Spikes in quantum trajectories

A quantum system subjected to a strong continuous monitoring undergoes quantum jumps. This very-well-known fact hides a neglected subtlety: sharp scale-invariant fluctuations invariably decorate the jump process, even in the limit where the measurement rate is very large. This article is devoted to the quantitative study of these remaining fluctuations, which we call spikes, and to a discussion of their physical status. We start by introducing a classical model where the origin of these fluctuations is more intuitive, and then jump to the quantum realm where their existence is less intuitive. We compute the exact distribution of the spikes for a continuously monitored qubit. We conclude by discussing their physical and operational relevance.

Figure 1

Schematics of the classical model considered. The particle jumps are recorded in a variable $R_t$; an imperfect camera yields blurry photos from which a binary variable ${\delta }_i$ is extracted. An estimate $Q_t$ of $R_t$ is constructed from the ${\delta }_u$'s for $u<t$. The obtained $Q_t$ is expected to be close to $R_t$ when the camera takes many images per unit time.

Figure 2

Emergence of the spikes: $Q_t$ for various choices of $\gamma$ (we take $\lambda =1$ hence $t$ is dimensionless). Top left $\gamma ={10}^{-2}$, the information flow is weak and $Q_t$ fluctuates around $1/2$. As $\gamma$ increases, $Q_t$ gets closer to $R_t$ but sharp excursions persist. No qualitative difference can be seen with the naked eye between $\gamma ={10}^4$ and $\gamma \to +\infty$.

Figure 3

Counting of the spikes in an interval where $R_t=0$. The number of spikes (here three) in the domain $D$ is quantified by the proposition.

Figure 4

The smoothed estimate $Q_t^s=\mathbb{E}\left[R_t\right|{\delta }_u,u\in \mathbb{R}]$ of $R_t$ shows no spikes for the same values of the parameters.

Figure 5

Repeated interaction scheme. A collection of probes, typically two-level systems, successively interact unitarily with the system before being projectively measured by a perfect detector. The probability $Q_t=\langle \psi |{\rho }_t|\psi \rangle$ for the system to be in some state $|\psi \rangle$ can be computed using the measurement results and the standard rules of quantum mechanics. Notice the analogy with the classical scheme of Fig. 1. In this setting, the randomness in the evolution does not come from experimental imperfections but is purely of quantum origin.

Figure 6

Emergence of the spikes: $Q_t$ for various choices of $\gamma$. Again, no qualitative difference can be seen with the naked eye between $\gamma ={10}^4$ and $\gamma \to +\infty$. We take $\mathrm{\Omega }=2+\sqrt{\gamma }$, i.e., $\mathrm{\Omega }$ not exactly linear in $\gamma$ to keep a nontrivial evolution (some Rabi oscillations) at $\gamma \simeq 0$. Again, $t$ is dimensionless. Notice that this time there is no underlying jump process $R_t$.

Figure 7

Two trajectories of $Q_{\tau }$ starting from $q$.

Figure 8

On the left, a trajectory starting from $q$, going up to $Q$, and eventually going down below $q_{\delta }$. In the large $\gamma$ limit, this whole trajectory is followed almost instantaneously and looks like a single vertical spike (shown on the right).