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 un- der 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 sys- tem 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] sys- tems. 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 j0i and j1i 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 2 (−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 j1i and j0i necessarily monitored via continuous partial measurements, quan- have a discontinuous counterpart, jumps between j0i and tum jumps can be anticipated, continuously monitored, j1i, 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 2 (0; π), we have Ω(θ) > 0, and the system evolves states j0i and j1i 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 j1i to j0i 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 j0i two possible readouts r = 0 (no-click) and 1 (click). The to j1i, instead, takes place via the region θ 2 (−π; 0) corresponding measurement back-action j i ! M (r) j i 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) p M = j0i h0j+ 1 − p j1i h1j ;M = p j1i h1j ; (1) evolves towards θ = −π; however, for λ > 1 there appear two fixed points, Ω(θ±) = 0, at where p 2 [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 j0i being frozen in one of the measurement eigenstates, where θ+ is a stable point, while θ− is an unstable one, or j1i. 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 j1i). 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 j (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) j (t + dt)i = M U j (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 j0i or j1i, its evo- (6) lution is constrained to the y–z section of the Bloch sphere and the state has the form j (t)i = j (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 j0i + i sin 2 j1i. 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 P1(θ) ≡ limt!1 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 j1i, while a “no-click” π s¯i(t) ≡ hσi(t)i = h (θ)j σi j (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.