Large-time limit of the quantum Zeno effect

TL;DR

This paper studies how the quantum Zeno effect transitions to diffusive behavior as measurement frequency and total observation time scale, revealing a critical threshold at  and dependence on spectral and arithmetic properties.

Contribution

It characterizes the large-time limit of the survival probability under different scaling regimes, identifying a sharp transition at  and exploring spectral and arithmetic sensitivities.

Findings

Survival probability jumps from 1 to 0 at  scaling.

Limit probability depends on spectral properties for  scaling.

Behavior transitions from quantum Zeno to diffusive at .

Abstract

If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, $t\propto N^\alpha$. The limit survival probability exhibits a sudden jump from 1 to 0 at $\alpha=1/2$, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for $\alpha\geq 1$ the limit probability becomes sensitive to the spectral properties of the initial state and to arithmetic properties of the measurement periods.

Equations

$$\mathcal{A}(t)=\langle\psi|\exp\Big{(}-\frac{\mathrm{i}t}{\hbar}H\Big{)}\psi\rangle$$

$$p(t)=\left|\mathcal{A}\left(t\right)\right|^{2}=\left|\langle\psi|\exp\Big{(}-\frac{\mathrm{i}t}{\hbar}H\Big{)}\psi\rangle\right|^{2}$$

$$p(t)=1-\frac{t^2}{{\hslash }^2}\left(⟨H\psi |H\psi ⟩-⟨\psi |H\psi {⟩}^2\right)+o\left(t^2\right),$$

$$p^{(N)}(t):=p\left(\frac{t}{N}\right)^{N}=\left|\langle\psi|\exp\Big{(}-\frac{\mathrm{i}t}{\hbar N}H\Big{)}\psi\rangle\right|^{2N}.$$

$$\underset{N\to +\infty }{\lim }{p}^{(N)}(t)=1,$$

$${\underset{t\to +\infty }{\lim }}_{N\to +\infty }{p}^{(N)}(t).$$

$$\mathop{\lim_{t\to+\infty}}_{N\to+\infty}p^{(N)}(t)=\mathop{\lim_{t\to+\infty}}_{N\to+\infty}\left|\langle\psi|\exp\Big{(}-\frac{\mathrm{i}t}{\hbar N}H\Big{)}\psi\rangle\right|^{2N}=\mathop{\lim_{\hbar\to 0}}_{N\to+\infty}\left|\langle\psi|\exp\Big{(}-\frac{\mathrm{i}t}{\hbar N}H\Big{)}\psi\rangle\right|^{2N}.$$

$$\underset{t\to +\infty }{\lim }\underset{N\to +\infty }{\lim }{p}^{(N)}(t)=1.$$

$$\underset{N\to +\infty }{\lim }\underset{t\to +\infty }{\lim }{p}^{(N)}(t)=0.$$

$$t=\tau N^{\alpha },$$

$$p_{N,\alpha }(\tau ):=p^{(N)}(\tau N^{\alpha })=p(\tau N^{\alpha -1}{)}^N={\left|⟨\psi |\exp \left(-\frac{\mathrm{i}\tau }{\hslash }{N}^{\alpha -1}H\right)\psi ⟩\right|}^{2N},$$

$$p_{\alpha }^{(\infty )}(\tau )=\underset{N\to +\infty }{\lim }{p}_{N,\alpha }(\tau ),$$

$$p_0^{(\infty )}(\tau )=\underset{N\to \infty }{\lim }p{\left(\frac{\tau }{N}\right)}^N,$$

$$p_1^{(\infty )}(\tau )=\underset{N\to \infty }{\lim }p(\tau {)}^N.$$

$$\hslash =\frac{{\hslash }_0}{N^{\alpha }},$$

$$p_{\alpha }^{(\infty )}(\tau )=1,$$

$$p_{1/2}^{(\infty )}(\tau )=\exp \left(-\frac{{\tau }^2}{{\tau }_Z^2}\right),$$

$${\tau }_Z^{-2}=\frac{1}{{\hslash }^2}\left(⟨H\psi |H\psi ⟩-⟨\psi |H\psi {⟩}^2\right);$$

$$p_{\alpha }^{(\infty )}(\tau )=0,$$

$$p_{N,\alpha }(\tau )=p{\left(\frac{\tau }{N^{1-\alpha }}\right)}^N,$$

$$p_{N,\alpha }(\tau )={\left[1-\frac{{\tau }^2}{{\tau }_Z^2{N}^{2(1-\alpha )}}+o\left(\frac{{\tau }^2}{N^{2(1-\alpha )}}\right)\right]}^N=1-\frac{{\tau }^2}{{\tau }_Z^2{N}^{1-2\alpha }}+o\left(\frac{{\tau }^2}{N^{1-2\alpha }}\right).$$

$$\underset{N\to +\infty }{\lim }{p}_{N,\alpha }(t)=1,$$

$$p_{N,\alpha }(\tau )={\left[1-\frac{{\tau }^2}{{\tau }_Z^{2}N}+o\left(\frac{{\tau }^2}{N}\right)\right]}^N,$$

$$\underset{N\to +\infty }{\lim }{p}_{N,\alpha }(\tau )=\exp \left(-\frac{{\tau }^2}{{\tau }_Z^2}\right)$$

$$p_{N,\alpha }(\tau )=\exp \left[N\log p(\tau /N^{1-\alpha })\right]$$

$$\log p(s)=\log \left(1-\frac{s^2}{{\tau }_Z^2}+o(s^2)\right)=-\frac{s^2}{{\tau }_Z^2}+o(s^2),$$

$$\log p(s)\le -\frac{s^2}{2{\tau }_Z^2},$$

$$\log p(\tau /N^{1-\alpha })\le -\frac{{\tau }^2}{2{\tau }_Z^2}\frac{1}{N^{2-2\alpha }},$$

$$p_{N,\alpha }(\tau )\le \exp \left(-\frac{{\tau }_{\mathrm{m}}^2}{2{\tau }_Z^2}{N}^{2\alpha -1}\right),$$

$$\underset{N\to +\infty }{\lim }{p}_{N,\alpha }(\tau )=0$$

Full text

I Introduction

The evolution of a quantum system is halted when very many measurements are performed in a finite time, in order to check whether the system is still in its initial state. This phenomenon is called Quantum Zeno Effect misra (QZE): the survival probability at a given time goes to one as the measurement frequency increases.

The survival probability after $N$ measurements in a time $t$ is expressed by a product formula depending on $t$ and $N$. In this paper we investigate the uniformity in time of the QZE, and study the behavior of the Zeno product formula for large $N$ and $t$ .

Let us first recall the basics of the QZE. Let a quantum system be prepared, at time $t=0$, in the state $\psi$, a normalized vector in the separable Hilbert space $\mathcal{H}$. We denote by $\langle\cdot|\cdot\rangle$ the scalar product in $\mathcal{H}$. The system evolves under the action of the Hamiltonian $H$, a self-adjoint operator on $\mathcal{H}$, through the unitary group $t\mapsto\exp(-\mathrm{i}tH/\hbar)$. The quantities

[TABLE]

and

[TABLE]

are called survival (or return) amplitude and probability, respectively, and represent the amplitude and probability that the quantum system is found back in the initial state $\psi$ at time $t$.

If the state $\psi$ is in the domain of the Hamiltonian $H$, we have for $t\to 0$

[TABLE]

where $\langle H\psi|H\psi\rangle-\langle\psi|H\psi\rangle^{2}$ is the variance of the Hamiltonian in the state $\psi$.

Let us now carry out $N$ measurements at time intervals $\tau=t/N$, in order to check whether the system remains in its initial state. If at each and every time the measurement has a positive outcome and the system is found in its initial state, the state “collapses” and the evolution starts anew from $\psi$. Thus the survival probability after $N$ measurements reads

[TABLE]

This is called Zeno product formula, and will be the subject of our investigation.

The limit of infinitely frequent measurements, $N\to+\infty$, of the Zeno product formula can be easily computed using the Taylor expansion in (3): if the initial state $\psi$ is in the domain of the Hamiltonian $H$ one gets

[TABLE]

uniformly in $t$ on compact subsets of $\mathbb{R}$, see ZenoMP ; exner ; artzeno . Therefore, if one performs frequent measurements on a quantum system in a given time interval $[0,t]$, a QZE takes place misra : the transitions to states different from the initial one are hindered, despite the action of the Hamiltonian (in general the state $\psi$ is not an eigenstate of the Hamiltonian $H$).

The QZE has been successfully demonstrated in a variety of physical systems, on experiments involving ionic hyperfine levels Itano90 , photons kwiat , nuclear spins Chapovsky , optical pumping molhave2000 , ultracold atoms raizenlatest , level dynamics of individual ions balzer2002 , Bose-Einstein condensates ketterle , optical systems hosten , and cavity quantum electrodynamics haroche . For a review on the mathematical and physical aspects of the subject see ZenoMP .

The QZE can be obtained both by pulsed and continuous measurements, as well as by a strong interaction Schulman97 ; Schulman98 ; zenopraga . Recently it has been realized that, by exploiting the quantum Zeno dynamics, one gets a powerful approach to control. The key idea is to engineer a given evolution by a rapid sequence of projections AA88 ; AP89 ; Bal2000 . This can yield a Berry phase berry or, more generally, non-Abelian geometric phases ShapereWilczek , a resource for holonomic quantum computation. Moreover, the QZE can be seen as an effective way of imposing constraints and boundary conditions regularize ; exner ; EINZ . Finally, notice that the QZE is a purely quantum phenomenon: in classical mechanics it is not observed, since the measurement process can be conceived so that it does not interfere with the evolution of the system.

In this article we want to investigate the behavior of the Zeno product formula (4) as the observation time becomes large, $t\to+\infty$, namely the double limit:

[TABLE]

Notice that, since the time dependence in (4) is given through the ratio $t/\hbar$, the long-time limit (6) is in fact a semiclassical limit, where the Planck constant $\hbar\to 0$, namely,

[TABLE]

In this respect, the limit (7) answers the following question: what happens to the evolution of the system when we compare the period between two successive measurements with the quantum scale given by $\hbar/E_{0}$, with $E_{0}$ being the relevant energy scale of the state?

For the analysis of the classical limit of the QZE see zenosemiclassic . By a semiclassical analysis on phase space tomolectures ; L2016 , it can be shown that the QZE vanishes at all orders in the Planck constant $\hbar$, in the limit $\hbar\to 0$, and thus it is a purely quantum phenomenon without classical analogue, at the same level of tunneling. (Notice, however, that at variance with zenosemiclassic , in the present situation the state $\psi$ and the Hamiltonian $H$ do not depend on $\hbar$.)

Heuristically, if we perform first the limit in $N$ and then the limit in $t$ we get QZE, namely

[TABLE]

Conversely, for a decaying system, if we invert the order of the two limits we obtain a classical behavior, namely

[TABLE]

Therefore, the limit (6) does not exist because it depends on the way in which it is computed.

In order to better understand the transition from (8) to (9), we look at the double limit (6) when $t$ diverges as a power of $N$, i.e.

[TABLE]

where $\tau>0$ is a fixed time interval and $\alpha\geq 0$, see the right panel of Fig. 1. In this case the survival probability $p^{(N)}(t)$ depends only on $N$, $\alpha$ and $\tau$. Therefore, in the following we will consider the product formula

[TABLE]

and investigate the limit

[TABLE]

for different values of $\alpha\geq 0$.

The value $\alpha=0$ will correspond to the QZE limit (5),

[TABLE]

while the value $\alpha=1$ will correspond to the large-time limit of an evolution stroboscopically measured with period $\tau$,

[TABLE]

The latter regime, describing a quantum system subject to periodic kicks, has become a paradigmatic example in the study of quantum chaos Berry79 ; DegliEspostiGraffi . It represents a standard test bed for the investigation of different features of quantum systems whose classical counterparts have a chaotic evolution. In Ref. qchaos the dynamics of a kicked quantum system undergoing repeated measurements of momentum has been investigated. A diffusive behavior has been obtained, even when the dynamics of the classical counterpart is not chaotic, and in general, the system has been shown to have an anomalous diffusive behavior, characteristic of intermittent classical dynamical systems and random walks in random environments BCLL2016 .

Thus as $\alpha$ ranges from 0 to 1, one goes from QZE to a kicked dynamics, and for a decaying system the limit probability $p^{(\infty)}_{\alpha}(\tau)$ goes from $1$ to [math]. We will show that the transition is abrupt with a threshold at $\alpha=1/2$, as shown in Fig. 1.

Moreover, we will also consider larger values of the exponent, i.e. $\alpha>1$, which correspond to stroboscopic measurements with a larger and larger period. In such a case the limit probability $p^{(\infty)}_{\alpha}(\tau)$ of a decaying system is obviously 0, but interesting exceptions will occur at particular values of $\alpha$ and $\tau$ for systems with recurrences.

Notice also that, as a consequence of the previous discussion, the limit (12) can be viewed as the semiclassical limit (7) when $\hbar$ goes to zero as a power of $N$, i.e.

[TABLE]

where $\hbar_{{0}}>0$ is a fixed constant. Therefore, for short in the following we will often refer to the regime with zero limit probability as the classical regime, as opposed to a nonzero limit probability characteristic of a quantum regime, see Fig 1.

The article is organized as follows: in Section II we discuss the case $\alpha<1$ and we show that $\alpha=1/2$ is the threshold exponent between quantum and classical behavior; in Section III we focus on the case $\alpha\geq 1$ and we prove that, essentially, the system exhibits always a classical behavior, but the limit becomes sensitive to the spectral properties of the state $\psi$ and to some interesting arithmetical properties of $\tau$ and $\alpha$.

II Threshold exponent between quantum and classical behavior

In this section we discuss the case $0\leq\alpha<1$ and we show that $\alpha=1/2$ is the threshold exponent between quantum and classical behavior.

Theorem 1**.**

Let $\psi$ be in the domain of $H$, i.e. $\|H\psi\|<+\infty$. The limit (12) of the product formula (11) has the following behavior:

1. (i)

If $0\leq\alpha<1/2$ then

[TABLE]

uniformly in $\tau$ on compact subsets of $\mathbb{R}$; 2. (ii)

If $\alpha=1/2$ then

[TABLE]

uniformly in $\tau$ on compact subsets of $\mathbb{R}$, where

[TABLE] 3. (iii)

If $1/2<\alpha<1$ and $\psi$ is not an eigenstate of $H$, then

[TABLE]

uniformly in $\tau$ on compact subsets of $\mathbb{R}\setminus\{0\}$.

Remark 1**.**

The time $\tau_{Z}>0$ defined by (18) characterizes the initial quadratic behavior of the survival probability and is known in the literature as the Zeno time ZenoMP . Notice that the variance of $H$ given by (18) is zero if and only if the state $\psi$ is an eigenstate of $H$. In such a case the Zeno time is $\tau_{Z}=+\infty$.

Proof.

If $0\leq\alpha<1/2$ then

[TABLE]

therefore using (3) we obtain that

[TABLE]

Since $1-2\alpha>0$, we immediately obtain that

[TABLE]

uniformly in $\tau$ on compact subsets of $\mathbb{R}$.

Following the same procedure we obtain that for $\alpha=1/2$

[TABLE]

therefore

[TABLE]

uniformly in $\tau$ on compact subsets of $\mathbb{R}$.

Finally we discuss the case $1/2<\alpha<1$. Notice that

[TABLE]

where $p$ is the survival probability (2). By (3), we have

[TABLE]

as $s\to 0$, with a finite $\tau_{Z}$, since $\psi$ is not an eigenstate of $H$. Therefore for $s$ sufficiently small, say $|s|\leq\sigma$, one gets

[TABLE]

whence

[TABLE]

for $N\geq(|\tau|/\sigma)^{1/(1-\alpha)}$.

It follows that for $\tau\in[\tau_{1},\tau_{2}]\subset\mathbb{R}\setminus\{0\}$ one gets

[TABLE]

for $N\geq(\tau_{\mathrm{M}}/\sigma)^{1/(1-\alpha)}$, where $\tau_{\mathrm{m}}=\min\{|\tau_{1}|,|\tau_{2}|\}>0$ and $\tau_{\mathrm{M}}=\max\{|\tau_{1}|,|\tau_{2}|\}$. Therefore,

[TABLE]

uniformly in $\tau$ on compact subsets of $\mathbb{R}\setminus\{0\}$. ∎

III Sensitivity to the spectral properties of the initial state

In this section we discuss the case $\alpha\geq 1$. We will show that in this regime the limit (12) exhibits always a classical behavior, but it becomes sensitive to the spectral properties of the state $\psi$ and to the arithmetical nature of $\alpha$.

We recall here some basic aspects of spectral theory; see, e.g., deoliv . Let $\varphi$ be a vector in the Hilbert space $\mathcal{H}$, and let $H$ be a self-adjoint operator. By the spectral theorem there exists a unique Borel measure $\mu_{\varphi}$ on $\mathbb{R}$ such that

[TABLE]

for all $f\in C_{\mathrm{b}}(\mathbb{R})$, where $\sigma(H)$ denotes the spectrum of $H$ and $C_{\mathrm{b}}(\mathbb{R})$ denotes the space of bounded and continuous functions on $\mathbb{R}$ with complex values. The spectral properties of the Hamiltonian induce a canonical decomposition of the Hilbert space $\mathcal{H}$ into the direct sum

[TABLE]

where

[TABLE]

is the continuous subspace,

[TABLE]

is the pure point subspace,

[TABLE]

is the absolutely continuous subspace, and

[TABLE]

is the singular continuous subspace.

We recall that a Borel measure $\mu_{\mathrm{c}}$ on $\mathbb{R}$ is continuous if it does not concentrate at any point, that is if

[TABLE]

while a measure $\mu_{\mathrm{pp}}$ is pure point (or discrete) if

[TABLE]

for all measurable sets $B\subset\mathbb{R}$. Moreover a measure $\mu_{\mathrm{ac}}$ is absolutely continuous (with respect to the Lebesgue measure $\mathrm{d}x$) if it has a density function $\rho$ locally integrable so that

[TABLE]

Finally, a singular continuous measure $\mu_{\mathrm{sc}}$ is continuous but not absolutely continuous; a paradigmatic example is the Cantor measure deoliv .

In this section we investigate how the limit (12) changes if the initial state $\psi$ belongs to the spectral subspaces $\mathcal{H}_{\mathrm{pp}}$, $\mathcal{H}_{\mathrm{ac}}$, and $\mathcal{H}_{\mathrm{sc}}$, which physically correspond to bound states (made up of eigenstates), scattering states, and recurring extended states, respectively.

In the following theorem we study the case $\alpha=1$.

Theorem 2**.**

If $\alpha=1$ the product formula (11) is given by

[TABLE]

and its limit

[TABLE]

has the following behavior:

1. (i)

If $p(\tau)<1$ for all $\tau\in\mathbb{R}\setminus\{0\}$ then

[TABLE]

uniformly in $\tau$ on compact subsets of $\mathbb{R}\setminus\{0\}$; 2. (ii)

If there exists $\tau_{0}\in\mathbb{R}\setminus\{0\}$ such that $p(\tau_{0})=1$, then $\psi\in\mathcal{\mathcal{H}_{\mathrm{pp}}}$ and there exists a positive integer $m$ such that

[TABLE]

Proof.

The assertion $(i)$ is obvious. Now assume that

[TABLE]

for some $\tau_{0}\in\mathbb{R}\setminus\{0\}$. Then we have that

[TABLE]

for some $a\in\mathbb{R}$. By the spectral theorem we have that

[TABLE]

and thus

[TABLE]

Since $(1-\cos(\tau_{0}(\lambda-a)/\hbar))\geq 0$, we have that

[TABLE]

almost everywhere with respect to $\mu_{\psi}$. Therefore, there exist $r\in\mathbb{N}\cup\{+\infty\}$ and a subset of integers $\{k_{j}:j=1,\dots,r\}\subset\mathbb{Z}$, such that the support of the spectral measure $\mu_{\psi}$ is the set

[TABLE]

Therefore the state $\psi\in\mathcal{H}_{\mathrm{pp}}$ and has the form

[TABLE]

where $\psi_{j}$ is a normalized eigenvector belonging to the eigenvalue $\lambda_{j}$ and $c_{j}\in\mathbb{C}\setminus\{0\}$, for all $j=1,\dots,r$; see deoliv . Using (48) it is easy to check that $p(\tau)$ can be written as follows

[TABLE]

where $p_{j}=|c_{j}|^{2}$, $j=1,\dots,r$, with

[TABLE]

since $\|\psi\|=1$. Consider now the greatest common divisor of the $k_{j}$s

[TABLE]

so that $k_{j}=m\tilde{k}_{j}$ with $\tilde{k}_{j}\in\mathbb{Z}$ for $j=1,\dots r$. We get

[TABLE]

Therefore, if $m\tau/\tau_{0}$ is an integer then all cosines are equal to one and $p_{N,1}(\tau)=1$ for all $N$. On the other hand, if $m\tau/\tau_{0}$ is not an integer then there exists at least a pair of integers $\tilde{k}_{j}$ and $\tilde{k}_{l}$ which are coprime, and thus $(\tilde{k}_{j}-\tilde{k}_{l})m\tau/\tau_{0}$ is not an integer and the corresponding cosine is not $1$. Therefore (41) holds. ∎

Remark 2**.**

Notice that $|\tau_{0}|/m$ is the first return time of the survival probability (2), that is $p(|\tau_{0}|/m)=1$ and $p(\tau)<1$ for $0<\tau<|\tau_{0}|/m$. Moreover, observe that in the proof of assertion $(ii)$ of the previous Theorem we have retraced the proof of a well known result in probability, see shir , Theorem 5 pag. 288.

Now we discuss the case $\alpha>1$.

Theorem 3**.**

If $\alpha>1$ the product formula (11) is given by

[TABLE]

and its limit (12) has the following behavior:

1. (i)

If $H$ is bounded from below then

[TABLE]

almost everywhere in $\tau\in\mathbb{R}$; 2. (ii)

If $\psi\in\mathcal{H}_{\mathrm{ac}}$ then the limit (54) holds uniformly in $\tau$ on compact subsets of $\mathbb{R}\setminus\{0\}$; 3. (iii)

If there exists $\tau_{0}\in\mathbb{R}\setminus\{0\}$ such that $p(\tau_{0})=1$, then $\psi\in\mathcal{\mathcal{H}_{\mathrm{pp}}}$ and

[TABLE]

for all $\alpha\in\mathbb{Q}$ and all $M\in\mathbb{Z}$.

Proof.

Notice first that if $H$ is bounded from below, say $H\geq E_{\mathrm{min}}$, then $\tilde{H}=H-E_{\mathrm{min}}+1\geq 1$ is a strictly positive self-adjoint operator, and

[TABLE]

so we can assume that the Hamiltonian is strictly positive with spectrum

[TABLE]

Then observe that

[TABLE]

where $\hat{\mu}$ denotes the Fourier transform of the measure $\mu$. Let $\mu_{\psi,N}$ be the spectral measure of the self adjoint operator $N^{\alpha-1}H$ in the state $\psi$. Notice that the spectrum of the operator $N^{\alpha-1}H$ is

[TABLE]

Using the property of the Fourier transform it is easy to see that

[TABLE]

where

[TABLE]

and $\ast$ denotes the convolution product, defined by

[TABLE]

for all $f\in C_{\mathrm{b}}(\mathbb{R})$.

We prove assertion $(i)$. First we prove that

[TABLE]

for all $f\in C_{0}(\mathbb{R})$, where $C_{0}(\mathbb{R})$ denotes the space of continuous functions vanishing at infinity. Indeed, if we fix a function $f\in C_{0}(\mathbb{R})$ we have that

[TABLE]

Therefore, by the mean value theorem, we have that

[TABLE]

where

[TABLE]

for some $(\tilde{\lambda}_{1},\dots,\tilde{\lambda}_{N})\in[1,+\infty)^{N}$ and thus $N^{\alpha}\xi_{N}\to\leavevmode\nobreak\ +\infty$ as $N\to+\infty$. Therefore, the limit (63) holds.

Now we recall that, by the Riemann-Lebesgue lemma, for all $\phi\in L^{1}(\mathbb{R})$ its Fourier transform $\hat{\phi}\in C_{0}(\mathbb{R})$, and thus

[TABLE]

as $N\to+\infty$. Therefore we have proved that

[TABLE]

almost everywhere in $\tau\in\mathbb{R}$, whence

[TABLE]

almost everywhere in $\tau\in\mathbb{R}$.

Now we prove assertion $(ii)$. If we assume that $\psi\in\mathcal{H}_{\mathrm{ac}}$, then $\mathrm{d}\mu_{\psi}(\lambda)=\rho(\lambda)\,\mathrm{d}\lambda$ (where the density $\rho(\lambda)=|\tilde{\psi}(\lambda)|^{2}\in L^{1}(\mathbb{R})$ is the squared wave function of $\psi$ in the energy representation), and thus

[TABLE]

as $t\to\pm\infty$, by the Riemann-Lebesgue lemma. Therefore we have that

[TABLE]

definitively in $N$, for $\tau\neq 0$, and thus

[TABLE]

uniformly in $\tau$ on compact sets of $\mathbb{R}\setminus\{0\}$.

Finally we prove $(iii)$. In the proof of Theorem 2 we have shown that if the survival probability

[TABLE]

for some $\tau_{0}\neq 0$, then $\psi\in\mathcal{H}_{\mathrm{pp}}$ and the survival probability has the form (49). Therefore, we have that

[TABLE]

with $\sum p_{j}=1$ and $k_{j}\in\mathbb{Z}$ for $j=1,\dots r$. Notice that if $\alpha\in\mathbb{Q}$, then $\alpha-1=n_{1}/n_{2}$ for some $n_{1},n_{2}\in\mathbb{N}\setminus\{0\}$. Therefore if we consider the subsequence $N_{m}:=m^{n_{2}}$ we have that

[TABLE]

for all $m\in\mathbb{N}$ and $M\in\mathbb{Z}$, therefore we have that

[TABLE]

Since $p_{N,\alpha}(\tau)\leq 1$ for all $\tau\in\mathbb{R}$ and $N\in\mathbb{N}$, we conclude that

[TABLE]

∎

IV Conclusions

Let us summarize the main results obtained in this article in more intuitive terms, by focusing on those quantities that are more directly related to physical intuition. We have analyzed the double limit

[TABLE]

in the case $t\propto N^{\alpha}$, for all possible values of $\alpha\geq 0$.

We have shown that if $0\leq\alpha<1/2$ the limit equals $1$, namely the system is frozen in its initial state and the QZE takes place. At $\alpha=1/2$ the limit (78) is strictly smaller than $1$ for all times and decays in time as a Gaussian (17). If $1/2<\alpha<1$ the limit (78) is equal to [math] for all times, thus we observe a classical behavior. See Fig. 1.

Moreover, if $\alpha\geq 1$ the limit probability is a strange beast and becomes sensitive to the spectral properties of the state $\psi$. In general, the limit (78) is [math] for almost all times, and if the state is decaying, $\psi\in\mathcal{H}_{\mathrm{ac}}$, the limit (78) is always [math] for all times.

The existence of times $t$ at which the limit (78) is nonzero has been clarified, at least for bound states, $\psi\in\mathcal{H}_{\mathrm{pp}}$. In fact there are bound states with a periodic dynamics. Thus, if one performs repeated measurements at the natural period, namely if one looks stroboscopically at the particle dynamics, the presence of the measurements becomes immaterial: the classical and the quantum behavior simply coincide and the limit (78) at that times is equal to $1$.

Concerning the existence of times at which the limit (78) is not [math] for states in the continuous singular spectrum $\psi\in\mathcal{H}_{\mathrm{sc}}$, i.e.  recurrent unbounded states, we can only say that the set of those times is negligible. It is more difficult to grasp this situation by physical intuition, and its full comprehension would require a further analysis which is beyond the scope of this paper.

Summarizing, we have unveiled the presence of two threshold exponents: the threshold between quantum and classical behavior at $\alpha=1/2$, and the threshold of sensitivity to the spectral properties of the initial state at $\alpha=1$.

Acknowledgements.

This work was supported by Cohesion and Development Fund 2007-2013 - APQ Research Puglia Region “Regional program supporting smart specialization and social and environmental sustainability - FutureInResearch”, by the Italian National Group of Mathematical Physics (GNFM-INdAM), and by Istituto Nazionale di Fisica Nucleare (INFN) through the project “QUANTUM”.