolution equation to the form
where , with . We have assumed the condition which is always fulfilled whenever the free Hamiltonian of the system can be written as a diagonal time-independent operator, with a time-dependent coefficient . Moreover, we observe that in NMR relaxation experiments all the required pulses to perform the necessary rotations are applied either at the beginning, to prepare the initial state , or at the end of the experiment, to implement the tomography of the evolved state. Between the applications of these pulses, the prepared state of the system, described by the diagonal Hamiltonian , evolves only under the action of the environment.
By its turn, the density operator in the rotating frame is given by where . By assuming a weak system-environment coupling and getting rid of the degrees of freedom of the spin lattice, we solve Eq. (2) up to second order of perturbation theory, obtaining
We considered that the interaction is a stochastic operator with null mean value (7); (8), which results in a zero first order term. Next, let us use the Markov approximation , being the reduced density operator of the spin system, , and the reduced density operator of the environmental spin lattice. This approximation means that the state of the lattice is not affected by the interaction with the system. In other words, it means that the lattice presents a sufficiently large heat capacity in order to remain in the thermal equilibrium state , with , being the environment temperature. Finally, inspired in NMR systems (8); (17), we are going to apply the high-temperature approximation, which takes into consideration systems were the energy gap between the spin levels, (where is the characteristic transition frequency among levels), is much smaller than the thermal energy, , of the system, i.e. . In this sense, the density operator of the system can be written as . We stress that there is a crucial difference between high and infinite temperature limits; differently from the latter case, in the former there is still a population difference between the spin levels which accounts for the reminiscent equilibrium magnetization. Thus, applying the high-temperature approximation we obtain and, consequently
From the Heisenberg equation of motion for , we obtain
and, consequently, the evolution equation reads
By rewriting the spin-lattice interaction as , where models the lattice stochastic fluctuation and stems for an operator acting on the spin system space, we verify straightforwardly that the first term of the r.h.s. of Eq. (4) is null, in accordance with the assumption . Moreover, with the above definition for the spin-lattice interaction, we verify that and, consequently, . Integrating by parts the third term of the r.h.s. of Eq. (4) and considering, as usual, that the time oscillations of the operator is much faster than that of , we apply the rotating wave approximation to conclude that this term is also null. The fact that this is indeed the case, can be seen as follows: the operator oscillates with the spin-lattice coupling frequency, while the operator oscillates with the bare spin frequency which (in the assumed system-environment weak coupling regime) is much higher than the interaction frequency. Putting all this together, we finally obtain the simplified equation of motion for the spin system
which, in accordance with the high-temperature approximation, where and , becomes
where we have defined the operator and substituted the trace over the lattice degrees of freedom by the ensemble average over stochastic realizations, represented by the over bar. We have thus obtained the Redfield equation for a time-dependent spin system and we note, in spite of the -number character of the environment degrees of freedom, its resemblance with the master equation to be presented below.
It is important to stress that the high-temperature approximation, allowing us to define the latter operator , indicates a relaxation to the highly mixed thermal Gibbs state. However, we mention that in the whole calculation to obtain the Redfield equation (5) it is not necessary to impose such approximation. It was only done because it is characterisitic of NMR systems, on which we focus in the present work.
Towards the definition of the lattice spectral density we next introduce, through the eigenvalue equation , the spin basis . Taking the matrix element of Eq. (5) and back to Schrödinger picture where , we obtain the Redfield equations for the evolution of the density matrix elements
where we have used the short-hand notation , , , and defined the relaxation matrix elements
with the environment spectral densities given by
|To simplify the notation we have omitted the explicit time dependence of all functions in Eq. (6).|
|For the particular case of a time-independent system, we obtain , and the above Redfield equations reduce to the well-know text book result (7); (8)|
We observe that, although we have focused on a spin system, the equations obtained here are completely general, being valid for whichever the Hamiltonian , provided that the three following conditions are met: , system-environment weak coupling regime (Markovian environment), and iii) high-temperature approximation (specifically, to derive Eqs. (6), (7), and (8)). The restrictions and the validity of these approximations will be discussed in the conclusions of the article. Let us now turn to the master equation approach.
Iii The master equation approach
In this section, in contrast to the semiclassical approach of the Redfield formalism, we derive the master equation governing the dynamics of the dissipative time-dependent spin system where the environment is assumed to be modelled within the quantum formalism. We start from Eq. (3), such that
Now, instead of assuming a classical environment leading to the above defined spin-lattice interaction as , we consider two distinct quantum environments, to be defined below as the amplitude- and the phase-damping channels, each one being modelled by an infinite collection of decoupled harmonic oscillators, described by the Hamiltonian where labels the environments while stands for the infinity set of oscillators whose frequencies are denoted by . () represents the creation (annihilation) operator for the th mode of the th environment. The action of these environments on the spin system is modelled by the interaction
where and with being the phase factor coming from the transformation to the interaction picture. It is worth mentioning that the time-dependence of the system-environment coupling in the Schrödinger picture, , comes from the assumption of a time-dependent spin system Hamiltonian, . In fact, the coupling strength leads to the decay rate of the master equation which plays the role of the time-dependent relaxation matrix in the Redfield equation (6).
where we have defined the functions
For the environments considered in this work it follows that: , and , being the thermal average excitation of the th environment. These relations, of course, depend on the state of the environment. We observe that this master equation describes the Markovian evolution of a general time-dependent system, provided that the conditions and of the last section are satisfied.
Iv The characteristic relaxation times
In this section, restricting us to the case of spin- systems, we aim to derive the Bloch equations for the evolution of the magnetization components of non-interacting spins. First, we obtain the Bloch equations from the Redfield formalism, relating the characteristic relaxation times with the properties of the associated classical stochastic environment. Next, computing the evolution of the average magnetization from the master equation formalism, we are able to link the characteristic relaxation times with the properties of the quantum environment.
iv.1 From the Redfield to the Bloch equations
Let us consider here a spin- system placed in a constant magnetic field in direction. The frequency gap between the two Zeeman levels defines the Larmor frequency , with and representing the frequencies of the excited and the ground state, respectively. The modulation of these frequencies are due to some external influence, like an additional time-dependent magnetic field. The bare Hamiltonian of the spin- system is then given by . The action of the environment over the system is modelled by the spin-lattice Hamiltonian
where is the gyromagnetic factor, labels the orthogonal Cartesian directions , refers to the lattice stochastic fluctuation in direction, and stands for the spin (Pauli) operator.
For this system, the spectral density given in Eqs. (8) becomes
being a mean value depending on the specific nature of the spin system and the environment correlation time, measuring the rate of flips between the bath spins due to a specific anisotropic spin interaction (chemical shift, dipolar coupling, etc. ) (20). Note that we have assumed that the mean values of the coupling are not affected due to the time-dependence of the system. This is quite reasonable since we are modelling the environment as a stochastic noise source. Remembering Sec. II, we have
which is just the integral of the Larmor frequency with a positive or negative signal, depending on the difference ().
which enable us to compute the evolution of the mean value of the magnetization in an arbitrary direction:
Since, for spin- systems, , the above equation can be separated into the longitudinal and transverse field components,
where . We stress that we have neglected the free-evolution term in the above equation because we are only interested in the effect of the environment induced dynamics. We also note that if and only if and when , which explains why the term is a time-dependent function while is a constant. Writing the last equation in terms of the longitudinal and transversal magnetizations, defined as and , respectively, we obtain
being the equilibrium longitudinal magnetization. Now, comparing these results with the phenomenological Bloch equations
|the characteristic relaxation times, in terms of the time-dependent decay rates in the classical stochastic environment, is defined as|
|Equations (16) and (17) show that, in contrast to the longitudinal rate , the transverse decay rate is related to an energy conserving process, affecting only the quantum coherence of the system. This fact justify the choice of the amplitude- and phase-damping channels for the quantum description of the spin system. It is worth mentioning that can be controlled through the time-dependent parameter while can only be partially controlled since the decay rate does not depend on . Finally, from Eqs. (17) we obtain the well known relation between both characteristic times|
iv.2 From the master equation to the Bloch equations
In this section we consider the same system as before, but instead of a classical noise the spin system interacts with two quantum environments. In order to compute the evolution of the average magnetization from the master equation (11), which takes into account the amplitude- and phase-damping channels, we first address the decay rates and defined in the end of Sec. III. When considering the amplitude-damping channel () we associate () with the lowering (raising) spin operators whereas in the case of phase-damping () we define as the Hermitian number excitation operator. We then set and . For both cases we set the mean values for the environment operators and, consequently, , being the thermal average excitation of the th mode of the th environment. Considering that the environment frequencies are very closely spaced to allow a continuum summation, such that , being the spectral density of the environment, we obtain, for the amplitude damping case, the effective time-dependent decay rates
|where, remembering the time-dependence of the system-environment coupling in the Schrödinger picture ,|
To obtain the effective decay rates in Eq. (19) it was assumed, as usual, that the thermal average excitation of the environment modes vary slowly around the range of variation of the spin system frequency. This is a good approximation when the environment is in a thermal state, as the present case (9).
For the phase-damping channel, the effective decay rates are given by
|where we defined|
Due to the diagonal system-environment coupling associated with the phase-damping case, we are able to compute without having the explicit form of . In fact, assuming that the spectral density as well as the system-environment coupling vary slowly around , we obtain the time-independent parameter
From this result we see that, in contrast to the case of the amplitude-damping channel, the decay rates and do not acquire a time dependence due to the modulation of the system frequency, resembling the result obtained in the Redfield formalism. Finally, the master equation (11) becomes
Computing the evolution of the mean value of the magnetization in an arbitrary direction we obtain for the longitudinal and transversal magnetizations
As in the preceding subsection, we compose these equations with the Bloch equations (16), to obtain
|and, consequently, the relation|
|These equations show the connections between the semi-classical and quantum approaches to open system dynamics. In the next two sections we will construct the Kraus and the phenomenological operators for the time-dependent system studied in this section.|
with the Kraus operators satisfying the following relation
Our goal in this section is to construct the operators for both channels studied in the last section. To achieve this we will consider the density operator evolution equations, which follows from the Redfield or the master equation formalisms. Then, we compare these equations with those shown in Eq. (24) to obtain the time dependence of the Kraus operators.
In the next two subsections we will adopt the basis defined in Sec. IV, that diagonalize the spin operator , . Since both channels studied here are independent, let us then consider each one of them separately.
v.1 Phase-damping channel
Considering only the phase-damping (), the master equation (21) leads us to the following set of differential equations satisfied by the elements of the density operator
|where we have defined . Note that, as expected, the populations are not affected by this noisy channel. We assume that the Kraus operators for this case are given by (2)|
|with being the parameter to be determined. Starting from the initial density operator|
|and substituting Eq. (26) into Eq. (24), we thus obtain|
By imposing that the time derivative of the elements of the above evolved density operator must be identical to those in Eqs. (25), we derive the following differential equation for the parameter :
with the initial condition , arising from the fact that the Kraus operators must reduce to the identity at the initial time. The solution for is thus given by
which finally defines the Kraus operators in Eqs. (26).
v.2 Amplitude-damping channel
For the case of amplitude damping we assume that the Kraus operators are given by (2)
|where is the Boltzmann factor, being the energy gap of the spin- levels and the partition function. By analogy with the preceding subsection our aim is to obtain the differential equation obeyed by the parameter|