PHYSICAL REVIEW RESEARCH 2, 043107 (2020)

Dark states of quantum search cause imperfect detection

Felix Thiel ,* Itay Mualem, Dror Meidan , Eli Barkai, and David A. Kessler Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel

(Received 20 June 2019; revised 24 August 2020; accepted 23 September 2020; published 20 October 2020)

We consider a quantum walk where a detector repeatedly probes the system with fixed rate 1/τ until the walker

is detected. This is a quantum version of the first-passage problem. We focus on the total probability Pdet that the particle is eventually detected in some target state, for example, on a node rd on a graph, after an arbitrary number of detection attempts. Analyzing the dark and bright states for finite graphs and more generally for systems with a

discrete spectrum, we provide an explicit formula for Pdet in terms of the energy eigenstates which is generically τ independent. We find that disorder in the underlying Hamiltonian renders perfect detection, Pdet = 1, and then expose the role of symmetry with respect to suboptimal detection. Specifically, we give a simple upper bound

for Pdet that is controlled by the number of equivalent (with respect to the detection) states in the system. We also extend our results to infinite systems, for example, the detection probability of a quantum walk on a line, which is τ dependent and less than half, well below Polya’s optimal detection for a classical random walk.

DOI: 10.1103/PhysRevResearch.2.043107

I. INTRODUCTION on the statistics of this event, which is inherently random by the basic laws of . Our approach explicitly Recently, the first detection problem for quantum dynam- incorporates repeated strong measurements into the definition ics has attracted increasing interest [1–23], due in part to of the first detected arrival and therein differs from other its potential relevance for the readout of certain quantum quantum search setups [1,35–52] and from the time-of-arrival computations. More fundamentally, it sheds light on hitting problem [53–61]. time processes and measurement theory in quantum the- In some cases, the quantum search is by far more efficient ory [24–28]. The classical counterpart of this topic is the than its classical counterpart. In particular, for the hypercube first-passage-time problem, which has a vast number of appli- and for certain trees it was shown that the quantum search can cations in many fields of science [29–34]. In its simplest guise, be exponentially faster than possible classically [2,62–64]. In- a classical random walker initially located on a particular deed, while the classical random walker repeatedly resamples vertex of a graph is considered, and the question of interest its trajectory, a quantum walker may benefit from the con- is the following: When will the particle arrive at a target state, structive interference of its . This mechanism for instance, another vertex of the graph, for the first time? enables a quantum walker to achieve much faster detection For the quantum system, we investigate unitary evolution on times than its classical counterpart. In the same way, however, a graph, with a particle in an initial state |ψ , which could in certain initial conditions suffer from destructive interference be, e.g., a localized state on a vertex |r . This evolution is in such that the desired state is never detected and yields a repeatedly perturbed by detection attempts for another state diverging mean detection time [2–4,6,18]. We call such ini- |ψ called the detection state (for example, another localized d tial conditions dark states. The terminology dark states is vertex state of the system |r , discussed below). In this sit- d borrowed from and where it uation, the concept of first arrival is not meaningful, but we describes forbidden transitions or nonemissive states [65,66]. can register the first detected arrival time. The protocol of The classical random walk, if the process is ergodic, does not measurement (i.e., the epochs of the detection attempts) cru- possess dark states and hence in this sense performs “better,” cially determines this first detection time [9,18]. We consider since detection on a finite graph is guaranteed. In the quan- a stroboscopic protocol, i.e., a sequence of identical mea- tum problem, the presence of the detector splits up the total surements with fixed inter-attempt time τ, continued until the into a dark space [2,7,67] and its complement. first successful detection (see definitions below). One of the These play the roles of the ergodic components in a classical general aims in this direction of research is to gain information random walk. In contrast to the classical situation, they are not generated by a separation of state space (alone) but rather by destructive interference. *[email protected] The main focus of this paper is the total detection probabil- ity Pdet. This is the probability to detect the particle eventually, Published by the American Physical Society under the terms of the namely, the detection probability in principle after an infi- Creative Commons Attribution 4.0 International license. Further nite number of attempts (though the measurement process is distribution of this work must maintain attribution to the author(s) stopped once the particle is detected). In a finite system, if the and the published article’s title, journal citation, and DOI. initial and detection states coincide, Pdet is always unity [9].

2643-1564/2020/2(4)/043107(16) 043107-1 Published by the American Physical Society FELIX THIEL et al. PHYSICAL REVIEW RESEARCH 2, 043107 (2020)

However, when the initial state differs from the detection state, the initial state can have an overlap with some dark states, which are undetectable, and the overlap of the initial state with the dark space gives no contribution to Pdet. We derive an explicit formula for Pdet in terms of the eigenstates |El,m of the unitary propagator Uˆ (τ ):= e−iτHˆ /h¯ (or the Hamiltonian Hˆ ), the mth state with quasienergy λl = τEl /h¯ mod 2π, such −iλ that Uˆ (τ )|El,m=e l |El,m: gl 2 = ψd|El,mEl,m|ψin P (ψ ) = m1 . (1) det in gl 2 |E , |ψ | l m=1 l m d This is the first main result of this paper. For most values of τ, there is a one-to-one correspondence between the set of values of El and those of λl . Only if there are one or more pairs of energies {Ek, El }, such that the resonance condition

(Ek − El )τ/h¯ = 0mod2π (2) is satisfied, are there fewer quasienergies than energies. The primed sum runs over those distinct quasienergy sectors that have nonzero overlap with the detected state (i.e., excluding any completely dark sectors) with the inner sum running over the gl degenerate states of quasienergy level λl . Thus, except for the zero-measure set of resonant values of τ, Pdet does not depend on the detection period at all. However, at these resonant τ, Pdet changes dramatically [68]. This formula is obviously invariant under a change of basis within any quasienergy sector. We give two different derivations of this FIG. 1. Upper bound for some simple graphs showing the de- formula, one based on the quantum renewal formula [9,18] viations of detection probability Pdet from the classical counterpart, for the probability of detection after n measurements and a which is unity. The numbers represent the bound for Pdet from more elementary proof based on an explicit formula for the Eq. (47). An open circle denotes the detection site rd, and any other bright and dark states in terms of the propagator’s eigenstates. node is a possible localized initial state |ψin=|rin. The quantum This proof requires showing that, on a finite graph, all states particle resides on the nodes of these graphs and travels along its that are orthogonal to the dark space are bright, that is, they links, which are all identical (Hˆ is the adjacency matrix of the graph). are detected with probability one, and so Pdet is equal to the In all graphs, the on-site energies are equal to zero. Shown from initial state’s overlap with the bright space. Both our renewal left to right and top to bottom are the ring of size 6; the hypercube formula derivation and this latter proof break down in the of 3; a two-dimensional simple cubic lattice; the square cases of infinite graphs, where the spectrum has a continuous graph with a detection site in the center, in a corner, and in a corner part. We will discuss this point in detail in the context of the with one modified link; the binary tree graph in two generations with infinite line. detection in the root, middle, and leaves; the complete graph with While Eq. (1) is the exact solution to the problem, insight is eight sites; the Star-of-David graph; and the tree-of-life graph. The found addressing the symmetry of H and for systems whose infinite line is shown at the very bottom. The numbers are upper Hamiltonian is given by an adjacency matrix, the symmetry bounds for Pdet which are easy to obtain and can be compared with the exact result (1). of the graph. Krovi and Brun [2] already showed that the total detection probability is suboptimal, or in their language the hitting time is infinite, if the system exhibits symmetry. Indeed splitting of the Hilbert space into bright and dark subspaces in as indicated by Eq. (2), plays an important role Sec. IV and use this in Sec. V to rederive our main formula. in the evaluation of Pdet and hence the symmetry properties Examples are discussed in Sec. VI. The case of the infinite of the underlying unitary are crucial. In Sec. VIII line is analyzed in Sec. VII. In Sec. VIII we provide an of this paper, we obtain a remarkably simple upper bound upper bound exploiting the symmetry of the underlying graph. Pdet  1/ν, where ν is the number of distinct sites of the A brief summary of some of our results was presented in graph equivalent to the initial localized initial state. Briefly, Ref. [21]. that means that we search for ν graph nodes that are equivalent with respect to the detector (which is also localized on a node) II. STROBOSCOPIC DETECTION PROTOCOL and this number yields the mentioned bound on the detection probability (see Fig. 1 for details). More general symmetry One does not simply observe the first arrival of a quan- consideration will follow. tum particle in some target state |ψd, because it does not The rest of this paper is organized as follows. In Sec. II we have a trajectory in the classical sense. The measurement, will introduce our model. Then we derive our main result, first i.e., the attempted detection, must be explicitly incorporated using the renewal formula in Sec. III. After that we discuss the into the dynamics. Following Refs. [9,10,12,13,16–20], this

043107-2 DARK STATES OF QUANTUM SEARCH CAUSE IMPERFECT … PHYSICAL REVIEW RESEARCH 2, 043107 (2020) can be done by adhering to the stroboscopic detection pro- Sn(ψin) → 0asn →∞. Of course we may also have states tocol, where detection in state |ψd is attempted at times that are neither dark nor bright. τ,2τ,3τ,.... The detection period τ between the detection Our theory is developed in generality, valid for any finite- attempts is a parameter of the experimentalist’s choice. The dimensional Hamiltonian Hˆ . Besides the initial and detection experiment we have in mind follows this protocol. states, Hˆ and the detection period τ are the ingredients that 1. Prepare the system in state |ψ at time t = 0 and set enter the stroboscopic detection protocol via the evolution in ˆ τ = −iτHˆ /h¯ n = 0. operator U ( ): e . We assume that a diagonalization 2. The system evolves unitarily for time τ with the evo- of the latter is available: −iτHˆ /h¯ lution operator Uˆ (τ ):= e ; the wave function is then gl − −iλl |ψ(nτ + τ )=Uˆ (τ )|ψ(nτ ).[The− (+) superscript de- Uˆ (τ ) = e Pˆl , Pˆl := |El,mEl,m|. (7) notes the limit from below (above)]. Increase n by one, n = l m=1 n + 1. | 3. Attempt to detect the system in the state |ψd with a Here El,m are the eigenstates and Pˆl are the eigenspace pro- strong collapsing measurement. jectors, which gather all eigenstates of the gl -fold degenerate − quasienergy level λ = τE /h¯ mod 2π, so that all λ in Eq. (7) a. With conditional probability Dˆ |ψ(nτ )2 = l l l − 2 are distinct. This form is easily obtained from a similar diago- |ψd|ψ(nτ )| , the system is successfully detected. Here gl nalization of the Hamiltonian Hˆ = El = |E , E , |. Dˆ =|ψ ψ | (3) l m 1 l m l m d d For convenience, we label the states by their energy El , rather than by their quasienergy λl , since we focus on nonresonant is the projector onto the detection state. The detection time is τ t = nτ and the experiment ends. values of . Although the only effect of τ on Pdet is possible discontin- b. Otherwise, the measurement fails to detect the system uous changes at resonant τs, the detection period has a more in the target state. The wave function is instantaneously pro- profound effect on other quantities such as the mean detection jected to the state that has no overlap with the detection state time [18,70]. It is an important parameter of the stroboscopic |ψ d . This is the collapse postulate [69]. Mathematically, the detection protocol. In the limit τ → 0, when the system is wave function directly after the unsuccessful measurement observed almost continuously, the Zeno effect freezes the |ψ τ + = 1 − |ψ τ − is equal to (n ) Nn[ Dˆ ] (n ) , where Nn is a dynamics [71–74] and detection can become impossible. normalization constant and 1 is the identity operator. After constructing the new wave function, jump back to step 2. This III. FROM THE RENEWAL EQUATION TO THE TOTAL loop is repeated until the system is finally detected in step 3a. DETECTION PROBABILITY After following this procedure many times, one may con- One proof of Eq. (1) starts from the first detection am- struct a histogram for the first successful detection number n. plitudes ϕn [17], which in turn yield the first detection As shown by Dhar et al. [12], the overall probability of =|ϕ |2 = ∞ probabilities Fn n , and so Pdet n=1 Fn. The gener- detection at measurement n is ϕ = ∞ nϕ ating function for these amplitudes (z) n=1 z n can be n−1 2 Fn =DˆUˆ (τ )[(1 − Dˆ )Uˆ (τ )] |ψin (4) expressed in terms of Uˆ (τ ) (for details, see Refs. [17,18]): and the probability of no detection in the first n measurements Uˆ (τ ) ψd| |ψin ϕ = 1−zUˆ (τ ) . is (z) z 1 (8) n ψ | |ψ d 1−zUˆ (τ ) d n 2 Sn(ψin) = 1 − Fm =[(1 − Dˆ )Uˆ (τ )] |ψin m=1 This formula is obtained from the quantum renewal equation: n 2 a basic tool for the derivation of the detection amplitude =Sˆ |ψin , (5) [9,18]. As shown in [18], Pdet can be obtained from ϕ(z), ˆ = 1 − ˆ ˆ τ where S : ( D)U ( ) is the survival operator. The de- ∞ 2π ∞ ∗ 1 ∗ iθ(n−m) pendence on the initial state is stressed in the notation. P = δ , ϕ ϕ = dθ ϕ ϕ e det m n m n 2π m n The dependence on the detection state, however, will be m,n=1 0 m,n=1 suppressed throughout the article. Clearly, Sn involves n com- 1 2π pound steps of unitary evolution followed by unsuccessful = dθ|ϕ(eiθ )|2, (9) detection. The main focus of this paper is the total detection 2π 0 probability, the probability to be eventually detected, i.e., the probability to “not survive”: i.e., as the integral of the generating function’s modulus on the ∞ unit circle. Using the quasienergy representation of the evolution oper- Pdet(ψin) = Fn = 1 − lim Sn(ψin). (6) n→∞ ator (7), the generating function (8) is expressed as the fraction n=1 of two expressions An initial state |ψin that is never detected is called a dark − τ / |ψ ψ | ˆ |ψ ze i( El h¯) state with respect to the detection state d ; for these states d Pl in −i(τE /h¯) (Kμν)(z) ϕ = l 1−ze l = , Pdet(ψin) = 0 and Sn(ψin) = 1 for all n. Similarly, a bright (z) z (10) ψd|Pˆl |ψd μ − τ / (K )(z) state is detected with probability one, i.e., Pdet(ψin) = 1, and l 1−ze i( El h¯)

043107-3 FELIX THIEL et al. PHYSICAL REVIEW RESEARCH 2, 043107 (2020) where (Kμ)(z) is the so-called Cauchy transform of the func- Reference [20] discusses μ(λ) in depth for systems with a tion μ(θ ), 0  θ<2π, defined by [75] continuous spectrum. [Note that this reference defines ν(λ)in 1 2π μ(θ ) a slightly different way.] This proof does not make any explicit (Kμ)(z) = dθ . (11) reference to dark or bright states (except for the exclusion of 2π 1 − ze−iθ 0 dark levels) and so is not physically transparent. We therefore In Ref. [20] we showed that the denominator of ϕ(z)is present an alternate proof in the next section. a Cauchy transform of the so-called wrapped measurement spectral density of states, IV. PARTITION OF HILBERT SPACE μ θ = πψ |δ −iθ − τ |ψ ( ): 2 d (e Uˆ ( )) d A. Dark states In this section we discuss the partition of the Hilbert space = 2π ψ |Pˆ |ψ δ(θ − λ ), (12) d l d l into a bright and a dark part. In Sec. II we defined dark states l as those with F = 0 and S = 1 for all n. In particular, we fo- λ n n where l is the phase corresponding to the lth quasienergy cus on stationary dark states, which are invariant under unitary λ = τ / π λ level: l : El h¯ mod 2 .All l are distinct. evolution as well as under the detection attempts. During the The numerator is a Cauchy transform as well, but of a course of the detection protocol, only their phase is affected. product of functions μ(θ )ν(θ ): Hence they remain dark for all times. −iλ ψ |Pˆ |ψ e l In view of the diagonalization (7), there are two ways dark μ θ ν θ = ψ | ˆ |ψ δ θ − λ d l in . ( ) ( ): d Pl d ( l ) states can arise. ψd|Pˆl |ψd l (i) Completely dark quasienergy levels. Consider a =:ν(λl ) quasienergy level El that has no overlap in the detection (13) state. That means none of the level’s eigenstates overlaps Almost all values of the functions ν(θ ) are irrelevant, except with |ψd, i.e., Pˆl |ψd=0orEl,m|ψd=0form = 1,...,gl . at the specific arguments λl , where they are fixed, ν(λl ) = Alternatively, one can write Dˆ |El,m=0. We also define − λ i l |δ =| e ψd|Pˆl |ψin/ψd|Pˆl |ψd.[Duetotheδ functions in μ(θ ), these states as l,m El,m . If we take one of these as these are the only significant values of ν(θ ).] The primed an initial state, we have Sˆ |δl,m=(1 − Dˆ )Uˆ (τ )|δl,m=(1 − −iτE /h¯ −iτE /h¯ sum excludes all completely dark energy levels, i.e., those Dˆ )e l |δl,m=e l |δl,m. Thus, |δl,m is an eigenstate for which Pˆl |ψd=0, so ψd|Pˆl |ψd does not vanish in the of the survival operator with an eigenvalue on the unit circle, n 2 denominator of ν(θ ). Putting the numerator and denomi- and so Sn(δl,m ) =Sˆ |δl,m = 1. Thus, all the |δl,m are nator together, we find that ϕ(z) = z(Kμν)(z)/(Kμ)(z) =: dark states. We call quasienergy levels that do not appear z(Vμν)(z) is a so-called normalized Cauchy transform [75]. in a decomposition of the detection state completely dark For objects like this, Aleksandrov’s theorem allows one to quasienergy levels. All gl associated eigenstates are dark. compute the total detection probability (see [75], Proposition (ii) Degenerate energy levels. Consider now an quasienergy 10.2.3) level El that does overlap with the detection state, but which 2π 2π > 1 1 is degenerate, such that gl 1. Construct the projection of the = λ| V ν iλ |2 = λμ λ |ν λ |2, Pdet d ( μ )(e ) d ( ) ( ) detection state on this sector, 2π 0 2π 0 (14) ˆ |ψ |β = Pl d . where we used Eq. (9) for the first equality. An explicit ex- l : (17) ψ |Pˆ |ψ ample of how Aleksandrov’s theorem works in practice for a d l d two-level system in given in Appendix A. It will turn out that |βl is a bright state. All states within Since μ(θ )isjustasumofδ functions at the quasienergies this sector which are orthogonal to |βl are orthogonal to the λl and ν(λl ) is known, the required integral is easily deter- detector, since if |δl is such a state, mined: ψd|δl =ψd|Pˆl |δl ∝βl |δl =0 (18) 1 2π P = dλμ(λ)|ν(λ)|2 − det 2π and so are dark. These states constitute a (gl 1)-dimensional 0 subspace of this quasienergy sector, any state lying within 2 ψ |Pˆ |ψ which is dark. It can be convenient to have an explicit basis −iλl d l in = ψd|Pˆl |ψde ψ |Pˆ |ψ for this subspace, which can be generated by a determinantal l d l d formula similar to that used in the Gram-Schmidt procedure. gl 2 | ψ |E , E , |ψ | We define = m=1 d l m l m in . (15) gl ψ | |ψ = d El,m El,m d |E1|E2 ··· |E j+1 l m 1 ψ | ψ | ··· ψ | We have thus obtained Eq. (1). d E1 d E2 d E j+1 It is important to note that Aleksandrov’s theorem in this |δ = δ |E δ |E ··· δ |E + , l, j Nj 1 1 1 2 1 j 1 form only applies to operators with a point spectrum. If the . . . . . . .. . spectrum has a continuous component, then Aleksandrov’s theorem turns into an inequality (see [75], Proposition 10.2.3) δ j−1|E1δ j−1|E2 ··· δ j−1|E j+1 and we have (19) 2π 1 2 where Nj is a normalization factor and the subscript l was Pdet  dλμ(λ)|ν(λ)| . (16) 2π 0 omitted on the right-hand side. This determinant has the ob-

043107-4 DARK STATES OF QUANTUM SEARCH CAUSE IMPERFECT … PHYSICAL REVIEW RESEARCH 2, 043107 (2020) vious properties that (i) it yields an eigenstate of Uˆ (τ ); (ii) This relation shows that zp is a (finite) pole of ϕ(z) if and only it is orthogonal to |ψd, since ψd|δl, j is a determinant with if ζl = 1/zp is an eigenvalue of Sˆ , but not of Uˆ (τ ). These are two identical rows; and (iii) it is orthogonal to all |δl,m with exactly the eigenvalues of Sˆ that lie inside the unit disk. By m < j. Likewise, δl,m|δl, j is a determinant with two identical construction, ϕ(z) is analytic in the unit disk. Therefore, all of rows and so vanishes. its poles must lie outside the unit disk and have |zp| > 1. Con- This formula gives sequently, all of these nontrivial eigenvalues of Sˆ , which are the poles’ reciprocals, must lie inside the unit circle, |ζl | < 1. |δ , =N (ψ |E , |E , −ψ |E , |E , ) (20) l 1 1 d l 2 l 1 d l 1 l 2 The corresponding eigenvectors belong to the complement of and then recursively computes |δl,m from all previous |δl, j. the dark space, i.e., the bright space. These vectors span the The result is a set of gl − 1 normalized and mutually or- bright space, and it is clear that any superposition of bright |β˜ = |β thogonal stationary dark states. As |δl,m is constructed only eigenstates l bl l will yield an exponentially decay- β˜ = from the eigenstates |El,1, |El,2,...,|El,m+1, we find that ing survival probability. So S∞( ) 0 and the state is bright, many of the δl,m|El, j in Eq. (19) vanish, namely, all those as claimed. The decay rate is determined by the eigenvalue with j > m + 1. Consequently, the matrix in Eq. (19)islower ζmax closest to the unit circle. triangular except for the first rows. This enables us to compute There are two subtleties in this argument. One must allow the determinant explicitly: for the possibility that there are degenerate eigenvalues of Sˆ and the operator is not diagonalizable. Showing that |β˜ is m 2 ∗ [|α , | |E , + −α α , |E , ] j=1 l j l m 1 l,m+1 l j l j bright in this case is a bit technical, and we present the argu- |δl,m= . (21) m m+1 ment in Appendix B. The other point to be raised is that if the |α |2|α |2 j=1 j=1 l, j l, j system is infinite, the eigenvalues might approach arbitrarily closely to the unit circle and there might not be an exponential Here we abbreviated αl,m =El,m|ψd. Equation (21) contains Eq. (20) as a special case. decay of the survival probability. This indeed happens, and in such systems, states in the complement to the dark space are not necessarily bright. This will be discussed in more detail in B. Bright states Sec. VII. {| }gl |β We have seen above how dark states arise and constructed The just-described transition from El,m m=1 to l and − {|δ }gl 1 a set of stationary dark states, which not only are energy l,m m=1 is a change from one orthonormal basis to another. eigenstates but (unit modulus eigenvalue) eigenstates of the Still, all involved states are eigenstates and are thus invariant survival operator. We have also seen that each degenerate level under Uˆ (τ ). The special feature of the new representation that was not completely dark yielded an energy eigenstate |βl is that each individual stationary dark state and all bright which we claimed was not only not dark, but was bright. Sim- eigenstates together are additionally invariant under the de- ilarly, every nondegenerate level that is not completely dark tection process. In the language of Refs. [7,67], HB and HD also turns out to be bright, so |El =|βl . These bright states, are so-called invariant subspaces. The action of the survival while energy eigenstates, are not in general eigenstates of the operator Sˆ may change a particular superposition of bright |β |β˜ = |β survival operator. Together, the l states, one arising from eigenstates l bl l into some other superposition of each not-totally-dark level, span the orthogonal complement |βl , but it can never generate a dark state. of the dark space. We now demonstrate that they are indeed bright. It was V. Pdet CALCULATED FROM THE BRIGHT SPACE already mentioned that the stationary dark state |δl,m is a − τ / (right) eigenstate of Sˆ with eigenvalue e i El h¯ , lying on the We see from the above discussion that, as noted by Krovi unit circle. It is easy to see that these are the only eigenvalues and Brun [2], the limit limn→∞ Sn(ψin) of an arbitrary initial on the unit circle, since Dˆ must annihilate it. Since Sˆ is the state’s survival probability is equal to its overlap with the ˆ |ψ 2 product of a projector with a unitary matrix, it can have no dark space, PHD in , at least for a finite system. Comple- eigenvalues outside the unit disk, and so all other eigenvalues mentarily, the total detection probability must be equal to the ζ must lie inside the unit disk, i.e., |ζ | < 1. overlap with the bright space. ˆ = In fact, these eigenvalues are directly related to the poles The projector onto the bright space has the form PHB ϕ |β β | of the generating function (z) discussed above. We may l l l , where the sum excludes all completely dark en- ˆ = 1 − ˆ ˆ τ now use the definition S : ( D)U ( ) of the survival ergy levels, so that they do not possess a bright state, and |βl ζ 1 − operator and rewrite its characteristic polynomial as det[ are the bright eigenstates of Eq. (17). Identifying Pdet(ψin) = ˆ τ +|ψ ψ | ˆ τ ψ | ˆ |ψ U ( ) d d U ( )]. An application of the matrix determi- in PHB in , we find nant lemma yields |ψ | ˆ |ψ |2 −1 d Pl in det[ζ 1 − Sˆ ] = det[ζ 1 − Uˆ (τ )]ψ |[ζ 1 − Uˆ (τ )] |ψ . Pdet(ψin) = . (24) d d ψ |Pˆ |ψ (22) l d l d ϕ The last term can be identified with the denominator of (z) Plugging the definition of the eigenspace projectors Pˆl that of Eq. (8). Inverting the equation yields is found in Eq. (7) into Eq. (24) yields Eq. (1). Note that the sum only runs over those quasienergy levels El which are 1 det[ζ 1 − Sˆ ] ψ | |ψ =ζ . (23) not completely dark, so the denominator in Eq. (1) can never d 1 − 1 τ d ζ 1 − τ ζ Uˆ ( ) det[ Uˆ ( )] vanish.

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We can now make the following observations. Picking the localized states |rin and |rd=|L as initial and (a) When the initial state coincides with the detection state, detection states, we find, using Eq. (1), P = we get det 1, in agreement with Ref. [9]. This means that L/2−1 |ψ 2 2 2πlr r = L , L the detection state d itself is always bright. = + 2 in = 1forin 2 Pdet(rin) cos 1 . (b) The total detection probability does not depend on the L L L 2 otherwise τ l=1 sampling rate; Pdet is independent. This is true except for (28) τ the resonant detection periods c, defined in Eq. (2). At these For almost all sites, we find Pdet = 1/2 except when rin and points the number of bright states can suddenly change and rd coincide or are on opposing sides of the ring. This is in Pdet may exhibit a discontinuous drop. accordance with our results from Ref. [18]. The same result (c) In systems with only nondegenerate quasienergy levels appears for rings of odd sizes, although it is not possible to |ψ = that all overlap with the detection state, i.e., El d 0, any place initial and detection sites on opposite sites of the ring. initial state is detected with probability one. In this case there The energy basis bright states are given by the cosine are no dark states; all eigenstates are bright. Such behavior waves, whereas the stationary dark states are given by sine is classical, in the sense that a classical ergodic random walk waves also finds its target with probability one. It implies that some L 2πlr 2 L 2πlr disorder in the system, which removes all the degeneracy, |β =N cos |r, |δ = sin |r, may increase the detection probability provided none of the l l L l L L r=1 r=1 eigenstates is orthogonal to the detection state. In systems √ √ (29) with degenerate quasienergy levels, one may still be able to where Nl = 1/L for l = 0, L/2 and Nl = 2/L otherwise; find initial states which are detected with probability one, but |δ0 and |δL/2 are not defined. Thus, in this case, the devi- this will not be true generically for all initial states. ation of Pdet from unity is solely due to the degeneracies of the energy spectrum, which of course arise from the symmetry of the system. VI. EXAMPLES

Let us now consider some examples that demonstrate the B. Ring with a magnetic field relation of P to the dark and bright states. The examples all det In the ring, as we saw, the degeneracy of an consist of graphs of nodes connected by links, which represent gives rise to dark states, which in turn lead to a deficit in the hopping between the connected nodes, and we consider only total detection probability. What happens if this degeneracy is initial and detection states which are localized on individual lifted? To explore this, we add to the ring model of Eq. (25)a nodes. In all cases but the ring with a magnetic field, we take magnetic field the hopping strengths to be uniform, with value γ , and zero on-site energies. We also assume that τ avoids the resonant L iα −iα condition (2). Hˆ =−γ (e |rr + 1|+e |rr − 1|). (30) r=1 The magnetic field strength is proportional to α and its vector A. Ring is normal to the plane in which the ring lies. This field splits Let us now consider a ring with L (even) sides. Its Hamil- the twofold degeneracy of the energy levels. The eigenstates tonian takes the form are still the free wave states of Eq. (27), although the index l now runs from zero to L − 1. The new levels are L ˆ =−γ | + |+| − | . 2πl H [ r r 1 r r 1 ] (25) El =−2γ cos + α . (31) r=1 L Except for special values of α, namely, when α is an integer We employ periodic boundary conditions and identify |r + multiple of 2π/L, these are all distinct. Similarly, a degenerate L=|r. quasienergy level can only appear when the condition (2)for The free wave states diagonalize the Hamiltonian such that resonant τ is fulfilled. Since a localized detection state |rd the energy levels are given by has overlap with all eigenstates and since all energy levels are nondegenerate, we find 2πl El =−2γ cos , (26) P (r ) = 1, (32) L det in with the exception of special combinations of τ and α.This with l = 0, 1,...,L/2. Except for E and E / , which are 0 L 2 is nicely demonstrated in Fig. 2, Pdet as a function of τ and nondegenerate, all energy levels possess two eigenstates α, as well as the energy levels as a function of α for a ring with L = 6. We find that the total detection probability is L |r L (−1)r|r almost everywhere unity except along some lines in (α,τ) |E0= √ , |EL/2= √ , L L space which parametrize the resonant combinations at which r=1 r=1 − (27) degeneracy occurs. What is presented is 1 S50, rather than L π / | L − π / − ei(2 lr L) r e i(2 lr L)|r 1 S∞. Thus, the lines have finite width. Nevertheless, the |El,1= √ , |El,2= √ . extreme sensitivity to the presence of any symmetry breaking L L r=1 r=1 is apparent.

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FIG. 2. Total detection probability for a ring with a magnetic = − = field. Here Pdet was approximated by Pdet 1 SN , with N 50. FIG. 4. First detection probability Fn for an L = 6 ring with = = − − The initial and detection states are rin 1andrd 0, respectively. magnetic field parameter α = 0, 10 4, and 10 3. The behavior for Shown on the bottom are the energy levels of Eq. (31) as a function small n is indistinguishable in the three cases. Past some nc (α), as of α. Degeneracy occurs when they cross, which occurs for special marked in the figure, the finite α cases diverge from that of α = 0, α values of . Shown on top is the total detection probability as a decaying at a much slower rate. Here rd = 0, rin = 1, and γτ = h¯. function of τ and α. We usually find Pdet = 1, but a deficit occurs when there is a degeneracy in the quasienergy levels which appears as α dark lines in the figure. This happens for certain values of (vertical Fn, as depicted in Fig. 4. We see that, for the small values of α lines, coinciding with the intersection of two energy levels in the depicted, Fn is essentially independent of α for not too large n. τ α bottom) and for special combinations of and .AsN is taken to For small finite α and large n, Sn decays very slowly, as a result infinity, the dark lines will become infinitely thin. of the presence of slow modes of Sˆ . These modes, which for α = 0 were dark states with unit modulus eigenvalue, now are at a distance of order α2 from the unit circle, leading to This can be quantified by considering the value of n / for 1 2 a decay rate of order α2. These modes have to contribute a which S falls below its α = 0 limit of 1/2 (for the detection n total of 1/2toP . Given their slow O(α2 ) exponential decay, site not identical or diametrically opposite the initial site). det α = once this slow decay sets in (at nc, say), Fn must behave as This is plotted in Fig. 3, where even for the tiny value of ∼ − α2 − α −6 Fn Fnc exp[ d (n nc )], for some -independent d, with 10 the presence of a magnetic field can be unequivocally ∼ α−2 α Fnc ,togivean -independent sum. Thus, in Fig. 3 we detected after a series of runs of 50 measurement attempts. −4 see that Fn for α = 10 is roughly a factor of 100 smaller We see that n / grows only logarithmically with diminishing −c 1 2 than for α 3. Since, for n < n , the decay rate of F is set by α, so detection of very small magnetic fields is in principle c n the slowest nondark state (with an α-independent decay rate possible. This can be understood by looking at the behavior of ∼ − d0), we get that Fnc exp( d0n), so

nc ∝−ln α. (33) 50 Since, for α = 0, Pdet = 1/2, 1 − Sn will first exceed this value at an n1/2 of order nc. 40

C. Square with center node 30 1/2 We next consider a square with a single node in the middle n connected to all the others. The states localized on the corners 20 of the square are denoted by |1, |2, |3, and |4 and the additional node in the center of the diagonals is state |0.The Hamiltonian in matrix form is 10 ⎛ ⎞ 01111 -6 -5 -4 -3 -2 ⎜ ⎟ 10 10 10 10 10 ⎜10101⎟ ˆ =−γ ⎜ ⎟. α H ⎝11010⎠ (34) 10101 11010 FIG. 3. Plot of n1/2,thefirstn for which Sn(α) < Sn(α = 0) = 1/2, as a function of α, for a ring of size L = 6, with rd = 0, rin = 1, √ √ and γτ = h¯. A straight line is shown indicating the linear behavior The energy levels are E1 =−(1 + 5)γ , E2 = ( 5 − 1)γ , in ln α for small α. E3 = 2γ , and E4 = 0 and E4 is twofold degenerate. To find

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(a) (b) (c) (d) (e) (f)

FIG. 5. Total detection probability for graphs. A quantum particle is put on a graph whose node’s describe the particle’s possible position states. The links describe allowed transitions, each with equal strength. The particle starts localized on some node of the graph (closed circles) and we attempt to detect it on the node with the open circle. The numbers denote Pdet for this initial state. (a) Square with a center node, (b) ring of L sites, (c) complete graph with L sites, (d) star graph with L sites in the periphery, (e) hypercube (here of dimension d = 3, and (f) binary tree with two generations.

P we obtain the eigenvectors and the numbers next to them represent the corresponding det √ √ total detection probability. It is interesting to note that in all T |E1=( 5 − 1, 1, 1, 1) / 10 − 2 5, these simple cases, P is rational and, except for the tree, is det √ √ simply one over an integer. The detection probability for some T |E2=(−1 − 5, 1, 1, 1, 1) / 10 + 2 5, larger systems is discussed in Ref. [21]. (35) We note that after infinitely many unsuccessful detection |E =(0, −1, 1, −1, 1)T /2, 3 attempts only the dark component of the initial wave function T |E4,1=(0, 1, −1, −1, 1) /2, survives. The dark space is invariant under unitary evolution and strong detection [67,76]. Similar behavior can be found | = , , , − , − T / . E4,2 (0 1 1 1 1) 2 in the long-time behavior of general open Notice that, as opposed to the ring, where there were no [77]. An interesting question would be which of the features described here will carry over when one employs weak mea- completely dark levels, here there are two energy levels E3 and surements or open quantum dynamics [23,24,27,78–80]. E4 which are completely dark with respect to a measurement in the center. These sectors are dark because these states are π/ | not invariant under rotation by 2 (where E3 transforms to VII. INFINITE LINE −|E3 and |E4,1 transforms to −|E4,2), whereas the center is, and so there can be no overlap with the center. As per our We have seen above that finite systems are very different general discussion, this gives rise to a deviation of Pdet from from infinite systems, at least as far as the properties of Pdet unity when measuring at exactly this node, i.e., |rd=|0. are concerned. Grünbaum et al. [9] showed that when the The bright eigenstates are given by the projections of the detection and initial states coincide, Pdet is unity for every detection state into the energy subspaces; hence |β1=|E1 finite system; for an infinite system with a band of continuous and |β2=|E2. As mentioned, the energy levels E3 and E4 energies, such as the infinite line, it is always less than unity are completely dark and will be excluded in the sum of Eq. (1) [18]. We wish here to amplify this point by comparing the which yields finite ring of size L with its infinite counterpart. Above we exhibited the dark states for the finite ring. It is clear that the 1, r = 0 P (r ) = in (36) dark states remain dark even in the infinite-L limit. However, det in 1 , r = 0. 4 in the bright states of the finite ring, while remaining orthogonal For other detection states, say, rd = 1, there are no dark to every dark state, are not detected with unit probability in the energy levels, as other initial states are the same up to a infinite-L limit. Instead, they are “dim,” such that 0 < Pdet < 1 rotation, but E4 possesses√ one dark state due to degeneracy: for these states. Furthermore, Pdet for these states depends τ |δ4 := (|E4,1−|E4,2)/ 2. All energy levels participate in essentially on . To get insight into this phenomenon, we first consider the Eq. (1): 1, r = 0, 1, 3 space-time picture of the undetected probability density (i.e., P (r ) = in (37) the position distribution at time nτ, normalized to S ). We det in 1 , r = 2, 4. n 2 in start with the particle at x = L/2 on a ring of large length L, measuring at the same point. As time progresses, the density D. Additional examples spreads out ballistically from the initial location in both di- We have worked out a number of additional examples rections. As long as not enough time has passed to allow the where Pdet < 1 and the associated dark states for other simple two “wings” of the distribution to meet at the opposite end geometries of graphs. These include a complete graph, a star of the ring (x = 0 = L), a time of order L, the density has a graph, a hypercube, and a tree graph. The results are presented overall amplitude proportional to 1/t so that, given the ex- in Appendix C and summarized together with the previous tension over a distance proportional to t, the total undetected examples in Fig. 5. The detection node is represented by an density is of order unity. It should be noted that this is the open circle; all other (closed) circles are possible initial states same exact scaling behavior of the measurement-free density.

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2.5 that symmetry plays an important role. Take the example in Fig. 5(b), the ring. The probability to detect the particle once n = 238 n = 476 starting at one of the nearest neighbors of the detected site is 2 1/2. This, as we claim below, is related to the fact that we have two nearest neighbors on the ring structure. On the other 1.5 hand, starting on the site opposing the detected site yields = (x) Pdet 1. We claim that this is due to the absence of another n equivalent initial state in the system. This is similar for the

n P 1 example in Fig. 5(e), the cube. Here, starting on one of the three nearest neighbors to the detector, we find Pdet = 1/3. In 0.5 the remaining part of the paper we wish to further explore this relation between symmetry and Pdet. In this section we assume that the initial state is localized: 0 -2 0 2 |ψin=|rin. (38) x / n We already discussed the first detection amplitude given by [12,13,18] FIG. 6. Surviving (i.e., as yet undetected) particle probability ϕ ψ =ψ | ˆ τ 1 − ˆ ˆ τ n−1|ψ . density Pn(x)aftern = 238 and 576 measurements, in a ring of n( in) d U ( )[( D)U ( )] in (39) = γτ/ = length L 2000 and a detection interval of h¯ 1. For clarity, Reading the equation right to left, we see that the initial state is only every fifth point is plotted. subject to n − 1 compound steps of unitary evolution and un- successful detection, until finally, after the nth evolution step, This situation is presented in Fig. 6, where the scaled density detection is successful. As mentioned, the focus of this paper ψ = ∞ ψ = is presented after 238 and 476 steps. The total undetected is the total detection probability Pdet( in) n=1 Fn( in) ∞ |ϕ ψ |2 probability Sn is 0.6006 at both times, the difference being n=1 n( in) . −6 only of order 4 × 10 , while the average distance to the From Eq. (39) we see that the detection amplitude ϕn is origin of the surviving particle has doubled from 386 to 770. linear with respect to the initial state, and thus obeys a super- Thus, in the limit of large L, the total undetected probability position principle. This allows us to obtain an upper bound for remains of order unity, spreading out to infinity, for all times Pdet(rin) without the need of detailed calculations. To see this, less than the huge system traversal time of order L. consider localized initial and detection states |rd and |rin, An alternative picture arises from considering the station- respectively. Now we introduce an auxiliary initial state which ary bright states which are the eigenvectors of the survival is a of any two different localized states operator with eigenvalues lying inside the unit circle. For 1 iα large finite L, these eigenvalues span the range in magnitude |uα=√ (|rin+e |r ), (40) from 0 to very near unity. For times Nτ shorter than the 2 time to propagate across the system (i.e., for the wings of the where rin|r =0 and α is an arbitrary relative phase. As the distribution to meet, as described above) the relatively quickly first detection amplitudes are linear in the initial state [see decaying stationary bright states with eigenvalues with small Eq. (39)] we find magnitude contribute to  Fn = 1 − SN , while the states n N 1 iα with eigenvalues near the unit circle do not. These slow states ϕn(uα ) = √ [ϕn(rin) + e ϕn(r )]. (41) then are essentially dark over this timescale. We find that for 2 L = 1000, there are 267 fast eigenvalues (out of 501) further We can use this to find a very useful upper bound on the than a distance 0.002 from the unit circle, and for L = 2000, detection probability, provided there is a symmetry relation there are 531 fast eigenvalues (out of 1001) further than a between rin and r , namely, when |rin and |r are physically distance 0.001 from the unit circle (for γτ = h¯), i.e., twice equivalent. Mathematically, two orthogonal initial states |ψin |ψ as many. Examining the eigenvectors, they extend over the and in which yield identical transition amplitudes to the entire system, so an initial state localized near the origin will detection have a squared overlap with each bright mode proportional to r |Uˆ (t )|ψ =r |Uˆ (t )|ψ = 0 (42) 1/L. Therefore, the number of effectively dark states, those d in d in that do not contribute in time Nτ to Pdet, scales with L. Thus, are called physically equivalent to each other. As usual, the the undetected probability is of order 1, since it arises from amplitudes on the left- and right-hand sides of this equation λ order L modes times a number of order 1/L overlap for each may have irrelevant phase factors ei . More importantly, such mode. states only appear in systems with a certain degree of symme- try. From these two states√ we can construct the superposition state (|ψ −|ψ )/ 2, which must be a dark state. Hence, VIII. FROM SYMMETRY TO DETECTION PROBABILITY in in any pair of physically equivalent states yields a dark state. As mentioned in the Introduction, the symmetry of the This reveals dark states as an interference phenomenon [2]. underlying graph controls the deviations of the detection prob- Certain initial states result in permanent destructive interfer- ability from its classical counterpart, which is unity [2]. This ence and thus vanishing in the detection is also evident in the examples in Fig. 5, which clearly show state.

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Let us consider the example in the ring system of Fig. 1. d with periodic boundary conditions one finds The two sites left and right of the detection node are equivalent Pdet  1/2d for sites on the main horizontal, vertical, and due to reflection invariance. Clearly, when |rin and |r are main diagonals, but smaller values for sites which are off these physically equivalent then also ϕn(rin) = ϕn(r ). Under such main axes. The Star-of-David graph with a detector on one of circumstances, for the superposition |uα in Eq. (40), we have the tips yields Pdet  1/2, due to reflection symmetry, except α for the opposing tip. Similarly, yet less obvious, P  1/2in 1 + ei det ϕn(uα ) = √ ϕn(rin) (43) the tree-of-life graph. For a square graph with an additional 2 node in the center, we have ν = 4 for a transition from one and so of the corners to the center. When the detector is in one of the square’s corners, the neighboring corners are equivalent, =|ϕ |2 = + α . Fn(uα ) n(uα ) (1 cos )Fn(rin) (44) yielding ν = 2, but all other sites are unique. Also for a tree, The relative phase α affects the first detection statistics. The the number of physical equivalent sites and thus the upper aforementioned choice α = π obviously yields a dark state, bound varies with the position of the detector. The square, as can be seen. the ring, the complete graph, the hypercube, and the tree have A useful bound on Pdet can be found by summing over all been discussed above, where we computed Pdet exactly. With = ∞  one exception, the exact values of these examples actually n and using Pdet(uα ) n=1 Fn(uα ) 1: coincide with the upper bound. The only exception is the tree,  = + α . 1 Pdet(uα ) (1 cos )Pdet(rin) (45) when the detector is not placed on the root node. Choosing α = 0, we obtain, for the originally considered tran- Finally, the infinite line demands special attention. The  / sition from rin to rd, upper bound yields Pdet 1 2 for every nondetection site and is correct. However, since it is an infinite system, the exact P (r )  1 . (46) det in 2 formula (1) does not apply. In particular, one finds that Pdet Thus, if r has a physically equivalent partner, the total de- has a complicated dependence on τ (see Refs. [18,19], where in τ tection probability cannot be unity, unlike for the classical this model was investigated in detail). However, the Pdet( ) random walk. curves stay below the upper bound 1/2. This bound can be easily generalized to other structures, To gain further physical insight, consider the cross struc- for example, the cube of Fig. 1. Consider the transition to ture presented in Fig. 7. We detect on the center of the cross, one vertex, denoted by |0, from one of its nearest neighbors, at node |0, and start on one of the outer nodes, for exam- | say, |1. Let us denote the neighbors of |0 by |1, |2,√ and ple, on state 1 . This initial state can be decomposed into |3 and define an auxiliary state |u := (|1+|2+|3)/ 3. a linear combination of four states, out of which three are (This would be the generalization of |u from before.) Using easily understood as being dark states. For example, the state 0 |δ˜ = | −| +| −| / the same procedure as before, we find that Pdet(1)  1/3. 1 ( 1 2 3 4 ) 2 is dark since it is not inject- This trick can be easily extended and is summarized in the ing probability current into state |0. Destructive interference following proposition. erases all amplitude in the detection state. The uniform state Proposition 1. The total detection probability for the transi- |u=(|1+|2+|3+|4)/2, which is a normalized sum of tion from a localized initial state |rin to a localized detection all the equivalent states in the system, is the fourth state. This state |r is bounded by the reciprocal of the number ν of state gives a constructive interference pattern at the detected d | →| nodes physically equivalent to rin: state. Returning to the transition 1 0 , we decompose the initial condition into a superposition of the four states, as 1 P r  . shown in Fig. 7. Since three components√ are dark and the over- det( in) ν (47) lap of initial and uniform states is 1/ 4, we find Pdet(1)  Recall the definition of physically equivalent states from 1/4. In fact, a straightforward calculation using Eq. (1)shows Eq. (42): They have identical transition amplitudes to the Pdet(u) = 1 and thus Pdet(1) = 1/4. This is clearly in accord detection state at all times. Such states are indistinguishable with ν = 4. in the first detection problem. For a given system and a Let us formalize our result. Consider a localized initial given transition rin → rd from one localized detection state state |rin and assume that in the system we have a total ν { }ν−1 ν {| }ν−1 | =| to another, we can find a set of initial nodes r j j=0 of physically equivalent states r j j=0, where r0 rin = | =δ E = (where r0 rin) which are physically equivalent in this sense. and ri r j i, j. These states span the subspace |rin They can be identified from elementary symmetry considera- Span[|r j], i.e., the space of all possible superpositions of the tions. This number ν of physically equivalent states is what physically equivalent states |r j. We define appears in Eq. (47). The strongest bound is given by the maximal number of physically equivalent states, i.e., by the ν−1 largest possible value for ν. However, Eq. (47) still holds 1 |u(r ) := √ |r (48) even when this maximal number cannot be identified beyond in ν j j=0 doubt. Let us discuss the remaining examples of Fig. 1.Forthe | cube, we find Pdet  1/3, except for the diametrically opposed to be the auxiliary uniform state (AUS); u(rin) is one par-  E node, for which Pdet 1. For the complete graph with L sites, ticular state in the subspace |rin. We can find a new basis {| , |δ ,...,|δ } E all nodes besides the detection site are equivalent and we find u(rin) 1 ν−1 for |rin, which consists of the AUS Pdet  1/(L − 1) (here L = 8). For simple cubic lattices in and ν − 1 dark states. These dark states can be found similarly

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AUS is bright, then Pdet(u(rin)) = 1 and the upper bound sat- urates, i.e., it becomes an equality. We also note that Eq. (51) holds as well in infinite systems. A rigorous group-theoretic investigation of the problem is left for future work. A lower bound was also recently presented, including the discussion of hypercube in dimension d, i.e., the extension of this work to certain large systems with what we called a shell structure was considered [21].

IX. SUMMARY We have investigated herein the total probability of detec- tion Pdet in a quantum system that is stroboscopically probed in its detection state. An explicit formula for this in terms of the energy eigenstates was produced via the renewal equation [9,18] previously derived for the generating function for the detection amplitude, with the help of the Aleksandrov theorem for Cauchy transforms. An alternate derivation for the formula was also given via an analysis of the dark and bright subspaces that comprise the total Hilbert space. The dark states are those energy eigenstates that have no overlap with the detector and thus are never detected. There were found to be two classes of dark states: those that belong to completely dark energy levels, where every eigenstate of the sector is orthogonal to the detection state, and those which perforce appear in degenerate energy levels, whose sectors possess a nonzero projection of the detection state. The bright states are those eigenstates which are detected with probability unity and were shown to FIG. 7. Decomposition of an initial state |1, localized on an constitute, in a finite system, the orthogonal complement to exterior node of a cross, into three dark components, and the aux- the dark subspace. They were shown to belong to the spectrum iliary uniform state. Detection is attempted on the node in the center of the survival operator Sˆ inside the unit disk. An explicit set (open circle). We find ν = 4 from the π/2 degree rotation symmetry of basis states for the subspaces was constructed. From this,  / about the detection site. Consequently, Pdet(1) 1 4 from the simple Pdet was calculated as the overlap of the initial state with the bound (47). bright space, reproducing the original result. We considered several examples, showing in particular how lifting the de- to the stationary dark states in Sec. IV. The solution reads generacy in the energy spectrum discontinuously changes Pdet. The breakdown of our formula for P in an infinite system | − j−1 | det j r j m=0 rm was discussed in the context of the infinite line. |δ j= √ , (49) j( j + 1) The τ independence of the total detection probability where j = 1, 2,...,ν− 1. One quickly verifies that the states will survive when irregularities are introduced in the sam- |δ are normalized and orthogonal to each other and to pling times. The lack of total control over the detector can j be modeled by a random sequence of interdetection times the AUS. Furthermore, they are dark, because the transition {τ ,τ ,...} amplitudes to the detection state from any state |r are the 1 2 , as has been done, e.g., in Ref. [6], where the j sampling times were given by a Poisson process. A strong same. Consequently, the multiplication with r |Uˆ (t ) annihi- d finding of ours is that P actually does not depend on τ at all lates |δ , because it yields r |Uˆ (t )|δ =r |Uˆ (t )|r [ j − det √ j d j d in for systems with a discrete spectrum. As the set of resonant j]/ j( j + 1) = 0. The original initial state |r =|r is now in 0 detection periods [defined by Eq. (2)] has zero measure, all of rewritten as our results are expected to hold for nonstroboscopic sampling ν−1 1 |δ j as well. |rin=√ |u(rin)− √ . (50) ν j( j + 1) We also investigated the influence of the system’s symme- j=1 tries on Pdet. Whenever one has found ν physically equivalent Knowing about the dark states in |rin, we can directly obtain initial states, one will find that the total detection probability is  /ν Pdet(rin), because ϕn(δ j ) = 0. We immediately find bounded by Pdet 1 . Two states are physically equivalent when they yield the same transition amplitudes to the detec- 1 Pdet(rin) = Pdet(u(rin)). (51) tion state for all times. Any pair of equivalent states can be ν seen to yield a dark state from their negative superposition. Since Pdet(u)  1, we obtain the upper bound (47). It is im- The AUS defined by Eq. (48) is the positive superposition portant to keep in mind that ν is not a global property of the of all equivalent states, contains all bright components of the system but depends on the transition rin → rd. Clearly, if the original initial state, and gives Pdet via Eq. (51).

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ACKNOWLEDGMENTS equality, we consider the case when there are only two differ- ent energy levels E and E . The Hamiltonian of such a system Support from Israel Science Foundation through Grant No. 1 2 would read Hˆ = Pˆ E + Pˆ E . It is not necessary to specify 1898/17 is acknowledged. F.T. was supported by DFG (Ger- 1 1 2 2 the eigenstates and degeneracies, as long as the overlap of many) through Grant No. TH 2192/1-1. E.B. thanks Avihai each eigenspace with the detection state is given. For simplic- Didi for comments on the manuscript and Klaus Ziegler for ity, we assume that ψ |Pˆ |ψ =ψ |Pˆ |ψ =1/2. (In fact, fruitful discussions. d 1 d d 2 d the Aleksandrov ansatz does not “see” individual eigenstates |El,m, but just the overlap with the eigenspace projectors Pˆl .) We can replace ψd|Pˆ1|ψin=ν(λ1)ψd|Pˆ1|ψd=ν(λ1)/2, APPENDIX A: ALEKSANDROV’S THEOREM IN A and similarly with ν(λ2 ). Then ϕ(z) reads −iλ −iλ TWO-LEVEL SYSTEM ze 1 ν(λ1 ) ze 2 ν(λ2 ) − λ + − λ ϕ = 1−ze i 1 1−ze i 2 . A proof of the powerful equation (14) lies far outside the (z) 1 1 (A1) − λ + − λ scope of this paper; the interested reader is referred to [75]. 1−ze i 1 1−ze i 2 In order to nevertheless help to understand the origin of the Its absolute value on the unit circle equals

∗ λ λ θ ∗ λ λ θ − λ θ− λ − λ θ− λ ν(λ ) ei 1 (ei 2 − ei ) + ν(λ ) ei 2 (ei 1 − ei ) ν(λ )e i 1 (1 − ei i 2 ) + ν(λ )e i 2 (1 − ei i 1 ) |ϕ iθ |2 = 1 2 1 2 . (e ) λ λ θ θ − λ − λ (A2) ei 1 + ei 2 − 2ei 2 − ei (e i 1 + e i 2 )

− τ / When this expression is plugged into Eq. (14), the left-hand- correspond to eigenvalues e i El h¯ on the unit circle. In par- θ side integral can be treated by a variable change z = ei and ticular, these eigenvalues coincide with eigenvalues of Uˆ (τ ). dθ = dz/iz. The result is a complex contour integral which is Next Eq. (22), which relates the poles of the generating func- performed by residue inspection. The integrand has only two tion ϕ(z) and the spectrum of Sˆ , is generally true as well. simple poles inside the unit circle. One lies at the origin z = 0 Its algebraic eigenvalues ζ are given by the zeros of the and the other is defined by the linear term in the denominator characteristic polynomial det[ζ 1 − Sˆ ]. of the first line in Eq. (A2). After some lengthy algebra, we It remains to be shown that the survival probability ν λ ∗ν λ n find that the prefactors of the cross terms ( 1) ( 2 ) and Sn(β˜ ) =Sˆ |β˜ of some bright state decays to zero as n → ∗ ν(λ2 ) ν(λ1) vanish. Collecting the residues of the diagonal ∞. To see this, consider the restriction of Sˆ to the bright space terms, however, we find one half: ˆ = ˆ ˆ ˆ SB PHB SPHB . This removes all trivial eigenvalues that lie  λ − λ ˆ i 2 − i 2 on the unit circle from S. This operator is defective in general dz e (1 ze ) = 1. λ λ − λ − λ and cannot be put into a diagonal form. It can be put into a π i 1 + i 2 − − i 1 + i 2 |z|=1 2 iz [e e 2z][2 z(e e )] 2 Jordan normal form, though (A3) We obtain the same result for the integral proportional to ˆ = ζ + . 2 SB [ l Mˆ l Nˆl ] (B1) |ν(λ2 )| . Hence l 1 2 2 Pdet = [|ν(λ1)| +|ν(λ2 )| ], (A4) 2 Here ζl are the algebraic eigenvalues of Sˆ B that we deter- ϕ ˆ ˆ which is the result of Eqs. (14) and (1) for the chosen example mined from (z), and the matrices Ml and Nl act on the ζ ˆ system. A demonstration with more than two energy levels (generalized) eigenspaces of l ; Nl is a nilpotent matrix (i.e., ˆ pl = > ˆ using the same method first becomes tedious and soon unfea- Nl 0, for some integer pl 0) and Ml is the projector ˆ 2 = ˆ sible. In a system with w bright states there are w different to the corresponding eigenspace (hence Ml Ml ). The rank phases λl in μ(θ ) and w poles in the integrand of Eq. (9)after of Mˆ l is equal to the algebraic of ζl , Mˆ l and Nˆl iθ switching again to z = e . Aleksandrov’s theorem “magi- commute, and Mˆ l Nˆl = Nˆl Mˆ l = δl,l Nˆl . Using these proper- cally” ensures that the residues of these poles are exactly given ties, one finds, for n > pl , by |ν(λ )|2ψ |Pˆ |ψ , thus giving Eq. (1). l d l d p −1   l n (ζ Mˆ + Nˆ )n = ζ nMˆ + ζ n−mNˆ m, (B2) l l l l l m l l APPENDIX B: ADDITIONAL PROPERTIES OF THE m=1 SURVIVAL OPERATOR and therefore for any superposition |β˜ of bright states ˆ Here we will give a more detailed discussion of S’s proper- n 2 2n ties. In the main text we already mentioned that the stationary S (β˜) =Sˆ |β˜ = |ζ | C , (β˜ ). (B3) ˆ n B l l n dark states are eigenstates of S corresponding to eigenvalues l on the unit circle and that the stationary bright states belong to the spectrum inside the unit disk. In view of the fact that Sˆ is The coefficients are given by not a diagonalizable matrix in general, these statements need −1 n 2 C , (β˜ ):=[Mˆ + ζ Nˆ ] |β˜ . (B4) to be refined. l n l l l −m First, we note that the stationary dark states |δl,m of Using Eq. (B2 ), the  triangle inequality, as well as |ζl | < −p n n p Sˆ |ζl | l and   n l /pl !forn > 2pl and m < pl , one Eq. (21) are in general the right eigenstates of that m pl

043107-12 DARK STATES OF QUANTUM SEARCH CAUSE IMPERFECT … PHYSICAL REVIEW RESEARCH 2, 043107 (2020) can bound these coefficients with APPENDIX C: ADDITIONAL EXAMPLE GRAPHS AND THEIR Pdet AND DARK STATES −   pl 1 2 2 n −2m m 2 In this Appendix we present the calculation for the spec- Cl,n(β˜ )  Mˆ l |β˜ + |ζl | Nˆ |β˜ m l trum, eigenstates, Pdet, and dark states for the collection of m=1 graphs illustrated in Figs. 5(c)–(f),Figs.5(a) and 5(b) having − −1 pl 1 (n|ζ |)2pl been presented in the main text.  Mˆ |β˜ 2 + l Nˆ m|β˜ 2. (B5) l 2 l pl ! = m 1 1. Complete graph

2p We now consider a graph with L nodes, in which each node Hence Cl,n(β˜ )  O(n l ). Considering the largest of all pl ζ is connected to each other node [see Fig. 8(c)]. Its Hamilto- denoted by p∗ and the eigenvalue ∗ closest to the unit circle, nian reads we find that the survival probability will eventually decay L exponentially Hˆ =−γ (1 − δr,r )|rr |. (C1) r,r=1 2p∗ 2n Sn(β˜ )  O(n |ζ∗| ). (B6) We pick one node as the detection node, say, rd. The system has only two energy levels, namely, E = From writing [1 − Dˆ ] in the energy basis and inspecting 1 −γ (L − 1) and E2 = γ , the latter being (L − 1)-fold degen- the definition of Sˆ B, it is evident that each ζl has to be a convex τ / erate. The eigenstates are sum of the phase factors ei El h¯ .Infact,Ref.[9] shows that the ζ L L nontrivial eigenvalues l can be obtained from the stationary |r ei(2πmr/L)|r points of a certain two-dimensional Coulomb force field and |E1=|j := √ , |E2,m= √ , (C2) must lie inside the unit disk. This approach is investigated in r=1 L r=1 L detail in Ref. [68]. Therefore, we find another reason why the = , ,..., − | |ζ | < where m 1 2 L 1 and j is the uniform state over nontrivial eigenvalues obey l 1. all nodes. Therefore, none of these energy levels is completely This is easily seen for a system with only two bright states dark with respect to a localized detection state. The two bright |β1 and |β2 corresponding to two energies E1 and E2.The |ψ = eigenstates are detection state is decomposed into the bright states via d √ |β + |β | |2 +| |2 = ˆ L a 1 b 2 , where a b 1. In this case, SB reads, 1 L|r −|j |β =|j=√ |r, |β = √ d . (C3) in matrix notation, 1 2 − L r=1 L 1    ∗ − τ / 1 −|a|2 ab e i( E1 h¯) 0 Computing the overlap with the bright space, we find Sˆ = ∗ − τ / B ba 1 −|b|2 0 e i( E2 h¯)   1, rin = rd − τ / ∗ − τ / = | |2 i( E1 h¯) i( E2 h¯) Pdet(rin) 1 , = . (C4) b e ab e − rin rd = ∗ − τ / − τ / . (B7) L 1 ba e i( E1 h¯) |a|2e i( E2 h¯) 2. Star This matrix has one vanishing eigenvalue ζ1 = 0. The other | − τ / − τ / The next example is a star graph with a center node 0 ζ =| |2 i E2 h¯ +| |2 i E1 h¯ one is equal to 2 a e b e , i.e., it is a and L nodes in the periphery [see Fig. 8(d)]. The Hamiltonian convex sum of the phase factors, which lies inside the unit reads circle. Similar reasoning applies to systems with more than two bright states. L ˆ =−γ | |+| | . In systems with finite-dimensional Hilbert space we are H [ 0 r r 0 ] (C5) λ∗ = r=1 finished, because there is some minimum decay rate √ min {−2ln|ζ |} > 0. Infinite-dimensional systems may be- l l The system√ has three energy levels E1 =−γ L, E2 = 0, have more subtly. In the thermodynamic limit the eigenvalues and E = γ L, of which E is (L − 1)-fold degenerate. The ζ 3 2 l can get infinitely close to the unit circle. In that case eigenstates are unity is an accumulation point of the sequence {|ζl |} and ∗ | +| | −| λ = infl {−2ln|ζl |} = 0. In the view of the spectral theorem 0 j 0 j |E1= √ , |E3= √ , of Ref. [9], there are two options: Either one is lucky and Sn 2 2 decays to zero, albeit slower than exponentially fast, or not. In (C6) 1 L the former case Sn → 0 will still hold. The spectral theorem i(2π/L)mr |E2,m=√ e |r, of Ref. [9] says that this is the case when the spectrum of L r=1 Uˆ (τ ) has no absolutely continuous part. If such a part of the √ = , ,..., − | = L | / spectrum is present, on the other hand, then Sn will converge to where m 1 2 L 1. In addition, j : r=1 r L some positive value and we find Pdet(β˜ ) < 1. This is obviously is the uniform state over the periphery. only possible in infinite-dimensional systems. Some more If the detection takes place on the center node, the energy mathematical details about Sˆ B can be found in Appendix A level E2 is completely dark. If detection were to take place of Ref. [10]. in the periphery, there would be no completely dark energy

043107-13 FELIX THIEL et al. PHYSICAL REVIEW RESEARCH 2, 043107 (2020)

(a) (b) (c) (d) (e) (f)

FIG. 8. Notation used for the examples considered in Sec. VI: (a) square with a center node, (b) ring of L sites, (c) complete graph with L sites, (d) star graph with L sites in the periphery, (e) hypercube (here of dimension d = 3, and (f) binary tree with two generations.

{ } |{ }| = levels. The Pdet is computed from Eq. (1). The calculations are where the sum runs over all combinations x j , where x j similar to the ones of the complete graph: d = − j=1 x j 2l d. ⎧ d We now pick any localized initial state |rin=⊗= |bin, j, ⎪1, rd = rin j 1 ⎪ which differs from |rd in exactly ξ bits. (ξ is the so-called ⎨ 1 , = , = rd 0 rin 0 Hamming distance between both states.) The P only de- P (r ) = L (C7) det det in ⎪ , = , = pends on ξ and can be obtained from Eq. (1) and from ⎪1 rd 0 rin 0 ⎩⎪ Eq. (C11) and some complicated combinatoric computation. 1 , r = 0, r = 0. L−1 d in We do not present this here and instead refer to Ref. [45], where Pdet was computed in a different way: 3. Hypercube 1 P (r ) =  . (C12) The next example is the d-dimensional hypercube. An det in d example graph for d = 3 is depicted in Fig. 8(e).Inthis ξ system there are 2d different states which can be represented This shows that only two transitions in a hypercube are actu- by a d-length string consisting only of zeros and ones, i.e., ally reliable, in the sense that Pdet = 1: the return to the initial |ψ=⊗d | = , a string of d bits, j=1 b j , where b j 0 j 1 j is the node and the traversal to the opposing node [see Fig. 5(e)]. jth bit. The allowed transitions correspond to bit flips and the Hamiltonian can be written as a sum of Pauliσ ˆx matrices 4. Tree graph d Similar to the hypercube, trees are important examples ˆ =−γ σ ( j), H ˆx (C8) from quantum computation, in particular because they feature j=1 an exponential speedup [62,63] relative to classical algo- rithms. We will consider here only a small binary tree with ( j) ( j) whereσ ˆx only acts on |b j and σx =|0 j0 j|1 j + two generations [see Fig. 8(f)]. The top of the tree is called | | σ (1)| =| σ (2)| = 1 j 1 j 0 j. (As an example, ˆx 110 010 ,ˆx 110 the root; the bottom nodes are called leaves of the tree. Its |100, andσ ˆ (3)|110=|111.) The relation to a word of Hamiltonian reads x ⎛ ⎞ d bits with bit-flip transitions make this model particularly 0110000 relevant for the quantum computation field [2,64]. The en- ⎜1001100⎟ =−γ − = , ,..., ⎜ ⎟ ergy  levels are El (2l d), l 0 1 d, and each is ⎜1000011⎟ d ⎜ ⎟ -fold degenerate. Let x j =±1 and define |x j := (|0 j+ ˆ =−γ ⎜ ⎟. l √ H ⎜0100000⎟ (C13) x |1 )/ 2; thenσ ˆ ( j)|x =x |x . Hence the energy eigen- ⎜0100000⎟ j j x j j j ⎝ ⎠ states have the form 0010000 0010000 d |El,m=⊗= |x j (C9) √ j 1 =− γ =− γ   The corresponding energy levels are E1 2 , E2 √ 2 , d d = = γ such that = x j = 2l − d and m enumerates all combi- and E3 0, being thrice degenerate, as well as E4 2 and j 1 l = γ nations. In each bit, we have the important relation E5 2 . The energy eigenstates are |E =(2, 2, 2, 1, 1, 1, 1)T /4, 1 − / 1 b |x =√ (−1)b j [(1 x j ) 2], (C10) √ √ √ j j T 2 |E2=(0, − 2, 2, −1, −1, 1, 1) / 8, √ = =− T which has a negative sign when b j 1 and x j 1 and a |E3,1=(0, 0, 0, 1, −1, 0, 0) / 2, positive sign otherwise. √ | = , , , , , , − T / , We try to detect the system in the node with all bits equal E3,2 (0 0 0 0 0 1 1) 2 (C14) √ to zero, i.e., |rd=⊗j|0 j.FromEq.(C10) we find that all T |E3,3=(−2, 0, 0, 1, 1, 1, 1) / 8, energy eigenstates have the same overlap with the detection √ √ √ −d/2 | , = T state: rd El m 2 . Therefore, one finds the bright eigen- |E4=(0, 2, − 2, −1, −1, 1, 1) / 8, states | = , − , − , , , , T / . E5 (2 2 2 1 1 1 1) 4 |β =1 ⊗d | , l   j=1 x j (C11) We see that some of the energy levels may be completely dark d |{ }|= − l x j 2l d depending on the detection state.

043107-14 DARK STATES OF QUANTUM SEARCH CAUSE IMPERFECT … PHYSICAL REVIEW RESEARCH 2, 043107 (2020)

a. Detection on the root c. Detection in the leaves

The choice |rd=|0 renders E2 and E4 completely dark. We choose |rd=|3. Then there are two dark state in the The third energy level yields the bright eigenstate |β3= energy level E3. The bright state in this level is |E3,3. Doing the calculations with Eq. (1), we find the total detection probability equal to 1/2 for nodes |1 and |2 and 2|0−5|3+3|4−|5−|6 equal to 1/4 for the leaves, i.e., |β = √ . (C17) ⎧ 3 = 40 ⎨⎪1forrin 0 1 P (r ) = for rin = 1, 2 (C15) det in ⎩⎪ 2 The remaining bright states are equal to the eigenstates. This 1 . 4 otherwise results in

b. Detection in the middle ⎧ ⎪ 3 , = , ⎨ 5 rin 0 4 Now we choose |rd=|1. Then E3 is completely dark. All = , = , , remaining energy levels are nondegenerate; hence there are no Pdet(rin) 1 rin 1 2 3 (C18) ⎩⎪ other dark states. We find, from Eq. (1), 2 , r = 5, 6. ⎧ 5 in ⎪ 1 = ⎨ 2 for rin 0 P (r ) = 1forrin = 1, 2 (C16) All these results are summarized in Fig. 5. det in ⎩⎪ 3 . 8 otherwise

[1] E. Bach, S. Coppersmith, M. P. Goldschen, R. [21] F. Thiel, I. Mualem, D. A. Kessler, and E. Barkai, Phys. Rev. Joynt, and J. Watrous, J. Comput. Syst. Sci. 69, 562 Research 2, 023392 (2020). (2004). [22] F. Thiel, D. A. Kessler, and E. Barkai, Phys. Rev. A 102, 022210 [2] H. Krovi and T. A. Brun, Phys. Rev. A 73, 032341 (2006). (2020). [3] H. Krovi and T. A. Brun, Phys. Rev. A 74, 042334 (2006). [23] F. Thiel and D. A. Kessler, Phys. Rev. A 102, 012218 (2020). [4] H. Krovi and T. A. Brun, Phys. Rev. A 75, 062332 (2007). [24] S. Gurvitz, Fortschr. Phys. 65, 1600065 (2017). [5] M. Štefanák,ˇ I. Jex, and T. Kiss, Phys. Rev. Lett. 100, 020501 [25] Y. Ashida, K. Saito, and M. Ueda, Phys. Rev. Lett. 121, 170402 (2008). (2018). [6] M. Varbanov, H. Krovi, and T. A. Brun, Phys.Rev.A78, [26] L. Buffoni, A. Solfanelli, P. Verrucchi, A. Cuccoli, and M. 022324 (2008). Campisi, Phys.Rev.Lett.122, 070603 (2019). [7] F. Caruso, A. W. Chin, A. Datta, S. F. Huelga, and M. B. Plenio, [27] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Phys. J. Chem. Phys. 131, 105106 (2009). Rev. B 99, 224307 (2019). [8] E. Agliari, O. Mülken, and A. Blumen, Int. J. Bifurcat. Chaos [28] B. Skinner, J. Ruhman, and A. Nahum, Phys.Rev.X9, 031009 20, 271 (2010). (2019). [9] F. A. Grünbaum, L. Velázquez, A. H. Werner, and R. F. Werner, [29] S. Redner, A Guide to First-Passage Processes, 1st ed. (Cam- Commun. Math. Phys. 320, 543 (2013). bridge University Press, Cambridge, 2007). [10] J. Bourgain, F. A. Grünbaum, L. Velázquez, and J. Wilkening, [30] E. P. Raposo, S. V. Buldyrev, M. G. E. da Luz, G. M. Commun. Math. Phys. 329, 1031 (2014). Viswanathan, and H. E. Stanley, J. Phys. A: Math. Theor. 42, [11] P. L. Krapivsky, J. M. Luck, and K. Mallick, J. Stat. Phys. 154, 434003 (2009). 1430 (2014). [31] O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Rev. [12] S. Dhar, S. Dasgupta, and A. Dhar, J. Phys. A: Math. Theor. 48, Mod. Phys. 83, 81 (2011). 115304 (2015). [32] V. V. Palyulin, A. V. Chechkin, R. Klages, and R. Metzler, [13] S. Dhar, S. Dasgupta, A. Dhar, and D. Sen, Phys.Rev.A91, J. Phys. A: Math. Theor. 49, 394002 (2016). 062115 (2015). [33] A. Godec and R. Metzler, Sci. Rep. 6, 20349 (2016). [14] P. Sinkovicz, Z. Kurucz, T. Kiss, and J. K. Asbóth, Phys. Rev. [34] A. Godec and R. Metzler, Phys.Rev.X6, 041037 (2016). A 91, 042108 (2015). [35] L. K. Grover, Phys. Rev. Lett. 79, 325 (1997). [15] P. Sinkovicz, T. Kiss, and J. K. Asbóth, Phys.Rev.A93, [36] S. Aaronson and A. Ambainis, in Proceedings of the 44th 050101(R) (2016). Annual IEEE Symposium on Foundations of Computer Science [16] S. Lahiri and A. Dhar, Phys. Rev. A 99, 012101 (2019). (IEEE, Piscataway, 2003), pp. 200–209. [17] H. Friedman, D. A. Kessler, and E. Barkai, J. Phys. A: Math. [37] A. M. Childs and J. Goldstone, Phys.Rev.A70, 022314 (2004). Theor. 50, 04LT01 (2017). [38] O. Mülken and A. Blumen, Phys. Rev. E 73, 066117 (2006). [18] H. Friedman, D. A. Kessler, and E. Barkai, Phys. Rev. E 95, [39] H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and 032141 (2017). Y. Silberberg, Phys.Rev.Lett.100, 170506 (2008). [19] F. Thiel, E. Barkai, and D. A. Kessler, Phys. Rev. Lett. 120, [40] M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. 040502 (2018). Meschede, and A. Widera, Science 325, 174 (2009). [20] F. Thiel, D. A. Kessler, and E. Barkai, Phys. Rev. A 97, 062105 [41] F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, (2018). andC.F.Roos,Phys. Rev. Lett. 104, 100503 (2010).

043107-15 FELIX THIEL et al. PHYSICAL REVIEW RESEARCH 2, 043107 (2020)

[42] F. Magniez, A. Nayak, J. Roland, and M. Santha, SIAM J. [60] D. L. Sombillo and E. A. Galapon, Phys.Rev.A90, 032115 Comput. 40, 142 (2011). (2014). [43] O. Mülken and A. Blumen, Phys. Rep. 502, 37 (2011). [61] D. L. B. Sombillo and E. A. Galapon, Ann. Phys. (NY) 364, [44] S. R. Jackson, T. J. Khoo, and F. W. Strauch, Phys.Rev.A86, 261 (2016). 022335 (2012). [62] E. Farhi and S. Gutmann, Phys. Rev. A 58, 915 (1998). [45] L. Novo, S. Chakraborty, M. Mohseni, H. Neven, and Y. Omar, [63] A. M. Childs, E. Farhi, and S. Gutmann, Quantum Inf. Process. Sci. Rep. 5, 13304 (2015). 1, 35 (2002). [46] S. Chakraborty, L. Novo, A. Ambainis, and Y. Omar, Phys. Rev. [64] J. Kempe, Probab. Theory Relat. Fields 133, 215 (2005). Lett. 116, 100501 (2016). [65] M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 (1998). [47] S. Boettcher, S. Falkner, and R. Portugal, Phys. Rev. A 91, [66] F. D. Stefani, J. P. Hoogenboom, and E. Barkai, Phys. Today 052330 (2015). 62(2), 34 (2009). [48] P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. [67] P. Facchi and S. Pascazio, J. Phys. Soc. Jpn. 72, 30 (2003). Zupancic, Y. Lahini, R. Islam, and M. Greiner, Science 347, [68] R. Yin, K. Ziegler, F. Thiel, and E. Barkai, Phys. Rev. Research 1229 (2015). 1, 033086 (2019). [49] P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. [69] C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantenmechanik Sanders, Phys.Rev.Lett.114, 140502 (2015). (de Gruyter, Berlin, 2009), Vol. 1. [50] S. Li and S. Boettcher, Phys.Rev.A95, 032301 (2017). [70] Q. Liu, R. Yin, K. Ziegler, and E. Barkai, Phys. Rev. Research [51] B. Mukherjee, K. Sengupta, and S. N. Majumdar, Phys.Rev.B 2, 033113 (2020). 98, 104309 (2018). [71] B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977). [52] D. C. Rose, H. Touchette, I. Lesanovsky, and J. P. Garrahan, [72] W. M. Itano, D. J. Heinzen, J. J.Bollinger, and D. J. Wineland, Phys. Rev. E 98, 022129 (2018). Phys. Rev. A 41, 2295 (1990). [53] G. R. Allcock, Ann. Phys. (NY) 53, 253 (1969). [73] T. J. Elliott and V. Vedral, Phys.Rev.A94, 012118 (2016). [54] J. Kijowski, Rep. Math. Phys. 6, 361 (1974). [74] Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B 98, 205136 [55] Y. Aharonov, J. Oppenheim, S. Popescu, B. Reznik, and W. G. (2018). Unruh, Phys.Rev.A57, 4130 (1998). [75] J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy Trans- [56] J. A. Damborenea, I. L. Egusquiza, G. C. Hegerfeldt, and J. G. form (American Mathematical Society, Providence, 2006). Muga, Phys.Rev.A66, 052104 (2002). [76] F. Schäfer, I. Herrera, S. Cherukattil, C. Lovecchio, F. S. [57] C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 Cataliotti, F. Caruso, and A. Smerzi, Nat. Commun. 5, 3194 (2006). (2014). [58] J. J. Halliwell and J. M. Yearsley, Phys.Rev.A79, 062101 [77] J. Novotný, G. Alber, and I. Jex, J. Phys. A: Math. Theor. 45, (2009). 485301 (2012). [59] A. Ruschhaupt, J. G. Muga, and G. C. Hegerfeldt, in Time [78] B. Svensson, Quanta 2, 18 (2013). in Quantum Mechanics - Vol. 2,editedbyG.Muga,A. [79] B. Tamir and E. Cohen, Quanta 2, 7 (2013). Ruschhaupt, and A. del Campo, Lecture Notes in Physics, Vol. [80] K. Snizhko, P. Kumar, and A. Romito, Phys. Rev. Research 2, 789 (Springer, Berlin, 2009), Chap. 4, pp. 65–96. 033512 (2020).

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