Quantum Zeno Effect Appears in Stages

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, United Kingdom 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 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 in- trinsic 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 [5–7] systems. Tracking quantum trajectories has been exploited Figure 1. (a) The system. A Hamiltonian induces oscillations as a tool to engineering quantum states via continu- between levels |0i and |1i of a qubit, which is continuously ous feedback control [8–10] and entanglement distillation measured by a detector weakly coupled to one of the levels. [11, 12]. It has been used to observe fundamental proper- (b) Dynamical flow (red and blue arrows) of θ(t) from Eq. (3) ties of quantum measurements [13–17] and, recently, to under “no-click” postselected dynamics. For sufficiently weak predict topological transitions in measurement-induced measurements, λ < 1, (left) the dynamics is oscillatory; for geometric phases [18–21] and many-body entanglement λ > 1 (right), a stable and an unstable fixed points (θ+ and θ− respectively) emerge. The states in the interval θ ∈ (−π, θ+) phase transitions in random unitary circuits, invisible to are inaccessible to the system under both the “no-click” and the average dynamics [22–25]. Monitoring quantum tra- the full stochastic dynamics. jectories has also made possible anticipating and correct- ing quantum jumps in superconducting qubits [26]. The above-mentioned transitions stem from the ba- dynamics of the detector signal [42], average [43–47], or sic physics of the quantum Zeno effect [27, 28]. In this postselected [48, 49] state evolution. regime, as a result of repeated measurements, the sys- Here we study the transition between the regimes of tem state is mostly frozen next to one of the measure- coherent oscillations and Zeno-like dynamics in a qubit ment eigenstates, yet rarely performs quantum jumps be- subject to continuous partial measurements, cf. Fig.1(a),

arXiv:2003.10476v2 [quant-ph] 5 Oct 2020 tween them. The crossover between coherent oscillations a model directly describing some recent experiments [26]. and the Zeno regime is controlled by the frequency of By investigating the full stochastic dynamics of quantum the measurement and has been extensively explored both trajectories, we show that the quantum Zeno regime is theoretically [29–34] and experimentally [35–39]. Beyond established via a cascade of transitions in the system dy- projective measurements, the onset of the Zeno regime is namics, some being invisible to the average dynamics. richer [40, 41], and quantum jumps appear as part of con- Furthermore, we find that, in the Zeno regime, catchable tinuous stochastic dynamics. For example, in a system continuous jumps between states |1i and |0i necessarily monitored via continuous partial measurements, quan- have a discontinuous counterpart, jumps between |0i and tum jumps can be anticipated, continuously monitored, |1i, which are inherently unpredictable in individual real- and reverted [26], a task which is fundamentally impos- izations. Our results provide a unified picture of the onset sible with projective measurements. Moreover, the onset of the Zeno regime arising from continuous partial mea- of the Zeno regime with non-projective measurements is surements and demonstrate that investigating individual more convoluted and has been characterized by different quantum trajectories can uncover drastically new physics measurement strengths and phenomenology based on the even in simple and well-studied systems. Our findings 2 may be relevant for quantum error correction protocols evolution is governed by the the differential equation employing continuous partial measurements [50, 51]. θ˙ = −Ω(θ). The corresponding flow of the variable θ Model and post-selected dynamics.—We consider a is shown in Fig.1(b). Since Ωs > 0 and λ > 0, for any qubit performing coherent quantum oscillations between θ ∈ (0, π), we have Ω(θ) > 0, and the system evolves states |0i and |1i due to the Hamiltonian Hs = Ωsσx, continuously towards θ = 0. Notably, this is the only where Ωs > 0; at the same time the qubit is monitored way for the system state to evolve from |1i to |0i and it by a sequence of measurements at intervals dt 1/Ωs corresponds to the quantum jumps that have been con- – cf. Fig.1(a). Each measurement is characterized by tinuously monitored in Ref. [26]. The transition from |0i two possible readouts r = 0 (no-click) and 1 (click). The to |1i, instead, takes place via the region θ ∈ (−π, 0) corresponding measurement back-action |ψi → M (r) |ψi and has richer dynamics controlled by the measurement is given by the operators strength. For sufficiently weak measurements, 0 ≤ λ < 1, one has Ω(θ) > 0 for any θ, and the system monotonously (0) p (1) √ M = |0i h0|+ 1 − p |1i h1| ,M = p |1i h1| , (1) evolves towards θ = −π; however, for λ > 1 there appear two fixed points, Ω(θ±) = 0, at where p ∈ [0, 1] controls the measurement strength. For p = 1 p 2 , each measurement is projective and this induces θ± = 2 arctan −λ ± λ − 1 , (5) the conventional quantum Zeno effect with the system |0i being frozen in one of the measurement eigenstates, where θ+ is a stable point, while θ− is an unstable one, or |1i. In the opposite limit, p = 0, essentially no mea- as shown in Fig.1(b). Under the r = 0 postselected surement takes place, and the system performs Rabi os- dynamics for λ > 1, the system will eventually flow to H cillations under s. We investigate the intermediate case θ = θ+ [48] (where it remains until the occurrence of a of p = αdt with dt → 0, and α ≥ 0 controlling the ef- click, which collapses the system to |1i). fective measurement strength over a finite time interval. Stochastic evolution and dynamical transitions.— A physical model of this measurement process is realized Beyond the postselected r = 0 quantum trajectory, the by coupling the system to a two-level system detector stochastic dynamics of the system is described by the that, in turn, is subject to projective measurements, see probability density Pt(θ) of being in the state |ψ(θ)i at AppendixA for details. time t. Using Eqs. (3–4), one derives the master equation In each infinitesimal step the measurement and the for Pt(θ) system evolution add up to give the combined evolution (r) dPt(θ) |ψ(t + dt)i = M U |ψ(t)i , = ∂θ (Ω(θ)Pt(θ)) (2) dt

−iHsdt 2π ˜ where U = e ≈ 1 − iHsdt is the Hamiltonian 2 θ ˜ 2 θ ˜ −4Ωsλ sin Pt(θ)+4Ωsλδ(θ−π) dθ sin Pt(θ) . unitary evolution over an infinitesimal time interval dt. 2 ˆ0 2 When the system is initialized in |0i or |1i, its evo- (6) lution is constrained to the y–z section of the Bloch sphere and the state has the form |ψ(t)i = |ψ(θ(t))i = Here, the first term on the r.h.s. describes the “no- θ(t) θ(t) click” evolution, the second term describes the reduc- cos 2 |0i + i sin 2 |1i. Eq. (2) translates onto tion of Pt(θ) due to clicks that happen with probability ( p = 4Ω λ sin2 θ dt θ(t) − Ω(θ(t)) dt if r = 0 1 s 2 , cf. Eq. (4), while the last term ac- θ(t + dt) = , (3) counts for the clicks bringing the states from any θ to π if r = 1 θ = π. α Two experimentally accessible quantities directly re- where Ω(θ) = 2Ωs [1 + λ sin θ] and λ = sets the 4Ωs lated to Pt(θ) capture the main physics: the steady-state strength of the measurement relative to the Hamilto- distribution P∞(θ) ≡ limt→∞ Pt(θ), and the average nian. A measurement yielding readout r = 1 immedi- “polarization” of the qubit, s¯(t) ≡ (¯sy(t), s¯z(t)), where ately projects the system onto state |1i, while a “no-click” π s¯i(t) ≡ hσi(t)i = hψ(θ)| σi |ψ(θ)i Pt(θ) dθ, i = y, z. r = 0 −π readout implies an infinitesimal evolution of the Both quantities are´ plotted in Fig.2. They showcase Ω(θ) state with angular velocity . The probabilities of three qualitative transitions in the dynamics as function the two possible readouts are given by of the measurement strength. We can readily present the key physics of these transi- 2 θ(t) pr=1 ≡ p1 = αdt sin , pr=0 ≡ p0 = 1 − p1 (4) tions 2 before entering all the features in due details. For sufficiently small λ, the qubit can be found in any state and depend on the qubit state, i.e. on θ(t). with finite probability density P∞(θ) 6= 0. In particu- For understanding the full stochastic dynamics, it is lar, it is possible to evolve from |0i to |1i via trajectories instructive to review its continuous “no-click” part, pre- involving a detector click as well as no clicks from the viously analyzed in Ref. [48]. In this case the state detector. Instead, the evolution from |1i to |0i happens 3

Figure 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 trajectory of the expectation values s¯y,z(t), for the system initialized in |0i at t = 0, are shown with the dashed magenta lines. Each numerical simulation involved 10000 stochastic trajectory realizations, tracing the evolution until t = 10, with ◦ measurements yielding random outcomes happening at intervals dt = 0.01, using Ωs = 1; θ is binned in intervals of size 5 . only via no click sequences, as noted above. The first and for λ < 1, while for λ > 1 the expression reads most drastic transition happens at λ = 1, above which  √ λ θ 2 √ θ ∈ (−π; θ ] P (θ) = 0 tan +λ− λ −1 λ2−1 there opens a region of + where ∞ . In  λ 2 √ , θ ∈ (θ ; π], (1+λ sin θ)2 tan θ +λ+ λ2−1 + fact, this region is inaccessible for the qubit at any time P∞(θ) = 2 t, hence, for λ > 1 all quantum trajectories from |0i to 0, θ ∈ (−π; θ+]. |1i must involve a detector click. Generically, the click (8) may occur when the qubit has not reached θ+, which In Fig.2, the analytical results in Eqs. (7,8) are com- is typically√ the case. The second transition happens pared with Monte Carlo numerical simulations of individ- at λ = 2/ 3, above which P∞(θ) diverges at θ = θ+. ual quantum trajectories, showing excellent agreement. This indicates that the system initialized in |0i typically The first two transitions in the system dynamics high- reaches the vicinity of θ+, and spends a long time there, lighted above, are evident from P∞(θ) – cf. Fig.2. The before the click and the corresponding jump to θ = π opening of the forbidden region (−π; θ+] appears discon- take place. So far, the population imbalance between |0i tinuously rather than opening smoothly, since θ+ = θ− = and |1i, s¯z(t), exhibits oscillations, which are reflected −π/2 at λ = 1. It manifests itself in the non-analytic be- in the oscillations of the average state polarization, s¯(t). havior of P∞(θ) as a function of λ at λ = 1. For λ > 1, The third and final transition at λ = 2 marks the end of the second transition shows up at θ ≈ θ+, where the oscillations, so that, for λ > 2, s¯z(t) steadily decays √ λ −2 in time, completing the final onset of Zeno-like dynam- θ θ+ λ2−1 P∞(θ) ∝ tan − tan , (9) ics. These transitions set the overall picture of the onset 2 2 of Zeno regime in the system, and constitute the main √ findings of our work. which diverges at θ = θ+ for λ > 2/ 3. The physics un- derpinning this transition is evinced by Eq. (6) at θ = θ+. P (θ )∂ Ω(θ ) To analyze these transitions and their implications in The first two terms in the r.h.s., t + θ+ + and 2 θ+ some detail, consider first the non-trivial steady state, −4Ωsλ sin 2 Pt(θ+), describe the rate of accumulation P∞(θ). From the condition dPt→∞(θ)/dt = 0, Eq. (6) of probability for states at θ ≈ θ+ due to no-click dynam- gives ics 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 polarization s¯(t)) behavior and in the h λ+tan θ i λ exp √ 2λ arctan √ 2 − π post-selected r = 0 dynamics. 1−λ2 1−λ2 2 P∞(θ) = h i , (7) While the steady-state properties of P∞(θ) showcase (1 + λ sin θ)2 1 − exp − √2πλ 1−λ2 these transitions, their role in the onset of the Zeno 4 √ √ and diverges for λ > 2/ 3. Therefore, for λ < 2/ 3, the system typically jumps to θ =√ π via a detector click before it reaches θ+. For λ > 2/ 3, the system is likely to reach a close vicinity of θ+ before a click happens. To observe the last transition, λ = 2, one needs to consider the average state polarization, s¯(t) = (¯sy(t), s¯z(t)), and in particular the population imbalance, s¯z(t). When the system is initialized in state |0i at time t = 0, using Eq. (6), one finds, cf. AppendixB, √ ! sinh Ω t λ2 − 4 −Ωsλt p 2 s s¯z(t) = e cosh Ωst λ − 4 + λ √ . λ2 − 4 (12) One sees that λ = 2 marks a transition from oscillatory λ < 2 λ > 2 ¯ (at ) to non-oscillatory (at ) dynamics. The Figure 3. The decay rates ζ(λ) (solid) and ζ(λ) (dashed) s¯ (t) P (0)(t) characterizing respectively the probability to observe no clicks same transition is observed in y . Similarly to −2Ωsζ¯(λ)t (0) in Eq. (10), s¯z(t → +∞) ∝ e with the decay P (t) and the survival probability (1 +s ¯z(t))/2. Note the √ respective decay rate maxima at λ = 1 and λ = 2. Inset—The rate ζ¯(λ) = Re λ − λ2 − 4 /2. Therefore, the decay time dependence of P (0)(t) for λ = 0.5 (solid) and λ = 1.25 rate exhibits a maximum at λ = 2, cf. Fig.3. Impor- (0) (dashed). At long times P (t) decays exponentially with tantly, s¯z(t) characterizes not only the population imbal- (without) superimposed oscillations for λ < 1 (λ > 1). ance but also the survival probability (1 +s ¯z(t))/2, i.e., the probability to find the system in state |0i when performing a projective measurement at time t. The decay dynam- regime is fully unveiled only in the full stochastic rate behavior implies that the long-time survival proba- ics P (0)(t) . To appreciate that, consider the probability bility increases with increasing λ when λ ≥ 2, and de- r = 0 of obtaining a sequence of no clicks ( readouts) of creases otherwise [52]. On this ground, one marks the t duration , cf. Fig.3 (inset). We obtain that at long-times transition at λ = 2 as the final onset of the Zeno-like P (0)(t) decays exponentially as dynamics. Notably, the λ = 2 transition is also reflected P (0)(t) ∝ e−2Ωsζ(λ)t [A(λ) + B(λ) cos(εt + ϕ(λ))] , (10) in a topological transition in the statistics of the detector clicks [42]. where B(λ) = 0 for λ > 1,√ the frequency of the oscillatory We would like to note that while we used different 2 term for λ < 1 is ε = 2Ωs 1 − λ , and the decay rate is quantities to showcase each of the three dynamical tran- √ ζ(λ) = Re λ − λ2 − 1 ; see AppendixB for the deriva- sitions, all three of them can be inferred by looking at a tion. The value λ = 1 is special in two respects. For single quantity: the eigenmode spectrum of Eq. (6). This λ < 1, the system state rotates between |0i and |1i under more mathematical identification of the transitions and no-click dynamics. However, the probability of observing its relation to the physics described here is discussed in a click is different for different θ — thence the oscillations AppendixC. of P (0)(t). With the appearance of the forbidden region, Observing the transitions experimentally.—The above the evolution under no clicks readout is frozen at θ = θ+, physics can be readily observed in the setup of re- hence the oscillations of P (0)(t) disappear. The less obvi- cent experiments [26] by adjusting the measurement ous effect is that at λ = 1 the decay rate ζ(λ) is maximal, strength/Rabi frequency. The simplest transition to ob- cf. Fig.3. Therefore, the probability of observing a long serve is that at λ = 2, which is apparent in routine ex- sequence of “no-clicks” increases with the measurement periments measuring the average polarization or survival strength for λ ≥ 1, while it decreases for λ ≤ 1. probability√ in |0i. Observing the transitions at λ = 1 and (0) Consider now the probability P (θ) to reach a par- λ = 2/ 3 requires sampling the distribution P∞(θ) by ticular value of θ under no-click dynamics. P (0)(θ) is tracking individual quantum trajectories for sufficiently obtained from P (0)(t) and the no-click evolution evolu- long times and performing quantum tomography on the ˙ (0) (0) tion θ(t) = −Ω(θ(t)) via P (θ0) ≡ P (t0) with t0 final states. This is possible with state-of-the-art experi- satisfying θ(t0) = θ0. In proximity of θ = θ+, one has, mental techniques [5, 26], though laborious. A somewhat cf. AppendixB, less laborious alternative to observe the λ = 1 transition is to measure the probability to observe no clicks for a √ λ −1 2 (0) (0) θ θ+ λ −1 given time, P (t). While this requires recording every P (θ ≈ θ+) ∝ tan − tan . (11) 2 2 measurement outcome, it does not require knowing√ the qubit state at time t. For the transtion at λ = 2/ 3, (0) Note that P (θ) vanishes at θ = θ+ for any finite λ√ > one can measure the probability to reach a specific θ by (0) (0) 1. 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Appendix A: A physical model of the measurement

The system under consideration in the manuscript is a qubit (|0i, |1i) evolving under its own Hamiltonian and being measured by a two-state detector (|0di, |1di) at intervals dt. The system’s Hamiltonian is

(s) Hs = Ωsσx . (A1)

We consider a system-detector Hamiltonian given by

J (s) (d) Hs−d = (1 − σ )σ , (A2) 2 z y where the detector is also assumed to be a two-level system. The detector is initially prepared in the state |0di 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

H = Hs + Hs−d, (A3) for time dt, after which the detector is read out with readouts r = 0, 1 corresponding to it being in |rdi. p We consider the scaling limit of continuous measurements deﬁned as dt → 0, J 2dt → α = const (i.e., J = α/dt). In this limit, the measurement and the system evolution do not intermix in a single step, therefore,

(r) |ψ(t + dt)i = M Us |ψ(t)i . (A4)

The unitary evolution due to the system’s Hamiltonian is −iHsdt (s) 1 −iΩsdt 2 Us = e = cos Ωsdt − iσx sin Ωsdt = + O(dt ). −iΩsdt 1

The measurement back action matrices, combining the eﬀects of the system-detector evolution and the readout in the state |rdi, deﬁned as

(r) −iHs−ddt M = hrd| e |0di , (A5) are (0) 1 0 1 0 2 M = = 1 + O(dt ), (A6) 0 cos Jdt 0 1 − 2 αdt

0 0 0 0 M (1) = = √ + O(dt3/2). (A7) 0 sin Jdt 0 αdt

These are the operators in Eq. (1) of the manuscript describing the eﬀect of the measurements.

Appendix B: Calculation of postselection and survival probabilities

In the main text 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 polarization s¯(t). Here we derive the results stated in the main text.

1. Probabilities P (0)(t) and P (0)(θ)

To determine the probability of the r = 0-postselected trajectory, we start by solving the state evolution under a sequence of 0 readouts. The corresponding equation for θ (cf. Eq. (3) in the manuscript) is dθ = −2Ωs(1 + λ sin θ). (B1) dt 8

The solution is: θ p 2 p 2 tan = λ − 1 tanh Ωs λ − 1(t − t0) − λ. (B2) 2 Setting the initial condition θ(t = 0) = 0 and simplifying the expression, we arrive to ! tan θ+ θ(t) p 2 p 2 1 2 1 tan = λ − 1 coth Ωst λ − 1 − ln − λ = − √ √ . (B3) 2 2 θ− λ + λ2 − 1 coth Ω t λ2 − 1 tan 2 s √1 √ λ For λ > 1, this expression describes the evolution of θ(t) from π at t = − 2 arccoth 2 to θ+ at t = +∞. Ωs λ −1 λ −1 For λ < 1, Eq. (B3) becomes θ(t) 1 tan = − √ √ 2 2 (B4) 2 λ + 1 − λ cot Ωst 1 − λ

√π and describes the periodic evolution of θ(t) with period T = 2 . Ωs λ −1 We are interested in the the probability of having zero readout at time t, P0(t). Knowing the probability of obtaining r = 0 in each inﬁnitesimal step, the equation for P0(t) is readily determined: dP (0)(t) p θ(t) = − 1 P (0)(t) = −α sin2 P (0)(t) dt dt 2 √ 2 θ(t) 2 2 tan sinh (Ωst λ − 1) = −α 2 P (0)(t) = −α √ √ √ . (B5) 2 θ(t) λ2 cosh(2Ω t λ2 − 1) − 1 + λ λ2 − 1 sinh(2Ω t λ2 − 1) 1 + tan 2 s s

Integrating the equation and demanding P0(t = 0) = 1, one obtains, √ √ √ 2 2 2 2 λ cosh(2Ωst λ − 1) − 1 + λ λ − 1 sinh(2Ωst λ − 1) P (0)(t) = e−2Ωsλt . (B6) λ2 − 1

2 2 −2Ωst This expression is not singular at λ = 1, where it becomes P0(t) = (1 + 2Ωst + 2Ωst )e . For λ < 1, it reduces to √ √ √ 2 2 2 2 λ cos(2Ωst 1 − λ ) − 1 − λ 1 − λ sin(2Ωst 1 − λ ) P (0)(t) = e−2Ωsλt . (B7) λ2 − 1 From Eq. (B7), one can directly derive the long-time behavior of P (0)(t) reported in the manuscript [cf. Eq. (10) therein], which is (0) P (t → +∞) ∝ exp (−2Ωsλt) × oscillating function, (B8) for λ < 1, and h i (0) p 2 P (t → +∞) ∝ exp −2Ωs 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 2 θ dP0(θ) dP0(θ(t)) dθ 2λ sin 2 = / = P0(θ). (B10) dθ dt dt 1 + λ sin θ The solution with P (0)(θ = 0) = 1 is

λ λ θ ! √ θ !− √ 1 tan θ − tan + λ2−1 tan + λ2−1 P (0)(θ) = 2 2 2 1 + λ sin θ θ θ− θ− tan 2 − tan 2 tan 2 " #! 1 2λ λ + tan θ λ = exp √ arctan √ 2 − arctan √ . (B11) 1 + λ sin θ 1 − λ2 1 − λ2 1 − λ2

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 √ λ −1 2 (0) θ θ+ λ −1 P (θ ≈ θ+) ∝ tan − tan , (B12) 2 2 and in the properties of its derivatives discussed in the manuscript. 9

2. 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) ≡ (¯sy(t), s¯z(t)), which is related to the survival probability in the initial state |0i, P(t) = (1+¯sz(t))/2, i.e. the probability to measure the system in |0i upon a projective measurement at time t. These quantities require averaging over all the state trajectories, and no postselection is required. The average s¯y and s¯z components of the polarization can be expressed through the state distribution Pt(θ):

π s¯y(t) = dθPt(θ) sin θ, (B13) ˆ−π π s¯z(t) = dθPt(θ) cos θ. (B14) ˆ−π

It then follows from Eq. (6) that d s¯ − α −2Ω s¯ y = 2 s y . (B15) dt s¯z 2Ωs 0 s¯z √ 2 The evolution has two eigenvalues, Ωs(−λ ± λ − 4). When the system is initialized in state |0i at t = 0, the evolution of s¯(t) is given by √ sinh Ω t λ2 − 4 −Ωsλt s s¯y(t) = −2e √ , (B16) λ2 − 4 λ −Ωsλt p 2 p 2 s¯z(t) = e cosh Ωst λ − 4 + √ sinh Ωst λ − 4 . (B17) λ2 − 4 Its long-time behaviour is given by ( exp (−Ωsλt) × oscillating function, for λ < 2, s¯ (t) ∝ √ y,z 2 (B18) exp −Ωs λ − λ − 4 t , for λ > 2.

Therefore, at long times, the survival probability P(t) decays to the steady state value P(t → ∞) = 1/2. The decay √ √ 2 λ− λ2−4 rate is, Ωs λ − λ − 4 = 2Ωs 2 . It exhibits a maximum at λ = 2, resembling the maximum of the decay rate of P0(t) at λ = 1. √ Ω λ ± λ2 − 4 Note that the equation for the polarization evolution, yielding two eigenvalues,√ s , has been obtained γ = 1 −λ ± λ2 − 4 λ from Eq. (6). Thus, Eq. (6) “knows” about the eigenvalues 2 for any values√ of . At the same time, these eigenvalues correspond to normalizable eigenmodes of Eq. (6) only when λ < 2/ 3, cf. the discussion in AppendixC3.

Appendix C: Derivation and the eigenspectrum of the master equation

1. Derivation of Eq. (6)

The stochastic dynamics of the system is described by the probability density Pt(θ) of being in the state |ψ(θ)i at time t for the stochastic variable θ. The probability of the system being in an interval of states [θ1, θ2] at time t, θ2 Pt([θ1; θ2]) = dθPt(θ) θ1 obeys the evolution ´ θ2 θ˜2 2π ˜ ˜ ˜ ˜ ˜ ˜ dθPt+dt(θ) = dθPt(θ)p0(θ) + ΘH (π ∈ [θ1; θ2]) dθp1(θ)Pt(θ), (C1) ˆθ1 ˆθ˜1 ˆ0 where ΘH (x ∈ [a; b]) is 1 if x ∈ [a, b] and 0 otherwise, and pr(θ) are the probabilities of obtaining the readout r in a measurement,

2 θ(t) pr=1 ≡ p1 = αdt sin , pr=0 ≡ p0 = 1 − p1. (C2) 2 10

0 0 The first term on the r.h.s. of Eq. (C1) describes the change of Pt+dt([θ1; θ2]) 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 ˜ ˜ ˜ θ1,2 are defined via the self-consistent condition θ1,2 − Ω(θ1,2)dt = θ1,2, where Ω(θ) = 2Ωs [1 + λ sin θ], cf. Eq. (3) in 2 the manuscript. A differential equation for Pt(θ) is obtained by solving the self consistent equation to order (dt) , differentiating Eq. (C1) over θ2, and retaining the terms of order dt. With the explicit expressions for pr and Ω(θ), we get Eq. (6). The integro-differential master equation (6) needs to be supplemented by the boundary conditions. They are simple:

Pt(θ = 0) = Pt(θ = 2π). (C3)

In other words, Pt(θ) = Pt(θ +2π). With this condition, it is easy to check that the normalization by total probability 2π 0 dθPt(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, Eq. (6) is equivalent to

dPt(θ) 2 θ α = −α sin Pt(θ) + ∂θ (2Ωs + sin θ)Pt(θ) (C4) dt 2 2 at θ 6= π, supplemented with boundary condition

2π 2 θ 2Ωs (Pt(π + 0) − Pt(π − 0)) = −α dθ sin Pt(θ). (C5) ˆ0 2

2. Eigenmodes of Eq. (6)

2Ωsγt We now derive the eigenmode solutions of Eq. (6), i.e., ﬁnd all the solutions of the form Pt(θ) = e fγ (θ). The reader may skip the details and look at the result in AppendixC3. Before diving into the derivation, it is useful to analyze the expectations from the solution. Due to the normalization 2π condition, for every eigenmode with γ 6= 0, we should have 0 dθfγ (θ) = 0. Since Pt(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+ 2 θ+ 2 θ+ 2 θ+ 2 θ+ is p1(θ)/dt = α sin , for λ > 1 we expect an eigenmode with 2Ωsγ ≥ −α sin = −α tan / 1 + tan = √ 2 √ 2 2 2 2 2 2Ωs λ − 1 − λ , i.e., γ ≥ λ − 1 − λ. We also expect a steady state solution with γ = 0 to exist.

a. The functional dependence

Away from θ = π, the equation for the eigenmodes is

2 θ α − α sin fγ (θ) + ∂θ (2Ωs + sin θ)fγ (θ) = 2Ωsγfγ (θ). (C6) 2 2 Equivalently,

α 0 hα i (2Ωs + sin θ)f (θ) = (1 − 2 cos θ) + 2Ωsγ fγ (θ). (C7) 2 γ 2

Note that for α ≥ 4Ωs, at θ = θ±, the equation becomes singular as the factor multiplying the highest (and only) deriva- tive vanishes. These singular points require special treatment. However, this simply means that fγ (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

√λ+γ " θ #! θ θ+ ! λ2−1 C λ + γ λ + tan 2 π C tan 2 − tan 2 fγ (θ) = exp 2√ arctan √ − = , (C8) (1 + λ sin θ)2 1 − λ2 1 − λ2 2 (1 + λ sin θ)2 θ θ− tan 2 − tan 2 11 where λ = α/(4Ωs). The equivalence of the two expressions follows from 1 1 + ix arctan x = ln , (C9) 2i 1 − ix so one can write !y √ !y " !# tan θ − tan θ+ tan θ + λ − λ2 − 1 λ + tan θ π 2 2 = 2 √ = exp 2y arctan √ 2 − . θ θ− tan θ + λ + λ2 − 1 1 − λ2 2 tan 2 − tan 2 2

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, for computational purposes one can use either. Note the singularities at θ = θ+ and θ = θ− for λ > 1 (for θ± θ θ± θ λ < 1, tan 2 − tan 2 6= 0 for any θ since tan 2 has a non-zero imaginary part while tan 2 is a real function). From the equality

θ θ θ 2 θ tan θ − tan + tan θ − tan − 1 + 2λ tan 2 + tan 2 2 2 2 2 1 + λ sin θ = 2 θ = 2 θ , (C10) 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−

dθf(θ) < ∞, dθf(θ) < ∞. (C13) ˆ ˆ for Eqs. (C11, 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 6= 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 fγ (θ ∈ [−π; θ+]) = 0. Putting these results together, we have the general expression for the eigenmodes, which reads, for λ > 1,

 √λ+γ 2 2 θ C(1+tan θ ) tan θ −tan + λ2−1  2 2 2 θ 2 θ 2 θ , for θ ∈ [θ+; π], θ + θ − tan θ −tan − fγ (θ) = tan 2 −tan 2 tan 2 −tan 2 2 2 (C16)  0, for θ ∈ [−π; θ+]; and, for λ < 1,

" θ #! C λ + γ λ + tan 2 π fγ (θ) = exp 2√ arctan √ − , for θ ∈ (−π; π). (C17) (1 + λ sin θ)2 1 − λ2 1 − λ2 2 12

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 2 θ 2Ωs [fγ (π − 0) − fγ (π + 0)] = α dθ sin fγ (θ). (C18) ˆ−π 2 Note that this condition is independent of C, hence it determines the spectrum of eigenmodes γ. The normalization condition is ( 2π 1, for γ = 0, dθfγ (θ) = (C19) ˆ0 0, for γ 6= 0.

This should be used to determine C for the steady state (γ = 0) and should be satisﬁed automatically for all γ 6= 0 eigenmodes. The integrals can be calculated analytically

√λ+γ 2 θ+ ! λ2−1 1 + t t − tan 2 dθf(θ) = 2C dt 2 2 θ ˆ ˆ θ θ − + − t − tan 2 t − tan 2 t − tan 2 √λ+γ   θ+ ! λ2−1 2 C t − tan 2 λ (2λ + γ)(1 − 2γt) − γt = θ 1 +  , (C20) λ + γ t − tan − θ+ θ− θ+ θ− 2 t − tan 2 t − tan 2 γ − tan 2 γ − tan 2

θ where t = tan 2 . Similarly,

√λ+γ 2 θ+ ! λ2−1 2 θ t t − tan 2 dθ sin f(θ) = 2C dt 2 2 θ ˆ 2 ˆ θ θ − + − t − tan 2 t − tan 2 t − tan 2   √λ+γ 2 θ+ ! λ2−1 C 2 (γt − 1) 1 t − tan 2 = t +  θ . (C21) 2(λ + γ) θ+ θ− θ+ θ− t − tan − γ − tan 2 γ − tan 2 t − tan 2 t − tan 2 2

c. The solution for λ > 1 and θ ∈ [θ+; π].

λ√+Re γ > 1 Assuming λ2−1 , the boundary condition yields,

π 2 2 θ αC(2γ + 2λγ + 1) 2 2ΩsC = α dθ sin fγ (θ) = 2 ⇐⇒ Cγ(γ + γλ + 1) = 0. (C22) ˆθ+ 2 2(λ + γ)(γ + 2λγ + 1) This ﬁxes the possible γ, thus giving us three eigenmodes: 1 p γ = 0, γ = −λ ± λ2 − 4 . (C23) 2 The norm of the eigenmodes is then π C γλ C γ2 + γλ + 1 dθf(θ) = 1 − 2 = 2 , (C24) ˆθ+ λ + γ (γ + 2λγ + 1) λ + γ γ + 2λγ + 1 √ 1 2 which implies that the eigenfunctions with γ = 2 −λ ± λ − 4 integrate to 0 as expected. For the steady state, γ = 0 π dθf(θ) = 1 , the normalisation condition θ+ yields ´ C = λ, (C25) 13

1/ cos θ+ 2 θ 2 θ θ+ ! λ 1 + tan 2 tan 2 − tan 2 fγ=0(θ) = 2 2 θ θ− θ θ+ θ θ− tan 2 − tan 2 tan 2 − tan 2 tan 2 − tan 2 √ 2 √ !λ/ λ2−1 1 + tan2 θ tan θ + λ − λ2 − 1 = λ 2 2 √ . (C26) 2 θ θ 2 tan θ + λ + λ2 − 1 tan 2 + 2λ tan 2 + 1 2 Finally, we check the conditions in Eqs. (C11, C12) λ + Re γ p √ > 1 ⇐⇒ Re γ > λ2 − 1 − λ. (C27) λ2 − 1

Interestingly, the condition actually requires that all the eigenmodes decay not slower than the decay√ rate at θ = θ+. 1 2 The steady state, γ = 0, always satisﬁes the condition for λ ∈ (1; +∞). For λ ≤ 2, Re 2 −λ ± λ − 4 = −λ/2. Then, the normalizability condition is equivalent to p √ λ > 2 λ2 − 1 ⇔ 3λ2 < 4 ⇔ λ < 2/ 3. (C28) √ √ √ λ > 2 1 −λ − λ2 − 4 < 1 −λ + λ2 − 4 < λ2 − 1 − λ For , the inequality√ 2 2 √ always holds. Therefore, the 1 2 eigenmodes γ = 2 −λ ± λ − 4 are only normalizable for λ < 2/ 3.

d. The solution for λ < 1. √ 2 θ± The same considerations hold for 0 ≤ λ < 1. In this range of parameters, λ − 1 and tan 2 become complex. Therefore, it is more convenient to use fγ (θ) in the first form in Eq. (C8). At the same time, the integrals for the norm and the boundary condition at θ = π are more conveniently calculated using the expressions in Eq. (C16) (see AppendixC2b). Using these expressions and the results of AppendixC2b, one finds that the boundary condition at θ = π yields C γ2 + γλ + 1 λ + γ 1 − exp −2π √ = 0. (C29) λ + γ γ2 + 2λγ + 1 1 − λ2 √ 1 2 Therefore, on top of the three eigenmodes, γ = 0 and γ = 2 −λ ± λ − 4 , there is also an infinite sequence of eigenvalues given by p γ = −λ + im 1 − λ2, m ∈ Z. (C30)

Since there are no special points for λ < 1, both this inﬁnite set of solutions and the three eigenmodes in Eq. (C23) correspond to valid normalizable eigenmodes. π For the steady state, the normalization −π dθf(θ) = 1 yields ´ λ C = , (C31) 1 − exp − √2πλ 1−λ2

" θ !# λ 1 2λ λ + tan 2 π fγ=0(θ) = exp √ arctan √ − . 2 2 2 (C32) 1 − exp − √2πλ (1 + λ sin θ) 1 − λ 1 − λ 2 1−λ2

For γ 6= 0, π C γ2 + γλ + 1 λ + γ dθf(θ) = 1 − exp −2π √ , 2 2 (C33) ˆ−π λ + γ γ + 2λγ + 1 1 − λ which vanishes due to the boundary condition above, as expected. 14

3. Summary of eigenvalues and eigenmodes of Eq. (6)

Putting together the results from AppendixC2, we can summarize the solutions of the master equation (6) as follows. The steady state, with eigenvalue γ = 0, is given for λ < 1 by

" θ !# λ 1 2λ λ + tan 2 π f0(θ) = exp √ arctan √ − , 2 2 2 (C34) 1 − exp − √2πλ (1 + λ sin θ) 1 − λ 1 − λ 2 1−λ2 while for λ > 1, it reads √  √ λ/ λ2−1 tan θ +λ− λ2−1  λ 2 √ (1+λ sin θ)2 θ 2 , for θ ∈ [θ+; π], f0(θ) = tan 2 +λ+ λ −1 (C35) 0, for θ ∈ [−π; θ+], √ θ± 2 with tan 2 = −λ ± λ − 1, λ = α/(4Ωs), and √ √ !λ/ λ2−1 " !# tan θ + λ − λ2 − 1 2λ λ + tan θ π 2 √ = exp √ arctan √ 2 − . (C36) θ 2 2 2 2 tan 2 + λ + λ − 1 1 − λ 1 − λ These are the expressions used in the main text – cf. Eqs. (7,8) therein. 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, ( h i 1 exp − 2 , θ ∈ [−π/2; π], (1+sin θ)2 1+tan θ f0(θ) = 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: √ • γ = −λ + im 1 − λ2, m ∈ Z with " θ #! C λ + tan 2 π 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. √ 1 2 • γ = 2 −λ ± λ − 4 with √ 2 " θ #! C λ ± i 4 − λ λ + tan 2 π fγ (θ) = exp √ arctan √ − , for λ < 1, (C39) (1 + λ sin θ)2 1 − λ2 1 − λ2 2

√ 2 λ±√ λ −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 √ These eigenmodes disappear at λ > 2/ 3. This disappearance coincides with the steady state starting diverging at θ = θ+ + . √ 1 2 Finally, note that the eigenvalues γ = 2 −λ ± λ − 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 [42] 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. (6) does know about these eigenvalues for any λ, as we show in AppendixB2.