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’d · E 0 ”shows that |CS | |CP |, i.e. |µ2 is dominantly like 2S1/2 , while |µ2 is
mainly like 2P1/2 . The states |µ1 and |µ3 of the simple model pictured in Fig. 18.1
are identi¬ed with 1S1/2 and 2P3/2 respectively.
Since neither |µ2 nor |µ2 has de¬nite parity, the dipole selection rules now allow
single-photon transitions 2 ’ 1 and 2 ’ 1. The rate for 2 ’ 1 is proportional
2
to |CP | , so a proper choice of |E 0 | will guarantee that the single-photon process
dominates the two-photon process, while still being slow compared to the rate “31 for
the Lyman-alpha transition.
By the same token, there are dipole-allowed transitions from 3 to both 2 and 2 .
The unwanted level 2 can be e¬ectively eliminated by applying a microwave ¬eld
resonant with the 3 ’ 2 transition, but not with the 3 ’ 2 transition. The strength
of this ¬eld can be adjusted so that the stimulated emission rate for 3 ’ 2 is large
compared to the spontaneous rates for 3 ’ 2 and 3 ’ 2, but small compared to
the stimulated and spontaneous rates for the 3 ’ 1 transition. These settings ensure
that the population of |µ2 will remain small at all times and that |µ2 is an e¬ective
shelving state.
The main practical di¬culty for this experiment is that the pump would have to
operate at the vacuum-UV wavelength, 30.38 nm, of the Lyman-alpha line of the He+
ion. One possible way around this di¬culty is to use the radiation from a synchrotron
light source.
The transition 3 ’ 2 is primarily due to the microwave-frequency transition
2P3/2 ’ 2S1/2 , which occurs at 44 GHz. Our assumption that the spontaneous emis-
sion rate for this transition is small compared to the transition rate for the Lyman-
alpha transition is justi¬ed by the rough estimate
3
A (Lyman alpha) ν (Lyman alpha)
∼ ∼ 1016 (18.129)
A 2P3/2 ’ 2S1/2 ν 2P3/2 ’ 2S1/2

(Bethe and Salpeter, 1977), which uses the values 2.47 — 1015 Hz and 11 GHz for the
Lyman-alpha transition and the 2P3/2 ’ 2S1/2 microwave transition in hydrogen
respectively.
The combination of the low rate for the 3 ’ 2 transition and the long lifetime of
the shelving level 2 will permit easy observation of interrupted resonance ¬‚uorescence
at the helium ion Lyman-alpha line, i.e. quantum jumps.
½
Quantum jumps

For hydrogen, the lifetime of the 2P3/2 state is 1.595 ns, so the estimate (18.129)
tells us that the lifetime for the microwave transition is approximately 2 — 107 s, i.e.
of the order of a year. The lifetime of the same transition for a hydrogenic ion scales
as Z ’4 , so for Z = 2 the microwave transition lifetime is 1.5 — 106 seconds, which is
about a month.
This is still a rather long time to wait for a quantum jump. The solution is to
adjust the strength of the resonant microwave ¬eld driving the 3 ” 2 transition to
bring this lifetime within the limits of the experimentalist™s patience.

C Rate equation analysis
The assumption that the driving ¬eld is incoherent allows us to extend the rate equa-
tion approximation (11.190) for two-level atoms to our simple model to get
dP3
= ’ (“3 + W31 + W32 ) P3 + W31 P1 + W32 P2 , (18.130)
dt
dP2
= ’ (“2 + W32 ) P2 + (“32 + W32 ) P3 , (18.131)
dt
dP1
= ’W31 P1 + (W31 + “31 ) P3 + “2 P2 . (18.132)
dt
Adding the equations shows that the sum of the three probabilities is constant:
P1 + P2 + P3 = 1 . (18.133)
The inequalities (18.125) suggest that the adiabatic elimination rule (11.187) can
be applied to the rate equations (18.130)“(18.132). To see how the rule works in this
case, it is useful to express the rate equations in terms of the probability P31 = P3 + P1
that the ionic state is in H13 , and the inversion D31 = P3 ’ P1 . The new form of the
rate equations is
d 1 1
D31 = ’ 2W31 + “31 + “32 + W32 D31
dt 2 2
1 1
’ “31 + “32 + W32 P31 + (W32 ’ “2 ) P2 , (18.134)
2 2
d 1 1
P31 = ’ (“32 + W32 ) P31 ’ (“32 + W32 ) D31 + (“2 + W32 ) P2 , (18.135)
dt 2 2
d 1 1
P2 = ’ (“2 + W32 ) P2 + (“2 + W32 ) P31 + (“2 + W32 ) D31 . (18.136)
dt 2 2
The rate multiplying D31 on the right side of eqn (18.134) is much larger than any
other rate in the equations; therefore, D31 (t) will rapidly decay to the steady-state
solution of eqn (18.134), i.e.

W32 ’ “2
“31 + 1 “32 + 1 W32
=’ 2 2
D31 P31 + P2 . (18.137)
2W31 + “31 + 1 “32 + 1 W32 2W31 + “31 + 1 “32 + 1 W32
2 2 2 2

The coe¬cients of the probabilities P31 and P2 are very small, so we can set D31 0
in the rest of the calculation.
¾ The master equation

With this approximation, the remaining rate equations are
dP2
= ’Ron P2 + Ro¬ P31 (18.138)
dt
and
dP31
= Ron P2 ’ Ro¬ P31 , (18.139)
dt
where Ron = “2 + W32 is the rate at which the ¬‚uorescence turns on, and Ro¬ =
(“32 + W32 ) /2 is the rate at which ¬‚uorescence turns o¬. Solving eqns (18.138) and
(18.139) for P31 (t) yields
Ron
P31 (t) = P31 (0) e’(Ron +Roff )t + 1 ’ e’(Ron +Roff )t . (18.140)
Ron + Ro¬
The ¬‚uorescent intensity IF (t) is proportional to P31 (t), so IF (t) evolves smoothly
from its initial value IF (0) to the steady-state value
Ron
IF ∝ . (18.141)
Ron + Ro¬
This result is completely at odds with the ¬‚ickering on-and-o¬ behavior predicted
above. The source of this discrepancy is the fact that the quantities P1 , P2 , and P3 in
the rate equations (18.138) and (18.139) are unconditional probabilities. This means
that P1 , for example, is the probability that the ion is in level 1 without regard to its
past history or any other conditions. Another way of saying this is that P1 refers to
an ensemble of ions which have reached level 1 in all possible ways.
Before the development of single-ion traps, resonance ¬‚uorescence experiments
dealt with dilute atomic gases, and the total ¬‚uorescence signal would be correctly
described by eqn (18.140). In this case, the on-and-o¬ behavior of the individual atoms
would be washed out by averaging over the random ¬‚uorescence of the atoms in the
gas. For a single trapped ion, the smooth behavior in eqn (18.140) can only be recov-
ered by averaging over many observations, all starting with the ion in the same state,
e.g. the ground state.
In addition to the inability of the rate equations to predict quantum jumps, it
is also the case that statistical properties”such as the distribution of waiting times
between jumps”are beyond their reach. Thus any improvement must involve putting
in some additional information; that is, reducing the size of the ensemble.
The ¬rst step in this direction was taken by Cook and Kimble (1985) who intro-
duced the conditional probability P31,n (t, t + T ) that the ion is in H13 after making
n transitions between H13 and |µ2 during the interval (t, t + T ). The number of tran-
sitions de¬nes a subensemble of ions with this history. The complementary object
P2,n (t, t + T ) is the probability that the ion is in level |µ2 after n transitions between
H13 and |µ2 during the interval (t, t + T ).
By using the approximations leading to eqns (18.138) and (18.139), it is possi-
ble to derive an in¬nite set of coupled rate-like equations for P31,n (t, t + T ) and
P2,n (t, t + T ), with n = 0, 1, . . .. This approach permits the calculation of various
statistical features of the quantum jumps, but it is not easy to connect it with the
more re¬ned quantum-jump theories to be developed later on.
¿
Quantum jumps

D A stochastic model
We will now consider a simple on-and-o¬ model which is qualitatively similar to the
more sophisticated quantum-jump theories. In this approach, the analytical treatment
based on conditional probabilities is replaced by an equivalent stochastic simulation.
We ¬rst assume that the ¬‚uorescent intensity can only have the values I = 0 (o¬)
or I = IF (on). If the signal is on at time t, then the probability that it will turn o¬
in the interval (t, t + ∆t) is ∆po¬ = Ro¬ ∆t. Conversely, if the signal is o¬ at time t,
then the probability that it will turn on in the interval (t, t + ∆t) is ∆pon = Ron ∆t.
For su¬ciently small ∆t, we can assume that only one of these events occurs.
The ¬‚uorescent intensities In at the discrete times tn = (n ’ 1) ∆t can then be
calculated by the following algorithm.

For In = 0 choose a random number r in (0, 1) ;
then set In+1 = 0 if ∆pon < r or In+1 = IF if ∆pon > r .
(18.142)
For In = IF choose a random number r in (0, 1) ;
then set In+1 = IF if ∆po¬ < r or In+1 = 0 if ∆po¬ > r .

The random choices in this algorithm are a special case of the rejection method (Press
et al., 1992, Sec. 7.3) for choosing random variables from a known distribution.
From a physical point of view, the algorithm is an approximate embodiment of the
collapse postulate for measurements in quantum theory. The value In is the outcome of
a measurement of the ¬‚uorescent intensity at t = tn , so it corresponds to a collapse of
the state vector of the ion into the state with the value In . If In+1 = In the subsequent
collapse at t = tn+1 is into the same state as at t = tn . For In+1 = In the collapse at
tn+1 is into the other state, so we see a quantum jump.
A typical2 sequence of quantum jumps is shown in Fig. 18.3. Random sequences of
binary choices (dots and dashes) of this kind are called random telegraph signals.
This plot exhibits the expected on-and-o¬ behavior for a single ion, but the smooth
¬‚uorescence curve predicted by the rate equations is nowhere to be seen.
In order to recover an approximation to eqn (18.140), we consider M experiments,
all starting with I1 = IF , and de¬ne the average ¬‚uorescent intensity at time tn by
M
1
In,av = In,j , (18.143)
M j=1

where In,j is the ¬‚uorescent intensity at time tn for the jth run. A comparison of In,av
with the values predicted by eqn (18.140) is shown in Fig. 18.4, for M = 100.

E Experimental evidence
We have demonstrated a simple model displaying quantum jumps and a plausible
experimental realization for it, but the question remains if any such phenomena have
2 Thestochastic algorithm (18.142) gives a di¬erent plot for each run with the same input para-
meters. The ˜typical™ plot shown here was chosen to illustrate the e¬ect most convincingly. This kind
of data selection is not unknown in experimental practice.
The master equation


1

1

0.8

0.6

0.4

0.2

J
5 10 15 20

Fig. 18.3 Normalized ¬‚uorescent intensity I/IF versus time (in units of the radiative lifetime
1/“b of the shelving state). In these units, Ron = 1.6, Ro¬ = 0.3, and ∆t = 0.1. The initial
intensity is I (0) = IF .

1av
1

0.8

0.6

0.4

0.2

J
2.5 5 7.5 10 12.5 15 17.5 20


Fig. 18.4 Fluorescent intensity (normalized to IF and averaged over 100 runs) versus time
(measured in units of the radiative lifetime of the shelving state). The initial intensity in each
run is I (0) = IF , and the parameter values are those used in Fig. 18.3.

been seen in reality. For this evidence we turn to an experiment in which intermittent
¬‚uorescence was observed from a single, laser-cooled Ba+ ion in a radio frequency trap
(Nagourney et al., 1986).
The complementary relation between theory and experiment is in full play in this
case, as seen by comparing the level diagram for this experiment”shown in Fig. 18.5”
with Fig. 18.1. Fortunately, the complications involved in the real experiment do not
change the essential nature of the e¬ect, which is seen in Fig. 18.6.
Quantum jumps


62 23/2
62 21/2
614.2 nm

493.4 nm
52 ,5/2
649.9 nm
455.4 nm
52 ,3/2


62 51/2


Fig. 18.5 Level structure of Ba+ . The states in the simple three-level model discussed in
¬ ¬ ¬
« « «
the text are |µ1 = ¬62 S1/2 , |µ2 = ¬62 P3/2 , and |µ3 = ¬52 D5/2 , which is the shelf state.
The remaining states are only involved in the laser cooling process indicated by the heavy
solid lines. The 1 ” 2 transition is driven by an incoherent source (a lamp) indicated by the
light solid line. (Reproduced from Nagourney et al. (1986).)
Fluorescence photon counts / second




2000
Lamp on
62 51/2


1000




52 ,5/2
0
0 100 200
Time (s)

Fig. 18.6 A typical trace of the 493 nm ¬‚uorescence from the 62 P1/2 -level showing the
quantum jumps after the hollow cathode lamp is turned on. The atom is de¬nitely known to
be in the shelf level during the low ¬‚uorescence periods. (Reproduced from Nagourney et al.
(1986).)

Quantum jumps and the master equation—
18.7.2

Many features of quantum-jump experiments are well described”at least semi-quantit-
atively”by the rate equation approximation for the conditional probabilities, e.g.
P31,n , or by the equivalent stochastic simulation; but the rate equation model has
de¬nite limitations. The most important of these is the restriction to incoherent ex-
citation of the atomic states. Many experiments employ laser excitation, which is
inherently coherent in character.
The master equation

The e¬ort required to incorporate coherence e¬ects eventually led to the creation of
several closely related approaches to the problem of quantum jumps. These techniques
are known by names like the Monte Carlo wave function method, quantum trajectories,
and quantum state di¬usion. Sorting out the relations between them is a complicated
story, which we will not attempt to tell in detail. For an authoritative account, we
recommend the excellent review article of Plenio and Knight (1998) which carries the
history up to 1999.
We will present a brief account of the Monte Carlo wave function technique for the
solution of the master equation. The other approaches mentioned above are technically
similar; but they di¬er in the original motivations leading to them, in their physical
interpretations, and in the kinds of experimental situations they can address.
There are two complementary views of these theoretical approaches. One may re-
gard them simply as algorithms for the solution of the master equation, or as concep-
tually distinct views of quantum theory. The discussion therefore involves both com-
putational and fundamental physics issues. We will ¬rst consider the computational
aspects of the Monte Carlo wave function technique, and then turn to the conceptual
relations between this method and the approaches based on quantum trajectories or
quantum state di¬usion.
The master equation (18.115) is a di¬erential equation describing the time evolu-
tion of the sample density operator. Except in highly idealized situations”for which
analytical solutions are known”the solution of the master equation requires numeri-
cal methods. Even for the apparently simple case of a single cavity mode, the sample
Hilbert space HS is in¬nite dimensional, so the annihilation operator a is represented
by an in¬nite matrix. A direct numerical attack would therefore require replacing
HS by a ¬nite-dimensional space, e.g. the subspace spanned by the number states
|0 , . . . , |M ’ 1 . This would entail representing the creation and annihilation opera-
tors and the density operator by M — M matrices.
In some situations, such as those discussed in Section 18.5.2, an alternative ap-
proach is to replace the in¬nite-dimensional space HS by the two-dimensional quan-
tum phase space, and to use”for a restricted class of problems”the Fokker“Planck
equation (18.61) or the equivalent classical Langevin equation (18.72). In general, this
method will fail if the di¬usion matrix D is not positive de¬nite.
The master equation for an atom can also be represented by a Fokker“Planck
equation on a ¬nite-dimensional phase space, but the collection of problems amenable
to this treatment is restricted by the same kind of considerations, e.g. a positive-de¬nite
di¬usion kernel, that apply to the radiation ¬eld. In many cases the center-of-mass
motion of the atom can be neglected”or at least treated classically”so the sample
Hilbert space is ¬nite dimensional. In this situation the master equation for a two-level
atom is simply a di¬erential equation for a 2 — 2 hermitian matrix. This is equivalent
to a set of four coupled ordinary di¬erential equations, so it is not computationally
onerous.
Unfortunately, in the real world of experimental physics, atoms often have more
than two relevant levels, or it may be necessary to consider more than one atom at a
time. In either case the computational di¬culty grows rapidly with the dimensionality
of the sample Hilbert space.
Quantum jumps

In general, a numerical simulation will take place in a sample Hilbert space with
some dimension M . The master equation is then an equation for an M — M matrix,
and the computational cost for solving the problem scales as M 2 . This is an impor-
tant consideration, since increasing the accuracy of the simulation typically requires
enlarging the Hilbert space. On the other hand, if one could work with a state vector
instead of the density operator, the cost of a solution would only scale as M . This gain
alone justi¬es the development of the Monte Carlo wave function technique described
below.

The Monte Carlo wave function method—
18.7.3
According to eqn (18.115), the change in the density operator over a time step ∆t is
∆t
[HS , ρS ] + ∆tLdis ρS + O ∆t2 .
ρS (t + ∆t) = ρS (t) + (18.144)
i
By combining the ¬rst two terms in eqn (18.117) for Ldis with the Hamiltonian term,
this can be rewritten as
K
i∆t i∆t † †
ρS (t + ∆t) = ρS (t) ’ Hdis ρS (t) + ρS (t) Hdis + ∆t Ck ρS (t) Ck , (18.145)
k=1

where the dissipative Hamiltonian is
K
i †
= HS ’
Hdis Ck Ck . (18.146)
2
k=1

This suggests de¬ning a dissipative, nonunitary time translation operator,
i∆t
Udis (∆t) = e’i∆tHdis / = 1 ’ Hdis + O ∆t2 , (18.147)

and then using it to rewrite eqn (18.145) as
K
† †
ρS (t + ∆t) = Udis (∆t) ρS (t) Udis (∆t) + ∆t Ck ρS (t) Ck , (18.148)
k=1

correct to O (∆t).
The ensemble de¬nition (2.116) of the density operator shows that this is equivalent
to

|Ψe (t + ∆t) Pe Ψe (t + ∆t)| = Pe Udis (∆t) |Ψe (t) Ψe (t)| Udis (∆t)
e e
K

Pe ∆t Ck |Ψe (t) Ψe (t)| Ck ,
+
e k=1
(18.149)
where the Pe s are the probabilities de¬ning the initial state, and |Ψe (0) = |˜e .
The ¬rst term on the right side of this equation evidently represents the dissipative
The master equation

evolution of each state in the ensemble. This is closely related to the Weisskopf“Wigner
approach to perturbation theory, which we used in Section 11.2.2 to derive the decay
of an excited atomic state by spontaneous emission.
This is all very well, but what is the meaning of the second term on the right side
of eqn (18.149)? One way to answer this question is to ¬x attention on a single state
in the ensemble, say |Ψe (t) , and to de¬ne the normalized states
Ck |Ψe (t)
|φek (t) = , k = 1, . . . , K . (18.150)

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