Measurement-induced crossover in quantum first-detection times

Abstract

The quantum first-detection problem concerns the statistics of the time at which a system, subject to repeated measurements, is observed in a prescribed target state for the first time. Unlike its classical counterpart, the measurement back action intrinsic to quantum mechanics may profoundly alter the system dynamics. Here we show that it induces a distinct change in the statistics of the first-detection time. For a quantum particle in one spatial dimension subject to stroboscopic measurements, we observe an algebraic decay of the probability of the first-detection time if the particle is free, an exponential decay in the presence of a confining potential, and a time-dependent crossover between these behaviors if the particle is partially confined. This crossover reflects the purely quantum nature of the detection process, which fundamentally distinguishes it from the first-passage problem in classical systems.

Figures (5)

• Figure 1: A particle (red circle) in a potential well $U(x)$ is subject to stroboscopic measurements that test whether it is in the positive half-line $x>0$ (colored background). The measurement back action injects energy into the system, thereby enhancing the probability that the particle escapes from the well (wiggly arrow). This, in turn, leads to a crossover in the statistics of the first-detection time.
• Figure 2: Probability $F_n$ of first detection in $x>0$ at the $n$-th measurement for a free particle with $x_0 = -1$, $\sigma=0.8$, and for various values of the measurement interval $\tau$ (ranging from 1 to 4) femto. The data clearly show an algebraic decay as a function of $n$: The dashed black line correspond to the numerical fit $F_n \propto n^{\mu}$, yielding $\mu \simeq -3$. The inset shows the effective exponent of the algebraic dependence of $F_n$ on $n$ for $n \gtrsim 6$, showing convergence towards $-3$. This behavior emerges independently of the value of $\tau$ and of the choice of $x_0$ and $\sigma$.
• Figure 3: First-detection probability $F_n$ for the harmonic oscillator, with initial condition given in Eq. \ref{['eq:iintial_free']} and parameters $x_0 = -1$, $\sigma = 0.8$, as a function of $n$ on a log-log scale and for various values of the harmonic potential strength $\omega^2$. A behavior $F_n \propto n^{-3}$ consistent with that of the free particle is observed at small values of $\omega \lesssim (\sigma^2\tau)^{-1/2}$, as highlighted by the dashed line; for values of $n\omega\tau = \kappa\pi$ with integer $\kappa =1, 2, \dots$, resonance due to the harmonic confinement emerge.
• Figure 4: First-detection probability $F_n$ for the harmonic oscillator as a function of $n$. In the left panel $F_n$ is shown for $\omega = 2.45$ and various (possibly resonant) duration of the interval $\tau$ between successive measurements. For resonant values of $\tau$, $F_n$ may either vanish identically when measured at multiples of the period (as in the case $\tau=2\pi/\omega$ here) or it vanishes after the first measurement when measured at half the period (case $\tau=\pi/\omega$). Otherwise, $F_n$ decays exponentially to zero upon increasing $n$ (case $\tau=\pi/(2 \omega)$), as highlighted in the log-linear plot on the right, with $F_n \propto e^{-\alpha n \tau}$ and $\alpha\simeq 2.0$
• Figure 5: Dependence on $n$ of the first-detection probability $F_n$ (upper panel), of the average energy $\langle H \rangle$ (lower panel, blue) and of the probability $P_b$ (lower panel, orange) of being in a bound state, for a particle moving in a potential well of finite depth $U_0 < 0$. The upper panel shows that an initial exponential decay (red dashed line) of $F_n$ at intermediate values of $n$ is followed by an algebraic decay $F_n\propto n^{\mu}$ with $\mu \propto 2.8$. The crossover between these two behaviors occurs when the average energy (lower panel) becomes positive and the delocalized states become more relevant in the particle’s wave function, as confirmed by the fact that probability $P_b$ of being in a bound state approaches zero. Simulations correspond to $a=5.0$, $U_0 = -2$, $x_0 = -2$, and $\sigma = 1$.