loophole in the general argument of Section 18.1.

The second di¬culty with the robustness of coherent states is that it seems to

violate the ¬‚uctuation dissipation theorem. The ¬eld su¬ers dissipation, but there is

no added noise. Consequently, it is a relief to realize that the strength of the noise

term in the equivalent classical Langevin equations (18.71) vanishes for ncav = 0.

Further reassurance comes from the operator Langevin approach, in particular eqn

(14.74), which shows that the strength of the Langevin noise operator also vanishes

for ncav = 0.

¿

Phase space methods

B Thermalization of an initial coherent state

The coordinates de¬ned by eqn (18.78) are also useful for solving eqn (18.64), the

Fokker“Planck equation with di¬usion. According to eqn (18.86), the transformation

from u to u is a rotation followed by scaling with exp (“t/2). The operator ∇2 on

the right side of eqn (18.64) is invariant under rotations, so ∇2 = e“t ∇ 2 and the

Fokker“Planck equation becomes

‚ D0 “t 2

e ∇ P (u ; t ) ,

P (u ; t ) = (18.93)

‚t 2

which is the di¬usion equation with a time-dependent di¬usion coe¬cient. The Fourier

transform,

d2 u P (u ; t ) e’iq ·u ,

P (q ; t ) = (18.94)

then satis¬es the ordinary di¬erential equation

d D0 “t 2

P (q ; t ) = e q P (q ; t ) , (18.95)

dt 2

which has the solution

D0 “t

e ’ 1 q 2 P 0 (q ) .

P (q ; t ) = exp (18.96)

2“

For the initial coherent state, P0 (u) = δ2 (u ’ u0 ), one ¬nds

P 0 (q ) = exp [’iq · u0 ] , (18.97)

and the inverse transform can be explicitly evaluated to yield

(u ’ u (t))2

1

exp ’ , (18.98)

P (u; t) =

πw (t) w (t)

where u (t) is given by eqn (18.90) and

w (t) = ncav 1 ’ e’“t . (18.99)

1/“) u (t) ’ 0, and the P -function approaches the thermal

For long times (t

distribution given by eqn (5.176); in other words, the ¬eld comes into equilibrium with

1/“, we see that w (t) ∼ ncav “t

the cavity walls as expected. At short times, t 1

and the initial delta function is recovered.

C A driven mode in a lossy cavity

In Section 5.2 we presented a simple model for generating a coherent state of a sin-

gle mode in a lossless cavity. We can be sure that losses will be present in any real

experiment, so we turn to the Fokker“Planck equation for a more realistic treatment.

The master equation

The o¬-resonant term in the Heisenberg equation (5.38) de¬ning our model can

safely be neglected, so the situation is adequately represented by the simpli¬ed Hamil-

tonian

HS (t) = ω0 a† a ’ W e’i„¦t a† ’ W — ei„¦t a , (18.100)

that leads to the Liouville operator

1 1

ω0 a† a, ρS (t) ’ W e’i„¦t a† + W — ei„¦t a, ρS (t) .

LS ρS (t) = (18.101)

i i

After including the new terms in the master equation and applying the rules (18.58),

one ¬nds an equation of the same form as eqn (18.61), except that the Z (±) function

is replaced by

“

+ iω0 ± ’ iW e’i„¦t .

Z (±) = (18.102)

2

Instead of directly solving the Fokker“Planck equation, it is instructive to use the

equivalent set of classical Langevin equations. Substituting the new Z (±) function

into the general result (18.72) yields

d± (t) “

+ iω0 ± (t) + iW e’i„¦t +

=’ 2D0 · (t) , (18.103)

dt 2

which has the solution

± (t) = ± (0) e’(iω0 +“/2)t + ±coh (t) + 2D0 ‘ (t) , (18.104)

where

iW

e’(iω0 +“/2)t e(i∆+“/2)t ’ 1

±coh (t) = (18.105)

i∆ + “/2

is a de¬nite (i.e. nonrandom) function, and

t

dt1 e’(iω0 +“/2)(t’t1 ) · (t1 ) .

‘ (t) = (18.106)

0

The initial value, ± (0), is a complex random variable, not a de¬nite complex number.

The average of any function f (± (0)) is given by

d2 ± (0) P0 (± (0) , ±— (0)) f (± (0)) ,

f (± (0)) = (18.107)

but special problems arise if the initial state is not classical. For a classical state”

i.e. P0 (±, ±— ) 0”standard methods can be used to draw ± (0) randomly from the

distribution, but these methods fail when P0 (±, ±— ) is negative. For these nonclas-

sical states, the c-number Langevin equations are of doubtful utility for numerical

simulations.

Phase space methods

For the problem at hand, the initial state is the vacuum, with the positive distri-

bution P0 (± (0) , ±— (0)) = δ (± (0)). The initial value ± (0) and the noise term ‘ (t)

both have vanishing averages, so the average value of ± (t) is given by

± (t) = ±coh (t) . (18.108)

For the nondissipative case, “ = D0 = 0, the average agrees with eqn (5.41); but,

when dissipation is present, the long time (t 1/“) solution approaches

iW

e’i„¦t .

± (t) = (18.109)

i∆ + “/2

Thus the decay of the average ¬eld due to dissipation”shown by eqn (18.91)”is

balanced by radiation from the classical current, and the average ¬eld amplitude has

a de¬nite phase determined by the phase of the classical current. This would also be

true if the sample were described by the coherent state ρcoh (t) = |±coh (t) ±coh (t)|,

so it will be necessary to evaluate second-order moments in order to see if eqn (18.104)

corresponds to a true coherent state.

We will ¬rst investigate the coherence properties of the state by using the explicit

solution (18.104) to get

2

’(iω0 +“/2)t

2

± (t) = ± (0) e + ±coh (t) + 2D0 ‘ (t)

= ±2 (t) + 2D0 ‘2 (t) . (18.110)

coh

The simple form of the second line depends on two facts: (i) ±coh (t) is a de¬nite

function; and (ii) the distribution of ± (0) is concentrated at ± (0) = 0. A further

simpli¬cation comes from using eqn (18.105) to evaluate ‘2 (t) . The result is a double

integral with integrand proportional to · (t1 ) · (t2 ) , but eqn (18.74) shows that this

average vanishes for all values of t1 and t2 . The ¬nal result is then

±2 (t) = ±2 (t) = ± (t) 2 , (18.111)

coh

which also agrees with the prediction for a true coherent state.

Before proclaiming that we have generated a true coherent state in a lossy cavity,

2

we must check the remaining second-order moment, |± (t)| , which represents the

average of the number operator a† a. Since ± (t) is concentrated at the origin, we can

simplify the calculation by setting ± (0) = 0 at the outset. This gives us

2 2 2

|± (t)| = |±coh (t)| + 2D0 |‘ (t)| . (18.112)

Combining eqns (18.106) and (18.74) leads to

t t

dt2 e’(’iω0 +“/2)(t’t1 ) e’(iω0 +“/2)(t’t2 ) · — (t1 ) · (t2 )

2

|‘ (t)| = dt1

0 0

1 ’ e’“t

t

’“(t’t1 )

dt1 e = , (18.113)

=

“

0

The master equation

so that

2D0

1 ’ e’“t = ncav 1 ’ e’“t ,

|± (t)|2 ’ |±coh (t)|2 = (18.114)

“

where we used eqn (18.65) to get the ¬nal result. The left side of this equation would

vanish for a true coherent state, so the state generated in a lossy cavity is only coherent

if ncav = 0, i.e. if the cavity walls are at zero temperature.

The Lindblad form of the master equation—

18.6

The master equations (18.31) and (18.42) share three important properties.

(a) The trace condition, Tr [ρS (t)] = 1, is conserved.

(b) The positivity of ρS is conserved, i.e. Ψ |ρS (t)| Ψ 0 for all states |Ψ and all

times t.

(c) The equations are derivable from a model of the sample interacting with a collec-

tion of reservoirs.

The most general linear, dissipative time evolution that satis¬es (a), (b), and (c)

is given by

‚ρS

= LS ρS + Ldis ρS , (18.115)

‚t

where

1

LS ρS = [HS (t) , ρS ] (18.116)

i

describes the Hamiltonian evolution of the sample, and the dissipative term has the

Lindblad form (Lindblad, 1976)

K

1 † † †

Ldis ρS = ’ Ck Ck ρS + ρS Ck Ck ’ 2Ck ρS Ck . (18.117)

2

k=1

Each of operators C1 , C2 , . . . , CK acts on the sample space HS and there can be a

¬nite or in¬nite number of them, depending on the sample under study.

One can see by inspection that there are two Lindblad operators, i.e. K = 2, for

the single-mode master equation (18.31):

“ncav a† .

C1 = “ (ncav + 1)a , C2 = (18.118)

A slightly longer calculation”see Exercise 18.6”shows that there are three operators

for the master equation (18.42) describing the two-level atom.

The Lindblad form (18.117) for the dissipative operator can be used to investigate

a variety of questions. For example, in Section 2.3.4 we introduced a quantitative

measure of the degree of mixing by de¬ning the purity of the state ρS as P (t) =

Tr ρ2 (t) 1. One can show from eqn (18.115) that the time derivative of the purity

S

is

K

d † †

P (t) = ’2 Tr ρS (t) Ck Ck ρS (t) ’ ρS (t) Ck ρS (t) Ck . (18.119)

dt

k=1

At ¬rst glance, it may seem natural to assume that interaction with the environment

can only cause further mixing of the sample state, so one might expect that the time

Quantum jumps

derivative of the purity is always negative. If ρS (0) is a mixed state this need not be

true. For example, the purity of a thermal state would be increased by interaction

with a colder reservoir, as seen in Exercise 18.7.

On the other hand, for a pure state there is no way to go but down; therefore, the

intuitive expectation of declining purity should be satis¬ed. In order to check this, we

evaluate eqn (18.119) for an initially pure state ρS (0) = |Ψ Ψ|, to ¬nd

K

d † †

P (t) = ’2 Ψ Ck Ck Ψ ’ Ψ Ck Ψ Ψ |Ck | Ψ

dt t=0 k=1

K

†

= ’2 Ψ δCk δCk Ψ 0, (18.120)

k=1

where

δCk = Ck ’ Ψ |Ck | Ψ . (18.121)

Thus the Lindblad form guarantees the physically essential result that initially pure

states cannot increase in purity (Gallis, 1996).

The appearance of an inequality like eqn (18.120) prompts the following question:

Are there any physical samples possessing states that saturate the inequality? We can

answer this question in one instance by studying the master equation (18.31) with a

√

zero-temperature reservoir. In this case eqn (18.118) gives us C2 = 0 and C1 = “a,

so that eqn (18.120) becomes

d †

P (t) = ’2“ Ψ (a ’ ±) (a ’ ±) Ψ 0, (18.122)

dt t=0

where ± = Ψ |a| Ψ . The inequality can only be saturated if |Ψ satis¬es

a |Ψ = ± |Ψ , (18.123)

i.e. when |Ψ is a coherent state. For all other pure states, interaction with a zero-

temperature reservoir will decrease the purity, i.e. the state becomes mixed.

18.7 Quantum jumps

18.7.1 An elementary description of quantum jumps

The notion of quantum jumps was a fundamental part of the earliest versions of

the quantum theory, but for most of the twentieth century it was assumed that the

phenomenon itself would always be unobservable, since there were no experimental

methods available for isolating and observing individual atoms, ions, or photons.

This situation began to change in the 1980s with Dehmelt™s proposal (Dehmelt,

1982) for an improvement in frequency standards based on observations of a single

ion, and the subsequent development of electromagnetic traps (Paul, 1990) that made

such observations possible.

The following years have seen a considerable improvement in both experimental

and theoretical techniques. The improved experimental methods have made possible

the direct observation of the quantum jumps postulated by the founders of quantum

theory.

The master equation

3

(Allowed)

Fig. 18.1 A three-level ion with dipole-al-

2

lowed transitions 3 ” 2 and 3 ” 1, indicated

by wavy arrows, and a dipole-forbidden tran-

(Allowed)

(Forbidden)

sition 2 ’ 1, indicated by the light dashed

arrow. The heavy double arrows denote strong

incoherent couplings on the 3 ” 1 and 3 ’ 2

1

transitions.

A A three-level model

It is always good to have a simple, concrete example in mind, so we will study a single,

trapped, three-level ion, with the level structure shown in Fig. 18.1. The dipole-allowed

transitions, 3 ’ 1 and 3 ’ 2, have Einstein-A coe¬cients “31 and “32 respectively, so

the total decay rate of level 3 is “3 = “31 + “32 . Since the dipole-forbidden transition,

2 ’ 1, has a unique ¬nal state, it is described by a single decay rate “2 , which is small

compared to both “31 and “32 .

In addition to the spontaneous emission processes, we assume that an incoher-

ent radiation source, at the frequency ω31 , drives the ion between levels 1 and 3 by

absorption and spontaneous emission. As explained in Section 1.2.2, both of these

processes occur with the rate W31 = B31 ρ (ω31 ), where ρ (ω31 ) is the energy density of

the external ¬eld and B31 is the Einstein-B coe¬cient.

When level 3 is occupied, the ion can isotropically emit ¬‚uorescent radiation, i.e.

radiation at frequency ω31 . Another way of saying this is that the ion scatters the pump

light in all directions. Consequently, observing the ¬‚uorescent intensity”say at right

angles to the direction of the pump radiation”e¬ectively measures the population of

level 3.

We will further assume that the levels 2 and 3 are closely spaced in energy, com-

pared to their separation from level 1, so that ω32 ω31 . From eqn (4.162) we know

that the Einstein-A coe¬cient is proportional to the cube of the transition frequency;

therefore, the transition rate for 3 ’ 2 will be small compared to the transition rate

for 3 ’ 1, i.e. “32 “31 . In some cases, the small size of “32 may cause an excessive

delay in the transition from 3 to 2, so we also allow for an incoherent driving ¬eld on

the 3 ” 2 transition such that

“32 W32 = B32 ρ (ω32 ) “31 . (18.124)

Under these conditions, the ion will spend most of its time shuttling between levels

1 and 3, with infrequent transitions from 3 to the intermediate level 2. The forbidden

transition 2 ’ 1 occurs very slowly compared to 3 ’ 1 and 3 ’ 2, so level 2 e¬ectively

traps the occupation probability for a relatively long time. When this happens the

¬‚uorescent signal will turn o¬, and it will not turn on again until the ion decays back

to level 1. We will refer to these transitions as quantum jumps.1

1 Itwould be equally correct”but not nearly as exciting”to refer to this phenomenon as ˜inter-

rupted ¬‚uorescence™.

Quantum jumps

During the dark interval the ion is said to be shelved and |µ2 is called a shelving

state. The shelving e¬ect is emphasized when the 1 ” 3 transition is strongly saturated

and the state |µ2 is long-lived compared to |µ3 , i.e. when

W31 “3 “2 . (18.125)

During the bright periods when ¬‚uorescence is observed, the state vector |Ψion will

be a linear combination of |µ1 and |µ3 ; in other words, |Ψion is in the subspace H13 .

B A possible experimental realization

As a possible experimental realization of the three-level model, consider the intermit-

tent resonance ¬‚uorescence of the strong Lyman-alpha line, emitted by a singly-ionized

helium ion (He+ ) in a Paul trap. One advantage of this choice is that the spectrum is

hydrogenic, so that it can be calculated exactly.

The complementary relation between theory and experiment guarantees the pres-

ence of several real-world features that complicate the situation. The level diagram

in part (a) of Fig. 18.2 shows not one, but two intermediate states, 2S1/2 and 2P1/2 ,

that are separated in energy by the celebrated Lamb shift, ∆EL / = 14.043 GHz.

The 2S1/2 -level is a candidate for a shelving state, since there is no dipole-allowed

transition to the 1S1/2 ground state, but the 2P1/2 -level does have a dipole-allowed

transition, 2P1/2 ’ 1S1/2 . This adds unwanted complexity.

An additional theoretical di¬culty is caused by the fact that the dominant mech-

anism for the transition 2S1/2 ’ 1S1/2 is a two-photon decay. This is a problem,

because the reservoir model introduced in Section 14.1.1 is built on the emission or

absorption of single reservoir quanta; consequently, the standard reservoir model would

not apply directly to this case.

Fortunately, these complications can be exploited to achieve a closer match to our

simple model. The ¬rst step is to apply a weak DC electric ¬eld E 0 to the ion. In this

3

223/2

2

251/2

2

221/2

1

151/2

(a) (b)

Fig. 18.2 (a) Level diagram for the He+ ion. The spacing between the 2S1/2 and 2P1/2 levels

¬ ¬

« «

is exaggerated for clarity. (b) The unperturbed ¬2S1/2 and ¬2P1/2 states are replaced by the

Stark-mixed states |µ2 and |µ2 . The wavy arrows indicate dipole-allowed decays, while the

solid arrows indicate incoherent driving ¬elds. A dipole-allowed decay 2 ’ 1 is not shown,

since 2 is e¬ectively isolated by the method explained in the text.

¼ The master equation

application ˜weak™ means that the energy-level shift caused by the static ¬eld is small

compared to the Lamb shift, i.e.

2S1/2 |E 0 · d| 2P1/2 ∆EL . (18.126)

In this case there will be no ¬rst-order Stark shift, and the second-order Stark e¬ect

(Bethe and Salpeter, 1977) mixes the 2S1/2 and 2P1/2 states to produce two new

states,

|µ2 = CS 2S1/2 + CP 2P1/2 , (18.127)

|µ2 = CS 2S1/2 + CP 2P1/2 , (18.128)

as illustrated in part (b) of Fig. 18.2.

A second-order perturbation calculation”using the Stark interaction HStark =