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A weak value of an observable is studied for a quantum system which is placed under the influence of an environment, where a quantum system irreversibly evolves from a pre-selected state to a post-selected state. A general expression for a weak value influenced by an environment is provided. For a Markovian environment, the weak value is calculated in terms of the predictive and retrodictive density matrices, or by means of the quantum regression theorem. For a non-Markovian environment, a weak value is examined by making use of exactly solvable models. It is found that although the anomalous property is significantly suppressed by a Markovian environment, it can survive a non-Markovian environment.

One of the most characteristic features of quantum mechanics lies in a measurement process which provides some information about an observable of a quantum system to be measured [

In the most of the previous works on weak values, dynamics or time evolution of a system to be measured has been neglected. Only the interaction Hamiltonian between a system and a measuring device has been taken into account. However, since a measured system in a real world is unavoidably influenced by an environment, we have to consider the effect of the environment on the weak value as well as intrinsic dynamics of the system. Hence it is interesting to investigate the decoherence of weak values during the irreversible time evolution of a system from pre-selected state to a post-selected state. The irreversible time evolution of a system caused by an interaction with an environment is usually studied by means of the quantum master equation [17,18]. However, the post-selection of the system that is essential for weak values makes it very difficult to investigate the irreversible time evolution by the usual method when an environment is non-Markovian. Therefore, in this paper, we will consider the effect of the irreversible time evolution of the system on the weak value of an observable. In Section 2, we provides a general expression of a weak value during the irreversible time evolution of a quantum system between preand post-selection. We will find that the weak value can be calculated by the quantum master equation or by the quantum regression theorem [

We suppose that a quantum system to be measured is placed under the influence of an environment and is initially prepared or pre-selected in a quantum state at time. When there is no initial correlation between them, the equality holds, where is an equilibrium state of the environment. To measure a system observable, we prepare a measuring device in an appropriate quantum state. The interaction Hamiltonian between the system and the measuring device is assumed to be , where stands for the measurement time and is a momentum operator of the measuring device, which is canonically conjugate to a position operator (a pointer observable). The system and environment evolve until the measurement is performed at time while the measurement device remains unchanged. We denote as the unitary operator which describes such time evolution. Then the quantum state of the total system just before the measurement is given by the density operatorand it becomesjust after the interaction with the measuring device. After the interaction, the system and environment further evolves until the post-selection is performed on the system at time. Hence we obtain the quantum state just before the post-selection,

The post-selection performed on the system is, in general, described by means of probability operatorvalued measure which is denoted as. We obtain the joint probability that the post-selection is succeeded and the measuring device exhibits the value q of the pointer observable,

where is the eigenstate of the pointer observable such that and stands for the trace operator over the Hilbert spaces of the system and the environment. Using the Bayes theorem [

with

When the post-selection is succeeded, the average value of the pointer observable is given by

which will yield the weak value of the observable under the influence of the environment.

In the weak measurement, the strength of the interaction between the system and the measuring device is sufficiently small and only the terms up to the first order with respect to the coupling constant is taken into account. Then we obtain from Equation (1.1)

which yields the joint probability of and from Equation (1.3),

When we assume that the probability current density of the measurement device vanishes, the equalityholds [

where is the weak value of the observable influenced by the environment,

The probability that the post-selection is succeeded is given by

This is independent of the measuring device, which is characteristic of the weak measurement. Thus we obtain the probability of the measurement outcome,

which yields with .

We consider the property of the weak value given by Equation (1.9). Since the operator which represents the post-selection of the system is independent of the environment, the weak value becomes

It is obvious that the denominator is the average of the operator by the reduced density operator of the system,

where and represents the quantum channel for the system [

Thus when or, we can calculate the weak value by means of the quantum channel and otherwise calculating the weak value becomes much more difficult.

We assume that the environment is Markovian and the influence of the system on the environment is negligible. In this case, the reduced time evolution of the system has the semi-group property [18,20] and we can approximate as [

where represents the equilibrium state of the environment and is time evolution generator of the system, which is derived by solving the quantum master equation in a Lindblad form [18,20]. Then we find the weak value from Equation (1.12),

which is equivalent to that obtained by the quantum trajectory method [

where and are the predictive and retrodictive density matrices of the system [

which are derived by solving the predictive and retrodictive quantum master equations. On the other hand, since we have

we obtain the weak value,

Then if the environment is Markovian, using the quantum regression theorem [

In this section, using an exactly solvable model, we investigate the weak value of an observable influenced by a non-Markovian environment. For this purpose, we use the Kubo-Anderson model [22,23], where the quantum system to be measured is a two-level system or a qubit and the environment causes the stochastic dephasing of the system [

where is the z-component of a spin-1/2 and is a classical stochastic variable with zero mean. The unitary operator that describes the time evolution is given by

In this case, since the trace operation over the environmental Hilbert space in Equation (1.12) is replaced with the stochastic average, we obtain the weak value of a system observable,

where stands for the stochastic average and is the initial state of the qubit. Here we note that the approximation given by Equation (1.16) is equivalent to

which is valid only in the narrowing limit of the dephasing. To calculate the weak value given by Equation (1.25), we expand the initial state, the observable and the measurement operator as , and, where is an eigenstate of such that and. Then after some calculation, we obtain from Equation (1.25),

with

and

where is the characteristic function of the stochastic variable,

We can see that the approximation given by Equation (1.26) is valid if and only if the equalityholds or equivalently the characteristic function is given bywhich is derived in the narrowing limit of the dephasing [

while we obtain for the stationary two-state Markov jump process (or equivalently the random telegraph noise) [24,25],

with. In these equation, represents the strength of the dephasing and is an inverse of the correlation time of the stochastic variable. Note that the Markovian stochastic process does not imply that the dephasing process of the system is Markovian.

Let us now consider the case that the system observable is the -component of the spin. Then the weak value given by Equation (1.27) becomes

In particular, when the system pre-selected in at the time is post-selected in a state, the weak value is simplified as

which is plotted as function of time in

It is found from the figure that the weak value lies in the spectral range of the spin-1/2 operator,, in the narrowing limit or equivalently the Markovian limit. This means that the Markovian environment significantly suppresses the anomalous property of the weak value.

We consider the weak value influenced by a quantum mechanical environment. Here we suppose that a qubit interacts with an environment consisting of harmonic oscillators [

where and are bosonic annihilation and creation operators of the kth oscillator of the environment. It is assumed that the environment is initially in the vacuum state with and it has the Lorentzian spectral density,

If the inequality is fulfilled, the environment is non-Markovian and otherwise it is Markovian [

where and the time-de-

pendent parameter is given by

with. In Equation (1.37), we set, where the coefficient is given by

Then the exact time evolution of the qubit and the environment is provided by Equations (1.37)-(1.39).

To find how the weak value is influenced by the bosonic environment, we suppose that the observable is the z-component of the spin-1/2 operator and the qubit is post-selected in the excited state at time. Then if, we can derive the weak value from Equations (1.12) and (1.37), please see Equation (1.40) below.

It is interesting to note that the weak value does not depend on and. In other words, the weak value is independent of the pre-selection of the system. If the bosonic environment is Markovian, we see that since the equalityis satisfied. If the equality holds, we obtain

which yields the inequality. In this case, the weak value is always greater than the maximum eigenvalue of the spin-1/2 operator. Furthermore, if the inequality is fulfilled, the weak value can take values inside and outside the spectral range of the spin-1/2 operator since becomes an oscillatory function,

When is sufficiently small, the weak value becomes large, though the success probability of the post selection is very small. The time dependence of the weak value is plotted in

In this paper, we have considered the weak value of an

observable of a system interacting with an environment and we have provided the general expression of the weak value influenced by an environment. Since the post-selection of the system is performed, it is, in general, very difficult to calculate the weak value. When the environment is Markovian, we can obtain the weak value in terms of the predictive and retrodictive density matrices of the system, which are derived by solving the quantum master equations, or by means of the quantum regression theorem. On the other hand, for the non-Markovian environment, we don’t know the systematic method for calculating weak values. Hence to investigate the weak value in the case of the non-Markovian environment, we have applied the two exactly solvable models. One is the stochastic dephasing model and the other is the single excitation multi-mode Jayes-Cummings model. We have found that the Markovian environment significantly suppresses the anomalous behavior of the weak value in comparison with the non-Markovian environment. Since we have used the specific models, we have to consider more general cases. For this purpose, however, it is necessary to develop a method for calculating weak values under the influence of the environment.

The author would greatly appreciate Prof. F. Shibata and Prof. S. Kitajima for their stimulating discussions.