The exponential that is only true in the middle

The exponential decay law is usually introduced as a property of the decaying thing: each nucleus has the same chance of going in the next second as it had yesterday, so the population falls by a fixed fraction per unit time and the survival curve is $e^{-\lambda t}$. The rung below this one makes that case and makes it well, and everything in it is right about the middle of the curve.

It is also not where the law comes from. A decay law is a statement about a quantum amplitude, and the amplitude is fixed by an integral nobody has to assume anything about.

The reason to care is not that the corrections are large. In the middle they are undetectable and the exponential is exact to four figures. The reason is that both departures are forced — they follow from properties every quantum state has — so any argument that treats the exponential as fundamental has assumed something false, and one of the consequences is an effect that has been measured.

A decay law is a Fourier transform #

Prepare a system in a state that is not an energy eigenstate — a superposition, which is the ordinary condition rather than a special one. It can then be written as a superposition of eigenstates with some weight $\omega(E)$, and each of those evolves by nothing more than a phase. The amplitude to still find the system in the state it started in is therefore

$$a(t) = \int \omega(E)\,e^{-iEt/\hbar}\,\mathrm{d}E,$$

and the survival probability is $|a(t)|^2$. Nothing has been assumed about the decay: the whole of it is contained in the shape of $\omega$, and the question “what is the decay law” has become the question “what does this distribution look like”.

A decay law is a Fourier transform, and the classical twin makes the shape familiar. A driven oscillator with light damping responds in a peak whose profile is a Lorentzian of width $\gamma$ — and a Lorentzian is exactly the transform of a decaying exponential. So the exponential decay and the shape of a spectral line are the same fact in two domains, and anything that modifies one modifies the other.

For a Breit–Wigner resonance,

$$\omega(E) \propto \frac{1}{(E-E_0)^2 + \Gamma^2/4},$$

the transform can be done in closed form if the integral runs over the whole real line, and the answer is $|a(t)|^2 = e^{-\Gamma t/\hbar}$ exactly, at every $t$ from zero to infinity. That is the origin of the exponential law. It is also the origin of its trouble, because the calculation has just used states at $E = -10^{40}$ joules, and no bound system has any.

Both edges are real, and both change the answer #

A real spectrum has a bottom: energies below the ground state of whatever the decay products are do not exist. It also falls off faster than a Lorentzian at the top, because a Lorentzian tail carries an infinite mean square energy and nothing does. Put the two edges in and the closed form is gone — but the integral is not, and it can be done to any accuracy at any time by closing the contour in the lower half plane.

Doing it that way separates the answer into three pieces that can be read off. The pole at $E_0 - i\Gamma/2$ contributes the exponential exactly, with no approximation at all. The lower edge of the band contributes a term whose integrand is smooth and positive, dominated at large $t$ by the region right at the edge, and falling as $1/t$. The upper edge contributes another. Adding the three gives the curve above, and the middle of it is the pole term with the other two negligible.

The head is flat, and it cannot be otherwise #

At small $t$, expand the exponential inside the integral. The first-order term is $-it\langle E\rangle/\hbar$, which is a pure phase and does not change $|a|^2$ at all. The survival probability therefore begins

$$|a(t)|^2 = 1 - \left(\frac{t}{\tau_z}\right)^2 + \dots, \qquad \tau_z = \frac{\hbar}{\Delta E},$$

with $\Delta E$ the spread of the state’s own energy distribution. The gradient at the origin is zero, for any state whose energy has a finite spread — which is to say, for any state at all.

The number that matters is not $\tau_z$ but where the quadratic stops. The parabola $(t/\tau_z)^2$ meets the exponential’s $\Gamma t$ at $t = \Gamma\tau_z^2$, and that is shorter than $\tau_z$ by the ratio of $\tau_z$ to the lifetime. For the case drawn it is 1.6 × 10⁻² lifetimes against a Zeno time of 0.125 — a factor of eight — and for a real excited atom, whose energy spread involves the whole transition rather than the linewidth, the factor is enormous.

What $\Delta E$ actually is. It is not the linewidth. The linewidth $\Gamma$ describes the shape near the peak; $\Delta E$ is dominated by the far tails of the distribution, which is why it depends on where the spectrum is cut off. That is not a defect of the model: it is the statement that how fast a state starts to decay is set by its widest features, and how fast it goes on decaying is set by its narrowest.

The head exists exactly when the energy has a variance #

The expansion above needs $\langle E^2\rangle$ to be finite, and that condition is not a technicality — it is the whole of why the exponential law and the flat head cannot both be exact.

An unbounded Lorentzian has $\omega(E)$ falling as $E^{-2}$, so $E^2\omega(E)$ does not fall at all and the second moment diverges. With no finite $\Delta E$ there is no $\tau_z$, no quadratic term, and the curve leaves the origin with a nonzero slope — which is precisely the exponential, exact from $t=0$. The two statements are the same statement. A spectrum with infinite energy spread gives a perfect exponential and no head; a spectrum with any cutoff whatever gives a finite spread and a head. There is no intermediate case in which the head is present and small in some adjustable sense.

That is why widening the band in the figures shortens the head rather than removing it. Every cutoff, however far out, makes the second moment finite; where it is put decides only how large $\Delta E$ is, and $\Delta E$ sets both $\tau_z$ and the instant the parabola gives out. Pushing the cutoff to infinity sends both to zero together, which is the limit in which the head shrinks to a point and the exponential becomes exact everywhere — the same limit that requires states of arbitrarily negative energy.

It also settles what would otherwise be a puzzle about the memorylessness argument. The exponential is the unique continuous law with no memory: a survival function satisfying $P(t+s) = P(t)P(s)$ for all $t$ and $s$ must be an exponential, and no other function will do. So a decay that departs from an exponential anywhere is a decay that remembers when it was prepared, and the head and the tail are the two places where that memory shows. What the middle of the curve demonstrates is not that the system has no memory but that its memory is confined to timescales far shorter and far longer than the one being watched.

What a flat head buys: freezing a decay by watching it #

A measurement that finds the system still undecayed puts it back at the top of its own curve, and the decay starts again from $t = 0$. On an exponential this achieves nothing whatever: the survival over $N$ intervals of $T/N$ is $\left(e^{-\Gamma T/N}\right)^N = e^{-\Gamma T}$, whatever $N$ is. Memorylessness is precisely the property that makes interruption useless.

On a quadratic head it achieves a great deal, because $(T/N\tau_z)^2$ raised to the $N$ is $\exp(-T^2/N\tau_z^2)$, which goes to one as $N$ grows.

And what it costs. The interval has to beat $\Gamma\tau_z^2$, and for spontaneous emission from an atom that is of order $10^{-15}$ of a lifetime — a measurement every few femtoseconds, continued for nanoseconds. That is why the Zeno effect has been demonstrated on transitions driven at radio frequencies, where the timescales are laboratory ones, and not on the natural decay of an excited atom.

The tail, and why a half-life stops being one #

The lower edge of the spectrum contributes an amplitude falling as $1/t$ — a step in the integrand transforms to an inverse power, in exactly the way a sharp edge in an aperture transforms to a slowly falling diffraction pattern. An inverse square in the probability loses to no exponential forever.

Why an inverse square and not something else: the contour integral picks up, at the lower edge of the band, a term proportional to the value of the spectrum right at that edge divided by $t$. Any spectrum that is nonzero at a sharp edge gives $1/t$ in the amplitude and $t^{-2}$ in the probability, and the coefficient is the spectral density at the edge — so the tail is weaker the further the resonance sits above the bottom of its band, which is the sense in which a well-bound state decays more nearly exponentially than a barely bound one.

The crossing moves out only as the logarithm of the resonance’s sharpness, which is the reason this is hard rather than hopeless. Thirty lifetimes is not an absurd number of lifetimes to wait; the problem is that $10^{-10}$ of the original sample is what has to be counted, against every background there is. A deviation from exponential decay at long times has been reported in the luminescence of organic molecules in solution, at around ten lifetimes, and the difficulty of the measurement is entirely in the last sentence rather than in the first.

The tail is a counting problem before it is a physics problem. Two thousand independently sampled nuclei against the smooth law: by eight half-lives fewer than eight of the original two thousand are left, and by twelve there is nothing to average. So the deviations the theory predicts at long times sit in a regime where the sample size has collapsed — which is why the tail has been so much harder to measure than the head.

A theorem, not an estimate #

The tail is easy to mistake for a feature of the particular spectrum used here — a hard step, a Breit–Wigner peak, a specific cutoff. It is not. That a bounded-below spectrum forbids a strictly exponential survival at long times is a theorem, and it comes from a piece of Fourier analysis with no physics in it whatever.

The amplitude $a(t)$ is the transform of a function supported on a half-line, so it is the boundary value of a function analytic in a half-plane, and a classical result about such functions constrains how fast they may vanish: the integral of $|\ln|a(t)||$ against $1/(1+t^2)$ must converge. An amplitude falling as $e^{-\Gamma t/2}$ has $|\ln|a||$ growing linearly, and that integral diverges. So no state built out of a spectrum with a lowest energy can decay exponentially for all time — not approximately, not for a well-chosen lineshape, not ever. Applied to decay in 1957 by Khalfin, it says the tail exists before anybody has decided what the resonance looks like.

What the theorem does not supply is the power. That the survival falls as $t^{-2}$ here is a consequence of the spectral density being nonzero at a hard edge, and a spectrum switching on more gently gives a different exponent. So the calculation and the theorem divide the work cleanly: the theorem says an exponential is impossible at long times whatever the lineshape, and the contour integral says what replaces it for this one.

It is worth noticing which of the two departures this covers. The head follows from the spectrum having a finite width; the tail follows from its having a bottom. Two independent properties of any bound system, two independent failures of the same law, at opposite ends of the curve, and neither removable by any choice of parameters.

And watching can also make it faster #

The Zeno argument above is usually presented as though frequent measurement always slows a decay, and the figures here are the case where it does. The general statement is less comfortable and follows from the same reasoning.

What repeated measurement really does is replace the decay rate by an average of the state’s spectrum against a window whose width is set by the measurement interval — the shorter the interval, the wider the window. Whether that average is smaller or larger than $\Gamma$ depends on whether the spectrum near the resonance is peaked or not. Measure often enough and the window covers the whole distribution, the average is dominated by the wings where there is very little, and the decay slows: that is the Zeno regime, and it is the one reached in the limit. But at intermediate intervals the window can be wide enough to reach a part of the spectrum richer than the peak — a nearby second resonance, a rising threshold — and then interrupting the decay accelerates it.

That is the anti-Zeno effect, and it is the more commonly reachable of the two, because the interval it needs is far longer than $\Gamma\tau_z^2$. Both are measurements of the shape of the spectrum rather than statements about observation, which is the useful way to hold them: nothing here is about a conscious observer, and everything is about what interrupting a coherent evolution does to the interference between its parts.

Where the model actually stops #

Three things the picture cannot show, and one it should not be blamed for.

The lineshape is an input. Everything above takes the Breit–Wigner form and cuts it off. A real spectrum near a threshold does not look like that — it rises as a power of the energy above the edge — and the exponent of the tail is decided by that behaviour rather than by the resonance. The inverse square drawn here belongs to a hard step; a threshold going as $\sqrt{E}$ gives a different power. What survives every choice is that the tail is a power and not an exponential.

A level with a lifetime is a level with a width, and the two are the same number in different units: $\Gamma = \hbar/\tau$. That is the object the whole calculation is about, and it is why the non-exponential edges are not an obscure correction — a strictly exponential decay would require a strictly Lorentzian line, which would require states at arbitrarily negative energy. The spectrum has a floor, so the decay cannot be exponential forever.

Nothing here is an ensemble. $|a(t)|^2$ is the probability that one system is still undecayed. The exponential curve a laboratory measures is a count of survivors, and the two agree only because the systems are independent. Where they are not — a dense sample, a cavity, a condensate — the decay is not this curve at all.

And “the state was prepared at $t=0$” is doing work. The quadratic head is a statement about a system that was definitely in the initial state at a definite moment. A nucleus formed in a reaction was not prepared like that, and the moment of preparation is itself uncertain by something like $\hbar/\Delta E$ — which is $\tau_z$, the very scale on which the head lives. That is not a reason to distrust the head; it is a reason to be careful about what counts as a measurement of it. An experiment that claims to have watched a decay start has had to define the starting instant to better than the width of the head, and defining an instant that sharply is itself an energy-spreading operation.

There is a fourth thing, and it is the one that makes the middle of the curve so robust. The exponential is not merely a good fit between the two departures: it is the exact contribution of a single pole, and the corrections to it are the two edge terms, which are separately computable and separately small. So the decay constant a laboratory measures is not an effective parameter standing in for something more complicated. It is the imaginary part of a pole position, and the two departures are additions to it rather than modifications of it — which is why fitting an exponential to data from the middle of the range returns Γ and not some average of Γ with whatever the ends are doing.

The classical ringdown has no such subtlety, and the contrast is worth drawing. A damped oscillator released from rest decays as a genuine exponential envelope for as long as anyone cares to watch, because nothing about it is bounded below. The quantum case differs precisely because the energy spectrum has a bottom, which is a statement about states rather than about damping.

The rung after this one #

Two of this ladder’s rungs now say that a decay rate is not a property of the decaying thing alone. The first was that the rate depends on a barrier, so a factor of two in energy is twenty-four decades in half-life. This one is that the rate is a feature of a spectrum, and that where the spectrum ends decides what happens at both ends of the curve.

The next rung is the obvious question that leaves: if the decay rate is set by the states available to decay into, then changing those states changes the rate. Put an excited atom between two mirrors a half-wavelength apart and there are no modes for it to emit into; put it in a cavity tuned to its transition and there are far more than in free space. The lifetime moves by orders of magnitude in both directions, and nothing about the atom has been touched.

A box allows only some frequencies, which is the same counting that gives a standing wave its discrete set — and it is where the next argument starts. The spectrum having a floor is what forbids a strict exponential, and the spectrum has a floor because the states are counted rather than continuous.