Nonclassical dynamics induced by a quantum meter

Conventionally, the effect of measurements on a quantum system is assumed to introduce decoherence, which renders the system classical-like. We consider here a microscopic meter, that is, an auxiliary essentially quantum system whose state is measured repeatedly, and show that it can be employed to induce transitions from classical states into inherently quantumlike states. The meter state is assumed to be lost in the environment and we derive a non-Markovian master equation for the dynamic system in the case of nondemolition coupling to the meter; this equation can be cast in the form of an $\left(N_a\right)$th-order differential equation in time, where $N_a$ is the dimension of the meter basis. We apply the approach to a harmonic oscillator coupled to a spin-$\frac{1}{2}$ meter and demonstrate how it can be used to engineer effective Hamiltonian evolution, subject to decoherence induced by the projective meter measurements.

Figure 1

Upper row, unitary evolution cycle of an initial coherent state under Kerr-type Hamiltonian, cf. Eq. (20). Lower row, generation and decoherence of a Schrödinger cat state for finite measurement intervals. Duration of one evolution cycle is $T=2\pi$ and the meter state has been measured with the time step $\mathrm{\Delta }t={10}^{-6}T$. Pictures correspond to the initial state and to the Schrödinger cat states after $1$, $10$, ${10}^2$, and ${10}^3$ evolution cycles. The initial coherent amplitude is $\beta =3$ and the other parameters are $\theta ={37}^{\circ }$, $a=1$, $b=0$, and $c=-a∕\cos \theta$. The states are represented with the help of the Wigner functions.