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)