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In this paper, we have estimated the temperature dependent path predictability for an electronic Mach-Zehnder interferometer. The increment of path predictability can directly be associated with stronger decoherence process. We have also theoretically predicted that placing two detectors in both the paths, which are at the same equilibrium temperature with the system, erases all the memory of path information and hence acts like a quantum eraser.

Quantum superposition is one of the elementary properties that distinguishes quantum mechanics from its classical counterpart. This specific property of quantum states gives rise to the phenomena of coherence, which though very much familiar in classical wave theory, appears in an apparently mysterious way since the effect arises from the summation of probability amplitudes rather than the physical field amplitudes. The amount of interference that can be observed in a quantum interference experiment, depends on the anonymity of the path that is taken by the particle whose interference property is under observation. The process of gaining information about the path is complimentary to the process of producing interference pattern, because gaining information about the path of the particle involves measurement processes, which destroy the superposition of the available states making the behavior of the particle classically probabilistic. If we discuss this issue in a more quantitative manner, then we must firstly specify a common measure giving the amount of interference, known as fringe visibility V, given by

where

In this paper, we will consider the “which way information” experimental situation from the backdrop of dissipative interaction. By dissipative interaction, here we mean the coupling of the concerning system with the meter state, which is, essentially considered to be a thermal magnetic field. We will study the effect of decoherence induced by the thermal magnetic field. Here the system under consideration is an electronic analogue of Mach-Zehnder interferometer as presented by Roulleau et al. [

To calculate the path predictability, we will consider the procedure developed by Krause et al. in a recent paper [

On the other hand, which way information may also be quantified by another parameter path predictability [

where

So increment in path predictability naturally results in the loss of visibility of the interference pattern. In other words we can say that with increment of path predictability the system becomes more and more classical. Krause et al. [

where

Here the magnitude of the wave vector is considered to be complex and which is given by

where

So this implies that the probability amplitude for a particle to travel a distance x can be given by

The second exponential term in the right hand side of Equation (9) represents dissipation. Here one has to keep in mind that the energy loss due to dissipation must be very small compared to the initial kinetic energy of the particle, i.e.

So the wave functions corresponding to the particles taking path

Now according to Krause et al. [

Putting the values of

Now we want to estimate the explicit expression of the decay parameter

For this purpose, we consider a two level atom with a vacuum thermal magnetic field. In the point contact region, the atom is interacting with the many modes of the vacuum radiation field, which is considered as a collection of harmonic oscillators. The total Hamiltonian of the system, filed and interaction is given by

where

here

The master equation for this two level atom interacting with a thermal magnetic field can be stated as [

where

It can be found that

It is the corrected result for Einstein A coefficient [

where

The solution of (16) can be found as

Using the this deacy parameter in Equation (12), we get the path predictability as

where

In

We also want to touch upon the issue of quantum erasing process. Erasing is the process by which we can protect the quantum interference pattern in the Welcher Weg type experiment. If by applying some method we can erase the path information of the particle, then we can regain the complimentary information of visibility; hence we get back the contrast of the fringe pattern. Let us now consider that there are two quantum point contact detector of length

For simplicity, we have assumed that the as of both the detectors to be the same. Now if the equilibrium temperatures of both the detectors are maintained to be the same, i.e.

In

In this paper, we have estimated the path predictability for a “which way information” experimental set up of an electronic Mach-Zehnder interferometer, in presence of decoherence. The result shows that, in presence of decoherence induced by the thermal magnetic field, the path predictability depends on the temperature of the system and the coupled magnetic environment. We must also mention here that we are considering the system in thermal equilibrium with the environment.

Samyadeb Bhattacharya thanks Prof. Guruprasad Kar of Indian Statistical Institute and Prof. Sibasish Ghosh of Institute of Mathematical Science, Chennai for useful discussions. Author also thanks Mr. Souvik Pramanik of Indian Statistical Institute for some technical helps.

Samyadeb Bhattacharya,Sisir Roy, (2016) Quantum Path Predictability for an Electronic Mach-Zehnder Interferometer in Presence of Environment Induced Decoherence and Quantum Erasing Process. Journal of Modern Physics,07,892-898. doi: 10.4236/jmp.2016.79080