(2020) Quantum Zeno Effect Appears in Stages

PHYSICAL REVIEW RESEARCH 2, 033512 (2020)

Quantum Zeno effect appears in stages

Kyrylo Snizhko ,1 Parveen Kumar ,1 and Alessandro Romito 2 1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel 2Department of Physics, Lancaster University, Lancaster LA1 4YB, England, United Kingdom

(Received 25 March 2020; accepted 13 August 2020; published 29 September 2020)

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. Some of these transitions are only apparent in the collective behavior of individual quantum trajectories and are invisible to the average dynamics. They include the appearance of a region of dynamically inaccessible states and of singularities in the steady-state probability distribution of states. These 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.

DOI: 10.1103/PhysRevResearch.2.033512

I. INTRODUCTION extensively explored both theoretically [29–34] and experimentally [35–39]. Beyond projective measurements, the onset The evolution of a quantum system under measurement is of the Zeno regime is richer [40,41], and quantum jumps ap- inherently stochastic due to the intrinsic quantum fluctuations pear as part of continuous stochastic dynamics. For example, of the detector [1]. If these fluctuations can be accurately in a system monitored via continuous partial measurements, monitored, measurements can be used to track the stochastic quantum jumps can be anticipated, continuously monitored, evolution of the system state, i.e., individual quantum trajec- and reverted [26], a task which is fundamentally impossible tories. From a theoretical tool to investigate open quantum with projective measurements. Moreover, the onset of the systems [2], quantum trajectories have become an observable Zeno regime with nonprojective measurements is more con- reality in experiments in optical [3,4] and solid-state [5–7] voluted and has been characterized by different measurement systems. Tracking quantum trajectories has been exploited strengths and phenomenology based on the dynamics of the as a tool to engineering quantum states via continuous feed- detector signal [42], average [43–47], or postselected [48,49] back control [8–10] and entanglement distillation [11,12]. It state evolution. has been used to observe fundamental properties of quantum Here we study the transition between the regimes of co- measurements [13–17] and, recently, to predict topological herent oscillations and Zeno-like dynamics in a qubit subject transitions in measurement-induced geometric phases [18–21] to continuous partial measurements [see Fig. 1(a)], a model and many-body entanglement phase transitions in random directly describing some recent experiments [26]. By investi- unitary circuits, invisible to the average dynamics [22–25]. gating the full stochastic dynamics of quantum trajectories, Monitoring quantum trajectories has also made possible an- we show that the quantum Zeno regime is established via ticipating and correcting quantum jumps in superconducting a cascade of transitions in the system dynamics, some be- qubits [26]. ing invisible to the average dynamics. Furthermore, we find The above-mentioned transitions stem from the basic that, in the Zeno regime, catchable continuous jumps between physics of the quantum Zeno effect [27,28]. In this regime, states |1 and |0 necessarily have a discontinuous counterpart, as a result of repeated measurements, the system state is jumps between |0 and |1, which are inherently unpredictable mostly frozen next to one of the measurement eigenstates, yet in individual realizations. Our results provide a unified picture rarely performs quantum jumps between them. The crossover of the onset of the Zeno regime arising from continuous partial between coherent oscillations and the Zeno regime is con- measurements and demonstrate that investigating individual trolled by the frequency of the measurement and has been quantum trajectories can uncover drastically new physics even in simple and well-studied systems. Our findings may be relevant for quantum error correction protocols employing continuous partial measurements [50,51].

Published by the American Physical Society under the terms of the II. MODEL AND POSTSELECTED DYNAMICS Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) We consider a qubit performing coherent quantum oscil- and the published article’s title, journal citation, and DOI. lations between states |0 and |1 due to the Hamiltonian

2643-1564/2020/2(3)/033512(12) 033512-1 Published by the American Physical Society SNIZHKO, KUMAR, AND ROMITO PHYSICAL REVIEW RESEARCH 2, 033512 (2020)

The probabilities of the two possible readouts are given by θ 2 (t ) p = ≡ p = αdt sin , p = ≡ p = 1 − p (4) r 1 1 2 r 0 0 1 and depend on the qubit state, i.e., on θ(t ). For understanding the full stochastic dynamics, it is in- structive to review its continuous no-click part, previously analyzed in Ref. [48]. In this case the state evolution is governed by the the differential equation θ˙ =−(θ ). The corresponding flow of the variable θ is shown in Fig. 1(b). FIG. 1. (a) The system. A Hamiltonian induces oscillations be- Since s > 0 and λ>0, for any θ ∈ (0,π), we have (θ ) > tween levels |0 and |1 of a qubit, which is continuously measured 0, and the system evolves continuously towards θ = 0. No- by a detector weakly coupled to one of the levels. (b) Dynamical tably, this is the only way for the system state to evolve from θ flow (red and blue arrows) of (t ) from Eq. (3) under “no-click” |1 to |0 and it corresponds to the quantum jumps that have λ< postselected dynamics. For sufficiently weak measurements, 1 been continuously monitored in Ref. [26]. The transition from (left), the dynamics is oscillatory; for λ>1 (right), stable and unsta- | | θ ∈ −π, θ θ 0 to 1 , instead, takes place via the region ( 0) and ble fixed points ( + and −, respectively) emerge. The states in the has richer dynamics controlled by the measurement strength. interval θ ∈ (−π,θ+ ) are inaccessible to the system under both the For sufficiently weak measurements, 0 λ<1, one has no-click and the full stochastic dynamics. (θ ) > 0 for any θ, and the system monotonously evolves towards θ =−π; however, for λ>1 there appear two fixed θ = Hs = sσx, where s > 0; at the same time the qubit is mon- points, ( ±) 0, at itored by a sequence of measurements at intervals dt 1/ s θ = −λ ± λ2 − , [see Fig. 1(a)]. Each measurement is characterized by two ± 2arctan( 1) (5) = possible readouts r 0 (“no-click”) and 1 (“click”). The cor- where θ+ is a stable point, while θ− is an unstable one, as |ψ→ (r)|ψ responding measurement back-action M is given shown in Fig. 1(b). Under the r = 0 postselected dynamics for by the operators λ>1, the system will eventually flow to θ = θ+ [48] (where √ it remains until the occurrence of a click, which collapses the (0) = | | + − | |, (1) = | |, M 0 0 1 p 1 1 M p 1 1 (1) system to |1). where p ∈ [0, 1] controls the measurement strength. For p = 1, each measurement is projective and this induces the con- III. STOCHASTIC EVOLUTION AND DYNAMICAL ventional quantum Zeno effect with the system being frozen in TRANSITIONS | | one of the measurement eigenstates, 0 or 1 . In the opposite Beyond the postselected r = 0 quantum trajectory, the = limit, p 0, essentially no measurement takes place, and the stochastic dynamics of the system is described by the prob- system performs Rabi oscillations under Hs. We investigate ability density P (θ ) of being in the state |ψ(θ ) at time t. = α → α t the intermediate case of p dt with dt 0, and 0 Using Eqs. (3) and (4), one derives the master equation for controlling the effective measurement strength over a finite Pt (θ ): time interval. A physical model of this measurement process θ θ is realized by coupling the system to a two-level system de- dPt ( ) 2 = ∂θ [(θ )P (θ )] − 4 λ sin P (θ ) tector that, in turn, is subject to projective measurements (see dt t s 2 t Appendix A for details). 2π θ˜ 2 In each infinitesimal step the measurement and the system + 4sλδ(θ − π ) dθ˜ sin Pt (θ˜ ) . (6) evolution add up to give the combined evolution 0 2 Here, the first term on the right-hand side describes the |ψ + = (r) |ψ , (t dt) M U (t ) (2) no-click evolution, the second term describes the reduction −iH dt of Pt (θ ) due to clicks that happen with probability p1 = where U = e s ≈ 1 − iH dt is the Hamiltonian unitary θ s 4 λ sin2 dt [see Eq. (4)], while the last term accounts for evolution over an infinitesimal time interval dt. When the s 2 the clicks bringing the states from any θ to θ = π. system is initialized in |0 or |1, its evolution is constrained to Two experimentally accessible quantities directly related the y-z section of the Bloch sphere and the state has the form θ |ψ =|ψ θ = θ(t ) | + θ(t ) | to Pt ( ) capture the main physics: the steady-state distribu- (t ) [ (t )] cos 2 0 i sin 2 1 . Equation (2) tion P∞(θ ) ≡ limt→∞ Pt (θ ) and the average “polarization” translates onto ≡ , ≡σ = of the qubit, s¯(t ) (¯sy(t ) s¯z(t )), wheres ¯i(t ) i(t ) π ψ θ |σ |ψ θ θ θ = , θ(t ) − [θ(t )] dt if r = 0 −π ( ) i ( ) Pt ( ) d , i y z. Both quantities are θ(t + dt) = , (3) π if r = 1 plotted in Fig. 2. They showcase three qualitative transitions in the dynamics as function of the measurement strength. α where (θ ) = 2s[1 + λ sin θ] and λ = sets the strength We can readily present the key physics of these transitions 4s of the measurement relative to the Hamiltonian. A measure- before entering all the features in due details. For sufficiently ment yielding readout r = 1 immediately projects the system small λ, the qubit can be found in any state with finite proba- onto state |1, while a no-click r = 0 readout implies an in- bility density P∞(θ ) = 0. In particular, it is possible to evolve finitesimal evolution of the state with angular velocity (θ ). from |0 to |1 via trajectories involving a detector click as

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FIG. 2. The stochastic√ dynamics of the qubit state in the y-z section of the Bloch sphere exhibits transitions at relative measurement strengths λ = 1, 2/ 3, and 2. The long-time probability distribution Pt=∞(θ ) at different values of λ is shown as the height above the unit circle for the analytic result (red solid line) and for the numerical simulations (green dots). The trajectories of the expectation valuess ¯y,z (t ), for the system initialized in |0 at t = 0, are shown with the dashed magenta lines. Each numerical simulation involved 10 000 stochastic trajectory realizations, tracing the evolution until t = 10, with measurements yielding random outcomes happening at intervals dt = 0.01, using s = 1; θ is binned in intervals of size 5◦. well as no clicks from the detector. Instead, the evolution from evident from P∞(θ ) (see Fig. 2). The opening of the forbidden |1 to |0 happens only via no-click sequences, as noted above. region (−π; θ+] appears discontinuously rather than opening The first and most drastic transition happens at λ = 1, above smoothly, since θ+ = θ− =−π/2atλ = 1. It manifests itself which there opens a region of θ ∈ (−π; θ+] where P∞(θ ) = 0. in the nonanalytic behavior of P∞(θ ) as a function of λ at In fact, this region is inaccessible for the qubit at any time λ = 1. For λ>1, the second transition shows up at θ ≈ θ+, t; hence, for λ>1 all quantum trajectories from |0 to |1 where λ must involve a detector click. Generically, the click may occur √ −2 θ θ 2 θ + λ −1 when the qubit has not reached +, which is√ typically the case. P∞(θ ) ∝ tan − tan , (9) The second transition happens at λ = 2/ 3, above which 2 2 √ P∞(θ ) diverges at θ = θ+. This indicates that the system ini- which diverges at θ = θ+ for λ>2/ 3. The physics un- tialized in |0 typically reaches the vicinity of θ+, and spends derpinning this transition is evinced by Eq. (6)atθ = θ+. a long time there, before the click and the corresponding The first two terms in the right-hand side, P (θ+)∂θ (θ+) jump to θ = π take place. So far, the population imbalance t + 2 θ+ − λ θ+ between |0 and |1,¯s (t ), exhibits oscillations, which are and 4 s sin 2 Pt ( ), describe the rate of accumulation z θ ≈ θ reflected in the oscillations of the average state polarization, of probability for states at + due to no-click dynamics s¯(t ). The third and final transition at λ = 2 marks the end and the loss of such√ probability due to detector clicks, respec- λ = / of the oscillations, so that, for λ>2,s ¯ (t ) steadily decays tively. At √ 2 3 the two terms balance each other, and√ z λ> / λ = / in time, completing the final onset of Zeno-like dynamics. for 2 3 the former dominates. Note that the 2 3 These transitions set the overall picture of the onset of the transition goes unnoticed in both the average [state polariza- = Zeno regime in the system, and constitute the main findings tion s¯(t )] behavior and in the postselected r 0 dynamics. θ of our paper. While the steady-state properties of P∞( ) showcase these To analyze these transitions and their implications in some transitions, their role in the onset of the Zeno regime is fully detail, consider first the nontrivial steady state, P∞(θ ). From unveiled only in the full stochastic dynamics. To appreciate that, consider the probability P(0)(t ) of obtaining a sequence the condition dPt→∞(θ )/dt = 0, Eq. (6)gives of no clicks (r = 0 readouts) of duration t [see Fig. 3 (inset)]. λ λ+tan θ π λ exp √ 2 arctan √ 2 − We obtain that at long times P(0)(t ) decays exponentially as 1−λ2 1−λ2 2 P∞(θ ) = , (7) − ζ λ πλ (0) ∝ 2 s ( )t { λ + λ ε + ϕ λ }, (1 + λ sin θ )2 1 − exp − √2 P (t ) e A( ) B( ) cos[ t ( )] (10) 1−λ2 λ = λ> λ< λ> where B( ) 0for 1,√ the frequency of the oscillatory for 1, while for 1 the expression reads 2 ⎧ term for λ<1isε = 2s 1 − λ , and the decay rate is √ λ √ θ √ ⎨ tan +λ− λ2−1 λ2− ζ λ = λ − λ2 − λ 2 √ 1 ( ) Re[ 1] (see Appendix B for the deriva- θ ,θ∈ (θ+; π], P∞(θ ) = (1+λ sin θ )2 tan +λ+ λ2−1 λ = λ< ⎩ 2 tion). The value 1 is special in two respects. For 1, 0,θ∈ (−π; θ+]. the system state rotates between |0 and |1 under no-click dynamics. However, the probability of observing a click is dif- (8) ferent for different θ—hence the oscillations of P(0)(t ). With In Fig. 2, the analytical results in Eqs. (7) and (8) are com- the appearance of the forbidden region, the evolution under pared with Monte Carlo numerical simulations of individual no-click readout is frozen at θ = θ+, hence the oscillations quantum trajectories, showing excellent agreement. The first of P(0)(t ) disappear. The less obvious effect is that at λ = 1 two transitions in the system dynamics highlighted above are the decay rate ζ (λ) is maximal (see Fig. 3). Therefore, the

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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 [42]. 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 Appendix C.

FIG. 3. The decay rates ζ (λ) (solid) and ζ¯ (λ) (dashed) charac- IV. OBSERVING THE TRANSITIONS EXPERIMENTALLY (0) terizing, respectively, the probability to observe no clicks P (t )and The above physics can be readily observed in the setup + / the survival probability [1 s¯z (t )] 2. Note the respective decay rate of recent experiments [26] by adjusting the measurement maxima at λ = 1 and 2. Inset: The time dependence of P(0)(t )for strength/Rabi frequency. The simplest transition to observe λ = 0.5(solid)andλ = 1.25 (dashed). At long times P(0)(t ) decays is that at λ = 2, which is apparent in routine experiments exponentially with (without) superimposed oscillations for λ<1 (λ>1). measuring the average polarization or survival probability√ in |0. Observing the transitions at λ = 1 and 2/ 3 requires sampling the distribution P∞(θ ) by tracking individual quan- probability of observing a long sequence of no clicks increases tum trajectories for sufficiently long times and performing with the measurement strength for λ 1, while it decreases quantum tomography on the final states. This is possible with for λ 1. state-of-the-art experimental techniques [5,26], though labo- Consider now the probability P(0)(θ ) to reach a particu- rious. A somewhat less laborious alternative to observe the lar value of θ under no-click dynamics. P(0)(θ ) is obtained λ = 1 transition is to measure the probability to observe no from P(0)(t ) and the no-click evolution θ˙(t ) =−[θ(t )] via clicks for a given time, P(0)(t ). While this requires recording (0) θ ≡ (0) θ = θ P ( 0 ) P (t0 ) with t0 satisfying (t0 ) 0. In proximity every measurement outcome, it does not require√ knowing the of θ = θ+, one has (see Appendix B) qubit state at time t. For the transition at λ = 2/ 3, one can λ θ √ −1 measure the probability to reach a specific by a sequence θ θ 2 (0) + λ −1 (0) θ P (θ ≈ θ+) ∝ tan − tan . (11) of no clicks, P ( ), which requires further tracing the qubit 2 2 stateuptotimet. This can be done either by inferring the θ (0) θ θ = θ λ> state from the theoretical dependence (t ) or via a tomogra- Note that P ( ) vanishes at + for any finite√ 1. = (0) phy of states postselected on r 0 readouts (which has been However, dP (θ )/dθ vanishes at θ+ for λ<2/ 3 and di- √ √ implemented in Ref. [26]forλ 1). verges for λ>2/ 3. Therefore, for λ<2/ 3, the system typically jumps√ to θ = π via a detector click before it reaches θ+.Forλ>2/ 3, the system is likely to reach a close vicinity of θ+ before a click happens. V. CONCLUSIONS To observe the last transition, λ = 2, one needs to consider the average state polarization, s¯(t ) = (¯sy(t ), s¯z(t )), and in par- Here we have studied the full stochastic dynamics of a sys- ticular the population imbalance,s ¯z(t ). When the system is tem subject to a constant Hamiltonian and a continuous partial initialized in state |0 at time t = 0, using Eq. (6), one finds measurement. We have shown that the onset of the Zeno-like (see Appendix B) regime is preceded by a number of drastic qualitative changes √ in the system dynamics. Each such transition introduces a dif- sinh t λ2 − 4 −sλt 2 s ferent feature of the fully localized dynamics, starting with the s¯z(t ) = e cosh st λ − 4 + λ √ . (12) λ2 − 4 opening up of a finite-size region of forbidden states, followed by a singularity in the steady-state probability distribution of One sees that λ = 2 marks a transition from oscillatory states, and ultimately a nonoscillatory dynamics of the qubit (at λ<2) to nonoscillatory (at λ>2) dynamics. The survival probability. We have proposed how to observe our (0) same transition is observed ins ¯y(t ). Similarly to P (t ) findings in current experiments. Strikingly, depending on the − ζ¯ λ →+∞ ∝ 2 s ( )t in Eq. (10),s ¯z(√t ) e with the decay rate definition of the “Zeno-like regime,” one could call each of the ζ¯ (λ) = Re[λ − λ2 − 4]/2. Therefore, the decay rate ex- transitions its onset. For example, the probability of observing hibits a maximum at λ = 2 (see Fig. 3). Importantly,s ¯z(t ) a long sequence of no clicks starts increasing with increasing characterizes not only the population imbalance but also the the measurement strength at λ = 1. The survival probability survival probability [1 + s¯z(t )]/2, i.e., the probability to find starts increasing with increasing the measurement strength the system in state |0 when performing a projective mea- only after the last transition at λ = 2. Some of our findings surement at time t. The decay rate behavior implies that may depend on the specific measurement model, making it the long-time survival probability increases with increasing λ of interest to study the onset of the Zeno regime beyond when λ 2, and decreases otherwise [52]. On this ground, continuous partial measurement.

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ACKNOWLEDGMENTS are We thank Serge Rosenblum and Fabien Lafont for their 10 10 M(0) = = + O(dt2 ), (A6) comments on the paper. K.S. and P.K. acknowledge fund- 0 cos Jdt 01− 1 αdt 2 ing by the Deutsche Forschungsgemeinschaft Grants No. 00 00 / / M(1) = = √ + O(dt3 2 ). (A7) 277101999 (TRR 183 Project No. C01) and No. EG 96 13-1, 0sinJdt 0 αdt as well as by the Israel Science Foundation. A.R. acknowl- edges Engineering and Physical Sciences Research Council These are the operators in Eq. (1) of the paper describing the (GB) Grant No. EP/P010180/1. effect of the measurements. K.S. and P.K. contributed equally to this work. APPENDIX B: CALCULATION OF POSTSELECTION AND APPENDIX A: A PHYSICAL MODEL OF THE SURVIVAL PROBABILITIES MEASUREMENT In the main text we have presented some of the observable (0) The system under consideration in the paper is a qubit (|0, signatures of the transition in terms of the probabilities P (t ) (0) |1) evolving under its own Hamiltonian and being measured to observe a no-click sequence of duration t and P (θ )to θ by a two-state detector (|0d , |1d ) at intervals dt. The sys- reach by a sequence of no clicks, as well as the behavior of tem’s Hamiltonian is the average state polarization s¯(t ). Here we derive the results stated in the main text. = σ (s). Hs s x (A1) (0) (0) We consider a system-detector Hamiltonian given by 1. Probabilities P (t)andP (θ) = J To determine the probability of the r 0 postselected = − σ (s) σ (d), Hs−d (1 z ) y (A2) trajectory, we start by solving the state evolution under a 2 sequence of zero readouts. The corresponding equation for θ where the detector is also assumed to be a two-level system. [see Eq. (3)] is The detector is initially prepared in the state |0 for each mea- d dθ surement, i.e., at the beginning of each time step. The system’s =−2s(1 + λ sin θ ). (B1) evolution under the combined effect of its Hamiltonian and the dt coupling to the detector is given by the unitary evolution due Thesolutionis to θ tan = λ2 − 1 tanh[ λ2 − 1(t − t )] − λ. (B2) 2 s 0 H = Hs + Hs−d , (A3) Setting the initial condition θ(t = 0) = 0 and simplifying the for time dt, after which the detector is read out with readouts expression, we arrive to r = 0, 1 corresponding to it being in |rd . θ+ We consider the scaling limit of continuous measurements θ(t ) 1 tan √ tan = λ2 − 1 coth t λ2 − 1 − ln 2 − λ → 2 → α = = α/ s θ− defined as dt 0, J dt const (i.e., J dt). In 2 2 tan this limit, the measurement and the system evolution do not 2 1 intermix in a single step; therefore, =− √ √ . (B3) 2 2 (r) λ + λ − 1 coth(st λ − 1) |ψ(t + dt) = M Us|ψ(t ). (A4) For λ>1, this expression describes the evolution of θ(t )from The unitary evolution due to the system’s Hamiltonian is π =− √1 √ λ θ =+∞ λ< at t 2 arccoth( 2 )to + at t .For s λ −1 λ −1 − iHsdt (s) 1, Eq. (B3) becomes Us = e = cos sdt − iσ sin sdt x − θ(t ) 1 = 1 i sdt + 2 . tan =− √ √ (B4) O(dt ) λ + − λ2 − λ2 −isdt 1 2 1 cot( st 1 ) θ = The measurement back-action matrices, combining the effects and describes the periodic evolution of (t ) with period T √π 2 . of the system-detector evolution and the readout in the state s λ −1 |r , defined as We are interested in the probability of having zero readout d = − at time t, P0(t ). Knowing the probability of obtaining r 0 (r) = | iHs−d dt| , M rd e 0d (A5) in each infinitesimal step, the equation for P0(t )isreadily determined:

θ dP(0)(t ) p θ(t ) tan2 (t ) =− 1 (0) =−α 2 (0) =−α 2 (0) P (t ) sin P (t ) θ P (t ) dt dt 2 1 + tan2 (t ) √ 2 sinh2( t λ2 − 1) =−α √ s √ √ . (B5) 2 2 2 2 λ cosh(2st λ − 1) − 1 + λ λ − 1sinh(2st λ − 1)

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Integrating the equation and demanding P0(t = 0) = 1, one obtains √ √ √ 2 2 2 2 − λ λ cosh(2st λ − 1) − 1 + λ λ − 1sinh(2st λ − 1) P(0)(t ) = e 2 s t . (B6) λ2 − 1 − λ = = + + 2 2 2 st λ< This expression is not singular at 1, where it becomes P0(t ) (1 2 st 2 s t )e .For 1, it reduces to √ √ √ 2 2 2 2 − λ λ cos(2st 1 − λ ) − 1 − λ 1 − λ sin(2st 1 − λ ) P(0)(t ) = e 2 s t . (B7) λ2 − 1 From Eq. (B7), one can directly derive the long-time behavior of P(0)(t ) reported in the paper [see Eq. (10)], which is (0) P (t →+∞) ∝ exp (−2sλt ) × oscillating function, (B8) for λ<1, and (0) 2 P (t →+∞) ∝ exp −2s λ − λ − 1 t (B9) for λ>1. The probability to reach state θ via a sequence of r = 0 readouts, P(0)(θ ), is obtained from the equation θ dP (θ ) dP [θ(t )] dθ 2λ sin2 0 = 0 = 2 P (θ ). (B10) dθ dt dt 1 + λ sin θ 0 The solution with P(0)(θ = 0) = 1is √ λ − √ λ θ θ+ θ+ 1 tan − tan λ2−1 tan λ2−1 (0) θ = 2 2 2 P ( ) θ θ 1 + λ sin θ tan θ − tan − tan − 2 2 2 θ 1 2λ λ + tan λ = exp √ arctan √ 2 − arctan √ . (B11) 1 + λ sin θ 1 − λ2 1 − λ2 1 − λ2

√ λ> θ = 2 When 1, starting at 0 it is possible to reach only the The evolution has two eigenvalues, s(−λ ± λ − 4). states with θ ∈ (θ+; 0], which is reﬂected in the vanishing of When the system is initialized in state |0 at t = 0, the evo-

λ lution of s¯(t ) is given by √ −1 θ θ λ2− √ (0) + 1 P (θ ≈ θ+) ∝ tan − tan , (B12) sinh t λ2 − 4 −sλt s 2 2 s¯y(t ) =−2e √ , (B16) λ2 − 4 and in the properties of its derivatives discussed in the paper. λ −sλt 2 2 s¯z(t ) = e cosh st λ − 4 +√ sinh st λ − 4 . λ2− 4 2. Survival probability and the average state polarization (B17)

The ﬁnal quantity used in the paper to describe the dy- Its long-time behavior is given by namics of the system is the average state polarization after time t, s¯(t ) ≡ (¯s (t ), s¯ (t )), which is related to the survival − λ × , λ< , y z exp ( s t ) √oscillating function for 2 probability in the initial state |0, P(t ) = [1 + s¯ (t )]/2, i.e., s¯ , (t ) ∝ z y z exp − λ − λ2 − 4 t , for λ>2. the probability to measure the system in |0 upon a projective s measurement at time t. These quantities require averaging (B18) over all the state trajectories, and no postselection is required. The averages ¯y ands ¯z components of the polarization can be Therefore, at long times, the survival probability P(t ) decays θ P →∞ = / expressed through the state distribution Pt ( ): to the steady-state√ value (t √ ) 1 2. The decay rate is 2 λ− λ2−4 π s(λ − λ − 4) = 2s( ). It exhibits a maximum at λ = 2 s¯y(t ) = dθPt (θ )sinθ, (B13) 2, resembling the maximum of the decay rate of P0(t )at −π λ = 1. π = θ θ θ. Note that the equation for the√ polarization evolution, yield- s¯z(t ) d Pt ( ) cos (B14) 2 −π ing two eigenvalues, s(λ ± λ − 4), has been obtained from Eq. (6).√ Thus, Eq. (6) “knows” about the eigenvalues It then follows from Eq. (6) that γ = 1 −λ ± λ2 − λ 2 ( 4) for any values of .Atthesametime, these eigenvalues correspond√ to normalizable eigenmodes of − α − d s¯y 2 s s¯y Eq. (6) only when λ<2/ 3 (see the discussion in Ap- = 2 . (B15) dt s¯z 2s 0 s¯z pendix C3).

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APPENDIX C: DERIVATION AND THE EIGENSPECTRUM at θ = π, supplemented with boundary condition OF THE MASTER EQUATION 2π θ π + − π − =−α θ 2 θ . 1. Derivation of Eq. (6) 2 s[Pt ( 0) Pt ( 0)] d sin Pt ( ) 0 2 The stochastic dynamics of the system is described by the (C5) probability density Pt (θ ) of being in the state |ψ(θ ) at time t for the stochastic variable θ. The probability of the system 2. Eigenmodes of Eq. (6) being in an interval of states [θ1,θ2] at time t, Pt ([θ1; θ2]) = θ We now derive the eigenmode solutions of Eq. (6), i.e., find 2 θ θ θ d Pt ( ), obeys the evolution 2sγ t 1 all the solutions of the form Pt (θ ) = e fγ (θ ). The reader θ2 θ˜2 may skip the details and look at the result in Appendix C3. dθPt+dt (θ ) = dθ˜Pt (θ˜ )p0(θ˜ ) + H (π ∈ [θ1; θ2]) Before diving into the derivation, it is useful to analyze θ θ˜ 1 1 the expectations from the solution. Due to the normalization 2π condition, for every eigenmode with γ = 0, we should × θ˜ θ˜ θ˜ , π d p1( )Pt ( ) (C1) 2 θ θ = θ have 0 d fγ ( ) 0. Since Pt ( ) 0, there can be no 0 γ> γ ∈ ∈ , eigenmodes with 0. Thus, only solutions with 0 where H (x [a; b]) is 1 if x [a b] and zero otherwise, θ θ are acceptable. Finally, since the decay rate of clicks at + and pr ( ) are the probabilities of obtaining the readout r in θ / = α 2 θ+ λ> is p1( ) dt sin 2 ,for 1 we expect an eigenmode a measurement: θ θ θ with 2 γ −α sin2 + =−α tan2 + /(1 + tan2 + ) = θ(t ) √ s 2 √ 2 2 2 2 2 pr=1 ≡ p1 = αdt sin , pr=0 ≡ p0 = 1 − p1. (C2) 2 ( λ − 1 − λ), i.e., γ λ − 1 − λ. We also expect a 2 s steady-state solution with γ = 0toexist. The first term on the right-hand side of Eq. (C1) describes θ θ the change of Pt+dt ([ 1; 2]) due to the the smooth evolution a. The functional dependence under r = 0 readout, while the second term accounts for θ = π = Away from θ = π, the equation for the eigenmodes is jumps to for the measurement outcome r 1. The variables θ˜1,2 are defined via the self-consistent condition θ α −α 2 θ + ∂ + θ θ = γ θ . θ˜ , − θ˜ , dt = θ , θ = + λ θ sin fγ ( ) θ 2 s sin fγ ( ) 2 s fγ ( ) 1 2 ( 1 2 ) 1 2, where ( ) 2 s[1 sin ][see 2 2 Eq. (3)]. A differential equation for Pt (θ ) is obtained by solving the self-consistent equation to order (dt)2, differentiating (C6) Eq. (C1) over θ , and retaining the terms of order dt. With the 2 Equivalently, explicit expressions for p and (θ ), we get Eq. (6). r The integrodifferential master equation (6) needs to be α α 2 + sin θ fγ (θ ) = (1 − 2 cos θ ) + 2 γ fγ (θ ). supplemented by the boundary conditions. They are simple: s 2 2 s (C7) Pt (θ = 0) = Pt (θ = 2π ). (C3) Note that for α 4 ,atθ = θ±, the equation becomes sin- θ = θ + π s In other words, Pt ( ) Pt ( 2 ). With this condition, it gular as the factor multiplying the highest (and only) deriva- is easy to check that the normalization by total probability π tive vanishes. These singular points require special treatment. 2 θ θ = θ ∈ −π θ = 0 d Pt ( ) 1 is preserved by this equation. Finally, it is However, this simply means that fγ ( [ ; +]) 0asthis useful for solving to eliminate the integral part of the equation. interval is inaccessible from the time evolution of any state This is easily done, as it only contributes at θ = π. Therefore, initially outside it, as shown in the analysis of the postselected Eq. (6) is equivalent to dynamics in the paper. We will come back to the issue of the θ θ α special points later. dPt ( ) 2 =−α sin P (θ ) + ∂θ 2 + sin θ P (θ ) Away from the special points, the equation admits an an- dt t s t 2 2 alytic solution, which can be expressed in two alternative (C4) forms:

√λ+γ θ θ θ+ λ + γ λ + π − λ2−1 C tan 2 C tan 2 tan 2 fγ (θ ) = exp 2√ arctan √ − = , (C8) (1 + λ sin θ )2 − λ2 − λ2 2 (1 + λ sin θ )2 θ − θ− 1 1 tan 2 tan 2 where λ = α/(4s ). The equivalence of the two expressions follows from 1 1 + ix arctan x = ln , (C9) 2i 1 − ix so one can write √ θ y y tan θ − tan + tan θ + λ − λ2 − 1 λ + tan θ π 2 2 = 2 √ = exp 2y arctan √ 2 − . θ − θ− θ + λ + λ2 − − λ2 2 tan 2 tan 2 tan 2 1 1

033512-7 SNIZHKO, KUMAR, AND ROMITO PHYSICAL REVIEW RESEARCH 2, 033512 (2020)

Obviously, the first form in Eq. (C8) is more convenient for λ<1, while the second one is the natural choice for λ>1. However, θ± θ θ = θ+ θ− λ> λ< − = for computational purposes one can use either. Note the singularities at and for 1(for 1, tan 2 tan 2 0 θ for any θ since tan ± has a nonzero imaginary part while tan θ is a real function). From the equality 2 2 θ θ θ θ θ 2 θ − + − − 1 + 2λ tan + tan tan 2 tan 2 tan 2 tan 2 1 + λ sin θ = 2 2 = , (C10) + 2 θ + 2 θ 1 tan 2 1 tan 2 we obtain the following behavior in proximity of the singularity points at θ = θ±: λ+γ √ −2 θ θ+ λ2−1 f (θ ≈ θ+) ∼ tan − tan , (C11) 2 2 λ+γ − √ −2 θ θ− λ2−1 f (θ ≈ θ−) ∼ tan − tan . (C12) 2 2 The normalizability conditions θ+ θ− dθ f (θ ) < ∞, dθ f (θ ) < ∞ (C13) for Eqs. (C11) and (C12) imply that, for λ>1, one must have λ + Re γ √ > 1 (vicinity of θ+), (C14) λ2 − 1 λ + Re γ √ < −1 (vicinity of θ−). (C15) λ2 − 1 These two conditions are incompatible. The apparent contradiction is resolved by choosing the normalization constant C in Eq. (C8) 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 = 0on(−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 fγ (θ ∈ [−π; θ+]) = 0. Putting these results together, we have the general expression for the eigenmodes, which reads, for λ>1, ⎧ √λ+γ ⎪ θ 2 θ θ+ ⎨ + 2 − λ2−1 C(1 tan 2 ) tan 2 tan 2 θ , for θ ∈ [θ+; π], θ = θ θ+ 2 θ θ− 2 θ − − fγ ( ) tan −tan tan −tan tan 2 tan 2 (C16) ⎩⎪ 2 2 2 2 0, for θ ∈ [−π; θ+], λ< and, for 1, λ + γ λ + θ π C tan 2 fγ (θ ) = exp 2√ arctan √ − , for θ ∈ (−π; π ). (C17) (1 + λ sin θ )2 1 − λ2 1 − λ2 2

b. The boundary conditions and normalization The above does not present the final solution. We have two boundary conditions and a normalization condition to satisfy. The first boundary condition, fγ (θ ) = fγ (θ + 2π ), is satisfied trivially. The second boundary condition, which needs to be addressed further, Eq. (C5), yields π θ 2 2s fγ (π − 0) − fγ (π + 0) = α dθ sin fγ (θ ). (C18) −π 2 Note that this condition is independent of C, hence it determines the spectrum of eigenmodes γ . The normalization condition is 2π , γ = , θ θ = 1 for 0 d fγ ( ) , γ = . (C19) 0 0 for 0 This should be used to determine C for the steady state (γ = 0) and should be satisfied automatically for all γ = 0 eigenmodes. The integrals can be calculated analytically: √λ+γ θ+ 1 + t 2 t − tan λ2−1 dθ f (θ ) = 2C dt 2 2 2 θ− θ+ θ− − − − t tan 2 t tan 2 t tan 2 ⎛ ⎞ √λ+γ θ+ C t − tan λ2−1 λ (2λ + γ )(1 − 2γ t ) − γ t 2 = 2 ⎝ + ⎠, θ 1 (C20) λ + γ t − tan − − θ+ − θ− γ − θ+ γ − θ− 2 t tan 2 t tan 2 tan 2 tan 2

033512-8 QUANTUM ZENO EFFECT APPEARS IN STAGES PHYSICAL REVIEW RESEARCH 2, 033512 (2020)

= θ where t tan 2 . Similarly, √λ+γ θ+ θ t 2 t − tan λ2−1 dθ sin2 f (θ ) = 2C dt 2 2 2 θ− 2 θ+ θ− − − − t tan 2 t tan 2 t tan 2 ⎛ ⎞ √λ+γ θ+ C (γ t − 1)2 1 t − tan λ2−1 = ⎝ 2 + ⎠ 2 . t θ (C21) 2(λ + γ ) γ − θ+ γ − θ− − θ+ − θ− t − tan − tan 2 tan 2 t tan 2 t tan 2 2

c. The solution for λ>1 and θ ∈ [θ+; π] λ+ γ Assuming √ Re > 1, the boundary condition yields λ2−1 π θ α γ 2 + λγ + 2 C(2 2 1) 2 2 C = α dθ sin fγ (θ ) = ⇐⇒ Cγ (γ + γλ+ 1) = 0. (C22) s λ + γ γ 2 + λγ + θ+ 2 2( )( 2 1) This ﬁxes the possible γ , thus giving us three eigenmodes: 1 γ = 0,γ = −λ ± λ2 − 4 . (C23) 2 The norm of the eigenmodes is then π C γλ C γ 2 + γλ+ 1 dθ f (θ ) = 1 − = , (C24) λ + γ γ 2 + λγ + λ + γ γ 2 + λγ + θ+ ( 2 1) 2 1 √ γ = 1 −λ ± λ2 − γ = which implies that the eigenfunctions with 2 ( 4) integrate to zero as expected. For the steady state, 0, the π θ θ = normalization condition θ+ d f ( ) 1 yields C = λ, (C25) 1/ cos θ+ λ + 2 θ 2 θ − θ+ 1 tan 2 tan 2 tan 2 fγ = (θ ) = 0 2 2 θ θ− θ θ+ θ θ− − − − tan 2 tan 2 tan 2 tan 2 tan 2 tan 2 √ √ 2 2 λ/ λ −1 1 + tan2 θ tan θ + λ − λ2 − 1 = λ 2 2 √ . θ (C26) 2 θ + λ θ + 2 tan + λ + λ2 − 1 tan 2 2 tan 2 1 2

Finally, we check the conditions in Eqs. (C11) and (C12): in Eq. (C8). At the same time, the integrals for the norm and the boundary condition at θ = π are more conveniently λ + Re γ √ > 1 ⇐⇒ Re γ> λ2 − 1 − λ. (C27) calculated using the expressions in Eq. (C16) (see Appendix λ2 − 1 C2b). Using these expressions and the results of Appendix θ = π Interestingly, the condition actually requires that all the eigen- C2b, one finds that the boundary condition at yields modes decay not slower than the decay rate at θ = θ+.The γ = λ ∈ steady state, 0, always satisfies√ the condition for (1; +∞). For λ 2, Re 1 (−λ ± λ2 − 4) =−λ/2. Then, C γ 2+ γλ+ λ + γ 2 1 − − π √ = . the normalizability condition is equivalent to 1 exp 2 0 (C29) λ + γ γ 2+ 2λγ + 1 1 − λ2 √ λ>2 λ2 − 1 ⇔ 3λ2 < 4 ⇔ λ<2/ 3. (C28) √ Therefore, on top of the three eigenmodes, γ = 0 and For λ>2, the inequality 1 (−λ − λ2 − 4) < √ √ √ 2 1 (−λ ± λ2 − 4), there is also an infinite sequence of eigen- 1 −λ + λ2 − < λ2 − − λ 2 2 ( 4) 1√ always holds. Therefore, values given by γ = 1 −λ ± λ2 − the eigenmodes√ 2 ( 4) are only normalizable for λ<2/ 3. γ =−λ + im 1 − λ2, m ∈ Z. (C30) d. The solution for λ<1 The same considerations√ hold for 0 λ<1. In this range Since there are no special points for λ<1, both this infinite λ2 − θ± of parameters, 1 and tan 2 become complex. There- set of solutions and the three eigenmodes in Eq. (C23) corre- fore, it is more convenient to use fγ (θ ) in the first form spond to valid normalizable eigenmodes.

033512-9 SNIZHKO, KUMAR, AND ROMITO PHYSICAL REVIEW RESEARCH 2, 033512 (2020) π γ = For the steady state, the normalization −π dθ f (θ ) = 1 For 0, yields π C γ 2 + γλ+ 1 λ dθ f (θ ) = = , −π λ + γ γ 2 + 2λγ + 1 C (C31) − − √2πλ 1 exp −λ2 λ + γ 1 × 1 − exp −2π √ , (C33) λ 1 1 − λ2 fγ =0(θ ) = πλ + λ θ 2 1 − exp − √2 (1 sin ) which vanishes due to the boundary condition above, as ex- 1−λ2 pected. λ λ + θ π 2 tan 2 × exp √ arctan √ − . 3. Summary of eigenvalues and eigenmodes of Eq. (6) 1 − λ2 1 − λ2 2 Putting together the results from Appendix C2, we can (C32) summarize the solutions of the master equation (6) as follows. The steady state, with eigenvalue γ = 0, is given for λ<1by

λ λ λ + θ π 1 2 tan 2 f0(θ ) = exp √ arctan √ − , (C34) πλ + λ θ 2 2 2 1 − exp − √2 (1 sin ) 1 − λ 1 − λ 2 1−λ2 while for λ>1 it reads ⎧ √ √ ⎨ θ λ/ λ2−1 tan +λ− λ2−1 λ 2 √ θ , for θ ∈ [θ+; π], f (θ ) = (1+λ sin θ )2 tan +λ+ λ2−1 (C35) 0 ⎩ 2 0, for θ ∈ [−π; θ+], √ θ± =−λ ± λ2 − λ = α/ with tan 2 1, (4 s ), and √ √ θ λ/ λ2−1 θ tan + λ − λ2 − 1 2λ λ + tan π 2 √ = exp √ arctan √ 2 − . (C36) θ + λ + λ2 − − λ2 − λ2 2 tan 2 1 1 1 These are the expressions used in the main text [see Eqs. (7) and (8)]. Several comments are in order. First, the expressions for both λ<1 and λ>1 are the same except for the normalization constant, which exhibits an essential singularity at λ = 1 − . Second, exactly at the transition λ = 1, 1 2 exp − θ ,θ∈ [−π/2; π], θ = (1+sin θ )2 1+tan f0( ) 2 (C37) 0,θ∈ (−π; −π/2), which displays an essential singularity at θ =−π/2. Due to the nature of this singularity, the transition at λ = 1 appears smooth when looking at the steady-state probability density. In this sense it could be regarded as a crossover. The eigenmode spectrum√ consists of two sets of eigenvalues. (1) γ =−λ + im 1 − λ2, m ∈ Z with λ + θ π C tan 2 fγ (θ ) = exp 2im arctan √ − . (C38) (1 + λ sin θ )2 1 − λ2 2 These eigenvalues exist for λ 1, become massively degenerate at λ = 1, and disappear at λ>1. This disappearance coincides with the opening of the forbidden region (−π; θ+). Note, thus, that the λ = 1 transition, which appears as a crossover in the steady-state behavior,√ presents a drastic change in the eigenmode spectrum. (2) γ = 1 (−λ ± λ2 − 4) with 2 √ λ ± − λ2 λ + θ π C i 4 tan 2 fγ (θ ) = exp √ arctan √ − , for λ<1, (C39) (1 + λ sin θ )2 1 − λ2 1 − λ2 2 √ λ±√ λ2−4 θ θ+ − 2 λ2−1 C tan 2 tan 2 fγ (θ ∈ [θ+; π]) = , for λ>1. (C40) (1 + λ sin θ )2 θ − θ− tan 2 tan 2

√ √ λ> / γ = 1 −λ ± λ2 − These eigenmodes disappear at 2 3. This disappear- Finally, note that the eigenvalues 2 ( 4) ance coincides with the steady state starting diverging at θ = for λ<2 correspond to solutions with an oscillatory be- θ+ + .

033512-10 QUANTUM ZENO EFFECT APPEARS IN STAGES PHYSICAL REVIEW RESEARCH 2, 033512 (2020) havior superimposed with a decay in time, while for λ> of the eigenvalues spectrum.√ Curiously, these eigenvalues 2 they give steadily decaying in time solutions. This tran- are nonphysical for λ>2/ 3, which is before the transition is identiﬁed in the paper in terms of the average sition is reached. At the same time, Eq. (6) does know qubit polarization, and has been identiﬁed in the detec- about these eigenvalues for any λ, as we show in Appendix tor’s signal [42] as well. Here it appears as a property B2.

[1] K. Jacobs, Quantum Measurement Theory and its Applications and Future Quantum States in Resonance Fluorescence, Phys. (Cambridge University, Cambridge, England, 2014). Rev. Lett. 112, 180402 (2014). [2] H. Carmichael, An Open Systems Approach to Quantum Optics, [15] D. Tan, S. J. Weber, I. Siddiqi, K. Mølmer, and K. W. Murch, Lecture Notes in Physics Monographs Vol. 18 (Springer-Verlag, Prediction and Retrodiction for a Continuously Monitored Su- Berlin, 1993), p. 179. perconducting Qubit, Phys. Rev. Lett. 114, 090403 (2015). [3] C. Guerlin, J. Bernu, S. Deléglise, C. Sayrin, S. Gleyzes, [16] M. Naghiloo, D. Tan, P. M. Harrington, J. J. Alonso, E. Lutz, S. Kuhr, M. Brune, Jean-michel Raimond, and S. Haroche, A. Romito, and K. W. Murch, Heat And Work Along Individual Progressive ﬁeld-state collapse and quantum non-demolition Trajectories of a Quantum Bit, Phys. Rev. Lett. 124, 110604 photon counting, Nature (London) 448, 889 (2007). (2020). [4] C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, [17] M. Naghiloo, J. J. Alonso, A. Romito, E. Lutz, and K. W. S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, Murch, Information Gain and Loss for a Quantum Maxwell’s J.-M. Raimond, and S. Haroche, Real-time quantum feedback Demon, Phys. Rev. Lett. 121, 030604 (2018). prepares and stabilizes photon number states, Nature (London) [18] Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. 477, 73 (2011). Lee, S. Moon, and Y.-H. Kim, Emergence of the geometric [5] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, Observing phase from quantum measurement back-action, Nat. Phys. 15, single quantum trajectories of a superconducting quantum bit, 665 (2019). Nature (London) 502, 211 (2013). [19] V. Gebhart, K. Snizhko, T. Wellens, A. Buchleitner, A. Romito, [6]K.W.Murch,S.J.Weber,K.M.Beck,E.Ginossar,andI. and Y. Gefen, Topological transition in measurement-induced Siddiqi, Reduction of the radiative decay of atomic coherence geometric phases, Proc. Natl. Acad. Sci. USA 117, 5706 in squeezed vacuum, Nature (London) 499, 62 (2013). (2020). [7] S. J. Weber, A. Chantasri, J. Dressel, A. N. Jordan, K. W. [20] K. Snizhko, P. Kumar, N. Rao, and Y. Gefen, Weak- Murch, and I. Siddiqi, Mapping the optimal route between two measurement-induced asymmetric dephasing: A topological quantum states, Nature (London) 511, 570 (2014). transition, arXiv:2006.13244 (2020). [8]R.Vijay,C.Macklin,D.H.Slichter,S.J.Weber,K.W.Murch, [21] K. Snizhko, N. Rao, P. Kumar, and Y. Gefen, Weak- R. Naik, A. N. Korotkov, and I. Siddiqi, Stabilizing Rabi os- measurement-induced phases and dephasing: broken symmetry cillations in a superconducting qubit using quantum feedback, of the geometric phase, arXiv:2006.14641 (2020). Nature (London) 490, 77 (2012). [22] Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect [9] M. S. Blok, C. Bonato, M. L. Markham, D. J. Twitchen, V. V. and the many-body entanglement transition, Phys. Rev. B 98, Dobrovitski, and R. Hanson, Manipulating a qubit through the 205136 (2018). backaction of sequential partial measurements and real-time [23] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary- feedback, Nat. Phys. 10, 189 (2014). projective entanglement dynamics, Phys. Rev. B 99, 224307 [10] G. de Lange, D. Ristè, M. J. Tiggelman, C. Eichler, L. Tornberg, (2019). G. Johansson, A. Wallraff, R. N. Schouten, and L. DiCarlo, [24] B. Skinner, J. Ruhman, and A. Nahum, Measurement-Induced Reversing Quantum Trajectories with Analog Feedback, Phys. Phase Transitions in the Dynamics of Entanglement, Phys. Rev. Rev. Lett. 112, 080501 (2014). X 9, 031009 (2019). [11] D. Ristè, M. Dukalski, C. A. Watson, G. de Lange, M. J. [25] M. Szyniszewski, A. Romito, and H. Schomerus, Entanglement Tiggelman, Ya. M. Blanter, K. W. Lehnert, R. N. Schouten, transition from variable-strength weak measurements, Phys. and L. DiCarlo, Deterministic entanglement of superconducting Rev. B 100, 064204 (2019). qubits by parity measurement and feedback, Nature (London) [26] Z. K. Minev, S. O. Mundhada, S. Shankar, P. Reinhold, R. 502, 350 (2013). Gutiérrez-Jáuregui, R. J. Schoelkopf, M. Mirrahimi, H. J. [12] N. Roch, M. E. Schwartz, F. Motzoi, C. Macklin, R. Vijay, Carmichael, and M. H. Devoret, To catch and reverse a quantum A. W. Eddins, A. N. Korotkov, K. B. Whaley, M. Sarovar, and jump mid-ﬂight, Nature (London) 570, 200 (2019). I. Siddiqi, Observation of Measurement-Induced Entanglement [27] B. Misra and E. C. G. Sudarshan, The Zeno’s paradox in quan- and Quantum Trajectories of Remote Superconducting Qubits, tum theory, J. Math. Phys. 18, 756 (1977). Phys. Rev. Lett. 112, 170501 (2014). [28] Asher Peres, Zeno paradox in quantum theory, Am. J. Phys. 48, [13] J. P. Groen, D. Ristè, L. Tornberg, J. Cramer, P. C. de Groot, 931 (1980). T. Picot, G. Johansson, and L. DiCarlo, Partial-Measurement [29] P. Facchi and S. Pascazio, Quantum Zeno Subspaces, Phys. Rev. Backaction and Nonclassical Weak Values in a Superconduct- Lett. 89, 080401 (2002). ing Circuit, Phys. Rev. Lett. 111, 090506 (2013). [30] P. Facchi, A. G. Klein, S. Pascazio, and L. S. Schulman, Berry [14] P. Campagne-Ibarcq, L. Bretheau, E. Flurin, A. Auffèves, F. phase from a quantum Zeno effect, Phys. Lett. A 257, 232 Mallet, and B. Huard, Observing Interferences between Past (1999).

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[31] D. Burgarth, P. Facchi, V. Giovannetti, H. Nakazato, S. [41] M. Zhang, C. Wu, Y. Xie, W. Wu, and P. Chen, Quantum Zeno Pascazio, and K. Yuasa, Non-Abelian phases from quantum effect by incomplete measurements, Quantum Inf. Process. 18, Zeno dynamics, Phys.Rev.A88, 042107 (2013). 97 (2019). [32] S. Gherardini, S. Gupta, F. S. Cataliotti, A. Smerzi, F. Caruso, [42] F. Li, J. Ren, and N. A. Sinitsyn, Quantum Zeno effect as a and S. Ruffo, Stochastic quantum Zeno by large deviation the- topological phase transition in full counting statistics and spin ory, New J. Phys. 18, 013048 (2016). noise spectroscopy, Europhysics Lett. 105, 27001 (2014). [33] T. J. Elliott and V. Vedral, Quantum quasi-Zeno dynamics: [43] C. Presilla, R. Onofrio, and U. Tambini, Measurement quantum Transitions mediated by frequent projective measurements near mechanics and experiments on quantum Zeno effect, Ann. Phys. the Zeno regime, Phys. Rev. A 94, 012118 (2016). (NY) 248, 95 (1996). [34] M. Majeed and A. Z. Chaudhry, The quantum Zeno and anti- [44] S. A. Gurvitz, L. Fedichkin, D. Mozyrsky, and G. P. Berman, Zeno effects with non-selective projective measurements, Sci. Relaxation and the Zeno Effect in Qubit Measurements, Phys. Rep. 8, 14887 (2018). Rev. Lett. 91, 066801 (2003). [35] P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, [45] K. Koshino and A. Shimizu, Quantum Zeno effect by general H. Weinfurter, and A. Zeilinger, High-Efficiency Quantum In- measurements, Phys. Rep. 412, 191 (2005). terrogation Measurements via the Quantum Zeno Effect, Phys. [46] A. Chantasri, J. Dressel, and A. N. Jordan, Action principle for Rev. Lett. 83, 4725 (1999). continuous quantum measurement, Phys. Rev. A 88, 042110 [36] M. C. Fischer, B. Gutiérrez-Medina, and M. G. Raizen, (2013). Observation of the Quantum Zeno and Anti-Zeno Ef- [47] P. Kumar, A. Romito, and K. Snizhko, The quantum fects in an Unstable System, Phys. Rev. Lett. 87, 040402 Zeno effect with partial measurement and noisy dynamics, (2001). arXiv:2006.13970 (2020). [37] J. Wolters, M. Strauß, R. S. Schoenfeld, and O. Benson, Quan- [48] R. Ruskov, A. Mizel, and A. N. Korotkov, Crossover of phase tum Zeno phenomenon on a single solid-state spin, Phys. Rev. qubit dynamics in the presence of a negative-result weak mea- A 88, 020101(R) (2013). surement, Phys.Rev.B75, 220501(R) (2007). [38] A. Signoles, A. Facon, D. Grosso, I. Dotsenko, S. Haroche, [49] J. Li, T. Wang, L. Luo, S. Vemuri, and Y. N. Joglekar, Uni- J.-M. Raimond, M. Brune, and S. Gleyzes, Confined quantum fication of quantum Zeno-anti Zeno effects and parity-time Zeno dynamics of a watched atomic arrow, Nat. Phys. 10, 715 symmetry breaking transitions, arXiv:2004.01364 (2020). (2014). [50] Y.-H. Chen and T. A. Brun, Continuous quantum error [39] F. Schäfer, I. Herrera, S. Cherukattil, C. Lovecchio, F.S. detection and suppression with pairwise local interactions, Cataliotti, F. Caruso, and A. Smerzi, Experimental realiza- arXiv:2004.07285 (2020). tion of quantum Zeno dynamics, Nat. Commun. 5, 3194 [51] Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, Protecting (2014). entanglement from decoherence using weak measurement and [40] D. Layden, E. Martín-Martínez, and A. Kempf, Perfect quantum measurement reversal, Nat. Phys. 8, 117 (2012).

Zeno-like effect through imperfect measurements at a ﬁnite [52] In fact, it follows from Eq. (12)that∂λs¯z (t ) 0foranyt 0 frequency, Phys.Rev.A91, 022106 (2015). when λ>2.

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