Deviations of the decay law from exponents are discussing for a long time, however, experimental proofs of such deviations are absent. Here in the general form is shown that the conclusions about non-exponential contributions are due to the disregarding of advanced interactions, i.e. at principally non-relativistic considerations. We consider decay processes in the frame of interactions duration of the quantum field theory. We show that at this basis the usual exponential decay has place.
The transformation of canonical decay laws of moving unstable quantum systems is studied by approximating, over intermediate times, the decay laws at rest with superpositions of exponential modes via the Prony analysis. The survival probability $\mathcal{P}_p(t)$, which is detected in the laboratory reference frame where the unstable system moves with constant linear momentum $p$, is represented by the transformed form $\mathcal{P}_0\left(φ_p(t)\right)$ of the survival probability at rest $\mathcal{P}_0(t)$. The transformation of the intermediate times, which is induced by the change of refere…
We report the first observation of the Quantum Zeno and Anti-Zeno effects in an unstable system. Cold sodium atoms are trapped in a far-detuned standing wave of light that is accelerated for a controlled duration. For a large acceleration the atoms can escape the trapping potential via tunneling. Initially the number of trapped atoms shows strong non-exponential decay features, evolving into the characteristic exponential decay behavior. We repeatedly measure the number of atoms remaining trapped during the initial period of non-exponential decay. Depending on the frequency of measurements we …
This paper is a pedagogical yet critical introduction to the quantum description of unstable systems, mostly at the level of a graduate quantum mechanics course. Quantum decays appear in many different fields of physics, and their description beyond the exponential approximation is the source of technical and conceptual challenges. In this article, we present both general methods that can be adapted to a large class of problems, and specific elementary models to describe phenomena like photo-emission, beta emission and tunneling-induced decays. We pay particular attention to the emergence of e…
We study the short-time and medium-time behavior of the survival probability in the frame of the $N$-level Friedrichs model. The time evolution of an arbitrary unstable initial state is determined. We show that the survival probability may oscillate significantly during the so-called exponential era. This result explains qualitatively the experimental observations of the NaI decay.
The decay of a bound state weakly-coupled to a non-Hermitian tight-binding unstable continuum, i.e. a continuum of states comprising energies with positive imaginary part, is theoretically investigated. As compared to quantum decay in an Hermitian continuum, in the non-Hermitian case a richer scenario can be found as a result of non-unitary dynamics. Different behaviors are observed depending on the kind of instability of the continuum. These include complete or fractional decay in convectively-unstable continua, the absence of quantum decay for a bound state with energy embedded in the contin…
We discuss deviations from the exponential decay law which originate when going beyond the Breit-Wigner distribution for an unstable state. In particular, we concentrate on an oscillating behavior, reminiscent of the Rabi-oscillations, in the short-time region. We propose that these oscillations can explain the GSI experiment, which measured superimposed oscillations on top of the exponential law for hydrogen-like nuclides decaying via electron-capture (the so-called GSI anomaly). Moreover, we discuss the possibility, that the deviations from the Breit-Wigner distribution in the case of the GS…
A new quantum effect connected with the late time behavior of decaying states is described and its possible observational consequences are analyzed: It is shown that charged unstable particles as well as neutral unstable particles with non--zero magnetic moment which live sufficiently long may emit electromagnetic radiation. This mechanism is due to the nonclassical behavior of unstable particles at late times (at the post exponential time region). Analyzing the transition times region between exponential and non-exponential form of the survival amplitude it is found that the instantaneous ene…
Not all astrophysical mechanisms of the emission of electromagnetic radiation including $X$-- and $γ$-- rays coming from the space are clear. We find that charged unstable particles as well as neutral unstable particles with non--zero magnetic moment which live sufficiently long may emit electromagnetic radiation. This new mechanism is connected with the properties of unstable particles at the post exponential time region. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude it is found that the instantaneous energy of the unstable particl…
We analyze the survival probability of unstable particles in the context of quantum field theory. After introducing the spectral function of resonances, we show that deviations from the exponential decay law occur at short times after the creation of the unstable particle. For hadronic decays, these deviations are sizable and could lead to observable effects.
Decay laws of moving unstable quantum systems with oscillating decay rates are analyzed over intermediate times. The transformations of the decay laws at rest and of the intermediate times at rest, which are induced by the change of reference frame, are obtained by decomposing the modulus of the survival amplitude at rest into purely exponential and exponentially damped oscillating modes. The mass distribution density is considered to be approximately symmetric with respect to the mass of resonance. Under determined conditions, the modal decay widths at rest, $Γ_j$, and the modal frequencies o…
We apply the Leggett-Garg inequalities (LGI) to the case of classical and quantum unstable systems. For classical systems the two assumptions of macroscopic realism and non-invasive measurements imply that the three-measurement string $K_3$ is identically equal to one. Also for quantum mechanical systems -- for which the two assumptions are in general not valid -- we find that $K_3=1$ for purely exponential decays ($K_3\leq1$ is the general LGI). On the other hand, the necessary deviations from the exponential decay law at short and long times predicted by quantum mechanics lead to values of $…
A study is made of the behavior of unstable states in simple models which nevertheless are realistic representations of situations occurring in nature. It is demonstrated that a non-exponential decay pattern will ultimately dominate decay due to a lower limit to the energy. The survival rate approaches zero faster than the inverse square of the time when the time goes to infinity.
An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as $t\to\infty$ more slowly than any exponential function of $t$. For a time-dependent decay rate $γ(t)$ Khalfin's result means that this $γ(t)$ is not a constant for large $t$ but that it tends to zero as $t\to\infty$. We find that a similar conclusion can be drawn for the instantaneous energy of the unstable state for a large class of models of unstable states: This energy tends to the minimal energy …
The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the non-exponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through a specific quantum mechanical examples which can be accurately solved, we study gen…
We study the decay process of an unstable quantum system, especially the deviation from the exponential decay law. We show that the exponential period no longer exists in the case of the s-wave decay with small $Q$ value, where the $Q$ value is the difference between the energy of the initially prepared state and the minimum energy of the continuous eigenstates in the system. We also derive the quantitative condition that this kind of decay process takes place and discuss what kind of system is suitable to observe the decay.
For over a decade several workers have argued for the existence of quantum deviations from the classical, Einstein dilation of the decay evolution of moving or Lorentz boosted unstable particles. While the general claim is correct, the discussions have been incomplete and, sometimes, misleading. The discussions have been of three kinds. Type 1 examines the time dependence of the survival probability for 3-momentum eigenstates of the unstable quanton (Khalfin). Type 2 does the same for velocity eigenstates, obtaining an outrageous result which then discredits velocity eigenstates (Shirokov / He…
By means of expressing volumes in phase space in terms of traces of quantum operators, a relationship between the Hamiltonian poles and the Lyapunov exponents in a non Hermitian quantum dynamics, is presented. We illustrate the formalism by characterizing the behavior of the Gamow model whose dissipative decay time, measured by its decoherence time, is found to be inversely proportional to the Lyapunov exponents of the unstable periodic orbits. The results are in agreement with those obtained by means of the semiclassical periodic-orbit approach in quantum resonances theory but using a simpler…
It is well known, both theoretically and experimentally, that the survival probability for an unstable quantum state, formed at $t=0,$ is not a simple exponential function, even if the latter is a good approximation for intermediate times. Typically, unstable quantum states/particles can decay in more than a single decay channel. In this work, the general expression for the probability that an unstable state decays into a certain $i$-th channel between the initial time $t=0$ and an arbitrary $t>0$ is provided, both for nonrelativistic quantum states and for relativistic particles. These partia…
We study the decay properties of Tonks-Girardeau gases escaping from a double-well potential. Initially, the gases are constrained between two infinite $δ$ barriers with an on-center $δ$-split potential. The strength of one of the obstacles is suddenly reduced, and the particles start to tunnel to the open space. Using the resonance expansion method, we derive the single-term approximate expression for the $N$-particle survival probability and demonstrate its effectiveness in both the exponential and nonexponential regimes. We also predict a parity effect and provide physical insights into its…
We theoretically propose a scheme for verification of quantum Zeno effect (QZE) to suppress a decay process with Nuclear Magnetic Resonance (NMR). Nuclear spins are affected by low frequency noise, and so one can naturally observe non-exponential decay behavior, which is prerequisite in observing QZE. We also describe that a key component for QZE, namely measurements on the nuclear spin, can be realized with NMR in the current technology
After reviewing the description of an unstable state in the framework of Lee Hamiltonians (valid both for Quantum Mechanics (QM) and Quantum Field Theory (QFT)), we consider some theoretical aspects of non-exponential decays: the case of two decay channels, the broadening of the energy spectrum at short times, the effect of an imperfect measurement, the link to QFT, and the decay of an unstable moving particle with definite momentum. All the presented effects were not confirmed in experiments, hence are at the present stage predictions.
We present a detailed non-perturbative analysis of the time-evolution of a well-known quantum-mechanical system - a particle between potential walls - describing the decay of unstable states. For sufficiently high barriers, corresponding to unstable particles with large lifetimes, we find an exponential decay for intermediate times, turning into an asymptotic power decay. We explicitly compute such power terms in time as a function of the coupling in the model. The same behavior is obtained with a repulsive as well as with an attractive potential, the latter case not being related to any tunne…
We study the decay of unstable states by formulating quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles at a detector located a distance L from the tunneling region. For this purpose, we use a Positive-Operator-Valued-Measure (POVM) for the time-of-arrival determined in quant-ph/0509020 [JMP 7, 122106 (2006)]. This only depends on the initial state, the Hamiltonian and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that the exp…
We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude $a(t)$ of the state considered is found. Our attention is focused on the late time properties of $a(t)$ and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as $t$ goes to the infinity more slowly than any exponential function of $t$. This effect can be described using a time-dependent decay rate $γ(t)$ and then the Khalfin result m…
We study the survival probability of moving relativistic unstable particles with definite momentum $\vec{p} \neq 0$. The amplitude of the survival probability of these particles is calculated using its integral representation. We found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-expo\-nen\-tial form of the survival probability, should occur much earlier than …
We study the deviations from the exponential decay law, both in quantum field theory (QFT) and quantum mechanics (QM), for an unstable particle which can decay in (at least) two decay channels. After a review of general properties of non-exponential decay in QFT and QM, we evaluate in both cases the decay probability that the unstable particle decays in a given channel in the time interval between $t$ and $t+dt.$ An important quantity is the ratio of the probability of decay into the first and the second channel: this ratio is constant in the Breit-Wigner limit (in which the decay law is expon…
Properties of unstable false vacuum states are analyzed from the point of view of the quantum theory of unstable states. Some of false vacuum states survive up to times when their survival probability has a non-exponential form. At times much latter than the transition time, when contributions to the survival probability of its exponential and non-exponential parts are comparable, the survival probability as a function of time t has an inverse power-like form. We show that at this time region the instantaneous energy of the false vacuum states tends to the energy of the true vacuum state as $1…
