1 Introduction

For many years the two level atom provided the simplest and the most natural framework for the investigation of resonance phenomena in the presence of damping. The natural description of the two-level atom in terms of the Bloch vector b⃗=(u, v, w) gives a simple, yet powerful, geometrical description of the coherent and the incoherent dynamics of a two-level atom. The classical by now textbook of Allen and Eberly [1] provides a detailed description of the two-level dynamics. In this description the coherent dynamics is represented by a rotation of the Bloch vector on a Bloch sphere defined as u ²+v ²+w ²=const, while the incoherent dynamics leads to a spiraling of the Bloch vector into a steady state that is not necessarily on the Bloch sphere. The physical source of damping characterizing the incoherent part of the dynamics has been attributed to quantum or stochastic fluctuations of various properties of the reservoir coupled to the two-level atom. Perhaps the best known source of decoherence is the spontaneous emission damping due to quantum vacuum fluctuations of the electromagnetic field [2]. Other possible sources of noise can involve stochastic phase, frequency or amplitude fluctuations. There is vast literature devoted to stochastic models of collisions, phase diffusion or frequency fluctuations [3]. In the framework of a theory involving a driven two-level atom by a multiplicative Gaussian white noise, the Bloch vector description is still valid, with damping rates given by the diffusion coefficients of the corresponding fluctuations.

A fundamental new milestone has been achieved, when it was recognized that two input bits in a channel can be replaced by a two-level atom forming a quantum bit of information called: qubit. In realistic setups the qubit is usually exposed to a noisy channel leading to decoherence of quantum information. Due to decoherence there is no such thing as a perfect qubit or a perfect quantum logical gate [4]. If a qubit is exposed to a noisy channel one can observe a loss of coherence and it is still an open question whether there is an intrinsic lower limit to the decoherence rate for an arbitrary qubit. The quantum or the stochastic noise is an undesirable but an unavoidable property of quantum logical gates.

The issue that stochastic noise of a qubit can dramatically change logical operations has become one of the central themes of error correction and noise stabilization in quantum computational schemes. The control of decoherence of a qubit is a central problem of various models of quantum computation, where an incoming qubit is transmitted through a noisy channel. Due to the noise the statistical properties of such a qubit can be changed. For example, pure states become mixed states. For a class of such noisy channels, one could ask the question of a possible optimalization or minimalization of the decoherence effect of a qubit. One fundamental goal is to seek those noises that minimize a fidelity or an entropy of such a channel. In order to perform such a task a general understanding of a wide class of noises acting on the qubit is required. This problem has attracted a lot of attention and has been studied in the framework of quantum information theory [5].

It is the purpose of this work to address the problem of stochastic decoherence of a qubit in the framework of a standard quantum optics description, involving such properties like the Bloch sphere, the optical Bloch equations, and the Master equation.

2 Decoherence of two level atoms

The most widely used description of a two-level atom with a coherent and incoherent dynamics is given by the Bloch equations:

The meaning of all the parameters is standard and is described in the classical textbook on this subject [1]. The incoherent dynamics is described by two lifetimes. The transverse lifetime T ₂ gives a decay rate of the atomic dipole moment characterized by u and v, while the longitudinal lifetime T ₁ gives the decay rate of the atomic inversion w into an equilibrium state w _eq. The Bloch equations have been applied as the most successful tool used in quantum optics to describe and to study the impact of incoherent effects on the two-level atom dynamics.

Using the Bloch equations one can obtain a simple geometrical picture of the two-level atom decoherence. In this picture, the Bloch vector and the Bloch sphere are the essential geometrical ingredients of the description.

In order to make contact with the language of quantum information theory, we shall denote the decoherence effects as a stochastic noise operation acting on the Bloch vector. We shall denote this stochastic noise by the following map:

and call it for short: stochastic map. In the following Sections the structure and the properties of this stochastic map acting on a qubit will be investigated in the framework of the two-level Bloch theory based on Eqs. (1).

3 Decoherence of qubits

In order to investigate the impact of the stochastic map given by Eq. (2) on a qubit we shall use the matrix representation of the Bloch vector in the form of the following linear combination of Pauli matrices:

Under the coherent evolution (no noise) the Bloch vector stays on the Bloch sphere defined by the property

This means that the coherent (unitary) evolution of the qubit leads to an orthogonal transformation of the Bloch vector, leaving Eq. (4) invariant.

Because of the stochastic noise (2), the B matrix is transformed as follows:

In general, the stochastic map (2) can transform the Bloch vector to a vector inside the Bloch ball which consists of the interior of the Bloch sphere. In order to be inside of the Bloch ball such a stochastic transformation has to be a contraction, i.e., it has to satisfy the condition:

The stochastic map given by Eq. (2), if applied to an arbitrary density operator of a qubit, leads to the following transformation:

Such a stochastic map of the density operator has to be trace preserving

and positivity preserving Φ(ρ)≥0. Such positive maps play a central role in quantum information theory [6].

In the following we present two examples of such stochastic maps given by Eq. (2) or by Eq. (5). These two examples provide a hint about the general nature of such transformations. We shall see that such transformations have a simple relation to the Bloch equations (1). In the next sections of this paper we provide an explicit construction of the stochastic map Φ(ρ) directly from the Bloch equations.

Stochastic map for spontaneous emission noise

It is well known that the spontaneous emission noise is described by $\frac{1}{T_1}=\frac{2}{T_2}=A$ and w _eq=-1. The damping terms are characterized by the Einstein A coefficient of spontaneous emission. In geometrical terms, the dynamics of a coherently driven qubit with spontaneous noise can be seen as the following three operations on the Bloch sphere. Due to coherent interaction, the first operation amounts to a rotation of the Bloch vector. The spontaneous emission noise leads to two additional operations: a damping and a linear shift of the Bloch vector. In this case we have Φ_A : B↦B ^′ with

where the matrix M=RN _A has been written as a product of a coherent rotation R and the spontaneous emission noise damping N _A . This damping noise matrix is diagonal

with the following damping eigenvalues

The important property of the spontaneous emission noise is that it leads to an additional linear translation of the Bloch vector by:

Stochastic map for frequency diffusion noise

As a different model of a stochastic map (5) we take the case of frequency (detuning) white-noise fluctuations [7]. In this case w _eq=0, i.e., the steady inversion becomes thermal. The damping rate $\frac{1}{T_1}=0$ and only $\frac{1}{T_2}=\Gamma$ is responsible for the frequency diffusion. For such stochastic fluctuations the stochastic map Φ_Γ leads to no translation of the Bloch vector, i.e., b⃗₀=0 and as in the case of the spontaneous emission noise M=RN _Γ with a diagonal damping matrix N _Γ with eigenvalues

The spontaneous emission noise or the frequency diffusion noise results from the coupling of the qubit to a quantum or stochastic reservoir of noise fluctuations. Such a coupling can be described by a Master equation [2]. With these two examples we are now in a position to write the general stochastic noise transformation of the Bloch vector based on the corresponding Master equation.

4 Arbitrary stochastic noise

A general stochastic map of the form given by Eq. (2) of a qubit can be written in the form

of an affine transformation of the Bloch vector. From the fact that the Bloch vector is real it follows that such a stochastic map is fully characterized by a real 3x3 matrix M and by 3 real shift parameters b⃗₀. This means that an arbitrary linear stochastic map is fully characterized by an affine transformation labeled by 12 real independent parameters.

As we have already noted the coherent evolution is given by a rotation matrix, and as a result we can write that, in general, M=RN, where the damping matrix N is symmetric and is characterized by 6 real parameters. The damping matrix N can be diagonalized by another orthogonal transformation T, leading to a diagonal matrix Λ with eigenvalues Λ₁, Λ₂, Λ₃. Each of the two orthogonal transformations R and T corresponds geometrically to a rotation (with inversion) of the Bloch vector on the Bloch sphere. The resulting matrix Λ describes the three fundamental decoherence damping eigenvalues of the qubit in some selected principal axes of the Bloch sphere.

The density operator of the qubit has the familiar form:

For the two-level atom, pure states are on the Bloch sphere b⃗·b⃗=1, while mixed states correspond to b⃗·b⃗≤1. Under the stochastic map (5) the density operator becomes

with the property that the affine shift leads to

It is well known that orthogonal transformations are represented by unitary operators acting on the density operators. Based on this property, we conjecture that a general stochastic map of the density operator can be written in the following formal way:

where the hermitian operators h_R and h_T correspond to the two orthogonal transformations (rotations) of the Bloch vector and S _L,R are some superoperators acting on the density operator. The form of these superoperators remains to be established. In the next section we provide an explicit construction of such a transformation using a Master equation.

From general properties of positive maps it is known that an arbitrary positive stochastic map can be written in Kraus form [8]:

with the condition

From the explicit form of the superoperators Eq. (18), one can derive the Kraus operators (19).

5 General stochastic transformation

We shall start the construction of the stochastic map (7) assuming that the density operator for the qubit satisfies a Master equation in the form:

The first term of the time evolution describes the standard coherent two-level dynamics and the last term accounts for the gain and the damping mechanisms. This term has the form of the Liouville superoperator which can be written in the Lindblad form [9]:

where F_i and $F_i^{†}$ form a collection of generalized atomic creation and annihilation operators characteristic for a particular problem. The Lindblad form of the Master equation guarantees that the interaction with the damping reservoir preserves the positivity of the density operator.

To study the impact of the noise on a qubit we shall use the theory of the damping basis for the Liouville operator developed in [10]. In this theory one looks for eigenoperators of the Liouville superoperator. Such an operator has right and left eigenoperators

The right and the left eigenoperators form an orthogonal basis with respect to the Hilbert-Schmidt scalar product:

The right and the left eigenstates of the Liouville operator form a damping basis for the corresponding Master equation given by Eq. (22). We shall use this basis to describe the time evolution of an arbitrary initial density operator ρ(0). Such a time evolution is given by

where the eigenvalues of a damping matrix are in the form of Λ_λ=e^λt and give the dynamical evolution of the density operator. From the properties of the damping basis (24) we obtain

With the help of these properties one can construct an explicit form of the stochastic map generated by the superoperators (18) using the relation:

This is the central result of this paper. In the following section we show how an explicit calculation of Eq. (27) can be performed for the Bloch equations.

6 Stochastic noise of bloch equations

In this section we give an explicit construction of the stochastic map (7) for a Master equation associated with the Bloch equations. The form of the corresponding Lindblad operators (22) for the two-level atom is well known and has the form [2]:

The right

and the left

eigenoperators of the Liouville operator have been calculated in [10]. These eigenoperators correspond to the following four eigenvalues of the damping matrix:

With these explicit formulas for the damping basis and the damping eigenvalues we can calculate the transformation (27).

If we assume that the initial density operator of the qubit is in the form of a matrix:

a simple calculation shows that the stochastic map (7) of this matrix is:

where

From the resulting formula (33) we see that the stochastic map is a contraction if the Bloch sphere u ²+v ²+w ²=1 is mapped into the interior of the Bloch ball. This is obtained if:

This condition leads to various inequalities between the parameters of the Bloch dynamics that guarantee the positivity of the stochastic map. Such relations have been investigated in [6] and [11]. It is worthwhile to point out that in the approach presented in this paper the positivity of the stochastic map is guaranteed by the Lindblad form of the Master equation (21).

From the formula (33) one can calculate the image of the set of pure state density matrices generated by the stochastic map. Such an image is given by a family of ellipsoids:

With the help of such an equation one can obtain a geometric visualization of the stochastic map of a set of pure states using the Bloch equations.

7 Conclusions

It has been the purpose of this paper to study the decoherence of a qubit with the help of a stochastic map that can be derived from the quantum optical Bloch equations. It is clear that despite the fact that we have used a specific Master equation, the general approach based on the damping basis is general and flexible to handle more complicated cases. It is worth pointing out that the use of a Master equation in the Lindblad form guarantees that the noise map of the qubit preserves the complete positivity of the density operator.