First Detected Arrival of a Quantum Walker on an Infinite Line

Abstract

The first detection of a quantum particle on a graph has been shown to depend sensitively on the distance ξ between detector and initial location of the particle, and on the sampling time τ . Here we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an infinite line, using a tight-binding lattice Hamiltonian with nearest-neighbor hops. Universal features of the first detection probability are uncovered and simple limiting cases are analyzed. These include the large ξ limit, the small τ limit and the power law decay with attempt number of the detection probability over which quantum oscillations are superimposed. For large ξ the first detection probability assumes a scaling form and when the sampling time is equal to the inverse of the energy band width, non-analytical behaviors arise, accompanied by a transition in the statistics. The maximum total detection probability is found to occur for τ close to this transition point. When the initial location of the particle is far from the detection node we find that the total detection probability attains a finite value which is distance independent.Introduction: Recent experimental advances have made it possible to measure quantum walks at the single particle level [1][2][3][4]. A related advance is the quantum first detection problem which has drawn considerable theoretical attention [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], as it deals with the basic issue of when the particle will first be detected in a target state. Originally the topic emerged in the context of quantum search algorithms. Given a graph, and an Hamiltonian H, the presence or absence of a particle starting on node |x i is recorded at a node |x d sampled with period τ . H, τ and the measurement process [23], define the problem, which differs markedly from the corresponding wellstudied classical first-passage-time problem [24][25][26][27][28].Recently, a quantum renewal equation which relates the statistics of first detection times to the quantum evolution operator of the measurement-free system was derived [21,22]. This equation was used investigate the statistics of the first detected return, i.e. when |x i = |x d . The present Letter focuses on the first detected arrival, |x i = |x d . The questions to be tackled are: Given τ and the tight-binding Hamiltonian on an infinite line, what are the basic properties of the first detection probability? More specifically: Will the particle always be detected? If not, then what is the optimal sampling rate for which the total detection probability is maximized? The existence of an optimum is expected, since the Zeno effect [29,30] suppresses detection for too frequent measurements, while a too large τ aids the escape from the detector. What is the general behavior of the detection probability at attempt number n, which we denote F n ? What is its asymptotics for small and large n and how does it depend on the initial distance ξ = |x d − x i | of the particle from the detector? ...

Running measurement protocol for the quantum first-detection problem

“…This behavior is similar to behavior for not too large n in the SMP arrival problem which was discussed in Ref. [11]. As there, we can find the value of n c via considerations of group velocity.…”

“…As we shall see, the conversion from ϕ(z) to ϕ n yields qualitatively different results in the fast and slow detector cases. We use the same contour integral procedure used for the stationary detector in [11]. where N = 1000, vs τ , the results converging as we increase N further (note that P det = 1 for these cases).…”

“…The slow-moving detector case has two branch-cuts, so the analysis is similar to that of the stationary detector discussed in Ref. [11] and we will not revisit it here in this paper.…”

Large fluctuations of the first detected quantum return time

“…A well investigated approach is to consider measurements on the original node, repeated stroboscopically until the first detection. The problem is to find the distribution of the number of attempts n till first detection [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. This sheds light on the backfire of measurement on unitary evolution, and on time processes in quantum mechanics.…”

Short-time behavior of continuous-time quantum walks on graphs

“…where D is the degree matrix of G and V (ω) is the diagonal matrix defined in Eq. (55). This determines L ω , the graph of L ω completely.…”