ńņš. 21 |

ĻW (t) = [HSE (t) , Ļenv (t)]

env

i (18.6)

W

ā‚t

in the environment picture. The transformed operators,

ā

Oenv (t) = UW 0 (t) OUW 0 (t) , (18.7)

satisfy the equations of motion

ā‚ env

O (t) = [Oenv (t) , HW 0 (t)] .

env

i (18.8)

ā‚t

env env env

One should keep in mind that HW 0 (t) = HS (t) + HE (t), and that the sample

env

Hamiltonian, HS (t), still contains all interaction terms between diļ¬erent degrees

of freedom in the sample. Only the sampleā“environment interaction is treated as a

perturbation. Thus the environment-picture sample operators obey the full Heisenberg

equations for the sample.

18.3 Averaging over the environment

In line with our general convention, we will now drop the identifying superscript ā˜envā™,

and replace it by the understanding that states and operators in the following discus-

sion are normally expressed in the environment picture. Exceptions to this rule will

Ā¼ The master equation

be explicitly identiļ¬ed. Our immediate task is to derive an equation of motion for the

reduced density operator ĻS . In pursuing this goal, we will generally follow Gardinerā™s

treatment (Gardiner, 1991, Chap. 5).

The ļ¬rst part of the argument corresponds to the formal elimination of the reser-

voir operators in Section 14.1.2, and we begin in a similar way by incorporating the

quantum Liouville equation (18.6) and the initial density operator ĻW (0) into the

equivalent integral equation,

t

i

ĻW (t) = ĻW (0) ā’ dt1 [HSE (t1 ) , ĻW (t1 )] . (18.9)

0

The assumption that HSE is weakā”compared to HS and HE ā”suggests solving

this equation by perturbation theory, but a perturbation expansion would only be

valid for a very short time. From Chapter 14, we know that typical sample correlation

functions decay exponentially:

Q1 (t + Ļ„ ) Q2 (t) ā¼ eā’Ī³Ļ„ , (18.10)

with a decay rate, Ī³ ā¼ g 2 , where g is the sampleā“environment coupling constant. This

exponential decay can not be recovered by an expansion of ĻW (t) to any ļ¬nite order

in g.

As the ļ¬rst step toward ļ¬nding a better approach, we iterate the integral equation

(18.9) twiceā”this is suggested by the fact that Ī³ is second order in gā”to ļ¬nd

t

i

ĻW (t) = ĻW (0) ā’ dt1 [HSE (t1 ) , ĻW (0)]

0

2 t t1

i

+ā’ dt1 dt2 [HSE (t1 ) , [HSE (t2 ) , ĻW (t2 )]] . (18.11)

0 0

Tracing over the environment then produces the exact equation

t

i

ĻS (t) = ĻS (0) ā’ dt1 TrE ([HSE (t1 ) , ĻW (0)])

0

2 t t1

i

+ā’ dt1 dt2 TrE ([HSE (t1 ) , [HSE (t2 ) , ĻW (t2 )]]) (18.12)

0 0

for the reduced density operator. Since our objective is an equation of motion for

ĻS (t), the next step is to diļ¬erentiate with respect to t to ļ¬nd

ā‚ i

ĻS (t) = ā’ TrE ([HSE (t) , ĻW (0)])

ā‚t

t

1

ā’2 dt TrE ([HSE (t) , [HSE (t ) , ĻW (t )]]) . (18.13)

0

This equation is exact, but it is useless as it stands, since the unknown world den-

sity operator ĻW (t ) appears on the right side. Further progress depends on ļ¬nding

Ā½

Averaging over the environment

approximations that will lead to a manageable equation for ĻS (t) alone. The ļ¬rst sim-

plifying assumption is that the sample and the environment are initially uncorrelated:

ĻW (0) = ĻS (0) ĻE (0). By combining the generic expression,

grĪ½ bā Qr ā’ grĪ½ Qā brĪ½ ,

ā—

HSE = i (18.14)

rĪ½ r

r Ī½

for the sampleā“environment interaction with the conventional assumption, brĪ½ = 0,

E

and the initial factorization condition, it is straightforward to show that

TrE ([HSE (t) , ĻW (0)]) = 0 . (18.15)

If one or more of the reservoirs has brĪ½ E = 0, one can still get this result by

writing HSE in terms of the ļ¬‚uctuation operators Ī“brĪ½ = brĪ½ ā’ brĪ½ E , and absorbing

the extra terms by suitably redeļ¬ning HS and HE , as in Exercise 18.1.

Thus the initial factorization assumption always allows eqn (18.13) to be replaced

by the simpler form

t

ā‚ 1

ĻS (t) = ā’ 2 dt TrE {[HSE (t) , [HSE (t ) , ĻW (t )]]} . (18.16)

ā‚t 0

Replacing ĻW (t ) by ĻW (0) = ĻS (0) ĻE (0) in eqn (18.16) would provide a perturba-

tive solution that is correct to second order, butā”as we have just seenā”this would

not correctly describe the asymptotic time dependence of the correlation functions.

The key to ļ¬nding a better approximation is to exploit the extreme asymmetry

between the sample and the environment. The environment is very much larger than

the sample; indeed, it includes the rest of the universe. It is therefore physically rea-

sonable to assume that the fractional change in the sample, caused by interaction

with the environment, is much larger than the fractional change in the environment,

caused by interaction with the sample. If this is the case, there will be no reciprocal

correlation between the sample and the environment, and the density operator ĻW (t )

will be approximately factorizable at all times.

This argument suggests the ansatz

ĻW (t ) ā ĻS (t ) ĻE (0) , (18.17)

and using this in eqn (18.16) produces the master equation:

t

ā‚ 1

ĻS (t) = ā’ 2 dt TrE {[HSE (t) , [HSE (t ) , ĻS (t ) ĻE (0)]]} . (18.18)

ā‚t 0

The double commutator in eqn (18.18) can be rewritten in a more convenient way

by exploiting the fact that typical interactions have the form

F (t) + Fā (t) .

HSE (t) = (18.19)

This in turn allows the double commutator to be written as

Ā¾ The master equation

1

C2 (t, t ) = [HSE (t) , [HSE (t ) , ĻS (t ) ĻE (0)]]

2

Fā (t) , G (t ) + HC ,

= {[F (t) , G (t )] + HC} + (18.20)

where

G (t ) = [F (t ) , ĻS (t ) ĻE (0)] . (18.21)

18.4 Examples of the master equation

In order to go on, it is necessary to assume an explicit form for HSE . For this purpose,

we will consider the two concrete examples that were studied in Chapter 14: the single

cavity mode and the two-level atom.

18.4.1 Single cavity mode

In the environment picture, the deļ¬nition (14.43) of systemā“reservoir interaction for

a single cavity mode becomes

v (ā„¦Ī½ ) aā (t) bĪ½ (t) ā’ HC .

HSE = i (18.22)

Ī½

Since a (t) and bĪ½ (t) are evaluated in the environment picture, they satisfy the Heisen-

berg equations

ā‚ 1

a (t) = [a (t) , HS (t)]

ā‚t i

1

= ā’iĻ0 a (t) + [a (t) , HS1 (t)] , (18.23)

i

and

ā‚

bĪ½ (t) = ā’iā„¦Ī½ bĪ½ (t) . (18.24)

ā‚t

By introducing the slowly-varying envelope operators a (t) = a (t) exp (iĻ0 t) and

bĪ½ (t) = bĪ½ (t) exp (iĻ0 t), we can express HSE (t) in the form (18.19), with

F (t) = ā’iĪ¾ ā (t) a (t) , (18.25)

where we recognize

v (ā„¦Ī½ ) bĪ½ (t0 ) eā’i(ā„¦Ī½ ā’Ļ0 )(tā’t0 )

Ī¾ (t) = (18.26)

Ī½

as the noise operator deļ¬ned in eqn (14.52).

The terms in [F (t) , G (t )] contain products of Ī¾ ā (t), Ī¾ ā (t ), and ĻE (0) in various

orders. When the partial trace over the environment states in eqn (18.18) is performed,

the cyclic invariance of the trace can be exploited to show that all terms are propor-

tional to

Ī¾ ā (t) Ī¾ ā (t ) E = TrE ĻE (0) Ī¾ ā (t) Ī¾ ā (t ) . (18.27)

Just as in Section 14.2, we will assume that ĻE (0) is diagonal in the reservoir

oscillator occupation numberā”this amounts to assuming that ĻE (0) is a stationary

Āæ

Examples of the master equation

distributionā”so that the correlation functions in eqn (18.27) all vanish. This assump-

tion is convenient, but it is not strictly necessary. A more general treatmentā”that

includes, for example, a reservoir described by a squeezed stateā”is given in Walls and

Milburn (1994, Sec. 6.1).

When ĻE (0) is stationary, [F (t) , G (t )] and its adjoint will not contribute to the

master equation. By contrast, the commutator Fā (t) , G (t ) is a sum of terms con-

taining products of Ī¾ (t), Ī¾ ā (t ), and ĻE (0) in various orders. In this case, the cyclic

invariance of the partial trace produces two kinds of terms, proportional respectively

to

Ī¾ ā (t) Ī¾ (t ) E = ncav ĪŗĪ“ (t ā’ t ) (18.28)

and

Ī¾ (t) Ī¾ ā (t ) = (ncav + 1) ĪŗĪ“ (t ā’ t ) , (18.29)

where ncav is the average number of reservoir oscillators at the cavity-mode frequency

Ļ0 .

The explicit expressions on the right sides of these equations come from eqns (14.74)

and (14.75), which were derived by using the Markov approximation. Thus the master

equation also depends on the Markov approximation, in particular on the assump-

tion that the envelope operator a (t) is essentially constant over the memory interval

(t ā’ Tmem /2, t + Tmem /2).

After evaluating the partial trace of the double commutator C2 (t, t )ā”see Exercise

18.2ā”the environment-picture form of the master equation for the ļ¬eld is found to be

ā‚ Īŗ

ĻS (t) = ā’ (ncav + 1) aā (t) a (t) ĻS (t) + ĻS (t) aā (t) a (t) ā’ 2a (t) ĻS (t) aā (t)

ā‚t 2

Īŗ

ā’ ncav a (t) aā (t) ĻS (t) + ĻS (t) a (t) aā (t) ā’ 2aā (t) ĻS (t) a (t) .

2

(18.30)

The slowly-varying envelope operators a (t) and aā (t) are paired in every term;

consequently, they can be replaced by the original environment-picture operators a (t)

and aā (t) without changing the form of the equation. The right side of the equation

of motion is therefore entirely expressed in terms of environment-picture operators, so

we can easily transform back to the SchrĀØdinger picture to ļ¬nd

o

ā‚

ĻS (t) = LS ĻS (t) + Ldis ĻS (t) . (18.31)

ā‚t

The Liouville operators LS ā”describing the free Hamiltonian evolution of the sam-

pleā”and Ldis ā”describing the dissipative eļ¬ects arising from coupling to the environ-

mentā”are respectively given by

1

Ļ0 aā a + HS1 (t) , ĻS (t)

LS ĻS (t) = (18.32)

i

and

Īŗ

Ldis ĻS (t) = ā’ (ncav + 1) aā , aĻS (t) + ĻS (t) aā , a

2 (18.33)

Īŗ

ā’ ncav a, aā ĻS (t) + ĻS (t) a, aā .

2

The master equation

The operators we are used to, such as the Hamiltonian or the creation and annihi-

lation operators, send one Hilbert-space vector to another. By contrast, the Liouville

operators send one operator to another operator. For this reason they are sometimes

called super operators.

A Thermal equilibrium again

In Exercise 14.2 it is demonstrated that the average photon number asymptotically

approaches the Planck distribution. With the aid of the master equation, we can study

this limit in more detail. In this case, HS1 (t) = 0, so we can expect the density operator

to be diagonal in photon number. The diagonal matrix elements of eqn (18.31) in the

number-state basis yield

dpn (t)

= ā’Īŗ {(ncav + 1) n + ncav (n + 1)} pn (t)

dt (18.34)

+ Īŗ (ncav + 1) (n + 1) pn+1 (t) + Īŗncav n pnā’1 (t) ,

where pn (t) = n |Ļ (t)| n . The ļ¬rst term on the right represents the rate of ļ¬‚ow

of probability from the n-photon state to all other states, while the second and third

terms represent the ļ¬‚ow of probability into the n-photon state from the (n + 1)-photon

state and the (n ā’ 1)-photon state respectively.

In order to study the approach to equilibrium, we write the equation as

dpn (t)

= Zn+1 (t) ā’ Zn (t) , (18.35)

dt

where

Zn (t) = nĪŗ {(ncav + 1) pn (t) ā’ ncav pnā’1 (t)} . (18.36)

The equilibrium condition is Zn+1 (ā) = Zn (ā), but this is the same as Zn (ā) = 0,

since Z0 (t) ā” 0. Thus equilibrium imposes the recursion relations

(ncav + 1) pn (ā) = ncav pnā’1 (ā) . (18.37)

This is an example of the principle of detailed balance; the rate of probability

ļ¬‚ow from the n-photon state to the (n ā’ 1)-photon state is the same as the rate of

probability ļ¬‚ow of the (n ā’ 1)-photon state to the n-photon state. The solution of this

recursion relation, subject to the normalization condition

ā

pn (ā) = 1 , (18.38)

n=0

is the Boseā“Einstein distribution

n

(ncav )

pn (ā) = . (18.39)

n+1

(ncav + 1)

Examples of the master equation

18.4.2 Two-level atom

For the two-level atom, the sampleā“reservoir interaction Hamiltonian HSE is given by

eqns (14.131)ā“(14.133). In this case, the operator F (t) in eqn (18.19) is the sum of two

terms: F (t) = Fsp (t) + Fpc (t), that are respectively given by

ā ā

Fsp (t) = i w21 bā (t) S 12 (t) = i w21 bā (t) Ļ ā’ (t) (18.40)

in in

and

ā ā

w11 cā (t) S 11 (t) + w22 cā (t) S 11 (t)

Fpc (t) = i 1,in 2,in

ā

ā 1 ā’ Ļ z (t) 1 + Ļ z (t)

w11 cā (t) w22 cā (t)

=i + .

1,in 2,in

2 2

(18.41)

The envelope operators Ļ ā’ (t) and Ļ z (t) are related to the environment-picture forms

by Ļ ā’ (t) = Ļā’ (t) exp (iĻ21 t) and Ļ z (t) = Ļz (t), while the operators bā (t) and cā (t)

in q,in

are the in-ļ¬elds deļ¬ned by eqns (14.146) and (14.147) respectively.

We will assume that the reservoirs are uncorrelated, i.e. ĻE (0) = Ļsp (0) Ļpc (0),

and that the individual reservoirs are stationary. These assumptions guarantee that

most of the possible terms in the double commutator will vanish when the partial trace

over the environment is carried out.

After performing the invigorating algebra suggested in Exercise 18.4.2, the surviv-

ing terms yield the SchrĀØdinger-picture master equation

o

ā‚

ĻS (t) = LS ĻS (t) + Ldis ĻS (t) , (18.42)

ā‚t

where the Hamiltonian part,

1 Ļ21

LS ĻS (t) = Ļz + HS1 (t) , ĻS (t) , (18.43)

i 2

includes HS1 (t). The dissipative part is

w21

Ldis ĻS (t) = ā’ (nsp + 1) {[Ļ+ , Ļā’ Ļ] + [ĻĻ+ , Ļā’ ]}

2

w21

ā’ nsp {[Ļā’ , Ļ+ Ļ] + [ĻĻā’ , Ļ+ ]}

2

wpc

ā’ [ĻĻz , Ļz ] , (18.44)

2

where nsp is the average number of reservoir excitations (photons) at the transition

frequency Ļ21 . The phase-changing rate in the last term is

2

1

wpc = (2npc,q + 1) wqq , (18.45)

2 q=1

where npc,q is the average number of excitations in the phase-perturbing reservoir

coupled to the atomic state |Īµq .

The master equation

18.5 Phase space methods

In Section 5.6.3 we have seen that the density operator for a single cavity mode can be

described in the Glauberā“Sudarshan P (Ī±) representation (5.165). As we will show be-

low, this representation provides a natural way to express the operator master equation

(18.31) as a diļ¬erential equation for the P (Ī±)-function. In single-mode applicationsā”

and also in some more complex situationsā”this equation is mathematically identical

to the Fokkerā“Planck equation studied in classical statistics (Risken, 1989, Sec. 4.7).

By deļ¬ning an atomic version of the P -functionā”as the Fourier transform of a

properly chosen quantum characteristic functionā”it is possible to apply the same

techniques to the master equation for atoms, but we will restrict ourselves to the

simpler case of a single mode of the radiation ļ¬eld. The application to atomic master

equations can be found, for example, in Haken (1984, Sec. IX.2) or Walls and Milburn

(1994, Chap. 13).

For the discussion of the master equation in terms of P (Ī±), it is better to use

the alternate convention in which P (Ī±) is regarded as a function, P (Ī±, Ī±ā— ), of the

independent variables Ī± and Ī±ā— . In this notation the P -representation is

d2 Ī±

|Ī± P (Ī±, Ī±ā— ; t) Ī±| .

ĻS (t) = (18.46)

Ļ

The function P (Ī±, Ī±ā— ; t) is real and satisļ¬es the normalization condition

d2 Ī±

P (Ī±, Ī±ā— ; t) = 1 , (18.47)

Ļ

but it cannot always be interpreted as a probability distribution. The trouble is that,

for nonclassical states, P (Ī±, Ī±ā— ; t) must take on negative values in some region of the

Ī±-plane.

18.5.1 The Fokkerā“Planck equation

In order to use the P -representation in the master equation, we must translate the

products of Fock space operators, e.g. a, aā , and Ļ, in the master equation into the

action of diļ¬erential operators on the c-number function P (Ī±, Ī±ā— ; t). For this purpose

it is useful to write the coherent state |Ī± as

2

|Ī± = eā’|Ī±| |Ī±; B ,

/2

(18.48)

where the Bargmann state |Ī±; B is

ā

Ī±n

ā |n .

|Ī±; B = (18.49)

n!

n=0

The virtue of the Bargmann states is that they are analytic functions of Ī±. More

precisely, for any ļ¬xed state |ĪØ the c-number function

ā

Ī±n

ā

ĪØ |Ī±; B = ĪØ |n (18.50)

n!

n=0

is analytic in Ī±. In the same sense, Ī±; B| is an analytic function of Ī±ā— , so it is inde-

pendent of Ī±.

Phase space methods

Since |Ī±; B is proportional to |Ī± , the action of a on the Bargmann states is just

a |Ī±; B = Ī± |Ī±; B . (18.51)

The action of aā is found by using eqn (18.49) to get

ā

Ī±n ā

ā

ā

a |Ī±; B = n + 1 |n + 1

n!

n=0

ā‚

|Ī±; B .

= (18.52)

ā‚Ī±

The adjoint of this rule is

ā‚

Ī±; B| a =

Ī±; B| . (18.53)

ā‚Ī±ā—

In the Bargmann notation, the P -representation of the density operator is

d2 Ī±

P (Ī±, Ī±ā— ; t) eā’|Ī±| |Ī±; B Ī±; B| .

2

ĻS (t) = (18.54)

Ļ

The rule (18.51) then gives

d2 Ī±

P (Ī±, Ī±ā— ; t) eā’|Ī±| Ī± |Ī±; B

2

aĻS (t) = Ī±; B|

Ļ

d2 Ī±

Ī±P (Ī±, Ī±ā— ; t) |Ī± Ī±| .

= (18.55)

Ļ

Applying the rule (18.52) yields

d2 Ī± ā‚

ā

P (Ī±, Ī±ā— ; t) eā’|Ī±|

2

|Ī±; B

a ĻS (t) = Ī±; B| , (18.56)

Ļ ā‚Ī±

but this is not expressed in terms of a diļ¬erential operator acting on P (Ī±, Ī±ā— ; t).

Integrating by parts on Ī± leads to the desired form:

d2 Ī± ā‚ ā—

aā ĻS (t) = ā’ P (Ī±, Ī±ā— ; t) eā’Ī±Ī± |Ī±; B Ī±; B|

Ļ ā‚Ī±

2

dĪ± ā‚

Ī±ā— ā’ P (Ī±, Ī±ā— ; t) |Ī± Ī±| .

= (18.57)

Ļ ā‚Ī±

This result depends on the fact that the normalization condition requires P (Ī±, Ī±ā— ; t)

to vanish as |Ī±| ā’ ā.

Combining eqns (18.55) and (18.57) with their adjoints gives us the translation

table

aĻS (t) ā” Ī±P (Ī±, Ī±ā— ; t) ĻS (t) a ā” (Ī± ā’ ā‚/ā‚Ī±ā— ) P (Ī±, Ī±ā— ; t)

(18.58)

aā ĻS (t) ā” (Ī±ā— ā’ ā‚/ā‚Ī±) P (Ī±, Ī±ā— ; t) ĻS (t) aā ā” Ī±ā— P (Ī±, Ī±ā— ; t) .

The master equation

Applying the rules in eqn (18.58) to eqn (18.32)ā”for the simple case with HS1 =

0ā”and to eqn (18.33) yields the translations

1 ā‚ ā‚ā—

Ļ0 aā a, ĻS (t) ā” iĻ0 Ī± P (Ī±, Ī±ā— ; t)

LS ĻS (t) = Ī±ā’ (18.59)

ā—

i ā‚Ī± ā‚Ī±

and

Ī“ ā‚ ā‚

[Ī±P (Ī±, Ī±ā— ; t)] + [Ī±ā— P (Ī±, Ī±ā— ; t)]

Ldis ĻS (t) ā” ā—

2 ā‚Ī± ā‚Ī±

ā‚2

P (Ī±, Ī±ā— ; t) ,

+ Ī“ncav (18.60)

ā—

ā‚Ī±ā‚Ī±

for the Hamiltonian and dissipative Liouville operators respectively.

In the course of carrying out these calculations, it is easy to get confused about

the correct order of operations. The reason is that products like aā aĻā”with operators

standing on the left of Ļā”and products like Ļaā aā”with operators standing to the

right of Ļā”are both translated into diļ¬erential operators acting from the left on the

function P (Ī±, Ī±ā— ; t).

Studying a simple example, e.g. carrying out a direct derivation of both aā aĻ and

Ļaā a, shows that the order of the diļ¬erential operators is reversed from the order of the

Fock space operators when the Fock space operators stand to the right of Ļ. Another

way of saying this is that one should work from the inside to the outside; the ļ¬rst

diļ¬erential operator acting on P corresponds to the Hilbert space operator closest to

Ļ. This rule gives the correct result for Fock space operators to the left or to the right

of Ļ.

The master equation for an otherwise unperturbed cavity mode is, therefore, rep-

resented by

ā‚ ā‚ ā‚

P (Ī±, Ī±ā— ; t) = [Z (Ī±) P (Ī±, Ī±ā— ; t)] + [Z ā— (Ī±) P (Ī±, Ī±ā— ; t)]

ā—

ā‚t ā‚Ī± ā‚Ī±

2

ā‚

P (Ī±, Ī±ā— ; t) ,

+ Ī“ncav (18.61)

ā—

ā‚Ī±ā‚Ī±

where

Ī“

+ iĻ0 Ī± . (18.62)

Z (Ī±) =

2

We can achieve a ļ¬rmer grip on the meaning of this equation by changing variables

from (Ī±, Ī±ā— ) to u = (u1 , u2 ), where u1 = Re Ī± and u2 = Im Ī±. In these variables,

P (Ī±, Ī±ā— ; t) = P (u; t), and the Ī±-derivative is

ā‚ 1 ā‚ ā‚

ā’i

= . (18.63)

ā‚Ī± 2 ā‚u1 ā‚u2

In this notation, the master equation takes the form of a classical Fokkerā“Planck

equation in two dimensions:

ā‚ D0 2

P (u; t) = ā’ā Ā· [F (u) P (u; t)] + ā P (u; t) , (18.64)

ā‚t 2

Phase space methods

where

D0 = Ī“ncav /2 (18.65)

is the diļ¬usion constant, and we have introduced the following shorthand notation:

ā‚X1 ā‚X2

āĀ·X = + ,

ā‚u1 ā‚u2

Ī“ Ī“

F (u) = (ā’ Re Z, ā’ Im Z) = Ļ0 u 2 ā’ u1 , ā’Ļ0 u1 ā’ u2 , (18.66)

2 2

2 2

ā‚ ā‚

ā=2

+ .

ā‚u1 ā‚u2

The ļ¬rst- and second-order diļ¬erential operators in eqn (18.64) are respectively called

the drift term and the diļ¬usion term.

A Classical Langevin equations

The Fokkerā“Planck equation (18.64) is a special case of a general family of equations

of the form

N N

ā‚ 1 ā‚ ā‚

P (u; t) = ā’ā Ā· [F (u, t) P (u; t)] + Dmn (u, t) P (u; t) , (18.67)

ā‚t 2 m=1 n=1 ā‚um ā‚un

where u = (u1 , . . . , uN ), F = (F1 , . . . , FN ), Dmn is the diļ¬usion matrix, and

X Ā· Y = X1 Y1 + Ā· Ā· Ā· + XN YN . (18.68)

For the two-component case, given by eqn (18.64), the diļ¬usion matrix is diagonal,

Dmn = D0 Ī“mn , so it has a single eigenvalue D0 > 0. The corresponding condition in

the general N -component case is that all eigenvalues of the diļ¬usion matrix D are

positive, i.e. D is a positive-deļ¬nite matrix. In this case D has a square root matrix

B that satisļ¬es D = BB T .

When D is positive deļ¬nite, then eqn (18.67) is exactly equivalent to the set of

classical Langevin equations (Gardiner, 1985, Sec. 4.3.5)

N

dun (t)

= Cn (u, t) + Bnm (u, t) wm (t) , (18.69)

dt m=1

where the un s are stochastic variables and the wm s are independent white noise sources

of unit strength, i.e. wm (t) = 0 and

wm (t) wn (t ) = Ī“mn Ī“ (t ā’ t ) . (18.70)

In particular, the Langevin equations corresponding to eqn (18.64) are

du (t)

= F (u) + D0 w (t) . (18.71)

dt

Ā¼ The master equation

These real Langevin equations are essential for numerical simulations, but for ana-

lytical work it is useful to write them in complex form. This is done by combining

Ī± = u1 + iu2 with eqns (18.62) and (18.66) to get

dĪ± (t)

= ā’Z (Ī± (t)) + 2D0 Ī· (t) , (18.72)

dt

where Ī± (t) is a complex stochastic variable, and

1

Ī· (t) = ā [w1 (t) + iw2 (t)] (18.73)

2

is a complex white noise source satisfying

Ī· ā— (t) Ī· (t ) = Ī“ (t ā’ t ) .

Ī· (t) = 0 , Ī· (t) Ī· (t ) = 0 , (18.74)

The equivalence of the Fokkerā“Planck equation and the classical Langevin equa-

tions for a positive-deļ¬nite diļ¬usion matrix is important in practice, since the nu-

merical simulation of the Langevin equations is usually much easier than the direct

numerical solution of the Fokkerā“Planck equation itself.

For some problemsā”e.g. when the appropriate reservoir is described by a squeezed

stateā”the diļ¬usion matrix derived from the Glauberā“Sudarshan P -function is not

positive deļ¬nite, so the Fokkerā“Planck equation is not equivalent to a set of classical

Langevin equations. In such cases, another representation of the density operator may

be more useful (Walls and Milburn, 1994, Sec. 6.3.1).

18.5.2 Applications of the Fokkerā“Planck equation

A Coherent states are robust

Let us begin with a simple example in which ncav = 0, so that the diļ¬usion term in eqn

(18.64) vanishes. If we interpret the reservoir oscillators as phonons in the cavity walls,

then this model describes the idealized situation of material walls at absolute zero.

Alternatively, the reservoir could be deļ¬ned by other modes of the electromagnetic

ļ¬eld, into which the particular mode of interest is scattered by a gas of nonresonant

atoms. In this case, it is natural to assume that the initial reservoir state is the vacuum.

In other words, the universe is big and dark and cold.

The terms remaining after setting ncav = 0 can be rearranged to produce

ā‚

P (u; t) + F (u) Ā· āP (u; t) = Ī“P (u; t) . (18.75)

ā‚t

Let us study the evolution of a ļ¬eld state initially deļ¬ned by P (u; 0) = P0 (u). The

general technique for solving linear, ļ¬rst-order, partial diļ¬erential equations like eqn

(18.75) is the method of characteristics (Zauderer, 1983, Sec. 2.2), but we will employ

an equivalent method that is well suited to the problem at hand.

The ļ¬rst step is to introduce an integrating factor, by setting

P (u; t) = P (u; t) eĪ“t , (18.76)

Ā½

Phase space methods

so that

ā‚

P (u; t) + F (u) Ā· āP (u; t) = 0 . (18.77)

ā‚t

The second step is to transform to new variables (u , t ) by

u = V (u, t) , t = t , (18.78)

where we require u = u at t = 0, and also assume that the function V (u, t) is linear

in u, i.e.

2

Vn (u, t) = Gnm (t) um . (18.79)

m=1

The reason for trying a linear transformation is that the coeļ¬cient vector,

ā’Ī“/2 Ļ0

, (18.80)

Fj (u) = Wjl ul , where W =

ā’Ļ0 ā’Ī“/2

l

is itself linear in u.

The chain rule calculation in Exercise 18.5 yields expressions for the operators ā‚/ā‚t

and ā‚/ā‚ul in terms of the new variables, so that eqn (18.77) becomes

ā‚ dG (t) ā‚

P (u ; t ) + G (t) W + ul P (u ; t) = 0 . (18.81)

ā‚t dt ā‚un

nl

n l

Choosing the matrix G (t) to satisfy

dG (t)

+ G (t) W = 0 (18.82)

dt

ensures that the coeļ¬cient of ā‚/ā‚un vanishes identically in u, and this in turn simpliļ¬es

the equation for P (u ; t ) to

ā‚

P (u ; t ) = 0 . (18.83)

ā‚t

Thus P (u ; t ) = P (u ; 0), but t = 0 is the same as t = 0, so P (u ; t ) = P0 (u ).

In this way the solution to the original problem is found to be

P (u, t) = eĪ“t P0 (V (u, t)) , (18.84)

and the only remaining problem is to evaluate V (u, t). This is most easily done by

writing G (t) as

b (t) b2 (t)

G (t) = 1 , (18.85)

c1 (t) c2 (t)

and substituting this form into eqn (18.82). This yields simple diļ¬erential equations

for the vectors b (t) and c (t), with initial conditions b (0) = (1, 0) and c (0) = (0, 1).

The solution of these auxiliary equations gives

Ā¾ The master equation

cos (Ļ0 t) ā’ sin (Ļ0 t)

G (t) = eĪ“t/2 R (t) = eĪ“t/2 , (18.86)

sin (Ļ0 t) cos (Ļ0 t)

so that

P (u; t) = P0 eĪ“t/2 R (t) u eĪ“t . (18.87)

Thus, in the absence of the diļ¬usive term, the shape of the distribution is un-

changed; the argument u is simply scaled by exp (Ī“t/2) and rotated by the angle Ļ0 t.

In the complex-Ī± description the solution is given by

P (Ī±, Ī±ā— ; t) = P0 e(Ī“/2+iĻ0 )t Ī±, e(Ī“/2ā’iĻ0 )t Ī±ā— eĪ“t . (18.88)

This result is particularly interesting if the ļ¬eld is initially in a coherent state |Ī±0 ,

i.e. the initial P -function is P0 (u) = Ī“2 (u ā’ u0 ). In this case, the standard properties

of the delta function lead to

P (u; t) = eĪ“t Ī“2 eĪ“t/2 R (t) u ā’ u0 = Ī“2 (u ā’ u (t)) , (18.89)

where

ā’1

u (t) = eā’Ī“t/2 [R (t)] u0 . (18.90)

The conclusion is that a coherent state interacting with a zero-temperature reservoir

will remain a coherent state, with a decaying amplitude

Ī± (t) = Ī±0 eā’Ī“t/2 eā’iĻ0 t . (18.91)

Consequently, the time-dependent joint variance of aā and a vanishes at all times:

V aā , a ; t = Ī± (t) aā a Ī± (t) ā’ Ī± (t) aā Ī± (t) Ī± (t) |a| Ī± (t) = 0 . (18.92)

In other words, coherent states are robust: scattering and absorption will not destroy

the coherence properties, as long as the environment is at zero temperature.

This apparently satisfactory result raises several puzzling questions. The ļ¬rst is that

the initially pure state remains pure, even after interaction with the environment. This

seems to contradict the general conclusion, established in Section 18.1, that interaction

with a reservoir inevitably produces a mixed state for the sample.

The resolution of this discrepancy is that the general argument is true for the exact

theory, while the master equation is derived with the aid of the approximationā”see

eqn (18.17)ā”that back-action of the sample on the reservoir can be neglected. This

means that the robustness property of the coherent states is only as strong as the

approximations leading to the master equation. Furthermore, we will see in Section

18.6 that coherent states are the only pure states that can take advantage of this

loophole in the general argument of Section 18.1.

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

violate the ļ¬‚uctuation dissipation theorem. The ļ¬eld suļ¬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.

Āæ

Phase space methods

B Thermalization of an initial coherent state

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

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

from u to u is a rotation followed by scaling with exp (Ī“t/2). The operator ā2 on

the right side of eqn (18.64) is invariant under rotations, so ā2 = eĪ“t ā 2 and the

Fokkerā“Planck equation becomes

ā‚ D0 Ī“t 2

e ā P (u ; t ) ,

P (u ; t ) = (18.93)

ā‚t 2

which is the diļ¬usion equation with a time-dependent diļ¬usion coeļ¬cient. The Fourier

transform,

d2 u P (u ; t ) eā’iq Ā·u ,

P (q ; t ) = (18.94)

then satisļ¬es the ordinary diļ¬erential equation

d D0 Ī“t 2

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

dt 2

which has the solution

D0 Ī“t

e ā’ 1 q 2 P 0 (q ) .

P (q ; t ) = exp (18.96)

2Ī“

For the initial coherent state, P0 (u) = Ī“2 (u ā’ u0 ), one ļ¬nds

P 0 (q ) = exp [ā’iq Ā· u0 ] , (18.97)

and the inverse transform can be explicitly evaluated to yield

(u ā’ u (t))2

1

exp ā’ , (18.98)

P (u; t) =

Ļw (t) w (t)

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

w (t) = ncav 1 ā’ eā’Ī“t . (18.99)

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

For long times (t

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

1/Ī“, we see that w (t) ā¼ ncav Ī“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 Fokkerā“Planck equation for a more realistic treatment.

The master equation

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

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

tonian

HS (t) = Ļ0 aā a ā’ W eā’iā„¦t aā ā’ W ā— eiā„¦t a , (18.100)

that leads to the Liouville operator

1 1

Ļ0 aā a, ĻS (t) ā’ W eā’iā„¦t aā + W ā— eiā„¦t a, ĻS (t) .

LS ĻS (t) = (18.101)

i i

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

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

is replaced by

Ī“

+ iĻ0 Ī± ā’ iW eā’iā„¦t .

Z (Ī±) = (18.102)

2

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

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

into the general result (18.72) yields

dĪ± (t) Ī“

+ iĻ0 Ī± (t) + iW eā’iā„¦t +

=ā’ 2D0 Ī· (t) , (18.103)

dt 2

which has the solution

Ī± (t) = Ī± (0) eā’(iĻ0 +Ī“/2)t + Ī±coh (t) + 2D0 Ļ‘ (t) , (18.104)

where

iW

eā’(iĻ0 +Ī“/2)t e(iā+Ī“/2)t ā’ 1

Ī±coh (t) = (18.105)

iā + Ī“/2

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

t

dt1 eā’(iĻ0 +Ī“/2)(tā’t1 ) Ī· (t1 ) .

Ļ‘ (t) = (18.106)

0

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

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

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

f (Ī± (0)) = (18.107)

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

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

distribution, but these methods fail when P0 (Ī±, Ī±ā— ) 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 (Ī± (0) , Ī±ā— (0)) = Ī“ (Ī± (0)). The initial value Ī± (0) and the noise term Ļ‘ (t)

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

Ī± (t) = Ī±coh (t) . (18.108)

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

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

iW

eā’iā„¦t .

Ī± (t) = (18.109)

iā + Ī“/2

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

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

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

true if the sample were described by the coherent state Ļcoh (t) = |Ī±coh (t) Ī±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 ļ¬rst investigate the coherence properties of the state by using the explicit

solution (18.104) to get

2

ā’(iĻ0 +Ī“/2)t

2

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

= Ī±2 (t) + 2D0 Ļ‘2 (t) . (18.110)

coh

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

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

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

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

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

Ī±2 (t) = Ī±2 (t) = Ī± (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, |Ī± (t)| , which represents the

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

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

2 2 2

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

Combining eqns (18.106) and (18.74) leads to

t t

dt2 eā’(ā’iĻ0 +Ī“/2)(tā’t1 ) eā’(iĻ0 +Ī“/2)(tā’t2 ) Ī· ā— (t1 ) Ī· (t2 )

2

|Ļ‘ (t)| = dt1

0 0

1 ā’ eā’Ī“t

t

ā’Ī“(tā’t1 )

dt1 e = , (18.113)

=

Ī“

0

The master equation

so that

2D0

1 ā’ eā’Ī“t = ncav 1 ā’ eā’Ī“t ,

|Ī± (t)|2 ā’ |Ī±coh (t)|2 = (18.114)

Ī“

where we used eqn (18.65) to get the ļ¬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 equationā—

18.6

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

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

(b) The positivity of ĻS is conserved, i.e. ĪØ |ĻS (t)| ĪØ 0 for all states |ĪØ 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 satisļ¬es (a), (b), and (c)

is given by

ā‚ĻS

= LS ĻS + Ldis ĻS , (18.115)

ā‚t

where

1

LS ĻS = [HS (t) , ĻS ] (18.116)

i

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

Lindblad form (Lindblad, 1976)

K

1 ā ā ā

Ldis ĻS = ā’ Ck Ck ĻS + ĻS Ck Ck ā’ 2Ck ĻS Ck . (18.117)

2

k=1

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

ļ¬nite or inļ¬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):

Ī“ncav aā .

C1 = Ī“ (ncav + 1)a , C2 = (18.118)

A slightly longer calculationā”see Exercise 18.6ā”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 deļ¬ning the purity of the state ĻS as P (t) =

Tr Ļ2 (t) 1. One can show from eqn (18.115) that the time derivative of the purity

S

is

K

d ā ā

P (t) = ā’2 Tr ĻS (t) Ck Ck ĻS (t) ā’ ĻS (t) Ck ĻS (t) Ck . (18.119)

dt

k=1

At ļ¬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 ĻS (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 satisļ¬ed. In order to check this, we

evaluate eqn (18.119) for an initially pure state ĻS (0) = |ĪØ ĪØ|, to ļ¬nd

K

d ā ā

P (t) = ā’2 ĪØ Ck Ck ĪØ ā’ ĪØ Ck ĪØ ĪØ |Ck | ĪØ

dt t=0 k=1

K

ā

= ā’2 ĪØ Ī“Ck Ī“Ck ĪØ 0, (18.120)

k=1

where

Ī“Ck = Ck ā’ ĪØ |Ck | ĪØ . (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

ā

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

so that eqn (18.120) becomes

d ā

P (t) = ā’2Ī“ ĪØ (a ā’ Ī±) (a ā’ Ī±) ĪØ 0, (18.122)

dt t=0

where Ī± = ĪØ |a| ĪØ . The inequality can only be saturated if |ĪØ satisļ¬es

a |ĪØ = Ī± |ĪØ , (18.123)

i.e. when |ĪØ 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 Dehmeltā™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 ā” 2 and 3 ā” 1, indicated

by wavy arrows, and a dipole-forbidden tran-

(Allowed)

(Forbidden)

sition 2 ā’ 1, indicated by the light dashed

arrow. The heavy double arrows denote strong

incoherent couplings on the 3 ā” 1 and 3 ā’ 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 ā’ 1 and 3 ā’ 2, have Einstein-A coeļ¬cients Ī“31 and Ī“32 respectively, so

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

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

compared to both Ī“31 and Ī“32 .

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

ent radiation source, at the frequency Ļ31 , 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 Ļ (Ļ31 ), where Ļ (Ļ31 ) is the energy density of

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

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

radiation at frequency Ļ31 . Another way of saying this is that the ion scatters the pump

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

angles to the direction of the pump radiationā”eļ¬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 Ļ32 Ļ31 . From eqn (4.162) we know

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

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

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

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

the 3 ā” 2 transition such that

Ī“32 W32 = B32 Ļ (Ļ32 ) Ī“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 ā’ 1 occurs very slowly compared to 3 ā’ 1 and 3 ā’ 2, so level 2 eļ¬ectively

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

ļ¬‚uorescent signal will turn oļ¬, 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 correctā”but not nearly as excitingā”to refer to this phenomenon as ā˜inter-

rupted ļ¬‚uorescenceā™.

Quantum jumps

During the dark interval the ion is said to be shelved and |Īµ2 is called a shelving

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

and the state |Īµ2 is long-lived compared to |Īµ3 , i.e. when

W31 Ī“3 Ī“2 . (18.125)

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

be a linear combination of |Īµ1 and |Īµ3 ; in other words, |ĪØion 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 ļ¬‚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, āEL / = 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 ā’ 1S1/2 . This adds unwanted complexity.

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

anism for the transition 2S1/2 ā’ 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 ļ¬rst step is to apply a weak DC electric ļ¬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

Ā¬ Ā¬

Ā« Ā«

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

Stark-mixed states |Īµ2 and |Īµ2 . The wavy arrows indicate dipole-allowed decays, while the

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

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

Ā¼ The master equation

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

compared to the Lamb shift, i.e.

2S1/2 |E 0 Ā· d| 2P1/2 āEL . (18.126)

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

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

states,

|Īµ2 = CS 2S1/2 + CP 2P1/2 , (18.127)

|Īµ2 = CS 2S1/2 + CP 2P1/2 , (18.128)

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

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

ā’d Ā· E 0 ā”shows that |CS | |CP |, i.e. |Īµ2 is dominantly like 2S1/2 , while |Īµ2 is

mainly like 2P1/2 . The states |Īµ1 and |Īµ3 of the simple model pictured in Fig. 18.1

are identiļ¬ed with 1S1/2 and 2P3/2 respectively.

Since neither |Īµ2 nor |Īµ2 has deļ¬nite parity, the dipole selection rules now allow

single-photon transitions 2 ā’ 1 and 2 ā’ 1. The rate for 2 ā’ 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 Ī“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 eļ¬ectively eliminated by applying a microwave ļ¬eld

resonant with the 3 ā’ 2 transition, but not with the 3 ā’ 2 transition. The strength

of this ļ¬eld can be adjusted so that the stimulated emission rate for 3 ā’ 2 is large

compared to the spontaneous rates for 3 ā’ 2 and 3 ā’ 2, but small compared to

the stimulated and spontaneous rates for the 3 ā’ 1 transition. These settings ensure

that the population of |Īµ2 will remain small at all times and that |Īµ2 is an eļ¬ective

shelving state.

The main practical diļ¬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 diļ¬culty is to use the radiation from a synchrotron

light source.

The transition 3 ā’ 2 is primarily due to the microwave-frequency transition

2P3/2 ā’ 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 justiļ¬ed by the rough estimate

3

A (Lyman alpha) Ī½ (Lyman alpha)

ā¼ ā¼ 1016 (18.129)

A 2P3/2 ā’ 2S1/2 Ī½ 2P3/2 ā’ 2S1/2

(Bethe and Salpeter, 1977), which uses the values 2.47 Ć— 1015 Hz and 11 GHz for the

Lyman-alpha transition and the 2P3/2 ā’ 2S1/2 microwave transition in hydrogen

respectively.

The combination of the low rate for the 3 ā’ 2 transition and the long lifetime of

the shelving level 2 will permit easy observation of interrupted resonance ļ¬‚uorescence

at the helium ion Lyman-alpha line, i.e. quantum jumps.

Ā½

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 Ć— 107 s, i.e.

of the order of a year. The lifetime of the same transition for a hydrogenic ion scales

as Z ā’4 , so for Z = 2 the microwave transition lifetime is 1.5 Ć— 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 ļ¬eld driving the 3 ā” 2 transition to

bring this lifetime within the limits of the experimentalistā™s patience.

C Rate equation analysis

The assumption that the driving ļ¬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

= ā’ (Ī“3 + W31 + W32 ) P3 + W31 P1 + W32 P2 , (18.130)

dt

dP2

= ā’ (Ī“2 + W32 ) P2 + (Ī“32 + W32 ) P3 , (18.131)

dt

dP1

= ā’W31 P1 + (W31 + Ī“31 ) P3 + Ī“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)ā“(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 ā’ P1 . The new form of the

rate equations is

d 1 1

D31 = ā’ 2W31 + Ī“31 + Ī“32 + W32 D31

dt 2 2

1 1

ā’ Ī“31 + Ī“32 + W32 P31 + (W32 ā’ Ī“2 ) P2 , (18.134)

2 2

d 1 1

P31 = ā’ (Ī“32 + W32 ) P31 ā’ (Ī“32 + W32 ) D31 + (Ī“2 + W32 ) P2 , (18.135)

dt 2 2

d 1 1

P2 = ā’ (Ī“2 + W32 ) P2 + (Ī“2 + W32 ) P31 + (Ī“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 ā’ Ī“2

Ī“31 + 1 Ī“32 + 1 W32

=ā’ 2 2

D31 P31 + P2 . (18.137)

2W31 + Ī“31 + 1 Ī“32 + 1 W32 2W31 + Ī“31 + 1 Ī“32 + 1 W32

2 2 2 2

The coeļ¬cients of the probabilities P31 and P2 are very small, so we can set D31 0

in the rest of the calculation.

Ā¾ The master equation

With this approximation, the remaining rate equations are

dP2

= ā’Ron P2 + Roļ¬ P31 (18.138)

dt

and

dP31

= Ron P2 ā’ Roļ¬ P31 , (18.139)

dt

where Ron = Ī“2 + W32 is the rate at which the ļ¬‚uorescence turns on, and Roļ¬ =

(Ī“32 + W32 ) /2 is the rate at which ļ¬‚uorescence turns oļ¬. Solving eqns (18.138) and

(18.139) for P31 (t) yields

Ron

P31 (t) = P31 (0) eā’(Ron +Roff )t + 1 ā’ eā’(Ron +Roff )t . (18.140)

Ron + Roļ¬

The ļ¬‚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 ā . (18.141)

Ron + Roļ¬

This result is completely at odds with the ļ¬‚ickering on-and-oļ¬ 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 ļ¬‚uorescence experiments

dealt with dilute atomic gases, and the total ļ¬‚uorescence signal would be correctly

described by eqn (18.140). In this case, the on-and-oļ¬ behavior of the individual atoms

would be washed out by averaging over the random ļ¬‚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 propertiesā”such as the distribution of waiting times

between jumpsā”are beyond their reach. Thus any improvement must involve putting

in some additional information; that is, reducing the size of the ensemble.

The ļ¬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 |Īµ2 during the interval (t, t + T ). The number of tran-

sitions deļ¬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 |Īµ2 after n transitions between

H13 and |Īµ2 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 inļ¬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 reļ¬ned quantum-jump theories to be developed later on.

Āæ

Quantum jumps

D A stochastic model

We will now consider a simple on-and-oļ¬ 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 ļ¬rst assume that the ļ¬‚uorescent intensity can only have the values I = 0 (oļ¬)

or I = IF (on). If the signal is on at time t, then the probability that it will turn oļ¬

in the interval (t, t + āt) is āpoļ¬ = Roļ¬ āt. Conversely, if the signal is oļ¬ at time t,

then the probability that it will turn on in the interval (t, t + āt) is āpon = Ron āt.

For suļ¬ciently small āt, we can assume that only one of these events occurs.

The ļ¬‚uorescent intensities In at the discrete times tn = (n ā’ 1) āt 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 āpon < r or In+1 = IF if āpon > r .

(18.142)

For In = IF choose a random number r in (0, 1) ;

then set In+1 = IF if āpoļ¬ < r or In+1 = 0 if āpoļ¬ > 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 ļ¬‚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-oļ¬ behavior for a single ion, but the smooth

ļ¬‚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 deļ¬ne the average ļ¬‚uorescent intensity at time tn by

M

1

In,av = In,j , (18.143)

M j=1

where In,j is the ļ¬‚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 diļ¬erent plot for each run with the same input para-

meters. The ā˜typicalā™ plot shown here was chosen to illustrate the eļ¬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 ļ¬‚uorescent intensity I/IF versus time (in units of the radiative lifetime

1/Ī“b of the shelving state). In these units, Ron = 1.6, Roļ¬ = 0.3, and āt = 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

ļ¬‚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 experimentā”shown in Fig. 18.5ā”

with Fig. 18.1. Fortunately, the complications involved in the real experiment do not

change the essential nature of the eļ¬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

Ā¬ Ā¬ Ā¬

Ā« Ā« Ā«

the text are |Īµ1 = Ā¬62 S1/2 , |Īµ2 = Ā¬62 P3/2 , and |Īµ3 = Ā¬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 ā” 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 ļ¬‚uorescence from the 62 P1/2 -level showing the

quantum jumps after the hollow cathode lamp is turned on. The atom is deļ¬nitely known to

be in the shelf level during the low ļ¬‚uorescence periods. (Reproduced from Nagourney et al.

(1986).)

Quantum jumps and the master equationā—

18.7.2

Many features of quantum-jump experiments are well describedā”at least semi-quantit-

ativelyā”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

deļ¬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 eļ¬ort required to incorporate coherence eļ¬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 diļ¬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 diļ¬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 ļ¬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 diļ¬usion.

The master equation (18.115) is a diļ¬erential equation describing the time evolu-

tion of the sample density operator. Except in highly idealized situationsā”for which

analytical solutions are knownā”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 inļ¬nite dimensional, so the annihilation operator a is represented

by an inļ¬nite matrix. A direct numerical attack would therefore require replacing

HS by a ļ¬nite-dimensional space, e.g. the subspace spanned by the number states

|0 , . . . , |M ā’ 1 . This would entail representing the creation and annihilation opera-

tors and the density operator by M Ć— M matrices.

In some situations, such as those discussed in Section 18.5.2, an alternative ap-

proach is to replace the inļ¬nite-dimensional space HS by the two-dimensional quan-

tum phase space, and to useā”for a restricted class of problemsā”the Fokkerā“Planck

equation (18.61) or the equivalent classical Langevin equation (18.72). In general, this

method will fail if the diļ¬usion matrix D is not positive deļ¬nite.

The master equation for an atom can also be represented by a Fokkerā“Planck

equation on a ļ¬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-deļ¬nite

diļ¬usion kernel, that apply to the radiation ļ¬eld. In many cases the center-of-mass

motion of the atom can be neglectedā”or at least treated classicallyā”so the sample

Hilbert space is ļ¬nite dimensional. In this situation the master equation for a two-level

atom is simply a diļ¬erential equation for a 2 Ć— 2 hermitian matrix. This is equivalent

to a set of four coupled ordinary diļ¬erential equations, so it is not computationally

onerous.

Unfortunately, in the real world of experimental physics, atoms often have more

ńņš. 21 |