Quantum Nondemolition Measurements of Moving Target States

We present a protocol for probing the state of a quantum system by its resonant coupling and entanglement with a meter system. By continuous measurement of a time evolving meter observable, we infer the evolution of the entangled systems and, ultimately, the state and dynamics of the system of interest. The photon number in a cavity field is thus resolved by simulated monitoring of the Rabi oscillations of a resonantly coupled two-level system, and we propose to regard this as a practical extension of quantum nondemolition measurements with applications in quantum metrology and quantum computing.

Figure 1

(a) Unconditioned dynamics of the excited state population $⟨{\widehat{\sigma }}_{ee}⟩$ of a two-level system resonantly coupled to a harmonic oscillator. The complex behavior is due to its composition as a sum with weight factors $p_n$ of regular quantum Rabi oscillations with different frequencies shown in panel (b). Panel (c) shows the stochastic dynamics of the expectation value $⟨{\widehat{\sigma }}_{ee}⟩$, conditioned on weak continuous measurements of ${\widehat{\sigma }}_{ee}$ and simulated by the stochastic master equation (2). The continuous probing of ${\widehat{\sigma }}_{ee}$ gradually identifies a single Rabi oscillation frequency and hence collapses the system from an initial thermal ensemble with mean excitation $⟨\widehat{n}⟩=3$ into a single energy subspace as shown by the evolution of the subspace probabilities $p_n$ in panel (d).

Figure 2

Quantum trajectory simulations of the dynamics of the Jaynes-Cummings system subject to weak continuous probing [$k=0.1g$, panels (a) and (b)], intermediate probing [$k=g$, panels (c) and (d)], and strong probing [$k=10g$, panels (e) and (f)] of the excited state population of the two-level probe system. The harmonic oscillator is prepared initially in a thermal state with mean excitation number $⟨\widehat{n}⟩=3$ and the qubit is prepared in the excited state. The upper panels show the conditional probabilities for the total (integer) number of excitations, while the lower panels show the qubit excited state population.

Figure 3

The average purity $⟨P{⟩}_{dW}$ of 200 simulated trajectories under weak probing ($k=0.1g$). The shaded area represents one standard deviation from the mean. The average time $\tau$, needed to measure the energy, is extracted by fitting the purity $P(t)=1-[1-P(t=0)]e^{-t/\tau }$. The inset shows the average time of distinction of the excitation of the harmonic oscillator, prepared with an initial thermal distribution with $⟨\widehat{n}⟩=3$, as a function of probing strength $k$.

Figure 4

Observation of a quantum jump. The orange curve in the upper (lower) panel shows the excited state population (average excitation of the oscillator) inferred from weak continuous measurements on the qubit meter. The oscillator is subject to a single quantum jump occurring at $gt=25$, and the blue curves show the inferred qubit excited state population and oscillator number of quanta, assuming the added knowledge of when the jump happened. Parameters used for the simulation are $k=0.1g$, $\gamma ={10}^{-3}g$, and $⟨\widehat{n}⟩=n_T=3$.

Figure 5

Observation of multiple quantum jumps. The harmonic oscillator starts in a thermal distribution with mean excitation number $⟨\widehat{n}⟩=3$. The probing strength $k=g$, such that it is close to the optimal value. The coupling to the bath is $\gamma ={10}^{-3}g$ and its mean excitation is $n_T=3$.