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The Zeno time has been calculated for a metastable two-level atom tunneling through a interacting thermal magnetic field. The process of weak measurement has been utilized for the estimation of the timescale. Zeno time has been shown to be temperature dependent. From the calculation it is evident that the Zeno time decreases with the increase of temperature. Moreover, the result restricts the Zeno time to a maximum limiting value, irrespective of how frequent the measurement process is.

Quantum Zeno effect is the phenomena of inhibition of transition between quantum states by the process of frequent non-selective measurement [

Quantum Zeno Effect (QZE) can be theoretically presented in a very simple way with the consideration of the short time behavior of the decaying quantum state [

as

For small time interval t, a power series expansion of the time evolution up to 2^{nd} order

So then the modified survival probability is given by

where

is the uncertainty in system energy measurement. There are many quantum mechanical states having survival probability which appears to be decreasing exponentially on ordinary time scales. The quadratic time dependence of Equation (1) is naturally inconsistent with those decaying states and implies that in such cases Equation (1) holds only for very short time intervals. Consider the survival probability

which approaches 1 as

which is called the “Zeno time” or the timescale within which Zeno effect can be observed. From Equation (3) we can easily infer that as long as the time interval is shorter than the Zeno time, the system is freezed in the initial state. If the interval between consecutive measurements on the system is smaller than

It follows that

The only condition to go from Equation (7) to Equation (8) is

But in practical situation, Nakazato et al. [

Based on the theoretical background we just discussed, we are now going to introduce the framework of weak measurement, which we are going to use to calculate the Zeno time. The reason behind using this particular measurement framework is that in our consideration the interaction between the thermal electromagnetic field and the system is sufficiently weak. So one single measurement or interaction on the system does not give any significant result; or in other words does not disturb the system in a considerable way. So numerous such interactions are taken into account to get an ensemble average for some quantum observable. In this work we will use the method originally developed by Davies [

The time evolution of the state is considered to be

where the time evolution operator

The expression of the time dependent weak value of a certain operator A pre-selected at time

If we now consider that the two level atom is in a barrier region of external magnetic field in the z direction. Then the system Hamiltonian can be represented as

where

If the initial pre-selected state at

and the associated projection operator

In case of the decay of any metastable state, the system of two level atom is considered to be coupled to 2M number of environmental bath modes, which are initially in their ground states. Because of the presence of the interaction with these bath modes, the system loses energy to them. For simplicity we consider that any arbitrary state

with the assumption that the reference atom is equally coupled to all the bath modes. Following Davies [

where a_{0} is the amplitude of the pre-selected initial state and

with the limit

From Equation (20) we can clearly see that

So Equation (20) gives the generalized weak decay law. At initial time

The total time interval

The population inversion decay parameter

In this section we will concentrate on the time evolution of the two level atom coupled to a thermal magnetic field to determine the decay parameter, using their corresponding master equation [

where

where

This is the magnetic field operator in the interaction picture containing a wide range of frequencies. Among those frequencies, we are only concerned with the ones almost in resonance with

Similarly

Now the master equation for the the reduced density operator

Considering a slightly more intricate situation, where we take a complex electromagnetic field

Now the modified master equation is given as

From this master equation the time evolution of the expectation value of the Pauli spin operators can be given as

From Equation (34) we can see that the system state denoted by

Equation (35) represents the rate of population inversion, which we take as the decay parameter. Again

The variation of Zeno time with temperature is shown in the

From the figure it can be clearly seen that the Zeno time decreases with increasing temperature. This is very much compatible with experimental situation. Because with increasing temperature the environmental interaction increases, making the decay process stronger. For that reason, it becomes harder to make Zeno effect realizable with increasing temperature. The maximum Zeno time is at zero temperature given by the expression

Now if N can be increased to a very large number, making the measurement procedure quasi-continuous, then the Zeno time at zero temperature becomes

which shows that even if we make the measurement quasi-continuous, Zeno time will only increase to a certain limiting value.

In this paper, we have formulated the expression of Zeno time as a function of temperature and other parameters of system-bath interaction, for a two-level atom tunneling through a thermal magnetic field. We have used the procedure of weak measurement for our calculation. The reason behind using this particular measurement framework is that in this type of measurement scheme, the interaction between the system and the measuring device (in this case thermal magnetic field) is made very small. This can be a useful way to restrain the environmental interaction, which is in turn helpful to control the decohering process. Other important feature of weak measurement process which should be mentioned here is that, in this measurement scheme an ensemble average of numerous observations is taken over the pre-selected and post-selected states. Here one single measurement interaction is not sufficient to bring out necessary information about the system. Since the Zeno process is initiated by frequent non-selective measurements, an ensemble average over many such measurement interactions is necessary to observe the dynamics over a finite period of time. So scheme of weak measurement is also compatible with the Zeno type measurement process. In this work, our calculation shows that the Zeno time decreases with the increase of temperature, which is compatible with practical situations. With the increase of temperature, the system-bath interaction increases. So the process of decoherence due to environmental dissipative interaction gets stronger. The essence of Zeno dynamics is that due to non-selective frequent measurements the total Hilbert space reduces to a quasi-isolated reduced subspace, within which the decay dynamics can be stopped or at least considerably slowed down. Now if the process of decoherence gets stronger, then isolating the system in the reduced Zeno subspace gets much harder. So with the increase of temperature as the decoherence process gets stronger compared to the Zeno process, the Zeno time decreases. Again from Equation (38) we can see that even quasi-continuous measurement cannot make the Zeno time infinitely large. The Zeno time can only be increased to a limiting value. So our result also imposes a restriction over the method of non-selec- tive frequent measurement procedure, giving a limiting maximum value of Zeno timescale irrespective of how frequent the measurement process is. The reason behind this limitation is that even at zero temperature, the coupling between the system and environment exists. So even at zero temperature, the reduced Zeno subspaces are vulnerable to the environment-induced decoherence process, which gives a restriction over the Zeno dynamics.

SamyadebBhattacharya,SisirRoy, (2015) Temperature Dependent Zeno Time for a Two Level Atom Traversing through a Thermal Magnetic Barrier in the Framework of Weak Measurement. Journal of Modern Physics,06,1261-1269. doi: 10.4236/jmp.2015.69131