Weak Values Influenced by Environment

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 nonMarkovian 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.


Introduction
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 [1].When an appropriately prepared measuring device is strongly coupled to a system, we can obtain one of eigenvalues, say a, of a measured observable Â from the value exhibited by a pointer observable of the measuring device.The result is obtained with probability a   , where i  is an initial state of the measured system and a is the corresponding eigenstate of Â .When we perform measurement on an ensemble of identically prepared systems, we derive the average value î A A i    of the observable from the measurement outcomes.It is obvious that the average value lies inside the spectral range of the observable Â .Hence what we can obtain by quantum measurement is the eigenvalue and average value of the observable.However this is not only the story.In a usual measurement process, the measured system is not referred after the interaction with the measuring device, though it is prepared in an initial state before the interaction.Only the pre-selection of the system is performed.In 1988, Aharonov, Albert and Vaidman [2] have found that if an interaction between a system and a measuring device is sufficiently weak and the measured system is post-selected in a state f  after the inter-action with the measuring device, the weak value ŵ  of an observable Â can be obtained from the measurement outcomes.It is surprising that the weak value may take a complex value or a value outside the range of the eigenvalues of an observable.After the discovery of the weak value of an observable, many works have been performed for understanding and generalizing weak values [3][4][5][6][7][8][9][10][11][12][13][14], and furthermore the weak value has been observed experimentally [15,16].
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 evo-lution 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 pre-and post-selection.We will find that the weak value can be calculated by the quantum master equation or by the quantum regression theorem [18] when the environment is Markovian.To investigate the weak value in the case of a non-Markovian environment, we consider the stochastic dephasing in Section 3 and the single excitation multi-mode Jayes-Cummings model in Section 4, where we can obtain the exact expressions of the weak values in both cases.We provide a brief summary in Section 5.

Dynamics of Weak Values Influenced by Environment
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 ˆSE i  at time i .When there is no initial correlation between them, the equality holds, where ˆE  is an equilibrium state of the environment.To measure a system observable ˆS A , we prepare a measuring device in an appropriate quantum state ˆD   .The interaction Hamiltonian between the system and the measuring device is assumed to be where m t stands for the measurement time and D P is a momentum operator of the measuring device, which is canonically conjugate to a position operator (a pointer observable) ˆD Q  t . The system and environment evolve until the measurement is performed at time m while the measurement device remains unchanged.We denote as m i U t t t   ˆSE  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 operator    just 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 f t t  .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 operator-valued measure which is denoted as ˆS f  .We obtain the joint probability that the post-selection is succeeded and the measuring device exhibits the value q of the pointer observable where q is the eigenstate of the pointer observable such that ˆD Q q q q  and stands for the trace operator over the Hilbert spaces of the system and the environment.Using the Bayes theorem [19], the conditional probability that the measurement outcome is if the post-selection is succeeded becomes When the post-selection is succeeded, the average value f Q of the pointer observable is given by which will yield the weak value of the observable ˆS A 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 g is taken into account.Then we obtain from Equation (1.1) which yields the joint probability of f and q from Equation (1.3), When we assume that the probability current density of the measurement device vanishes, the equality ˆˆˆ0 D D D D P q q P q   q   holds [6].Then we obtain after some calculation, A t is the weak value of the observable fluenced by the environment, The probability that the post-selection is succeeded is iven by (1.10)This is independent of the measuring device, which is characteristic of the weak measurement.Thus we obtain the probability of the measurement outcome

  w m
A t We consider the property of the weak value given by Equation (1.9).Since the operator ˆS f  which on of the system is independent represents the post-selecti of the environment, the weak value Tr Tr It is obvious that the denominator is the average of the operator  by the reduced density operator where [19] ved by represents the quantum e sy .The reduced density rato n be deri means of the quanm channel for th r

 
ˆS i t  ca ope tu master equation method [17,18].When the weak measurement is performed just after the pre-selection or just before the post-selection, the weak value is simplified as Thus when 0 k value beu We assume that the environment is Markovian and the influence of the sy em on the environment is negligible.In tem has pr st this case, the reduced time evolution of the sys the semi-group property [18,20] and we can ap oximate as [21]    where ˆE  represents the equ enviro ent and is time evolution generator of the system ved by solving the quantum master equation in a Lindblad form [18,20]. Then we find the weak value from Equation (1.12), ilibrium state of the nm , which which are derived by solving the predictive and retrodictive quantum master equa  tions.On the other hand, since we have (1.21) we obtain the weak value, ˆ, Then if the environment is M quantum regression theorem [18], weak value.Hence we can investigate the weak value .ˆm arkovian, using the we can calculate the influenced by the Markovian environment by Equations (1.17), (1.18) and (1.22).For the non-Markovian environment, however, these results cannot be used and the calculation becomes much more difficult.

Weak Values in Stochastic Dephasing
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 [24].The time evolution of the system is governed by a stochastic Hamiltonian, where ˆz S is the z-component of a spin-1/2 and   t  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 ˆS A , where  st ˆS ands for the stochastic average and i  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 ˆS i  , the observable ˆS A and the measurement operator ˆS f  as , where j is an eigenstate of ˆz S om Equation ( 1such that .Then after some calculation, we obtain fr .25),, e e where is the characteristic function of the We can see that the approximation given by Eq b t G   1.3 uation (1.26) is valid if and only if the equality holds or by which is derived in the narrowing [17].Assuming that the stochastic mit of the dephasing li dephasing is characterized by the stationary Gauss-Markov process, we obtain the characteristic function [17,24], while we obtain for the stationary two-s jump process (or equivalently the random tel [24,25], . In these equation,  represents the strength of the dephasing and  is an inverse of the correlation time of the stochastic variable   t  .
Note that ian sto ss does not imply that the process of t is Marko-the Markov chastic p e dephasing he system Let us now consider the case that the system observable is the roc vian.
x -component weak value z S ˆ of the spin.Then the given by uation (1.27) becomes Eq In particular, when the system pre-selected in , the weak value is simplified as which is plotted as function of time in Figure 1.
It is found from the figure that the weak the spectral range of the spin-1/2 operator,  , in the narrowing limit or equivalently 2 2 ronment signific ider the weak value influenced by a quantum mechanical environment.Here we suppose that a qubit interacts with an environment consisting oscillators [18].The Hamiltonian of the qubit and nvironment is given by the Markovian limit.This means that the Markovian envi antly suppresses the anomalous property of the weak value.

Weak Value in Bosonic Environment
We cons If the inequality 2   is fulfilled, the environment is non-Markovian and otherwise it is Markovian .We can obtain an exact time evolution of the qubit and the environment.Indeed, when we set the initial state [18]   [18,26], and the time-de- pendent parameter   g t is given by   . In Equation (1.37), we set 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 0

Figure 1 .
Figure 1.The weak value of ˆx S in the stochastic dephasin process and (b) the stationary two-state-jump Markov proces between the pre-selection at i t and the post-selection at f t , th g w aracterized by (a) the stati s s, where the weak measurement is performed at the middle at is, hich is ch onary Gau s-Markov   m f i t t t 2   .We set and scale time by the post all.The time dependence of the weak value is plotted in Figure2.

Figure 2 .
Figure 2. The weak value of ˆz S ows of influenced by the bosonic environment, where (a) sh the dependence on the post-selection time in the case i t 0   we can calculate the weak value by means of the quantum channel i t  28)