Quantum walks: the first detected passage time problem
Even after decades of research the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time , we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We demonstrate our results with tight binding quantum walk models. We examine a closed system, i.e. a ring, and reveal the intricate influence of the sampling time on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one dimensional quantum walk the probability of first detection decays like with superimposed oscillations, with exceptional behavior when the sampling period times the tunnelling rate is a multiple of . The amplitude of the power law decay is suppressed as due to the Zeno effect. Our work presented here, is an extended version of Friedman et al. arXiv:1603.02046 [cond-mat.stat-mech], and it predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like , as elucidated by Schrödinger in .
How long it takes a lion to find its prey, a particle to reach a domain or an electric signal to cross a certain threshold? These are all examples of the first passage time problem Redner (); Hughes (); MOR (); Benichou (). A century ago Schrödinger showed that a Brownian particle in one dimension, i.e. the continuous limit of the classical random walk, starting at , will eventually reach , with, however, a probability density function (PDF) of the first arrival time that is fat tailed, in such a way that the mean first passage time diverges Schro (). Ever since, the classical first passage time has been a well studied field of research. More recently, much work has been devoted to the analysis of quantum walks Aharonov1 (); Dorit (); Ambainis (); Childs (); Konno () (see Blumen () for a review). These exhibit interference patterns and ballistic scaling and in that sense exhibit behaviors drastically different from the classical random walk. While several variants of quantum walks exist Blumen (), for example discrete time walks, coin tossing walks, and tight-binding models, one line of inquiry addresses a question generally applicable to all these cases, namely the statistics of first passage or detection times of a quantum particle (to be defined precisely below). Quantum walk search algorithms which are supposed to perform better than classical walk search methods vitalized this line of research in recent years. A physical example might be the statistics of the time it takes a single electron, ion or atom to reach a detection device. This question, which at first sight appears well-defined and physically meaningful, has nevertheless been the subject of much controversy. The Schrödinger Eqn. and the standard postulates of quantum mechanics CT () do not give a ready-made recipe for calculating these statistics. There is no textbook quantum operators or wave function associated with the first passage time measurements (see Hauge (); Muga (); PHD () for related historical accounts). Actually, time is a non-quantum ingredient of quantum mechanics and is treated as a object detached from the probabilistic interpretation inherent to non-classical reality. From the non-deterministic nature of quantum mechanics, we may expect that the time it takes a single particle to reach a detection point or domain for a given Hamiltonian and initial condition should be random even in the absence of external noise, but how to precisely obtain the distribution of first detection times has remained in our opinion a controversial matter.
The key to the solution is that we must take into consideration the measurement process Bach (); Brun06 (); Brun07 (); Brun08 (); Dhar (); Dhar1 (). For example consider a zebra sitting at the origin waiting for a lion to arrive for the first and, unfortunately for her, the last time. At some rate, the zebra records: did the lion arrive or did it not? The outcome is a string of answers: e.g. no, no, no, …. and finally yes. If the lion is a quantum particle, then continuous attempts to detect it by the zebra will maintain the zebra’s life, since the wave function of the lion is collapsed in the vicinity of the zebra; this is the famous quantum Zeno effect Misra () (see more details below). On the other hand, if the zebra samples the arrival of the lion at a finite constant rate, its likelihood of death is much higher. In this sense the measurement of the time of first detection, which implies a set of null measurements for times prior to the final positive recording is very different than the familiar measurements of canonical variables like position and momentum. There the system is prepared at time in some initial state, it evolves free of measurement until time , at which point an instantaneous recording of some observable is made. Furthermore, we must distinguish between first arrival or first passage problems Muga () and first detection at a site. Note that even classically the first detection does not imply that the particle arrived at the site for the first time at the moment of detection if the sampling is not continuous in time. More importantly, arrival times are ill-defined in quantum problems Aharonov (), because we cannot have a complete record of the trajectory of a quantum particle, whereas the first detection problem under repeated stroboscopic measurements is a well-defined problem, and that is what we treat in this manuscript.
Here we investigate the first detection problem of quantum walks following Dhar et al. Dhar (); Dhar1 () who formulated the problem as a tight-binding quantum walk, with projective local measurements every units of time (see also Brun06 ()). Specifically, we consider a particle on a discrete graph, the quantum evolution determined by the time independent Hamiltonian . Initially the particle is localized so the state function is (some of our general results are not limited to this initial condition, see below). Detection attempts are performed locally at a site we call the origin which is denoted with . Measurements on the origin are stroboscopic with the sampling time , and as mentioned the measurement stops once the particle is detected after attempts, so is the random first detection time. We investigate the statistics of the random observable . The questions are: is the particle eventually detected? what is the probability of detection after attempts? what is the average of number of attempts of detection before a successful measurement? This we investigate both for closed systems and open ones. Below we present a physical derivation of the quantum renewal equation describing the probability amplitude of first detection for the transition . In classical stochastic theories this corresponds to Schrödinger’s renewal equation Schro () for the first time a particle starting on reaches (see details below). We show how the solution of Schrödinger’s wave equation free of measurement can be used to predict the quantum statistics of the first detection time. Previously Grünbaum et al. Grunbaum () considered the case where the starting point is also the detection site . A topological interpretation of the detection process was provided for that initial condition and among other things they showed that the expected time of first detection is an integer times or infinity. This integer is the winding number of the so called Schur function of the underlying scalar measure, the latter is determined by the initial state and the unitary dynamics. Hence the expectation of the first detection time is quantized Sinkovicz (). A vastly different behavior is found when we analyse the transition for Dhar1 (); Brun06 (). The average of is not an integer, neither is detection finally guaranteed. As demonstrated below for a ring geometry half dark states are observed in some cases while in others the average of exhibits divergences and non-analytical behaviours, for certain critical sampling times. Finally, we show that critical sampling, including slowing down is found even for an infinite system. Namely, for the quantum walk on the line, the first detection probability decays like a power law, with additional oscillations, where the amplitude of decay is not a continuous function of the sampling rate. Thus rich physical behaviors are found for the quantum first detection problem, if compared with the known results of the classical random walker.
The spatial quantum first detection problem is a timely subject. Current day experiments on quantum walks can be used to study these problems in the laboratory Yaron (); QWe2 (); QWe1 (); QWe (); Xue (). First passage time statistics in the classical domain are usually recorded based on single particle analysis. Namely, one takes, say, a Brownian particle, releases it from a certain position and then detects its time of arrival at some other location. This single particle experiment is repeated many times and then a histogram of the first passage time is reported. While in principle one could release simultaneously many particles from the same position, their mutual interaction will influence the statistics of first arrival and similarly statistics of identical particles, either bosons or fermions, alter the many particle statistics compared to the single particle case. Hence, measurement should be made on single particles, or in other words at least classically the first detection time is a property of the single particle path and hence its history. The recent advance of single particle quantum tracking and measurement, for systems where coherence is maintained for relatively long times, is clearly a reason to be optimistic with respect to possible first detection measurements. Such measurements could test our predictions as well as those of a variety of other theoretical approaches Muga (); Aharonov (); Lumpkin (); Grot (); Stefanak (); Shikano (); Krap (); Ranjith (); Das (); Miquel (), some of which are compared with our results towards the end of this manuscript.
The navigation map of this manuscript is as follows. We start with the presentation of the quantum walk model and the measurement process in Sec. II. In Sec. III the first detection wave function formalism is developed. The main tool for actual solution of the problem is based on the generating function formalism given in Sec. IV and in subsection IV.1 the quantum renewal equation is discussed. Sec. V presents the example of first detection on rings, with special emphasis on the peculiar statistics found on a benzene-like ring. In Sec. VI we obtain statistics of first detection times, for a one-dimensional quantum walk on an infinite lattice. We end with further discussion of previous results (Sec. VII) and a summary. A short account of part of our main results was recently published FKB ().
Ii Model and Basic Formalism
We consider a particle whose evolution is described by a time independent Hermitian Hamiltonian according to the Schrödinger Eqn. . The initial condition is denoted . For simplicity, we consider a discrete -space. As an example we shall later consider the tight binding model
on a lattice, though our general results are not limited to a specific Hamiltonian. We denote a subset of lattice points , and loosely speaking we are interested in the statistics of first passage times from the initial state to any site in the subset. More generally, could be any subset of orthogonal states. An example is when consists of a single lattice point, say and initially the particle is localized at some other lattice point . We then investigate the distribution of the first detection times. For that we must define the measurement process following Dhar (); Bach (); Ambainis ().
Measurements on the subset are made at discrete times and hence clearly the first recorded detection time is either or etc. The measurement provides two possible outcomes: either the particle is in or it is not. Consider the first measurement at time . At time with being positive, the wave function is
and as usual. In what follows, we set . The probability of finding the particle in is, according to the standard interpretation,
If the outcome of the measurement is positive, namely the particle is found in , the first detection time is . On the other hand, if the particle is not detected, which occurs with probability , the evolution of the quantum state will resume. According to collapse theory, following the measurement the particle’s wave function in is zero. Namely, a null measurement alters the wave function in such a way that the probability of detecting the particle in at time vanishes. In this sense we are considering projective measurements whose duration is very short, while between the measurements the evolution is according to the Schrödinger Eqn. Mathematically the measurement is a projection CT (), so that at time we have
where is the identity operator, and the constant is determined from the normalization condition. Here we have used the assumption of a perfect projective measurement that does not alter either the relative phases or magnitudes of the wave function not interacting with the measurement device, i.e., outside the observation domain the wave function is left unchanged beyond a global renormalization. This is the fifth postulate of quantum mechanics CT (), though clearly it should be the subject to continuing experimental tests. Since just prior to measurement the probability of finding the particle in not belonging to is we get
In sum, the measurement nullifies the wave functions on but maintains the relative amplitudes of finding the particles outside the spatial domain of measurement device, modifying only the normalization.
We now proceed in the same way to the second measurement. Between the first and second detection attempts we have . The probability of finding the particle in at the second measurement, conditioned on the quantum walker not having been found in the first attempt is
We define the projection operator
This iteration procedure is continued to find the probability of first detection in the -th measurement, conditioned on prior measurements not having detected the particle
In the numerator the operator appears times corresponding to the prior measurements. Similarly, in the denominator we find probabilities of null measurements . Following Dhar (); Dhar1 () we define the first detection wave function
or equivalently with the initial condition . The bra is defined only for the moments of detection , unlike which is a function of continuous time. With this definition
The main focus of this work is on the probability of first detection in the -th measurement, denoted . This is of course not the same as which as mentioned is a conditional probability, namely the probability of detection on the -th attempt given no previous detection. The conceptual measurement process for the calculation of is as follows. We start with an initial spatial wave function and evolve it until time when the detection of the particle in is attempted, and with probability the first measurement is also the first detection. Hence to simulate this process on a computer we toss a coin using an uniform random number generator and if the particle is detected the measurement time is . If the particle is not detected we compute . Then at time either the particle is detected with probability or not. This process is repeated until a measurement is recorded (see remark below) and that measurement constitutes the random first detection event. In order to gain statistics of the first detection time we return to the initial step and restart the process with the same initial condition. In this way, repeating this many times, we construct the first detection probability
Using Eq. (11) we obtain
We see that the first detection probability is the expectation value of the projection operator with respect to , which we term the first detection wave function.
Remark. We shall see that not all sequences of measurements, generated on a computer or in the lab, yield a detection in the long-time limit. This is not problematic since also classical random walks in say three dimensions are not recurrent and hence the total probability of detection is not necessarily unity. In many works one defines the survival probability, i.e., the probability that the particle is not detected in the first attempts,
The eventual survival probability can be equal zero or not. If the initial condition and the detection location are identical and the quantum walk is called recurrent. We will later investigate whether or not the quantum walk is recurrent, both for the cases of an infinite lattice and a finite ring.
Iii First detection amplitude
In this section, we solve the first detection time problem for quantum dynamics with a single detection site, which we label , so . We define the amplitude of the first detection as
so that . Using Eq. (10) , and a short calculation yields
In Appendix A, using induction we find our first main equation
We call this iteration rule the quantum renewal equation. It yields the amplitude in terms of a propagation free of measurement; i.e., is the amplitude for being at the origin at time in the absence of measurements, from which we subtract terms related to the previous history of the system. The physical interpretation of Eq. (17) is that the condition of non-detection in previous measurements translates into subtracting wave sources (hence the minus sign) at the detection site following the th detection attempt. This is due to the nullification of the wave function at the detection site in the th measurement. The evolution of that wave source from the th measurement onward is described by the free Hamiltonian, hence which gives the amplitude of return back to the origin, in the time interval .
We now consider the formal solution to the first detection problem for an initial condition on the origin hence and as mentioned the origin is also the point at which we perform the detection trials. Clearly in this case and since we get when which is expected. For where we use the short-hand notation . Similarly . The general solution is obtained by iteration using Eq. (17),
The double sum is over all partitions of , i.e. all -tuples of positive integers satisfying . For example for we have five partitions corresponding to , for the set in the second sum is , for we sum over and , for we use , for , and for we have one term . Hence
With a symbolic program like Mathematica one can obtain similar exact expressions for intermediate values of . However, to gain some insight we turn now to the generating function approach Brown ().
Iv Generating function Approach
is also called the generating function. Multiplying Eq. (17) by and summing over
Evaluating the first term on the right hand side we get
or more explicitly
This equation, relates the generating function to the Hamiltonian evolution between the initial condition and the detection attempt.
This approach is also valid for other types of measurements, repeatedly performed at times . For example the case where we measure a set of points is given in Eq. (100) in Appendix A. First detection measurements of general observables is also treated there.
iv.0.1 Relations between and , and .
As usual the amplitudes are given in terms of their -transforms by the inversion formula
where is a counter clockwise path that contains the origin and is entirely within the radius of convergence of .
The probability of being measured is also related to the generating function by
The latter is the average of only when the particle is detected with probability one, namely when . A shorthand notation of Eq. (28) is .
iv.1 Connection between first detection and spatial wave function
In classical random walk theory the key approach to the first passage time problem is to relate it to occupation probabilities Redner (). Let us unravel a similar relation in the quantum domain, connecting between first detection statistics and the corresponding wave packet, namely the time dependent solution of the Schrödinger equation in the absence of measurement (see also Grunbaum () for the transition). To that end, we first briefly review the classical random walk.
Consider a classical random walk in discrete time , for example a random walk on a cubic lattice in dimension with jumps to nearest neighbours. The main assumption is that the random walk is Markovian. Denote as the probability that the walker is at at time when initially the particle is at the origin and in the absence of any absorption. Let be the first passage probability: the probability that the random walk visits site for the first time at time with the same initial condition. Following the first equation in the first chapter in Redner (), and are related by
This equation Schro (); Montroll (); MontrollW (), sometimes called the renewal equation, is generally valid for Markov processes in the sense that it is not limited to discrete time and space models; in the continuum one needs only to replace summation with integration, and probabilities by probability densities. The idea behind Eq. (29) is that a particle on position at time must have either arrived there previously at time for the first time and then returned back or it arrived at exactly at time for the first time (the term) Schro (); Redner (). Using the transform the following equations are derived Redner ()
From this formula, various basic properties of random walks can be derived. One example is the Pólya theorem which answers the question: does a particle eventually return to its origin; i.e., whether the random walk is recurrent. A second is that in one dimension, for an open system without bias, the famous law is found for large first passage time and hence the first passage time has an infinite mean, as mentioned in the introduction. We will later find the quantum analogue to this well known behaviour.
At first glance this classical picture might not seem related to ours. However consider the case where we detect the particle at the origin, so and initially . Then Eq. (24) reads
We add and subtract one in the numerator and use and
to rewrite Eq. (31) as
Expanding in we get a geometric series
By definition the sum is the generating function of the amplitude of being at the origin retrieved from the solution of the Schrödinger equation without detection. Namely, let be the solution of the Schrödinger equation for the same initial condition (the subscript denotes a wave function free of measurement). The amplitude of being at the origin at time is and as usual. We define the generating function of this amplitude, for the sequence of measurements under consideration
and clearly , the subscript zero denotes the initial condition. Hence we get the appealing result reported already in Grunbaum ()
Thus the generating function of the first detection time is determined from the transform of the spatial wave function at the point of detection . This connection is the quantum analogue of the second line in the classical expression Eq. (30) since in both cases we start and detect at the origin.
Similarly for an initial condition initially localized at some site , so with detection at site we find
where is the transform of the wave function free of measurements initially localized on site , with . We see that the ratio of the generating functions of the amplitudes of finding the particle on for initial condition on and the location of measurement site , obtained from the measurement-free evolution, yields the generating function of the measurement process. This is the sought after quantum renewal equation, namely the amplitude analogue of the upper line of the classical Eq. (30).
Remark Our formalism is not limited to spatially homogeneous Hamiltonians. Note that in our classical discussion, following the textbook treatment Redner () and for the sake of simplicity, we have assumed translation invariant random walks. In non-translation invariant systems, one should replace in the left hand side of Eq. (29) with . Since the convolution structure of the equation remains, related to the Markovian hypothesis, Eq. (30) can be easily modified to include non-homogeneous effects.
Remark Sinkovicz et al. Sinkovicz1 () found a quantum Kac-Lemma for recurrence time, thus analogies between quantum and classical walks are not limited to the renewal equation under investigation.
iv.2 Zeno Effect
The amplitude of finding the particle at the origin in the first attempt, is given by the initial wave function projected on the origin, i.e. the probability amplitude of finding the particle at the origin at . Hence the above expression gives an obvious answer for the first measurement; the repeated measurements being very frequent do not allow the wave function to be built up at the origin, and hence for all . This means that we may investigate the problem for small relative to the time scales of the Hamiltonian, but we cannot take the limit if we wish to retain information on the measurement process beyond the initial state.
iv.3 Energy representation
Eq. (22) for a time-independent Hamiltonian yields
so that the operator is diagonal in the energy representation. Here is a stationary state of the Hamiltonian , namely . Clearly it is worthwhile presenting the solution in that basis. Consider the example of the measurement at the spatial origin corresponding to state . This state can be expanded in the energy representation with . Here as usual . Similarly the initial condition is expanded as . The matrix element
yields using Eq. (23). For the special case where we get and
Here as usual . It is easy to check that when we get since a particle starting at the origin is with probability one detected when .
For our explicit calculations, we will focus on tight binding models in one dimension Blumen (). The first model is a quantum walk on a ring of length :
This describes a quantum particle jumping between nearest neighbours on the ring. We use periodic boundary conditions and thus from the site labeled one may jump either to the origin or to the site labeled . In condensed matter physics the parameter is called the tunnelling rate.
v.1 Benzene-Type Ring
As our first example we consider the tight-binding model on a hexagonal ring presented in Fig. 1, namely a structure similar to the benzene molecule CT (); Feynman (). We consider the influence of initial states with on the statistics of first detection times for detection at site so . According to our theory, to find the generating function we need the energy levels of and its eigenstates. The six energy levels of the system are with and the eigenstates are with CT () ( is the transpose). Hence the coefficients , reflecting the symmetry of the problem.
v.1.1 Starting at
We use Eq. (42) and find
The nondegenerate energy levels are and while and are doubly degenerate, hence for real
It is interesting to note that the generating function satisfies the identity
an identity we will return to below when discussing and . Inserting Eq. (46) in Eq. (27) and integrating over gives . Thus the survival probability is zero in the long time limit. This behavior is classical in the sense that for finite systems a classical random walker is always detected. Note that for a quantum walker this conclusion is not generally valid. If we start at for example and measure at , and perform measurements on full revival periods, the particle is never recorded (see further details and other examples below). Hence, for a quantum particle the survival probability does not generally decay to zero as , even for finite systems.
For special values of we get exceptional behaviors. When is times an integer we get namely the measurement in the first attempt is made with probability , so the first detection time is , which is expected since the wave function is fully revived at these ’s in its initial state at the origin Blumen (). If we get . Inverting we find and for , thus the amplitude decays exponentially. It follows that the first detection probabilities are
The average number of detection attempts is . If we find which has simple poles and hence
where and . For this case . Similarly for and we get . The general feature of finite rings is an exponential decay of with a superimposed oscillation determined by the poles of the generating function. However the sampling times considered so far exhibit behaviors which are not typical, as we now show.
A surprising behavior is found for the average, with
for any sampling rate in the interval besides what we call the exceptional sampling times where as mentioned respectively, which is continued periodically (see Fig. 2). This result is derived below. As mentioned in the introduction the fact that is some integer was already pointed out rather generally by Grunbaum () and this is related to topological effects. Except for the exceptional points, the variance of is
so that the first detection time exhibits large fluctuations near these points. Thus for the to transition it is only the average that is nearly always not sensitive to the sampling rate, not the full distribution of first detection times.
There are numerous methods to find . For the exceptional points we used the exact solution for (as mentioned). For other sampling times we use two approaches: the first using Mathematica and is based on a Taylor expansion of and the second is an analytic calculation. The former approach is very general in the sense that it can be used in principle for general initial conditions and other problems beyond the benzene ring.
Specifically, we calculate exactly using the expansion of with symbolic programming on Mathematica. This is performed up to some large . We then calculate . Clearly , and increasing we see convergence towards except for the mentioned exceptional points. An example is shown in Fig. (2) for the cases and .
and with Eq. (28)
Rewriting we can proceed to show that
where (or ) is the number of zeros of for within (or on) the unit circle respectively. As explained in Appendix B, for otherwise we would find . For the exceptional values of we find , as follows:
This agrees with the values of we have found at the exceptional points. This exercise shows that mathematically, at least for this example, the exceptional points are those specific values of where some of the zeros of the polynomial are found to lie on the unit circle in the complex plane. We will soon find a by far more physical and explicit formula for these points, Eq. (58) below.
v.1.2 Half Dark states
Another peculiar behavior is found if the detection is at the origin and the starting point is with . The total probability of detection is found to be, by the method explained in Appendix B,
for all values of besides exceptional points which are listed in Table 1. The exceptions include the case when is the full revival time, for which case the probability of being detected is of course . The behaviour Eq. (55) was observed in Dhar (); Dhar1 () for even larger systems. It is remarkable that for certain initial conditions, the detection of the particle is not guaranteed, and only in half of the measurement processes we detect the particle, hence we call these initial conditions half dark states.
v.1.3 Starting on site measuring on
In contrast, if the starting state is the total probability of detection is found to be , if the measurement time is not the full revival time , or one of the exceptional sampling times listed in Table 1. In Appendix B, we find
an equation valid for all besides the exceptional points. The general behavior of is obviously quite different from the case when the initial location is , compare Fig. 2 and Fig. 3 indicating that the initial condition plays a crucial rule. As shown in Fig. 3 the average exhibits nontrivial behavior as it diverges as it approaches some of the exceptional points. These singularities are found near those exceptional sampling times where the total probability of measurement is not one. Interestingly the values of , conditioned on return, are finite at the exceptional points themselves.
An analytical calculation for the exceptional sampling times or finds . This sampling time is unique since the average exhibits a discontinuity: for in the vicinity of and we find using Eq. (56) (so at these points the equation is not valid). Similar to any discontinuity at a point, the discontinuity of at might not be detectable in experiment. However one finds critical slowing down, namely the convergence of for any point in the vicinity of these exceptional points is very slow, as demonstrated in Fig. 3.
v.2 Rings of Size
While the benzene ring is instructive, one must wonder how general are the main results. In Appendix B we derive the following four results:
For a ring of size and for a particle initially on site where the measurements are performed, the particle is detected with probability unity, and in this sense the motion is recurrent. We emphasize that this result is a property of the specified initial condition.
For the same initial condition, besides those isolated exceptional sampling times listed below, the average number of detection events is
This result is remarkable since the average is independent of the sampling time . Here we see that is the number of distinct energy levels of the system. In the language of Grunbaum () it is the winding number of the Schur function, or the effective dimension of the Hilbert space. For large systems grows linearly with the size of the system , while from classical random walk theory we naively expect diffusive scaling . In that sense the quantum walk is more efficient (see below further remarks).
The exceptional sampling times are given by the rule
where is an integer, and is the energy difference between pairs of eigenenergies of the underlying Hamiltonian . For example the stationary energies of the benzene ring are as mentioned, and hence Eq. (58) predicts the exceptional sampling times . The condition Eq. (58) implies a partial revival of the wave packet free of measurement, namely two modes of the system behave identically when strobed at the period . On these exceptional points the solution exhibits non-analytical behavior. This is manifested in discontinuities or diverging behavior of or the fluctuations of and also slow critical-like convergence to the asymptotic theory. The exact nature of the non-analytical behavior depends on the initial condition as we have demonstrated for the benzene ring.
For a particle starting on every time the condition (58) is met by a pair of energy levels we reduce the value of by unity. Thus Eq. (58) is the upper limit of for a system of fixed size . More specifically we find that
where are the energy levels of the system. For nearly any this is the same as the number of distinct energy levels, but of course for special sampling times, this integer is less than that.
v.3 Bose-Einstein Distribution
A curiosity is the fact that we may express the solution for the (starting point) to