CONTINUOUS VS DISCRETE PROCESSES THE PROBABILISTIC EVOLUTION OF SINGLE

CONTINUOUS VS DISCRETE PROCESSES THE PROBABILISTIC EVOLUTION OF SINGLE

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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 SiSj 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 13 be 1 per ms, corresponding to a
natural lifetime of 1 ms, and let the rate for the weak transition 23
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 ΨA and ΨB
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(AA), i.e. 1/5  1/2 = 1/10. The
probability of the second route is p(B|Ψ)  p(BA) = 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 ΨA. Similar calculations yield 19/30 for p(ΨB).
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 ΨA and ΨB 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(DA) and p(DB) 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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