The Quantum Zeno effect appears in stages

In the quantum Zeno effect, quantum measurements can block the coherent oscillation of a two-level system by freezing its state to one of the measurement eigenstates. The effect is conventionally controlled by the measurement frequency. Here we study the development of the Zeno regime as a function of the measurement strength for continuous partial measurement. We show that the onset of the Zeno regime is marked by a cascade of transitions in the system dynamics as the measurement strength is increased. They include the appearance of a region of dynamically inaccessible states and of singularities in the steady-state probability distribution of states. These newly predicted dynamical features, which can be readily observed in current experiments, show the coexistence of fundamentally unpredictable quantum jumps with those continuously monitored and reverted in recent experiments.

In the quantum Zeno effect, quantum measurements can block the coherent oscillation of a two level system by freezing its state to one of the measurement eigenstates. The effect is conventionally controlled by the measurement frequency. Here we study the development of the Zeno regime as a function of the measurement strength for a continuous partial measurement. We show that the onset of the Zeno regime is marked by a cascade of transitions in the system dynamics as the measurement strength is increased. They include the appearance of a region of dynamically inaccessible states and of singularities in the steady-state probability distribution of states. These newly predicted dynamical features, which can be readily observed in current experiments, show the coexistence of fundamentally unpredictable quantum jumps with those continuously monitored and reverted in recent experiments.
Introduction.-The evolution of a quantum system under measurement is inherently stochastic due to the intrinsic quantum fluctuations of the detector [1]. If these fluctuations can be accurately monitored, measurements can be used to track the stochastic evolution of the system state, i.e., individual quantum trajectories. From a theoretical tool to investigate open quantum systems [2], quantum trajectories have become an observable reality in experiments in optical [3,4] and solid state [5][6][7] systems. Tracking quantum trajectories has been exploited as a tool to engineering quantum states via continuous feedback control [8][9][10] and entanglement distillation [11,12]. It has been used to observe fundamental properties of quantum systems [13][14][15] and, recently, to predict a many-body entanglement phase transition invisible to the average dynamics [16][17][18][19]. Monitoring quantum trajectories has also made possible to anticipate and correct quantum jumps in superconducting qubits [20].
Quantum jumps are an important characteristic of the quantum Zeno regime [21,22]. In this regime, as a result of repeated measurements, the system state is mostly frozen next to one of the measurement eigenstates, yet rarely performs quantum jumps between them. The crossover between coherent oscillations and the Zeno regime is controlled by the frequency of the measurement and has been extensively explored both theoretically [23][24][25][26][27][28] and experimentally [29][30][31][32][33]. Beyond projective measurements, the onset of the Zeno regime is richer [34,35], and quantum jumps appear as part of continuous stochastic dynamics. For example, in a system monitored via continuous partial measurements, quantum jumps can be anticipated, continuously monitored, and reverted [20], a task which is fundamentally impossible with projective measurements. Moreover, the onset of the Zeno regime with non-projective measurements is more convoluted and has been characterized by different measurement strengths and phenomenology based on the dynamics of the detector signal [36], average [37,38], or postselected [39] state evolution.
Here we study the transition between the regimes of Figure 1. (a) The system. A Hamiltonian induces oscillations between levels |0 and |1 of a qubit, which is continuously measured by a detector weakly coupled to one of the levels.
(b) Dynamical flow (red and blue arrows) of θ(t) from Eq. (3) under "no-click" postselected dynamics. For sufficiently weak measurements, λ < 1, (left) the dynamics is oscillatory; for λ > 1 (right), a stable and an unstable fixed points (θ+ and θ− respectively) emerge. The states in the interval θ ∈ (−π, θ+) are inaccessible to the system under both the "no-click" and the full stochastic dynamics.
coherent oscillations and Zeno-like dynamics in a qubit subject to continuous partial measurement, cf. Fig. 1(a). Our model supports the catchable quantum jumps between measurement eigenstates, |1 and |0 , observed in recent experiments [20]. By investigating the full stochastic dynamics we are able to provide a unified picture of the onset of the Zeno regime, which is established via a cascade of transitions in the system dynamics. Each of the transitions brings in a different aspect of the Zeno-like behavior. Moreover, we find that, in the Zeno regime, the catchable continuous jumps between states |1 and |0 necessarily have a discontinuous counterpart, jumps between states |0 and |1 , which are inherently unpredictable in individual realizations.
Model and post-selected dynamics.-We consider a qubit performing coherent quantum oscillations between states |0 and |1 due to the Hamiltonian H s = Ω s σ x , where Ω s > 0; at the same time the qubit is monitored by a sequence of measurements at intervals dt 1/Ω s -cf. Fig. 1(a). Each measurement is characterized by arXiv:2003.10476v1 [quant-ph] 23 Mar 2020 two possible readouts r = 0 (no-click) and 1 (click). The corresponding measurement back-action |ψ → M (r) |ψ is given by the operators where p ∈ [0, 1] controls the measurement strength. For p = 1, each measurement is projective and this induces the conventional quantum Zeno effect with the system being frozen in one of the measurement eigenstates, |0 or |1 . In the opposite limit, p = 0, essentially no measurement takes place, and the system performs Rabi oscillations under H s . We investigate the intermediate case of p = αdt with dt → 0, and α ≥ 0 controlling the effective measurement strength over a finite time interval. A physical model of this measurement process is realized by coupling the system to a two-level system detector that, in turn, is subject to projective measurements [40].
In each infinitesimal step the measurement and the system evolution add up to give the combined evolution where U = e −iHsdt ≈ 1 − iH s dt is the Hamiltonian unitary evolution over an infinitesimal time interval dt.
When the system is initialized in |0 or |1 , its evolution is constrained to the y-z section of the Bloch sphere and the state has the form |ψ(t) = |ψ(θ(t)) = cos θ(t) where Ω(θ) = 2Ω s [1 + λ sin θ] and λ = α 4Ωs sets the strength of the measurement relative to the Hamiltonian. A measurement yielding readout r = 1 immediately projects the system onto state |1 , while a "no-click" r = 0 readout implies an infinitesimal evolution of the state with angular velocity Ω(θ). The probabilities of the two possible readouts are given by and depend on the qubit state, i.e. on θ(t).
For understanding the full stochastic dynamics, it is instructive to review its continuous "no-click" part, previously analyzed in Ref. [39]. In this case the state evolution is governed by the the differential equatioṅ θ = −Ω(θ). The corresponding flow of the variable θ is shown in Fig. 1(b). Since Ω s > 0 and λ > 0, for any θ ∈ (0, π), we have Ω(θ) > 0, and the system evolves continuously towards θ = 0. Notably, this is the only way for the system state to evolve from |1 to |0 and it corresponds to the quantum jumps that have been continuously monitored in Ref. [20]. The transition from |0 to |1 , instead, takes place via the region θ ∈ (−π, 0) and has richer dynamics controlled by the measurement strength. For sufficiently weak measurements, 0 ≤ λ < 1, one has Ω(θ) > 0 for any θ, and the system monotonously evolves towards θ = −π; however, for λ > 1 there appear two fixed points, Ω(θ ± ) = 0, at where θ + is a stable point, while θ − is an unstable one, as shown in Fig. 1(b). Under the r = 0 postselected dynamics for λ > 1, the system will eventually flow to θ = θ + [39] (where it remains until the occurrence of a click, which collapses the system to |1 ). Stochastic evolution and dynamical transitions.-Beyond the postselected r = 0 quantum trajectory, the stochastic dynamics of the system is described by the probability density P t (θ) of being in the state |ψ(θ) at time t. Using Eqs. (3)(4), one derives the master equation Here, the first term on the r.h.s. describes the "noclick" evolution, the second term describes the reduction of P t (θ) due to clicks that happen with probability p 1 = 4Ω s λ sin 2 θ 2 dt, cf. Eq. (4), while the last term accounts for the clicks bringing the states from any θ to θ = π.
Two experimentally accessible quantities directly related to P t (θ) capture the main physics: the steady-state distribution P ∞ (θ) ≡ lim t→∞ P t (θ), and the average "polarization" of the qubit,s(t) ≡ (s y (t),s z (t)), wherē Both quantities are plotted in Fig. 2. They showcase three qualitative transitions in the dynamics as function of the measurement strength.
We can readily present the key physics of these transitions before entering all the features in due details. For sufficiently small λ, the qubit can be found in any state with finite probability density P ∞ (θ) = 0. In particular, it is possible to evolve from |0 to |1 via trajectories involving a detector click as well as no clicks from the detector. Instead, the evolution from |1 to |0 happens only via no click sequences, as noted above. The first and most drastic transition happens at λ = 1, above which there opens a region of θ ∈ (−π; θ + ] where P ∞ (θ) = 0. In fact, this region is inaccessible for the qubit at any time t, hence, for λ > 1 all quantum trajectories from |0 to |1 must involve a detector click. Generically, the click may occur when the qubit has not reached θ + , which is typically the case. The second transition happens at λ = 2/ √ 3, above which P ∞ (θ) diverges at θ = θ + . This indicates that the system initialized in |0 typically reaches the vicinity of θ + , and spends a long time there, before the click and the corresponding jump to θ = π take place. So far, the population imbalance between |0 and |1 ,s z (t), exhibits oscillations, which are reflected in the oscillations of the average state polarization,s(t). The third and final transition at λ = 2 marks the end of the oscillations, so that, for λ > 2,s z (t) steadily decays in time, completing the final onset of Zeno-like dynamics. These transitions set the overall picture of the onset of Zeno regime in the system, and constitute the main findings of our work.
To analyze these transitions and their implications in some detail, consider first the non-trivial steady state, P ∞ (θ). From the condition dP t→∞ (θ)/dt = 0, Eq. (6) gives for λ < 1, while for λ > 1 the expression reads In Fig. 2, the analytical results in Eqs. (7,8) are compared with Monte Carlo numerical simulations of individual quantum trajectories, showing excellent agreement. The first two transitions in the system dynamics highlighted above, are evident from P ∞ (θ) -cf. Fig. 2. The opening of the forbidden region (−π; θ + ] appears discontinuously rather than opening smoothly, since θ + = θ − = −π/2 at λ = 1. It manifests itself in the non-analytic behavior of P ∞ (θ) as a function of λ at λ = 1. For λ > 1, the second transition shows up at θ ≈ θ + , where which diverges at θ = θ + for λ > 2/ √ 3. The physics underpinning this transition is evinced by Eq. (6) at θ = θ + . The first two terms in the r.h.s., P t (θ + )∂ θ+ Ω(θ + ) and −4Ω s λ sin 2 θ+ 2 P t (θ + ), describe the rate of accumulation of probability for states at θ ≈ θ + due to no-click dynamics and the loss of such probability due to detector clicks respectively. At λ = 2/ √ 3 the two terms balance each other, and for λ > 2/ √ 3 the former dominates. Note that the λ = 2/ √ 3 transition goes unnoticed in both the average (state polarizations(t)) behavior and in the post-selected r = 0 dynamics.
The value λ = 1 is special in two respects. For λ < 1, the system state rotates between |0 and |1 under no-click dynamics. However, the probability of observing a click is different for different θ -thence the oscillations of P (0) (t). With the appearance of the forbidden region, the evolution under no clicks readout is frozen at θ = θ + , hence the oscillations of P (0) (t) disappear. The less obvious effect is that at λ = 1 the decay rate ζ(λ) is maximal, cf. Fig. 3. Therefore, the probability of observing a long sequence of "no-clicks" increases with the measurement strength for λ ≥ 1, while it decreases for λ ≤ 1.
To observe the last transition, λ = 2, one needs to consider the average state polarization,s(t) = (s y (t),s z (t)), and in particular the population imbalance,s z (t). When the system is initialized in state |0 at time t = 0, using Eq. (6), one finds (12) One sees that λ = 2 marks a transition from oscillatory (at λ < 2) to non-oscillatory (at λ > 2) dynamics. The same transition is observed ins y (t). Similarly to P (0) (t) in Eq. (10),s z (t → +∞) ∝ e −2Ωsζ(λ)t with the decay rateζ(λ) = Re λ − √ λ 2 − 4 /2. Therefore, the decay rate exhibits a maximum at λ = 2, cf. Fig. 3. Importantly,s z (t) characterizes not only the population imbalance but also the survival probability (1 +s z (t))/2, i.e., the probability to find the system in state |0 when performing a projective measurement at time t. The decay rate behavior implies that the long-time survival probability increases with increasing λ when λ ≥ 2, and decreases otherwise [41]. On this ground, one marks the transition at λ = 2 as the final onset of the Zeno-like dynamics. Notably, the λ = 2 transition is also reflected in a topological transition in the statistics of the detector clicks [36].
We would like to note that while we used different quantities to showcase each of the three dynamical transitions, all three of them can be inferred by looking at a single quantity: the eigenmode spectrum of Eq. (6). This more mathematical identification of the transitions and its relation to the physics described here is discussed in the Supplementary Material [40].
Observing the transitions experimentally.-The quantum dynamics in Eq. (3) describes the physics realized in recent experiments [20] that, in principle, allow for tuning the measurement strength. Thus, the dynamical transitions can be induced therein or in analogous setups. The simplest transition to observe is that at λ = 2 which is apparent in routine experiments measuring the average polarization or survival probability in |0 . Observing the transitions at λ = 1 and λ = 2/ √ 3 requires sampling the distribution P ∞ (θ) by tracking individual quantum trajectories for sufficiently long times and performing quantum tomography on the final states. This is possible with state-of-the-art experimental techniques [5,20], though laborious. A somewhat less laborious alternative to observe the λ = 1 transition is to measure the probability to observe no clicks for a given time, P (0) (t). While it requires recording every measurement outcome, it does not require knowing the qubit state at time t. For the transtion at λ = 2/ √ 3, one can measure the probability to reach a specific θ by a sequence of no clicks, P (0) (θ), which requires further tracing the qubit state up to time t. This can be done either by inferring the state from the theoretical dependence θ(t) or via a tomography of states postselected on r = 0 readouts.
Conclusions.-Here we have studied the full stochastic dynamics of a system subject to a constant Hamiltonian and a continuous partial measurement. We have shown that the onset of Zeno-like regime is preceded by a number of drastic qualitative changes in the system dynamics. Each such transition introduces a different feature of the fully localized dynamics, starting with the opening up of a finite size region of forbidden states, followed by a singularity in the steady-state probability distribution of states, and ultimately a non-oscillatory dynamics of the qubit survival probability. We have proposed how to observe our findings in current experiments. Strikingly, depending on the definition of "Zeno-like regime", one could call each of the transitions its onset. For example, the probability of observing a long sequence of "no clicks" starts increasing with increasing the measurement strength at λ = 1. The survival probability starts increasing with increasing the measurement strength only after the last transition at λ = 2. These findings make it of interest to study how the onset of Zeno regime depends on the specifics of measurement schemes beyond the model considered here.
We thank Serge Rosenblum and Fabien Lafont for their comments on the manuscript. K. S. and P. K. have contributed equally to this work. K. S. and P. K.

Supplementary material
The main results of the manuscript "The Zeno effect appears in stages" are the identification of three qualitative changes in the stochastic dynamics of the system with the increase of the measurement strength. In the manuscript, the transitions are characterized in terms of some physically observable quantities. Here we provide supplementary material for the results in the manuscript. In Section I, we present a physical system-interaction model that gives rise to the partial measurement dynamics used in the manuscript, Eqs. (1, 2) therein. In Section II, we derive and solve the main equation describing the stochastic dynamics of the system, i.e., Eq. (6) in the manuscript. We identify the transitions in the dynamics with. qualitative changes in the eigenspectrum of the equation. Finally, in Section III, we derive and solve the equations governing the behavior of the probabilities P (0) (t) to observe a no-click sequence of duration t and P (0) (θ) to reach θ by a sequence of no clicks, as well as the behavior of the average state polarization s(t).

I. A PHYSICAL MODEL OF THE MEASUREMENT
The system under consideration in the manuscript is a qubit (|0 , |1 ) evolving under its own Hamiltonian and being measured by a two-state detector (|0 d , |1 d ) at intervals dt. The system's Hamiltonian is We consider a system-detector Hamiltonian given by where the detector is also assumed to be a two-level system. The detector is initially prepared in the state|0 d for each measurement, i.e. at the beginning of each time step. The system's evolution under the combined effect of its Hamiltonian and the coupling to the detector is given by the unitary evolution due to for time dt, after which the detector is read out with readouts r = 0, 1 corresponding to it being in |r d . We consider the scaling limit of continuous measurements defined as dt → 0, J 2 dt → α = const (i.e., J = α/dt). In this limit, the measurement and the system evolution do not intermix in a single step, therefore, The unitary evolution due to the system's Hamiltonian is The measurement back action matrices, combining the effects of the system-detector evolution and the readout in the state |r d , defined as are These are the operators in Eq. (1) of the manuscript describing the effect of the measurements.

II. DERIVATION AND THE EIGENSPECTRUM OF THE MASTER EQUATION
The stochastic dynamics of the system is described by the probability density P t (θ) of being in the state |ψ(θ) at time t for the stochastic variable θ. The probability of the system being in an interval of states [θ 1 , θ 2 ] at time t, P t ([θ 1 ; θ 2 ]) =´θ 2 θ1 dθP t (θ) obeys the evolution where and 0 otherwise, and p r (θ) are the probabilities of obtaining the readout r in a measurement, The first term on the r.h.s. of Eq. (20) describes the change of P t+dt ([θ 1 ; θ 2 ]) due to the the smooth evolution under r = 0 readout, while the second term accounts for jumps to θ = π for the measurement outcome r = 1. The variables θ 1,2 are defined via the self-consistent conditionθ 1,2 − Ω(θ 1,2 )dt = θ 1,2 , where Ω(θ) = 2Ω s [1 + λ sin θ], cf. Eq. (3) in the manuscript. A differential equation for P t (θ) is obtained by solving the self consistent equation to order (dt) 2 , differentiating Eq. (20) over θ 2 , and retaining the terms of order dt. With the explicit expressions for p r and Ω(θ), we get The above integro-differential equation (22) needs to be supplemented by the boundary conditions. They are simple: In other words, P t (θ) = P t (θ + 2π). With this condition, it is easy to check that the normalization by total probabilitý 2π 0 dθP t (θ) = 1 is preserved by this equation. Finally, it is useful for solving to eliminate the integral part of the equation. This is easily done, as it only contributes at θ = π. Therefore, the above Eq. (22) is equivalent to at θ = π, supplemented with boundary condition 2Ω s (P t (π + 0) − P t (π − 0)) = −αˆ2 π 0 dθ sin 2 θ 2 P t (θ).

A. Eigenmodes
We now derive the eigenmode solutions of Eq. (22), i.e., find all the solutions of the form P t (θ) = e 2Ωsγt f γ (θ). The reader may skip the details and look at the result in Sec. II B.
Before diving into the derivation, it is useful to analyze the expectations from the solution. Due to the normalization condition, for every eigenmode with γ = 0, we should have´2 π 0 dθf γ (θ) = 0. Since P t (θ) ≥ 0, there can be no eigenmodes with γ > 0. Thus, only solutions with γ ≤ 0 are acceptable. Finally, since the decay rate of clicks at θ + is p 1 (θ)/dt = α sin 2 θ+ 2 , for λ > 1 we expect an eigenmode with 2Ω s γ ≥ −α sin 2 θ+ 2 = −α tan 2 θ+ 2 / 1 + tan 2 θ+ We also expect a steady state solution with γ = 0 to exist. Equivalently, Note that for α ≥ 4Ω s , at θ = θ ± , the equation becomes singular as the factor multiplying the highest (and only) derivative vanishes. These singular points require special treatment. However, this simply means that f γ (θ ∈ [−π; θ + ]) = 0 as this interval is inaccessible from the time evolution of any state initially outside it, as shown in the analysis of the postselected dynamics in the manuscript. We will come back to the issue of the special points later. Away from the special points, the equation admits an analytic solution, which can be expressed in two alternative forms where λ = α/(4Ω s ). The equivalence of the two expressions follows from so one can write Obviously, the first form in Eq. (28) is more convenient for λ < 1, while the second one is the natural choice for λ > 1. However, for computational purposes one can use either. Note the singularities at θ = θ + and θ = θ − for λ > 1 (for λ < 1, tan θ± 2 − tan θ 2 = 0 for any θ since tan θ± 2 has a non-zero imaginary part while tan θ 2 is a real function). From the equality 1 + λ sin θ = 1 + 2λ tan θ 2 + tan 2 θ we obtain the following behavior in proximity of the singularity points at θ = θ ± : The normalizability conditions ˆθ for Eqs. (31,32) imply that, for λ > 1, one must have These two conditions are incompatible. The apparent contradiction is resolved by choosing the normalization constant C in Eq. (28) independently on intervals (−π; θ − ), (θ − ; θ + ), and (θ + ; π). Then for normalizable solutions, C = 0 either on the first two intervals or on the last two intervals. We do not investigate the case of C = 0 on (−π; θ − ), as these solutions (if exist) describe quick escape from the interval and cannot contribute if the system is initialized outside of it. Choosing C = 0 in the first two intervals, the eigenmodes are normalizable, and one recovers the expected property f γ (θ ∈ [−π; θ + ]) = 0.
Putting these results together, we have the general expression for the eigenmodes, which reads, for λ > 1, Therefore, on top of the three eigenmodes, γ = 0 and γ = 1 2 −λ ± √ λ 2 − 4 , there is also an infinite sequence of eigenvalues given by Since there are no special points for λ < 1, both this infinite set of solutions and the three eigenmodes in Eq. (43) correspond to valid normalizable eigenmodes. For the steady state, the normalization´π −π dθf (θ) = 1 yields For which vanishes due to the boundary condition above, as expected.
B. Summary of eigenvalues and eigenmodes Finally, note that the eigenvalues γ = 1 2 −λ ± √ λ 2 − 4 for λ < 2 correspond to solutions with an oscillatory behavior superimposed with a decay in time, while for λ > 2 they give steadily decaying in time solutions. This transition is identified in the manuscript in terms of the average qubit polarization, and has been identified in the detector's signal [36] as well. Here it appears as a property of the eigenvalues spectrum. Curiously, these eigenvalues are non-physical for λ > 2/ √ 3, which is before the transition is reached. At the same time, Eq. (22) does know about these eigenvalues for any λ, as we show in Sec. III B.

III. CALCULATION OF POSTSELECTED AND SURVIVAL PROBABILITIES
In the manuscript we have presented some of the observable signatures of the transition in terms of the probabilities P (0) (t) to observe a no-click sequence of duration t and P (0) (θ) to reach θ by a sequence of no clicks, as well as the behavior of the average state polarizations(t). Here we derive the results stated in the manuscript.
The solution with P (0) (θ = 0) = 1 is When λ > 1, starting at θ = 0 it is possible to reach only the states with θ ∈ (θ + ; 0], which is reflected in the vanishing of and in the properties of its derivatives discussed in the manuscript.

B. Survival probability and the average state polarization
The final quantity used in the manuscript to describe the dynamics of the system is the average state polarization after time t,s(t) ≡ (s y (t),s z (t)), which is related to the survival probability in the initial state |0 , P(t) = (1+s z (t))/2, i.e. the probability to measure the system in |0 upon a projective measurement at time t. These quantities require