(7045) Supervised evolution in a 2 site system

Consider a particle in a 2 site system. The particle is prepared at \( x=0 \). The subsequent dynamics is generated by \( H =c \sigma_1 \). A detector is used in order to measure the state of the system. The result of a statistical measurement provides the probabilities \( p_0 \) and \( p_1 \) to find the particle at \(x=0,1\). The evolution is commonly described by the function \( F(t)=p_0 \) or by \( M(t)=p_0-p_1 \).

In a supervised evolution, the time is divided into intervals of length \( \tau \). A measurement is performed at the end of each interval. Here we assume that the detector supervises the target site \( (x=1) \). We associate with it a projector \( P=|1\rangle\langle 1| \).

Formally a measurement operation is represented by the formula \(\rho := (1{-}P) \rho (1{-}P) \), or by the formula \(\rho := P \rho P + (1{-}P) \rho (1{-}P) \), depending on whether the particle is annihilated or not by the detector. Accordingly we distinguish between repeated detection scenario and repeated measurement scenario. In the latter case the \(P \rho P\) term assumes that the detected particle is re-injected into the target site. Otherwise, without this term, \( F(t)=\text{Tr}(\rho) \) is the survival probability in the initial site.

Note: The questions below can be addressed without reference to the "probability matrix", using the common phrasing "collapse" for describing the effect of detection. In a repeated detection scenario, each detection is a projection that describes "collapse" of the wavefucntion. The norm \( F(t) \) of the remaining wavefunction in such scenario is the probability that the particle survives in the system. Plotting this norm as a function of time provides the inverse cumulative distribution of the time-to-detection.

(1) Given \( c \), write what are \( F(t) \) and \( M(t) \) for coherent evolution.

(2) Given \( c \) and \( \tau \), write what is \( F(t) \) for a repeated detection scheme that consist of \( t/\tau \) (integer) steps. Assume that the particle is annihilated once detected. Explain why in this scenario \( F(t) \) decays exponentially.

(3) Given \( c \) and \( \tau \), write what are \( M(t) \) and \( F(t) \) for a repeated measurement scheme that consist of \( t/\tau \) (integer) steps. Assume that the particle is kept in the system. Describe the evolution using a Bloch picture. Using this picture explain why in this scenario \( M(t) \) decays exponentially.

(4) Write the results for (2) and (3) in terms of the probability \( p \) to detect the particle at the end of the first interval. Assume that \( \tau \) is very small the exponential decay can be written as \( \exp(-\Gamma t) \). Write what is \( \Gamma \) in both cases. Exploit the smallness of \( \tau \) in order to get a simple expression.

For \( \tau \rightarrow 0 \) we get \( \Gamma \propto \tau \rightarrow 0 \). This is known as the Quantum Zeno effect. It is not widely recognized that the same effect comes out also in the classical analysis...