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Physical Review A

Decay of a quantum-mechanical state described by a truncated Lorentzian energy distribution

Kathleen M. Sluis, Eric A. Gislason

Published May 1, 1991• 32 citations
Publisher

Abstract

The nondecay probability Q(t) for a Lorentzian energy distribution peaked at ${\mathit{E}}_{0}$ with full width at half maximum equal to \ensuremath{\Gamma} is given in atomic units by ${\mathit{Q}}_{\mathit{L}}$(t)=exp(-\ensuremath{\Gamma}t). However, the Lorentzian distribution includes energies in the range -\ensuremath{\infty}\ensuremath{\le}E\ensuremath{\le}\ensuremath{\infty}. On physical grounds, this distribution must be truncated at some minimum energy E=0. In this paper, Q(t) is derived for a properly normalized truncated Lorentzian energy distribution. To aid the analysis, the average lifetime \ensuremath{\tau}(${\mathit{t}}_{\mathrm{min}}$) for that fraction of the system that has not decayed by ${\mathit{t}}_{\mathrm{min}}$ is also calculated for this system. For a given value of the ratio ${\mathit{E}}_{0}$/\ensuremath{\Gamma}, the function Q(t) shows different behavior in different time regimes; this behavior is presented and the time periods are determined. It is shown that Q(t) falls off linearly at very small t, but at a rate only half as fast as for ${\mathit{Q}}_{\mathit{L}}$(t). Thus this system cannot exhibit the behavior known as Zeno's paradox, but rapid measurements on this system that collapse the wave function back to the original state could, in principle, reduce the decay rate by a factor of 2.

DOI
10.1103/physreva.43.4581

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The nondecay probability Q(t) for a Lorentzian energy distribution peaked at E0{\mathit{E}}_{0}E0​ with full width at half maximum equal to \ensuremath{\Gamma} is given in atomic units by QL{\mathit{Q}}_{\mathit{L}}QL​(t)=exp(-\ensuremath{\Gamma}t). However, the Lorentzian distribution includes energies in the range -\ensuremath{\infty}\ensuremath{\le}E\ensuremath{\le}\ensuremath{\infty}. On physical grounds, this distribution must be truncated at some minimum energy E=0. In this paper, Q(t) is derived for a properly normalized truncated Lorentzian energy distribution. To aid the analysis, the average lifetime \ensuremath{\tau}(tmin{\mathit{t}}_{\mathrm{min}}tmin​) for that fraction of the system that has not decayed by tmin{\mathit{t}}_{\mathrm{min}}tmin​ is also calculated for this system. For a given value of the ratio E0{\mathit{E}}_{0}E0​/\ensuremath{\Gamma}, the function Q(t) shows different behavior in different time regimes; this behavior is presented and the time periods are determined. It is shown that Q(t) falls off linearly at very small t, but at a rate only half as fast as for QL{\mathit{Q}}_{\mathit{L}}QL​(t). Thus this system cannot exhibit the behavior known as Zeno's paradox, but rapid measurements on this system that collapse the wave function back to the original state could, in principle, reduce the decay rate by a factor of 2.
The short-time behavior of an unstable particle is examined in a realistic field theoretical model. The onset of linear decreasing of the nondecay probability is shown to be extremely rapid, so rapid as to exclude any relevance of regimes quadratic in time (``Zeno'' quantum paradox). The result applies to super-renormalizable, renormalizable, and nonrenormalizable cases, particularly to the proton decay problem. We discuss also the deviations from the exponential decay law in small-Q-value decays.
It has been shown for various systems that the decay rate of an unstable quantum system deviates from exponential behavior in short- and long-time regimes associated with memory effects. In particular, it is widely believed that in the short-time regime, the decay is quadratic, inducing the quantum Zeno effect, in which the decay is suppressed by rapidly repeated measurements. In our study, we find that when the environment of the unstable system has an energy spectrum with a lower bound but without an upper one, the decay rate in both regimes is scaled in terms of the spatial dimension and the exponent of the energy dispersion of the environment. Surprisingly, we find that in the short-time regime the decay exhibits fractional scaling, which leads to a quantum Zeno effect with a different scaling of the Zeno time.
The deviation of the decay law from the exponential is a well known effect of quantum mechanics . Here we analyze the relativistic survival probabilities , S ( t , p ) , where p is the momentum of the decaying particle and provide analytical expressions for S ( t , p ) in the exponential (E) as well as the nonexponential (NE) regions at small and large times. Under minimal assumptions on the spectral density function, analytical expressions for the critical times of transition from the NE to the E at small times and the E to NE at large times are derived. The dependence of the decay law on the relativistic Lorentz factor, γ = 1 / 1 − v 2 / c 2 , reveals several interesting features. In the short time regime of the decay law, the critical time, τ s t , shows a steady increase with γ , thus implying a larger NE region for particles decaying in flight. Comparing S ( t , p ) with the well known time dilation formula, e − Γ t / γ , in the exponential region, an expression for the critical γ where S ( t , p ) deviates most from e − Γ t / γ is presented. This is a purely quantum correction. Under particular conditions on the resonance parameters, there also exists a critical γ at large times which decides if the NE region shifts backward or forward in time as compared to that for a particle at rest. All the above analytical results are supported by calculations involving realistic decays of hadrons and leptons .
The characteristic short-time behavior of the survival probability of a decaying state, which underlies the quantum Zeno effect (QZE), is shown to follow quite simply from the Schr\"odinger equation even when the spectrum of the Hamiltonian has no finite lower bound, provided the mean energy of the initial state is finite. The failure of the Breit-Wigner form of energy distribution to predict QZE is traced to the nonexistence of a well-defined mean energy. On the basis of a suitably modified distribution, it is explicitly demonstrated that the existence of a finite lower bound to the energy distribution only affects the asymptotic decay law. For short times, the survival probability depends quadratically on time if the variance of the distribution is also finite, not otherwise. For a sufficiently well-behaved energy distribution, the asymptotic decay must be as ${t}^{{\ensuremath{-}}^{n}}$ if the spectrum has a finite lower bound, or faster than any inverse power of t if there is no lower bound.
Instead of counting the number of nuclei ( ),tN it is easier to count the rate at which decays occur in the radioactive sample, and this is called the activity, defined as ( )0 tdN N e Ndt λλ λ−= = t (47.5) and follows the same exponential decay law (47.3). The SI unit for activity is the Becquerel (Bq), equal to one decay per second, and the common unit is the Curie (Ci), defined as It is also convenient to introduce the half-life as the time interval in which the activity is reduced to one-half .Bq107.3Ci1 10×= 2/1t 2ln2ln2/1 τλ ==t (47.6) The probability of radioactive decay through the direct transition of the nucleus from an initial state to a final state Ψi Ψ f , in time t, is given by (47.7) P t a tif f( ) | ( )|= 2 where the wave function amplitudes a are obtained from Eq.(37.11), which can be approximated by tf ( ) == /)()()()( tifiif ifetVtadt tda i εε −= (47.8) Since the probability of finding our decaying system in the initial state Ψi must decrease with time according to the radioactive decay law (47.3), which gives 2 2| ( ) | | | ti it e λ−Ψ = Ψ (47.9) we have to choose / 2( ) tia t e λ−= (47.10) rather than as considered in Eq.(37.13). Substituting this result into Eq.(47.8), and assuming a constant perturbation which is switched on at time t = 0, yields ai = 1, i εελ −+−= (47.11) Integrating, we get an oscillatory behaviour of the wave function amplitude, given by ( ) ( ) /2 V e a t i ε ε λ − +⎡ ⎤−⎣ ⎦= − + = == = (47.12) Figure 47.1. Energy spectrum of a decaying state of width Γ At the end of the nuclear transition we should consider the amplitude corresponding to ,∞→t which reads )( λεε = = i V a if fi f +−=∞ (47.13) and this yields the transition probability (47.7) in the form )( || )( Γ+−=∞ if fi if V P εε = (47.14) Hence, the probability to observe the nucleus with energy fε in the vicinity of iε follows a Lorentzian distribution, illustrated in Figure 47.1. By analogy with the atomic transitions, Γ = = 〈 〉= =λ / t gives the natural linewidth of the emitted radiation, which is the same as the width of the decaying nuclear state. Nuclear states which are populated in ordinary decays typically have lifetimes greater than corresponding to sec,10 12− ( ) τ/1066.0eV 15−×=Γ (47.15) which is of the order of Since this width is small compared with the energy spacing of nuclear levels, which is of the order of the nuclear decaying states may be regarded to be discrete quasi-stationary states.
This work is a continuation of papers presented at the Optics and Photonics Symposium 2009 and is devoted to the case of the high finesse nanocavity with the average photon escaping rate Γ = η c/R << g -coupling constant. The case is of special interest as possible pretender for a qubit in quantum computers. The probability distribution to find photon in the (ωk , t )-space, investigated in the interval 0≤Γ/4g <<1, has triplet structure with very low central peak and satellites at frequencies ≈ ( ωa ± g). The latter emerge as result of emission from two upper atomic split sublevels. The peak is produced by ensuing reabsorptions of satellites by atom through its upper sublevels. Oscillating as t2•cos(gt) and decaying fast, the peak is accompanied with the simultaneously arising satellites, When the peak disappears the satellites become stable. The stability is quenched with continuum of final states. The profile of structure consisting of two identical components has the time-dependence t • exp(−Γt/4) and the width of satellites is by order less than the distance between them. These components with frequencies (ωa ± g ) have the average photon energies equal to 1/2(ωa ± g ) where factor "1/2 "accounts for normalization condition. The satellites amplitudes reach maximum at Γ/4g = 0.05. The profile of the structure has the form Γt • exp (−Γt/4) with maximum attained for Γ/4g =0,05 and average photon cavity life-time proportional to 4lnΓ/Γ. We name the structure "entangled photon."