A Derivation of Eq. (8)

A Master equation approach to line shape in dissipative systems


We propose a formulation to obtain the line shape of a magnetic response with dissipative effects that directly reflects the nature of the environment. Making use of the fact that the time evolution of a response function is described by the same equation as the reduced density operator, we formulate a full description of the complex susceptibility. We describe the dynamics using the equation of motion for the reduced density operator, including the term for the initial correlation between the system and a thermal bath. In this formalism, we treat the full description of non-Markovian dynamics, including the initial correlation. We present an explicit and compact formula up to the second order of cumulants, which can be applied in a straightforward way to multiple spin systems. We also take into account the frequency shift by the system-bath interaction. We study the dependence of the line shape on the type of interaction between the system and the thermal bath. We demonstrate that the present formalism is a powerful tool for investigating various kinds of systems, and we show how it is applied to spin systems, including those with up to three spins. We distinguish the contributions of the initial correlation and the frequency shift, and make clear the role of each contribution in the Ohmic coupling spectral function. As examples of applications to multispin systems, we obtain the dependence of the line shape on the spatial orientation in relation to the direction of the static field (Nagata-Tazuke effect), including the effects of the thermal environment, in a two-spin system, along with the dependence on the arrangement of a triangle in a three-spin system.

Line shape, Magnetic response, Dissipation, line width, quantum master equation
67.10.Fj, 03.65.Yz, 71.70.-d, 76.20.+q

I Introduction

Recently, the quantum dynamics of microscopic systems have been observed due to the development of experimental methods. For example, the quantum mechanical magnetization processes of single molecular magnets (SMM) have attracted much interest. Various new aspects of quantum effects are seen in such systems(1); (2); (3); (4).

To investigate the energy level structures of SMM molecules, electron spin resonance (ESR) experiments have been conducted for (5). The quantum tunneling effect was monitored by a proton NMR in (6); (7) and the dynamics of each magnetic atom was studied using NMR on Mn atoms in (8). The temperature dependence of the ESR signal was also studied in (9).

Complex susceptibility has been studied for a long time, and the effects of the exchange and/or dipolar interactions between the numerous constituent spins have been clarified (10); (11); (12). In order to evaluate the ESR spectra for spatially-structured systems, theoretical approaches for obtaining line shapes from a microscopic view point have recently been proposed by focusing on the effects of the interactions between spins, using direct numerical evaluations of the Kubo formula (13); (14); (16); (15), along with a field theoretical approach(17); (18). In these previous works, the line width comes from the interactions between the spins, which are described by the Hamiltonian system, and the line shape is given by an ensemble of delta-functions. The effects of contact with the thermal bath have not been studied, even though the thermal effect has attracted interest in studies of microscopic processes. Thus, an approach becomes necessary to introduce the effects of the surroundings, which cause the temperature dependent width of each resonant peak in the complex susceptibility.

In order to take these effects into account, we have to study an extended system in contact with a thermal bath, and consider the dynamical effects from the thermal bath. For this purpose, the time-evolution of the reduced density operator is usually studied, which is obtained by projecting-out the degrees of freedom of the thermal bath. A standard formalism has been established for the equation of motion for the reduced density operator(19); (22); (20); (21), which is generally called the quantum master equation. This formalism has been successfully applied to various fields. For example, the natural line width of a two-level (spin) system has been estimated(22); (23); (24), and systems with interacting spins(25) and nonlinear spin relaxation(26) have been studied. A rapid thermal bath correlation was assumed in these studies, and analyses were therefore made in the Markovian limit.

However, the effect of the finite correlation time of the thermal bath becomes important when we are interested in phenomena where the time scale of the system is comparable to that of the thermal bath. Then, we have to take into account the time-correlation function of the thermal bath and the initial condition of the density operator in the above mentioned master equation. In the equilibrium state of the total system, which consists of the system, the bath, and the interaction between them, the density operator is not given by a decoupled form. Therefore, we need to take the contribution from it into account, even though this effect has often been ignored by assuming a factorized form of the density operator. In the regression theorem(27), we obtain the time evolution for the average of a quantity for the factorized initial condition and estimate the correlation function from it, which is good in the Markovian limit(28); (29); (30). However, for short time phenomena, we need to estimate the correlation function of the quantity in a non-Markovian evolution, treating the initial correlation correctly(31); (43); (34); (33); (35). For example, Tanimura(34) obtained an exact hierarchical formulation with a functional integral for the spectral distribution of an Ohmic form with a Lorentzian cutoff. Breuer and Petruccione(33) studied the effects of the initial correlation on the dynamics of a spin-boson system, and pointed out the importance of their contribution. However, they did not obtain an explicit form for the correlation function or the complex susceptibility as a function of the frequency.

In the present paper, we provide a formulation for the complex susceptibility by extending the Nakajima-Zwanzig type of master equation without discarding the non-Markovian effect and the initial correlation. We derive an equation for the motion of the response function. Then we consider the equation of motion of the quantity , where is the initial density operator for the total system and is a system operator. We include the initial correlation between the relevant system and the bath, which is called the “inhomogeneous term” of the master equation. Since the equation is described by a time-convolution(TC) type of equation for the non-Markovian dynamics, the Laplace transformation can be explicitly evaluated. Here we obtain a concrete form of the complex susceptibility. It should be noted that the obtained formula is easily evaluated, even in interacting spins, by a concrete numerical calculation. Moreover, by using the Hilbert-Schmidt (H-S) representation, the formula is compactly expressed. In the present formulation, we can include the frequency shift due to a system-bath interaction, which comes from the imaginary part of the memory term expressed by the principal value integral of the correlation function of the thermal bath operators. While we present the formula up to the second order of cumulants, it could easily be extended to the higher orders.

We apply the obtained formula to spin systems linearly interacting with a bosonic bath. For a single spin system, we study the dependence of the line shape on the type of system-bath coupling, e.g., the case of pure dephasing, in which only the diagonal component of the spin interacted with the bath, and the case of longitudinal relaxation, in which the off-diagonal components interacted with the bath (the non-adiabatic interaction). We find that the initial correlation and the frequency shift due to the memory kernel are more dominant in the pure dephasing case than in the non-adiabatic interaction case. Owing to the usage of the H-S representation, we could extend our formalism to multiple-spin systems in a straightforward way. For a linearly coupled spin chain, the dependence of the peak shift on the angle between the chain and the static field has been studied as the Nagata-Tazuke effect(36). As an example of an application to multispin systems, we study the dependence, including the effects of the thermal environment. We also study the relationship between the line shape and the geometrical configuration in a three-spin system on a triangle.

This paper is organized as follows: We provide a general formulation of susceptibility in Sec. 2. The application of the obtained formula to the linear spin-boson model is given in Sec. 3. Discussions and concluding remarks are given in Sec. 4.

Ii Formulation

In this section, we present a formulation of the complex susceptibility of a system in contact with a thermal bath. Generally, the linear response theory gives the complex susceptibility in the form(19)


which describes the response of the operator to an oscillating external field conjugate to the operator with the frequency . Here, and are components of the operators and , respectively, and denotes an equilibrium state. If we consider the response in a pure quantum state, the time evolution of is given by and is , where is the Hamiltonian of the system and is the partition function of the system at a temperature . On the other hand, to analyze the complex susceptibility under dissipation, we need to describe the time evolution of by taking into account the interaction between the relevant system and a thermal bath.

As will be shown in the next section, the dynamics in contact with a thermal bath are not only given by the quantum dynamics of the system, but are also affected by memory effects inherent in the contact with the thermal bath. The memory effect is often treated in the so-called Markovian approximation(37). This approximation is often used to study the time evolution of the reduced density operator of a system, which leads to the quantum master equation. As long as the equation has the so-called Lindblad-Kossakowski-Sudarshan form(37) as in the field of quantum optics, the density operator is positive definite. However, it has been pointed out that the Markovian approximation may violate the positivity of the density operator. In particular, in a spin-boson model, the breakdown of the positivity has been explicitly reported. A method to amend this breakdown has been proposed using a kind of slippage supplement in the initial conditions(38); (39); (40). Using the Markovian time evolution with this supplement enables us to simulate the time evolution of non-Markovian time evolution, but its validity is limited in a time region larger than the correlation time of the thermal bath(39); (40).

Since experimental development has accelerated in recent years, we need to formulate a line shape theory that can correctly describe the non-Markovian effect, including the region of the correlation time of the thermal bath. Moreover, the term for the system-bath correlation at an initial time in the equation for the time evolution of the density operator has often been ignored. However, the importance of this term has been pointed for obtaining a correct description of the dynamics(38); (41). Finally, we also need a compact formula that can be evaluated by a concrete numerical method. For this purpose, we will present a straightforward way to derive a complex susceptibility that includes the initial correlation as well as the non-Markovian effect.

ii.1 Formula of susceptibility

We suppose that a relevant system is in contact with a thermal bath and that the whole system is in an equilibrium state with temperature . Defining the density operator of the whole system as , the linear response theory is extended to give the susceptibility as


where denotes the trace operation for the whole system. When we denote as the Hamiltonians of the systems ,, and the system-bath interaction, the time evolution of an arbitrary operator for the relevant system is determined by the Heisenberg equation,


Defining the total Hamiltonian as (), and using the relation as


we can rewrite Eq. (2) in the form,




where denotes the trace operation over the thermal bath. With the Fourier-Laplace transform where is an appropriate function, we find that the susceptibility is given by


The above formulation shows that the procedure to obtain the complex susceptibility reduces to obtaining . As shown in Appendix A, the time evolution of is given in a form of a “ master” equation by using the projection operator technique. Here, we define the projection operator to be . Up to the second order of the system-bath interaction , we have


where the kernel and the inhomogeneous term are given by




respectively. In Eqs. (9) and (10), we used the following definitions , and and . Because the whole system is assumed to be in an equilibrium state, we have to take into account the contribution of the initial correlation between the system and the thermal bath, which is represented by the inhomogeneous term .

From Eq. (7) and Eq. (8), the susceptibility is given by


where we define with an arbitrary operator . Our remaining task is to obtain and . For this purpose, we give concrete expressions for and in the next subsection.

ii.2 Concrete expressions for and

For simplicity, we consider the case in which the interaction between the system and the thermal bath is given in the form


with the system operator and the thermal-bath operator . In this form, the second and third terms in Eq. (8) are given by


respectively. Here, we assumed that , and we used Eq. (86), and definitions with and


It might be convenient to use the eigenstates of an unperturbed relevant system to obtain the matrix elements of in the “master” equation, Eq. (8). We denote the eigenstates of the relevant system and for the energy eigenvalues as and . The component of is given by

We can obtain the elements for in a similar way. In order to evaluate the susceptibility, Eq. (11), we need to obtain the Fourier-Laplace transform of each element and solve the simultaneous equations for Eq. (8). We can express the equation by making use of the Hilbert-Schmidt (or Liouville) space, which we show in the next subsection.

ii.3 Transformation to Hilbert-Schmidt space

In order to evaluate the susceptibility, Eq. (11), it is convenient to transform operators of the relevant system into vectors that construct the H-S space. This is because the Liouville operators in the Hilbert space are written as a supermatrix in the H-S space, which makes the evaluations easier. Defining a scalar product between operators and as , the transformation from the Hilbert space to the H-S space is done by expanding an arbitrary operator in the Hilbert space with a set of orthonormal operators as


where the orthonormal condition of is written as . We can transform an operator in the Hilbert space to a vector in the H-S space with the set of . In the case where an operator is written as an dimensional matrix, the corresponding vector in the H-S space has elements.

Multiplication operations on a density operator in the Hilbert space are transformed to a supermatrix in the H-S space: When the arbitrary operators and are multiplied by the density operator as , the product is transformed into the H-S space as


where the supermatrix is symbolically expressed as


Here and correspond to matrices of the operators and , denotes the Kornecker product, and denotes the complex conjugate operation. When the density operator is written in an -dimensional matrix, is an -dimensional vector and is an -dimensional matrix.

Using Eqs.(18) and (19), we obtain the transformation of Eq. (8) into the H-S space as


which gives the susceptibility in a more straightforward way than using Eq. (LABEL:eqn:15n), since the kernel, is written in the H-S space as a matrix

The inhomogeneous term in the H-S space is given by


which is an -dimensional vector for the -dimensional density operator.

The -th component of Eq. (20) is expressed by


where with . It should be noted that the Fourier-Laplace transform of the memory kernel in Eq. (23) is given by


and, we have


Thus, we have




which corresponds to in Eq. (11). All of the matrix elements of are given in an explicit way, as will be shown below. The complex susceptibility in the H-S space is given in the form


ii.4 Concrete form of

Now we obtain the matrix elements of on the basis of the eigenstates of the relevant system,


with the eigenfrequency . A more explicit expression is obtained by introducing the spectrum of the thermal bath, as


Using the spectrum,


where we define and use the following relation,


The terms of the principal value represent the frequency shift that results from the system-bath interaction. While these terms have often been neglected, we can take them into account in the present formalism.

ii.5 Concrete form of

The inhomogeneous term, Eq. (LABEL:eqn:13n), is simply written in the H-S space by the multiplication of a matrix and the H-S vector of




The Fourier-Laplace transform of the inhomogeneous term is given as


where we define as


The complex susceptibility, Eq. (28), is now written in an explicit form, which can be applied to an arbitrary type of thermal bath by specifying . We will show a few examples of baths in the next subsection.

ii.6 Bath

When analyzing the relaxation phenomena of a relevant system, we often introduce a thermal bath that consists of an infinite number of bosons(42); (43); (44) or spins(45). This section discusses procedures to obtain the spectra for a bosonic bath . We use a bosonic bath for the relaxation phenomena caused by phonons in a medium or photons in a cavity. The Hamiltonian for the boson system is written as,


where () denotes an annihilation (creation) operator for the-th mode of a boson. As an example, we will consider a case in which the contribution to , Eq. (12), from the bath is given by,


Then, the correlation function for the bath is written as


where is the coupling constant between the relevant system and the -th mode of the boson. In order to evaluate the correlation function of the thermal bath, we need to introduce a coupling spectral function as


We can rewrite the weighted summation for an arbitrary function in the following form,


Using Eq. (41), is rewritten as


where is the boson distribution function given by . The spectrum of the thermal bath is obtained in the form


where denotes the step function.

Iii Applications

We are now in a position to apply the formalism presented in the previous section to the relaxation phenomena in a spin system. First, we evaluate the spectral line shape of a system where a single spin interacts with a bosonic bath. Although such a system is trivial, the evaluation shows the concrete procedure, which is essentially the same as in multiple-spin systems. Next, we demonstrate the Nagata-Tazuke effect for two and three spin systems, showing the dependence of the line shape on the angle between the spatial configuration and the direction of the static applied field. We include the initial correlation and the frequency shift of the line shapes.

iii.1 Spin-boson model

Suppose that a spin () linearly interacts with a thermal bath that consists of bosons. The Hamiltonian of the relevant system is written as


and the interaction operator in Eq. (12) is given by


where corresponds to the , and components of the relevant spin. In the following, we set and , since the generality is not lost when is a real number. We control the types of relaxation by the value of : the case of describes the pure dephasing phenomena of the spin. For other cases of , we can include the longitudinal relaxation in the transverse relaxation of the spin.

Equation (27) is now given as follows: The second term is written as


We can evaluate the third term of Eq. (27) concretely by using Eq. (31) as,