CONTINUOUS VS DISCRETE PROCESSES: THE PROBABILISTIC EVOLUTION OF

SINGLE TRAPPED IONS.

Storrs McCall (McGill University),

Andrew Whitaker (Queen's University Belfast)

Glyn George (Memorial University Newfoundland)

Paper presented at the 10th UK Conference on the Foundations of

Physics, Belfast, September 10-14, 2001.

Abstract

The evolution of a single trapped ion exhibiting intermittent

fluorescence and dark periods may be described either as a continuous

process, using differential rate equations, or discretely, as a Markov

process. The latter models the atom as making instantaneous

transitions from one energy eigenstate to another, and is open to the

objection that superpositions of energy states will form which are not

covered by the Markov process. The superposition objection is replied

to, and two new mathematical elements, Markov vectors and Markov

matrices, are proposed as additions to quantum theory. The paper

concludes by attributing the cause of dark periods in the ion's

history to instantaneous transitions in the ion itself, rather than to

photon detection or other components of measurement.

1. Introduction: shelving and dark periods.

In 1975 Hans Dehmelt, investigating methods of determining the

frequency of atomic line emissions to a very high standard of

accuracy, suggested that an extremely narrow line‑width would be

obtained if a single trapped ion's electron were driven to a

metastable energy level, and in Dehmelt's suggestive word "shelved"

there.

Ten years later, Cook and Kimble (1985) proposed that the concept of

shelving could be used to investigate a problem of theoretical

interest: the existence of "quantum jumps". A three‑level atom in the

"V" configuration (Cook and Kimble's figure 1, p. 1023) exhibits

resonance fluorescence when irradiated by laser light of the

appropriate frequency. The rapid oscillations between levels 1 and 3

give the appearance of constant radiation intensity. But when the weak

transition is also stimulated the atom has a small probability of

moving to level 2: when this happens the fluorescence is switched off

and the atom enters a dark period.

Cook and Kimble noted that, whereas the detection of a single photon

emitted by the weak transition is problematic, the beginning and

ending of dark periods in the fluorescence is detectable by the human

eye aided by a 10X magnifying glass. Their suggestion was therefore

that dark periods could provide a means of directly observing the

instantaneous atomic transitions that had been the cause of much

discussion between Bohr, Heisenberg and Schrödinger.

2. Rate equations for atomic change of state.

The three‑level system in the V configuration evolves according to

differential equations which govern changes in its density matrix

(Kimble, Cook and Wells (1986), p. 3191). The authors show that the

system is reducible to a two‑level system, the first level consisting

of a combination of levels 1 and 3 when fluorescence is occurring and

the electron is oscillating back and forth. The second level is the

"shelved" state 2.

Where A ("light") and B ("dark") are the levels of the new two‑level

system, g is the transition rate from A to B, h that from B to A, PA(t)

is the probability of the system being "light" and PB(t) that of being

"dark", the evolution of the fluorescing atom is described by the

following simple rate equations:

dPAdt = ‑gPA(t) + hPB(t) (1a)

dPBdt = +gPA(t) ‑ hPB(t). (1b)

Solution of these simultaneous equations, starting from initial

conditions PA(0) = 1, PB(0) = 0, yields PA and PB as explicit

functions of t:

PA(t) = (h + g exp(‑(g+h)t)) / (g+h) (2a)

PB(t) = g(1 ‑ exp(‑(g+h)t)) / (g+h). (2b)

PA(t) starts at 1 and decays asymptotically to h/(g+h).

PB(t) starts at 0 and increases asymptotically to g/(g+h).

At all times, PA(t) + PB(t) = 1.

These rate equations, as well as the density‑matrix equations,

describe change of state in the fluorescing atom as a smooth,

continuous process. The same holds for Itano's rate equations for the

evolution of a three‑level system in the lambda configuration (levels

1, 2, and 3/4 (=0) of fig. 1 of Itano, Bergquist and Wineland (1988),

p. 559). Where γ1, γ2 and γ0 are the decay rates of the levels 1, 2

and 0, and f1 and f2 (= 1 ‑ f1) give the branching ratio for the

transition from level 1 to levels 0 and 2 respectively, the

three‑level equations (p. 560) are:

dP1/dt = ‑γ1P1 + γ0P0 (3a)

dP2/dt = f2γ1P1 ‑ γ2P2 (3b)

dP0/dt = f1γ1P1 + γ2P2 ‑ γ0P0 (3c)

3. Discrete, non‑continuous atomic evolution.

The evolution of the two‑level system described by equations (1) and

(2), and of the three‑level system described by equations (3), is

continuous throughout. Nothing in these equations gives any hint of

discrete transitions from one state to another, or of the sudden

appearance and disappearance of dark periods. Although the rate

equations provide answers to questions such as "What is the

probability that the system is in a dark period at time t?", they

cannot answer questions like "What is the probability that a dark

period begins at t and ends at t'?" To address questions of this kind,

a mathematical description of the atom's evolution is needed which is

not based on differential equations, but introduces instantaneous

stochastic changes. These changes, though unpredictable, are subject

to strict probabilistic constraints, hence are not arbitrary or

"chance".

The need for a stochastic approach is recognized in Cook and Kimble

(1985) and in Kimble, Cook and Wells (1986). They note that the rate

equations (1) do not describe the discrete alternation between light

and dark periods of figure 2 of (1985, p. 1023). Their solution is to

"adopt a point of view in which time is coarse‑grained over intervals

which are long compared with 1/A1, but short compared with 1/A2",

where 1/A1 and 1/A2 are the residence times in levels 1 and 2

respectively (1986, p. 3190). This procedure "converts the temporal

variations of the strong fluorescence into a classical stochastic

process", so that the atomic fluorescence "resembles a classical

random telegraph signal".

4. Sequences of atomic transitions as a Markov process.

Let {S1, S2, ... Sm} be a set of states, which in the case of an atom

will be its energy levels, and imagine the atom's optical electron

successively occupying these states over a sequence of short time

intervals Δt1, Δt2, ... of equal duration. At each interval the atom

undergoes a change of state or transition (it being understood that

the transition may be from a state Si to itself), and each possible

transition is assigned a probability. A Markov process differs from a

Bernoulli or Poisson process in that in a Markov process the

probability of the transition SiSj in or at the end of interval Δtn

depends upon the state Si of the system in the interval. A system

undergoing a Markov process has a "memory", but the memory extends

only to its current state, not to the atom's previous history.

Once the division of time into intervals of constant length Δt is

established, the properties of the atomic Markov process are summed up

in its transition matrix. An example for a two‑level process such as

the light/dark alternation of a two-level atomic system is the

following (Matrix M):

From:

A B

To: A 1/2 1/3 (4)

B 1/2 2/3

When the system corresponding to Matrix M has run for a while, it

settles down into a steady state in which the probabilies of finding

it in states A and B are 2/5 and 3/5 respectively. Using matrix

multiplication, we have that

(1/2 1/3)(2/5) = (2/5)

(1/2 2/3)(3/5) (3/5),

which is to say that (2/5, 3/5) is the eigenvector of the transition

matrix M with eigenvalue 1. We return to this topic in section 7

below.

5. Transition matrix for the V configuration.

As an artificial example, consider a fluorescent ion in the V

configuration of figure 1 of Cook and Kimble (1985). Let the decay

rate of the strong transition 13 be 1 per ms, corresponding to a

natural lifetime of 1 ms, and let the rate for the weak transition 23

be 0.125 per ms, corresponding to a lifetime of 8 ms. We choose for

the purposes of this example Δt = 4 ms, which accords with Kimble,

Cook and Wells' proposal to make time coarse‑grained over intervals

long compared to the strong dwell‑time but short compared to the weak

dwell‑time. Where γ is the decay rate, both transitions are subject to

the law of exponential decay:

p(non‑decay by time t) = exp(‑γt).

We arrive at the following transition matrix:

1 2 3

1 .018 0 .999

2 0 .607 .001 (5)

3 .982 .393 0

This matrix describes the behaviour of the three‑level ion when time

is divided into intervals of 4 ms. It can be used e.g. to calculate

the probability that the ion will enter a dark period at time interval

Δt25 and remain in it until Δt40, at which time the fluorescence

recommences.

What remains to be proved concerning the Markov description of atomic

processes is that the probabilities of such things as the future

occurrence of dark periods of given lengths are stable over different

choices of Δt. Thus, for Δt = .04 ms, the probability value for the

dark period from Δt2500 to Δt4000 should be closely related to the

earlier value for Δt25 ‑ Δt40. In the limit, the most accurate

probability predictions would presumably result from Markov processes

based on very short time intervals, possibly of the order of 10‑40

sec.

6. The superposition objection.

Although the representation of atomic transitions as Markov processes

appears to capture the sudden appearance and disappearance of dark

periods, there is a powerful objection to the idea that the history of

an atom can be written as the successive occupancy of discrete states,

with instantaneous transitions in between. The objection is that

quantum mechanics permits, and on occasion requires, the atom to enter

into a coherent superposition of discrete energy states (see for

example Schenzle and Brewer (1986), p. 3128). If the atom is in a

linear combination of such states, with complex coefficients, how can

it be conceived to jump? Cook ((1990), p. 365) asserts that the

formation of coherent superpositions of states in effect "eliminates

the quantum‑jump picture as a useful interpretation of quantum

behaviour".

This objection is a serious one, but it can be answered. Even if the

temporal evolution of the atom carries it into a superposition of

energy states it can still make discontinuous transitions, and an

elegant method of calculating the probabilities for it to jump in

different ways can be constructed. This method rests on the addition,

in quantum theory, of "Markov vectors" and "Markov matrices" to the

traditional categories of vectors and operators in Hilbert space.

7. Markov vectors and Markov matrices.

A normalized vector is a vector of unit length; a "Markov" vector is a

vector the components of which are real non‑negative numbers which sum

to unity, e.g. (1/2, 1/3, 1/6). A vector may be a Markov vector

relative to one basis but not to another: e.g. (1, 0) is not a Markov

vector when written in a different basis as (1/2, 1/2).

To any normalized vector v = (c1, c2, ...) there corresponds a Markov

vector v+ = (c1c1*, c2c2*, ...), where c* is the complex conjugate of

c. Since v is normalized, c1c1* + c2c2* + ... = 1.

A Markov matrix is a square matrix, each column of which is a Markov

vector. Thus the transition matrices (4) and (5) above are Markov

matrices, relative to a basis. It is not difficult to generalize the

argument in section 4 to show that the product of a Markov matrix and

a Markov vector is always a Markov vector, and we conjecture that

every Markov matrix has, in the set of Markov vectors, a unique

eigenvector with eigenvalue 1, and no eigenvector with eigenvalue

other than 1.

8. Resolution of the superposition problem.

In this section we show that superpositions of atomic energy states

can be incorporated into Markov processes. Consider an arbitrary

linear combination

v = a1u1 + a2u2 + ...

where |v| = 1 and the basis vectors u1, u2, ... are chosen so that

each represents one of the atom's discrete energy levels. These

energy‑vectors are orthogonal. What the vector v represents is an

atomic state with a complex amplitude a1 of being at energy level u1,

an amplitude a2 of being at level u2, ... etc. The Markov vector

v+ = a1a1*u1 + a2a2*u2 + ...

yields a1a1* as the probability of an atom in state |v> being in state

|u1>, a2a2* as the probability of being in state |u2>, etc.

To show that atomic transitions of an atom in an arbitrary

superposition of energy states can be dealt with using Markov methods,

consider for simplicity a two‑level case. Let S be an atomic system

with energy levels A and B, let u1 = (1, 0) represent the state A, and

let u2 = (0, 1) represent B. Suppose that the Markov matrix M of

section 4 above represents the transition probabilities and hence the

dynamics of S. Suppose also that S evolves into a complex

superposition Ψ of its basis states, where in this example Ψ is

(represented by) the normalized vector 1/5(1, 2i). Then Ψ+ = (1/5,

4/5). Multiplying the matrix M and Ψ+ together yields the following:

(1/2 1/3)(1/5) = (1/10 + 4/15) = (11/30)

(1/2 2/3)(4/5) (1/10 + 8/15) (19/30).

We argue that the Markov vector (11/30, 19/30) gives the correct

probabilities 11/30 and 19/30 for the atomic transitions ΨA and ΨB

respectively, using the following reasoning. We note first that |<Ψ|u1>|2

= 1/5 and |<Ψ|u2>|2 = 4/5, which is to say that the conditional

probability of the atom being in state A, given that it is in state Ψ,

is 1/5, and the probability that it is in B, given that it is in Ψ, is

4/5. p(A|Ψ) = 1/5 and p(B|Ψ) = 4/5. These probabilities are not

transition probabilities, but reflect the A and B "aspects" of the

superposition Ψ.

Secondly, there are two ways or "routes" by which an atom in Ψ can

make the probabilistic transition to A. Either the "A‑aspect" of Ψ

makes the transition to A, or the "B‑aspect" does. The probability for

the atom to follow the first route is obtained by multiplying p(A|Ψ)

by the transition probability p(AA), i.e. 1/5 1/2 = 1/10. The

probability of the second route is p(B|Ψ) p(BA) = 4/5 1/3 = 4/15.

Since the two routes are mutually exclusive, we add 1/10 and 4/15 to

get the total probability 11/30 of an atom in state Ψ jumping to A,

i.e. for the transition ΨA. Similar calculations yield 19/30 for p(ΨB).

These are precisely the values assigned these transitions by the

Markov vector (11/30, 19/30).

The calculation of these probabilities using Markov vectors and

matrices opens up the possibility of an empirical test of the overall

hypothesis of this paper, which is that atoms exhibit a discrete

probabilistic evolution, not a continuous one. The test is this. Place

a single trapped atom or ion in a known mathematical superposition Ψ

of excited energy levels. Let A be the ground state, from which the

electron is quickly driven into the fluorescent strong transition, and

let B be the metastable "dark" state. Calculate the probabilities of

the transitions ΨA and ΨB using the Markov methods of this section,

and check in a series of trials whether the observed frequencies match

the predicted probabilities.

(If it is more convenient to place the test atom in a mixed state D

rather than a pure superposition, then the diagonal of the density

matrix will serve equally well as the required Markov vector, from

which the probabilities p(DA) and p(DB) can be obtained as before.)

9. Discrete atomic transitions: ontology or epistemology?

The present paper attempts to show that a consistent account can be

given of energy changes in single atoms as Markov processes. In the

course of the process the atom makes discrete transitions from one

level to another in accordance with the probabilities of a Markov

transition‑matrix. Many investigators on the other hand, who are of

the opinion that quantum mechanics precludes transitions of this kind

having any ontological status, interpret the transitions as

measurement‑induced, or as characterizing our knowledge of the atom

rather than the atom itself. See e.g. Javanainen (1986); Pegg, Loudon

and Knight (1986); Cook (1990), pp. 401‑407; Wiseman (1996), esp. pp.

205‑208; Wiseman and Toombes (1999). A review and discussion of this

approach is found in Home and Whitaker (1992), (1993).

Here are some quotes:

"The atomic evolution described by the density‑matrix elements is of

course continuous, but the detection of the emitted photon immediately

converts a priori probabilities into a posteriori probabilities,

giving rise to discontinuities or jumps associated with the detection

process." (Pegg, Loudon and Knight (1986), p. 4085)

"It is tempting to go beyond the picture presented by the measured

results and to ask questions such as, "When did the quantum jump

actually occur between the two measurements that gave different

results?" But such questions are undoubtedly inappropriate because it

is the measurement itself that projects the system into the new state.

... Hence, quantum jumps seem to be a property of the measurement

process." (Cook (1990), p. 399)

The question of whether the sources of quantum jumps are instantaneous

transitions in the atom, or alternatively the process of observation,

comes to a head in the case of dark periods. If during a period of

active fluorescence a dark period intervenes in which no photons are

detected, do the probabilities P1 and P2 of being in levels 1 and 2

change? Not for dark periods of short duration, according to Cook

(1990) p. 406, since it might be that the atom is in state |1> and

simply has not yet emitted a photon. But as the dark period lengthens,

there occurs a knowledge‑induced probability shift, a "Bayesian

transition", which gives the atom the new and longer expected lifetime

of level 2 in place of the shorter lifetime of level 1.

These Bayesian changes in probability are in Cook's words "as 'real'

as those caused by physical transitions". His claim is that, when an

atom initially in any superposition of excited states is projected

into the metastable state |2> by a Bayesian flow of probability, this

constitutes "the quantum mechanical explanation of Dehmelt's intuitive

shelving concept". Cook concludes (p. 407) that "It is interesting

that the quantum formalism attributes electron shelving to the lack of

fluorescence, whereas the intuitive picture of the process attributed

the lack of fluorescence to electron shelving".

The majority view seems to be that discrete atomic transitions are not

a feature of the physical world, but rather of our observation of the

world. To this thesis there are a number of possible replies, among

them the following.

(i) As is argued in this paper, it is possible to give a consistent

account of the evolution of atomic systems as Markov processes. The

appeal to photon detection as the source of discontinuous changes is

therefore unnecessary.

(ii) Intuitively, one would think that atomic transitions in an atom

cause photon emission, and photon emission causes a registration in a

detector. Conversely, lack of an atomic transition (i.e. "shelving")

causes lack of photon detection. To reverse these causal dependencies,

and argue that electron shelving should be attributed to absence of

photon observation, would be justified only if the intuitive idea that

atoms undergo discontinuous transitions were subject to severe

objections. However, the central thesis of this paper is that the

discrete‑transition model of atomic evolution is consistent, plausible

and physical.

(iii) As is pointed out in Home and Whitaker, detection of photons

emitted by an atom can take place at great distances away, and at

times long after the apparatus used to contain the atom has been

dismantled (1992, p. 2392; 1993, p. 115). One cannot "attribute

quantum jumps to detection" if the system to which the transitions are

attributed has ceased to exist before the detection takes place. In

particular, it would be inconsistent to suggest that the atomic

process itself is "driven by the act of observation" (1993, p. 115).

(iv) As is also noted in Home and Whitaker, those who claim that the

behaviour of a physical system depends on the participation of an

observer must be prepared to assert that the system would behave in a

different fashion if unobserved (1992, p. 2392). But this seems

implausible. How would one go about comparing the behaviour of an

observed with an unobserved system? The standard procedure is to

construct a theory of a system, e.g. a Markov model of atomic

transitions, and then test the model by comparing predicted with

observed effects, probability values, etc.

(v) Finally, one of the strongest arguments against the idea that

atomic transitions are the product of measurement, and in favour of

their ontological status, is the mathematical conclusion that Cook and

Kimble come to in their (1985). This is, that in the V‑configuration

fluorescent system of section 1, both the "on/light" and "off/dark"

times are distributed exponentially. This implies that a histogram

plot should show an inverse exponential relation between length of

period and number of periods of a given length. Furthermore, the

difference in decay rates for the "on" and "off" states must be due to

a process, such as spontaneous emission, which affects only the length

of the dark periods (Erber et al. (1989), pp. 259‑60). But if all this

is so, then the evidence for identifying "dark period" with "atomic

shelved state", as opposed merely to "absence of photon detection",

becomes very strong. What reason, other than decay of an atomic energy

state, could account for the precise exponential distribution of dark

times?

10. Conclusion.

The objective of the paper has been to suggest explicit recognition of

the presence of discontinuity in atomic evolution by giving an account

of atomic transitions as Markov processes, and to deal with the

objection that superpositions of energy states cannot be incorporated

into processes of this kind. Amongst other things, this view implies

that instantaneous transitions can be regarded as physical phenomena,

and not simply as by‑products of observational methods. The strongest

argument for the "ontological" status of atomic transitions is the

fact, noted in (v) of section 9 above, that the exponential

distribution of dark periods in fluorescence is evidence that such

periods are caused by exponentially decaying metastable states. In

fact, both Nagourney, Sandberg and Dehmelt (1986), and Bergquist,

Hulet, Itano and Wineland (1986) fit exponential curves to observed

dark‑time durations in order to obtain expected lifetime values of

metastable states in barium and mercury ions. Without the

identification of dark periods with the occupancy of "shelved" atomic

states, the central reasoning on which these papers are based would be

lost.

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