Protecting a quantum state from environmental noise by an incompatible finite-time measurement
We show that measurements of finite duration performed on an open two-state system can protect the initial state from a phase-noisy environment, provided the measured observable does not commute with the perturbing interaction. When the measured observable commutes with the environmental interaction, the finite-duration measurement accelerates the rate of decoherence induced by the phase noise. For the description of the measurement of an observable that is incompatible with the interaction between system and environment, we have found an approximate analytical expression, valid at zero temperature and weak coupling with the measuring device. We have tested the validity of the analytical predictions against an exact numerical approach, based on the superoperator-splitting method, that confirms the protection of the initial state of the system. When the coupling between the system and the measuring apparatus increases beyond the range of validity of the analytical approximation, the initial state is still protected by the finite-time measurement, according with the exact numerical calculations.
pacs:03.65.-w Quantum mechanics, 03.65.Ta Foundations of quantum mechanics; measurement theory, 02.60.-x Numerical approximation and analysis
The problem of measurement is fundamental to quantum theory key-1 (); key-2 (); key-3 (); key-4 (); key-31 (). According to the standard interpretation key-1 (); key-3 (); key-4 (), established after the pioneer contributions by Born key-3 (); key-44 (); key-45 (), Dirac key-46 (), and von Neumann key-3 (), the postulated collapse of the wave function implies an instantaneous measurement process. However, the de facto meaning of a quantum measurement is the result of a physical interaction between the system being measured and the measuring apparatus. Such an interaction can be described by a suitable formalism key-5 (); key-6 (), which, under the hypothesis that the measurement process is irreversible, leads to the Lindblad equation key-7 (); key-33 (); key-34 (). This equation describes the evolution of a measurement during a finite time interval by the stochastic term - Lindbladian. At last, the information of the probabilities associated to every possible result is acquired by the conventional way with the density operator diagonal elements - populations key-8 (); key-9 ().
Based on superoperator algebra and Nakajima-Zwanzig projectors key-11 (); key-12 (), we have been able to describe the dynamics of an arbitrary measurement that occurs during a finite time interval, while the system being measured interacts with the rest of the universe and, due to the consequent environmentally-induced noise, undergoes decoherence key-10 (). The assumption that the interaction of the measuring apparatus with the system is Markovian justifies a Lindbladian approach. However, to treat the noise introduced by the fact that, during the finite-duration measurement, the system is perturbed by the environment, a Markovian approximation is too restrictive, since non-Markovian noise effects can become non-negligible for time scales in which coarse graining is not a good approximation key-16 (); key-17 (); key-18 (); key-19 (); key-36 (); key-39 (). Instead, to develop a formalism able to include non-Markovian effects, we have used a Redfield approach to the dynamical description of the interaction between the system and its environment, excluding the apparatus. After tracing out the degrees of freedom introduced to describe the non-Markovian noise, the resulting master equation in the Born approximation (not a Born-Markov approximation), referred only to the system, can be used to investigate the effects of the environmental noise on the measurement dynamics. We do not use additional Lindblad channels to describe the environmental noise because such a formulation would preclude any attempt to treat non-Markovian effects. Since we are interested in describing the effects of the noise during the finite-time measurement, turning off the environmental perturbation during the action of the measuring apparatus is not a valid approximation. Therefore, our hybrid approach is the most economical dynamical description of a measurement that is simultaneous to non-Markovian environmental perturbations.
The Lindblad equation can be used to describe Markovian and non-Markovian environmental effects, with applications ranging from decoherence and dissipation analyses key-9 (); key-33 () to measurement processes key-5 (), particularly when we consider that the measurements have finite durations key-8 (); key-9 (). It follows as a mathematical consequence of a semi-group dynamics to describe irreversibility in quantum mechanics key-7 (). Therefore, it can only be used to describe Markovian processes, unless the number of dynamical variables describing the system be enlarged key-36 (), causing computational overhead that we must avoid. Incidentally, we point out that the extension of the Lindblad equation to non-Markovian systems key-36 () can be applied to non-semigroup dynamics as well, with time-dependent Hamiltonian and Lindbladian operators.
In the present paper, we analyze finite-time measurements that commute or do not commute with the interaction Hamiltonian. We study a two-state system interacting with a bath of harmonic oscillators, that emulates a phase-noisy environment. The approximate analytical solution agrees with the numerical results of the superoperator-splitting method key-41 () for weak coupling between the system and its environment. We find that the finite-time measurement can protect the measured state, if the observable being measured does not commute with the Hamiltonian describing the noisy environmental interaction with the system but, when the measurement commutes with the interaction, the environment only increases the coherence decay. For strong coupling between the system and the environment, our analytical solution fails, as shown by a thorough comparison of its predictions against the corresponding exact numerical results. In the case of strong coupling, the effect predicted by the numerical calculations is a more intense error as the coupling with environment increases.
Subsequent researchers may use the method presented here to study the effects of varying the temperature, the density of states, or even the system considered, increasing the number of quantum states. There are some interesting possible directions to follow to extend the present investigation, such as the study of non-Markovian systems key-16 (); key-17 (); key-18 (); key-19 (); key-36 (); key-39 (), time-dependent Nakajima-Zwanzig projectors key-20 (); key-30 (), higher-order approximations for the interaction between the system and its environment key-37 (); key-38 (), and effects of other kinds of environmental noise. One problem which the present theory can not address is the possibility that the measurement result be different from one of the eigenvalues of the meter key-40 ().
The paper is structured as follows: In Sec. II we briefly present the proposed formalism and state the problem to be analyzed. In Sec. III, we present the approximate analytical solution to the master equation, followed by a derivation of its numerical counterpart in Sec. IV. In Sec. V, we discuss the results obtained with the expressions in Sec. III and Sec. IV and its physical implications. In Sec. VI, we present some perspectives for expanding the material that we have presented here and a conclusion.
Ii The hybrid master equation
The Lindblad equation,
where is the density operator describing the system and its environment, is the total Hamiltonian and the are the Lindblad operators, which act only on the system, is the most general form for a master equation key-7 (); key-34 (). The second term on the right-hand side of Eq. (1),
is the Lindbladian operator acting on the density operator The Liouvillian operator acting on
accounts for the unitary portion of the propagation, before the environmental degrees of freedom are traced out, and the Lindbladian represents the Markovian measurement dynamics.
We begin with a system and its environment whose interaction is described by the Hamiltonian:
where, for each index , operates only on the system and only on the environment . The form of the interaction, Eq. (2), is general enough, assumed by both amplitude and phase damping models key-9 (). Here, to account for non-Markovian noise, we choose to treat the environmental interaction as part of the total Hamiltonian, appearing in the Liouvillian term of Eq. (1). Thus, we write
where is the environmental Hamiltonian and is the system Hamiltonian. All the information read by the measuring apparatus, which is assumed to be Markovian, will be accounted for by the Lindbladian term of Eq. (1) and, as usual, the Lindblads will act only on the Hilbert space of the system, since we are interested in measuring system observables only. Our aim is to obtain an equation for the time evolution of the reduced density matrix of the system, , that is,
For any density matrix operator , let the superoperators and be defined, respectively, as
It is easy to show that and .
for any and any initial time . With these projectors and algebraic manipulations, we obtain the general equation,
Evidently, according to Eq. (7), once is calculated, can be found by the action of on the reduced that is,
As we explain in the Introduction, here we consider a two-state system interacting with a bath of harmonic oscillators, that emulates a phase-noisy environment. Thus, we take
and phase-damping interaction key-9 (), that is,
where and the are real constants, and are the annihilation and creation bath operators, are complex coefficients. Here and next, we will use the Pauli matrices and ,
Hence, let us define the operator
Therefore, the interaction can be written in the simplified form:
In the case of a finite temperature, we take the initial state of the environment as given by
Iii Solution of the hybrid master equation
Now let us define the operator:
Therefore, to recover the reduced density operator of the system, we apply to
An unusual aspect that should be clarified is the action of the superoperator exponentials, and . Let us consider, initially, the time-independent operators and related by the operation :
When we take the time derivative of and use Eq. (14), we obtain
Now, if we consider the definition of the superoperator Eq. (3), we obtain the elementary Liouville-von Neumann equation, that is,
whose solution is easily determined for a time-independent as in the case of (9):
iii.1 Expanding the integrand that appears in the hybrid master equation
It is interesting to note that, in Eq. (17), the actions of and are completely separated in each term appearing on the right-hand side. This proves extremely valuable in the calculations that follow.
iii.2 Tracing out the environmental degrees of freedom
For the sake of convenience, let us analyze, firstly, the environment quantities appearing in Eq. (17). According to Appendix A, the partial trace over the environmental variables gives:
iii.3 Introducing a continuous density of states characterizing the environment
In Eq. (18), if we adopt the general definition of the density of states as
then the sum over the index can be replaced by an integral over a continuum of frequencies:
Here, to obtain an analytical solution, we choose the Ohmic density of states (19):
where and is the constant that gives the intensity of the coupling between the system and its environment.
iii.4 Reduced density operator describing the system
To obtain the action of the operator , it is necessary to solve Eq. (1) without the environment. Then, we take
and, in the Lindbladian, or .
In the case of , the solution is simple and can be found in Ref. key-8 ():
where the upper index indicates that the solution is written in the eigenbasis of
For , the solution is more complicated:
To analyze the result of a measurement it is natural to represent the density operator in the eigenbasis of the measuring apparatus. Accordingly, in the present case, we use the eigenstates of that is,
The change of basis is performed with the eigenvectors matrix
and the result is:
iii.5 The case of
Here, we consider the evolution of the system in contact with a thermal reservoir at arbitrary temperature, i.e., we assume that the initial condition of the environment is given by Eq. (11).
Let us write
where, for notational convenience, we take Then, using Eq. (22), we obtain
Substituting this into Eq. (20) and manipulating the terms according to Appendix B gives
iii.5.1 Obtaining the populations
According to Eq. (27), the populations are independent of and can be immediately evaluated, giving
where . Then, using the constraint that the trace of the density operator must equal unity, we obtain
iii.5.2 Obtaining the coherences
From Eq. (27) it follows that the non-diagonal elements satisfy
where and . From the procedure of Appendix C, we obtain
where denotes the gamma function. Simplifying Eq. (30), according to Appendix D, gives
iii.6 The case of
From Eq. (23) the action of on an operator
results in the final differential equation:
iii.6.1 The populations represented in the eigenbasis of
Analogously to the case of and , the populations do not depend on . From Eq. (35) we obtain
where . We notice that and, using Eq. (13), we can write:
iii.6.2 The coherences represented in the eigenbasis of , for and
Since is the complex conjugate of , we only need to calculate one of them. Let us introduce the new variable Hence, it follows from Eq. (35) that
There is no analytic solution for this equation at a finite temperature. However, as detailed in Appendix F, we have found the following result for and