🔬 condensed matter

Splitting Probability in Monitored Quantum Dynamics

This paper establishes a general renewal framework for calculating splitting probabilities in continuous-time quantum walks monitored by projective measurements on two targets, revealing that reflection symmetry induces interference effects and that sampling times relative to the system's energy bandwidth determine whether detection probabilities remain uniform or exhibit position-dependent oscillations and discontinuous jumps due to dark states.

Original authors: Prashant Singh, Sabine Tornow, David A Kessler, Eli Barkai

Published 2026-10-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Prashant Singh, Sabine Tornow, David A Kessler, Eli Barkai

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a particle moving through a maze, not as a solid object hopping from one spot to the next, but as a wave of possibility that can be in many places at once. In the world of classical physics, if you place two exits at opposite ends of such a maze, the chance of the particle leaving through one or the other depends entirely on where it started. If it begins closer to the left exit, it is far more likely to leave there; if it starts in the middle, the odds are a simple, predictable mix based on distance. This is a fundamental rule of chance known as the gambler's ruin, familiar to anyone who has watched a random walk or a drunkard stumbling home. But when physicists apply the strange rules of quantum mechanics to this scenario, the outcome changes in a way that defies intuition. The particle's wave-like nature allows it to interfere with itself, creating a competition between the two exits that behaves nothing like the classical version.

A team of researchers has now mapped out exactly how this quantum competition plays out when the particle is watched repeatedly. They studied a system where a quantum walker moves along a line of connected points, with detectors placed at the very beginning and the very end. The twist is that these detectors do not watch continuously; instead, they check the particle's position at fixed, regular intervals, like a camera taking snapshots every fraction of a second. Between these snapshots, the particle evolves freely as a wave. The researchers wanted to know: as the time between these snapshots changes, does the particle still favor the closer exit, or does the quantum nature of the system rewrite the rules of probability?

The study reveals a dramatic shift in behavior that depends entirely on how frequently the particle is observed. When the snapshots are taken very quickly, the system enters a regime where the particle is effectively frozen in place by the act of measurement itself. In this rapid-fire limit, the probability of the particle being detected at either end becomes exactly equal, regardless of where it started. If the particle begins in the middle of the line, it has a fifty-fifty chance of being found at the left or the right. This result is startling because it completely erases the "proximity effect" that governs classical particles; being closer to an exit no longer gives an advantage. The researchers found that this flat, equal probability holds true for any starting position that is not directly on top of a detector, creating a uniform landscape of chance across the entire system.

However, as the time between snapshots increases and crosses a specific critical threshold, the behavior changes abruptly. The smooth, flat probability curve collapses, and the system enters a fluctuating regime. In this new state, the chance of detection at one exit versus the other begins to oscillate wildly depending on the exact starting position of the particle. A tiny shift in where the particle begins can flip the odds from heavily favoring the left exit to heavily favoring the right. These oscillations are not random noise; they are a precise, rhythmic pattern driven by the wave nature of the particle. The researchers showed that these patterns emerge because the time between measurements allows different parts of the particle's wave to align or clash in specific ways, a phenomenon known as phase matching. When the snapshots are too slow, these waves can synchronize to create constructive or destructive interference, causing the probability to rise and fall in a predictable rhythm as the starting point moves along the line.

The team also discovered that this system is fragile in a very specific way. At certain precise intervals between snapshots, the system can get "stuck" in a state where the particle is never detected at all, no matter how long the experiment runs. This happens because the particle's wave function forms a dark state, a configuration that is invisible to the detectors at the moments they look. If the particle starts in a position that overlaps with this invisible state, it effectively disappears from the measurement process, causing the total probability of detection to drop below one hundred percent. This is a purely quantum effect with no classical equivalent, where the act of looking for the particle can sometimes guarantee that it remains hidden.

Furthermore, the researchers explored what happens when the particle starts not as a single point, but as a coherent blend of two locations at once. They found that the relationship between these two starting points—their distance apart and the phase relationship between them—can either preserve the flat, equal-probability behavior or destroy it. If the two starting points are separated by an even number of steps, the system retains its simple, flat behavior for a wide range of measurement speeds. But if they are separated by an odd number of steps, the system becomes highly sensitive to the phase between them, and the probability of detection can swing dramatically based on this subtle quantum relationship.

Finally, the study addressed a practical reality: in real-world experiments, scientists cannot take an infinite number of measurements. They must stop after a finite number of attempts. The researchers showed that this limitation smooths out the sharp, sudden jumps in probability that occur at the special "dark state" intervals. Instead of a sudden drop to zero, the probability dips gradually, creating a broadened valley rather than a sharp cliff. This finding is crucial for connecting the theoretical predictions of quantum mechanics with actual laboratory results, where finite observation times are the norm.

In essence, this work provides a complete map of how a quantum particle chooses between two exits when watched intermittently. It shows that the frequency of observation acts as a dial that can switch the system between a state of perfect equality and a state of complex, oscillating competition. The findings highlight that in the quantum world, the timing of a measurement is just as important as the measurement itself, capable of turning a simple probability into a rich, structured landscape of interference and resonance. This understanding not only deepens our grasp of quantum dynamics but also offers a blueprint for controlling quantum systems in future technologies, where managing the flow of information and the timing of observations will be key to success.

Technical Summary: Splitting Probability in Monitored Quantum Dynamics

Problem Statement
This paper investigates the quantum analogue of the classical splitting probability problem within the context of monitored quantum walks. In classical stochastic processes, the splitting probability quantifies the likelihood that a trajectory reaches one designated target before any other available targets (e.g., the gambler's ruin problem). The authors extend this concept to a continuous-time, discrete-space quantum system evolving under a time-independent Hermitian Hamiltonian $H$ . The system is subjected to stroboscopic projective measurements at two distinct, orthogonal target states, $|D_1\rangle$ and $|D_2\rangle$ , at regular time intervals $\tau$ . The goal is to determine the probability that the quantum walker is first detected at $|D_1\rangle$ versus $|D_2\rangle$ , and to understand how this competition differs qualitatively from its classical counterpart.

Methodology
The authors develop a general theoretical framework based on quantum renewal theory combined with generating function techniques.

Renewal Framework: They derive a quantum renewal equation for the first-detection amplitude $\phi^{(D_\alpha)}_n(\psi_0)$ at the $n$ -th measurement attempt. This amplitude is expressed in terms of the measurement-free unitary evolution operator $U(\tau) = e^{-iH\tau}$ and the projection operator $\hat{D} = |D_1\rangle\langle D_1| + |D_2\rangle\langle D_2|$ . The renewal equation (Eq. 6) relates the $n$ -th detection amplitude to the direct propagation amplitude minus correction terms representing histories where detection occurred at an earlier step.
Generating Functions: By applying a Z-transform to the renewal equation, the authors convert the recursive relation into a linear algebraic problem involving a matrix of generating functions. This allows for the exact calculation of the total splitting probabilities $P_{D_1}$ and $P_{D_2}$ via contour integration over the unit circle in the complex plane.
Symmetry Mapping: For Hamiltonians possessing reflection symmetry (parity) and targets localized at the boundaries of a one-dimensional chain ( $|D_1\rangle=|1\rangle, |D_2\rangle=|N\rangle$ ), the authors demonstrate a specific mapping. The dual-target problem is decomposed into two independent single-target first-detection problems involving auxiliary target states $|d_\pm\rangle = (|1\rangle \pm |N\rangle)/\sqrt{2}$ . This mapping reveals that the splitting probability is an interference effect between these two single-target processes.
Specific Model: The framework is applied to a one-dimensional tight-binding lattice with nearest-neighbor hopping. The authors analyze the system in the thermodynamic limit ( $N \to \infty$ ) and for finite $N$ , examining the dependence on the sampling time $\tau$ and the initial position $x_0$ .

Key Contributions and Results

Exact Expressions: The paper provides exact analytical expressions for splitting probabilities for arbitrary Hamiltonians with reflection symmetry, arbitrary sampling times, and arbitrary initial states.
Flat-to-Fluctuating Transition: A central finding is a non-analytic, phase-transition-like behavior in the splitting probabilities as a function of the sampling time $\tau$ $τ$ for large system sizes.
- Regime I ( $0 < \tau < \tau_c$ ): Below a critical sampling time $\tau_c = 2\pi/\Delta E$ (where $\Delta E$ is the energy bandwidth), the splitting probabilities are essentially equal to $1/2$ for all initial positions $x_0$ (excluding the boundaries). This "flat" profile is independent of the initial position and contrasts sharply with the linear dependence seen in classical random walks.
- Regime II ( $\tau \ge \tau_c$ ): Above the critical time, the probabilities deviate from $1/2$ and exhibit pronounced oscillations as a function of the initial position $x_0$ .
Mechanism of Oscillations: The authors identify the physical mechanism driving these oscillations as phase-matching events between different momentum modes of the quantum walk. They derive the dominant wavenumber $Q_{max}(\tau) = 2\sin^{-1}(r\tau_c/\tau)$ governing the spatial structure of the fluctuations. These oscillations only emerge when $\tau \ge \tau_c$ , as this is the condition required for real, phase-matched momentum pairs to exist.
Resonances and Dark States: At specific "exceptional" sampling times where quasienergies of states with the same parity become degenerate ( $e^{-iE_\ell \tau} = e^{-iE_{\ell'} \tau}$ ), the system can form dark states orthogonal to both detectors. If the initial state has an overlap with such a dark state, the total detection probability drops below unity ( $P_L + P_R < 1$ ), resulting in pointwise discontinuous jumps in the splitting probabilities.
Zeno Limit: In the limit of small but finite $\tau$ (the Zeno limit), the splitting probabilities depend only on the projection of the initial state onto the target sites. For an initial state localized in the interior, $P_L \approx P_R \approx 1/2$ . However, the limit $\tau \to 0$ is singular; for strictly $\tau=0$ , the dynamics freeze, and detection never occurs unless the walker starts at a boundary.
Coherent Initial States: The study extends to coherent superpositions of initial states. The splitting probability retains the flat-to-fluctuating transition but is modulated by the relative phase and spatial separation of the superposition components. The dependence on the relative phase $\Phi$ is shown to be $\sin \Phi$ or $\cos \Phi$ depending on whether the separation between components is odd or even.
Finite Measurements: The authors analyze the effect of a finite number of measurements ( $T_{max}$ ), a realistic constraint in experiments. They show that finite $T_{max}$ broadens the pointwise discontinuous jumps (resonances) into dips of finite width and modifies the normalization condition, though the ideal infinite-measurement results are recovered as $T_{max} \to \infty$ .

Significance and Claims
The paper claims to establish a systematic theoretical treatment of splitting probabilities in monitored quantum dynamics. Its primary significance lies in demonstrating that quantum interference and measurement-induced collapse lead to phenomena absent in classical stochastic processes:

Qualitative Difference: Unlike classical splitting probabilities which vary linearly with position, quantum splitting probabilities can be uniform (flat) over a wide range of parameters and exhibit complex spatial oscillations.
Interference as a Mechanism: The work explicitly frames the dual-target splitting problem as an interference effect between single-target processes, providing a physical interpretation for the observed transitions and oscillations.
Dark State Physics: The identification of resonances as a consequence of ergodicity breaking due to dark states offers a deeper understanding of how repeated measurements can suppress detection in quantum systems.
Experimental Relevance: The authors note that these protocols are implementable in existing experimental platforms such as quantum computers and optical waveguides, where the predicted resonances and transitions could be observed.

The paper concludes that the renewal framework developed here is a robust tool for analyzing first-passage phenomena in quantum systems and suggests future directions involving weak measurements, tracking protocols, and many-body systems.

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