√ √

and

κ2 b2,in (ω) + κ1 b1,in (ω)

a (ω) = . (14.121)

κC /2 ’ i (ω ’ ω0 )

The frequency-domain version of the scattering equation (14.109) for b1,out (ω),

where b1,out (ω) = b1,out (ω ’ ω0 ), combines with the explicit solution (14.121) to yield

the input“output equation

√

[(κ2 ’ κ1 ) /2 ’ i (ω ’ ω0 )] b1,in (ω) ’ κ1 κ2 b2,in (ω)

b1,out (ω) = . (14.122)

κC /2 ’ i (ω ’ ω0 )

For far o¬-resonance radiation, i.e. |ω ’ ω0 | κC /2, this relation reduces to

b1,out (ω) ≈ b1,in (ω) , (14.123)

which corresponds to complete re¬‚ection of the radiation incident on M1. For a sym-

metrical resonator, i.e. κ1 = κ2 = κC /2, the input“output relation simpli¬es to

¾ Quantum noise and dissipation

i (ω ’ ω0 ) b1,in (ω) + (κC /2) b2,in (ω)

b1,out (ω) = , (14.124)

i (ω ’ ω0 ) ’ κC /2

and for nearly resonant radiation, ω ≈ ω0 , this becomes

(κC /2) b2,in (ω)

b1,out (ω) = . (14.125)

i (ω ’ ω0 ) ’ κC /2

In this limit, the output ¬eld from mirror M1 is simply proportional to the input ¬eld

at mirror M2, i.e. there is essentially no re¬‚ection of radiation incident on mirror M2.

In this situation the cavity is called a Lorentzian ¬lter, since the output intensity,

2

(κC /2)

b† b† (t0 ) b2,ω (t0 ) ,

1,out (ω) b1,out (ω) = (14.126)

2,ω

2 2

(κC /2) + (ω ’ ω0 )

has a typical Lorentzian line shape.

14.4 Noise and dissipation for atoms

In Section 11.3.3 we obtained a dissipative form of the Bloch equation for a two-level

atom by adding phenomenological damping terms to the quantum Liouville equation

for the atomic density operator. The Liouville equation is de¬ned in the Schr¨dinger

o

picture, or sometimes in the interaction picture; consequently, the Bloch equation does

not immediately ¬t into the Heisenberg-picture formulation of the sample“reservoir

model employed above. In order to make the connection, we ¬rst recall that an N -

level atom is completely described by the transition operators, Sqp = |µq µp |, de¬ned

in Section 11.1.4. In particular, the matrix elements ρpq (t) = µp |ρ (t)| µq of the

density operator are given by

ρpq (t) = Tr ρ (t) Sqp . (14.127)

The trace is invariant under unitary transformations, so this result can equally well

be written as

ρpq (t) = Tr ρSqp (t) , (14.128)

where ρ and Sqp (t) are both expressed in the Heisenberg picture. Since the Heisenberg-

picture density operator ρ is time independent, the Bloch equation for the matrix

elements of the density operator is an immediate consequence of the Heisenberg equa-

tions of motion for the transition operators. For this reason, we will sometimes use the

name operator Bloch equation for these particular Heisenberg equations.

14.4.1 Two-level atoms

In order to avoid unnecessary complications, we will restrict the detailed discussion to

the simplest case of two-level atoms. With these results in hand, the generalization to

¿

Noise and dissipation for atoms

N -level atoms is straightforward. For a sample consisting of a single two-level atom,

the sample Hamiltonian is HS = HS0 + HS1 (t), where

ω21

[S22 ’ S11 ] .

HS0 = (14.129)

2

The terms in the Heisenberg equations of motion contributed by HS1 (t) play no role

in the following discussion, so we will omit them from the intermediate calculations

and restore them at the end to get the ¬nal form of the Langevin equations.

A Noise reservoirs

There are now two forms of dissipation to be considered: spontaneous emission (sp)

and phase-changing perturbations (pc). We already have the complete theory for

spontaneous emission, but in the present context it is more instructive to use the

schematic approach of Section 14.1.2. The creation and annihilation operators for the

reservoir excitations (photons) that are emitted and absorbed in the 2 ” 1 transition

are denoted by b† and b„¦ . The second form of dissipation is associated with the decay

„¦

of the atomic dipole, due to perturbations that do not cause real transitions between

the two levels. In the simplest case, the atom is excited from an initial state to a virtual

intermediate state and then returned to the original state. In a vapor, this e¬ect arises

primarily from collisions with other atoms. In a solid, phase-changing perturbations

are often caused by local ¬eld ¬‚uctuations. The phase-changing perturbations of the

two levels may arise from di¬erent mechanisms, so we need a reservoir for each level,

with creation and annihilation operators c† and cq„¦ (q = 1, 2).

q„¦

The environment Hamiltonian is therefore

∞ ∞

2

d„¦ d„¦

„¦b† b„¦ + „¦c† cq„¦ ,

HE = (14.130)

„¦ q„¦

2π 2π

0 0

q=1

and the sample“environment interaction Hamiltonian HSE is

HSE = Hsp + Hpc , (14.131)

where Hsp and Hpc are responsible for spontaneous emission and phase-changing per-

turbations respectively. The spontaneous emission Hamiltonian,

∞

D („¦)

v („¦) b† S12 ’ S21 b„¦ ,

Hsp = i d„¦ (14.132)

„¦

2π

0

is modeled directly on the RWA Hamiltonian of eqn (11.25), with the coupling constant

v („¦) playing the role of the dipole matrix element.

The simplest phase-changing perturbation is a second-order process in which the

atom starts and ends in the same state. The transition from an initial state |µq to

an intermediate state |µp is represented by the operator Spq , and the return to the

original state is described by Sqp ; consequently, the complete transition is described

by the product Sqp Spq = Sqq . Since there is no overall change in energy, the resonance

Quantum noise and dissipation

for this transition occurs at zero frequency. We model the phase-changing mechanism

by coupling the atom to two reservoirs according to

∞

2

D („¦)

uq („¦) c† Sqq ’ Sqq cq„¦ .

Hpc = i d„¦ (14.133)

q„¦

2π

0

q=1

Coupling to the zero-frequency resonance is enforced by assuming that the coupling

constant uq („¦) is proportional to the cut-o¬ function centered at zero frequency.

B Langevin equations

Since the sample and environment operators commute at equal times, the terms in the

total Hamiltonian can be written in any desired order. We chose to put them in normal

order with respect to the environment operators, so that the Heisenberg equations

d 1

Sqp (t) = [Sqp (t) , HE + HS + Hsp + Hpc ] (14.134)

dt i

are also normally ordered. The resonance frequencies for the interaction of the sample

with the spontaneous-emission and phase-changing reservoirs are ω = ω21 and ω = 0

respectively; therefore, we express eqn (14.134) in terms of the envelope operators

S 12 (t) = S12 (t) eiω21 t , S qq (t) = Sqq (t) ,

(14.135)

b„¦ (t) = b„¦ (t) eiω21 t , cq„¦ (t) = cq„¦ (t) ,

to ¬nd

d † †

S 12 (t) = S 22 (t) ’ S 11 (t) β (t) + γ2 (t) ’ γ1 (t) S 12 (t)

dt

’ S 12 (t) {γ2 (t) ’ γ1 (t)} , (14.136)

d

S 22 (t) = ’β † (t) S 12 (t) ’ S 21 (t) β (t) , (14.137)

dt

d

b„¦ (t) = ’i („¦ ’ ω21 ) b„¦ (t) + 2πD („¦) v („¦) S 12 (t) , (14.138)

dt

d

cq„¦ (t) = ’i„¦cq„¦ (t) + 2πD („¦) uq („¦) S qq (t) , (14.139)

dt

where

∞

D („¦)

β (t) = d„¦ v („¦) b„¦ (t) (14.140)

2π

0

and

∞

D („¦)

γq (t) = d„¦ uq („¦) cq„¦ (t) (q = 1, 2) . (14.141)

2π

0

The equation for S 11 (t) has been omitted, by virtue of the identity S 11 (t)+S 22 (t) = 1.

The Langevin equations for the atomic transition operators are derived by an ar-

gument similar to the one employed in Section 14.2.1. The formal solutions of eqns

Noise and dissipation for atoms

(14.138) and (14.139) for the reservoir operators are combined with the Markov con-

ditions

2πD („¦) |v („¦)|2 w21 = 2πD (ω21 ) |v (ω21 )|2 (14.142)

and

2 2

2πD („¦) |uq („¦)| wqq = 2πD (0) |uq (0)| , (14.143)

to get

√ w21

β (t) = w21 bin (t) + S 12 (t) (14.144)

2

and

√ wqq

γq (t) = wqq cq,in (t) + S qq (t) (q = 1, 2) . (14.145)

2

The in-¬elds for the reservoirs are given by

∞

d„¦

b„¦ (t0 ) e’i(„¦’ω21 )(t’t0 )

bin (t) = (14.146)

2π

0

and ∞

d„¦

cq„¦ (t0 ) e’i„¦(t’t0 ) .

cq,in (t) = (14.147)

2π

0

Substituting these results into eqns (14.136) and (14.137) yields the Langevin equations

for the transition operators:

d 1

S12 (t) = [iω12 ’ “12 ] S12 (t) + [S12 (t) , HS1 (t)] + ξ12 (t) , (14.148)

dt i

d 1

S22 (t) = ’w21 S22 (t) + [S22 (t) , HS1 (t)] + ξ22 (t) , (14.149)

dt i

d 1

S11 (t) = w21 S22 (t) + [S11 (t) , HS1 (t)] + ξ11 (t) , (14.150)

dt i

where w21 is the spontaneous decay rate for the 2 ’ 1 transition, w11 and w22 are the

rates of the phase-changing perturbations, and

1

“12 = (w21 + w22 + w11 ) (14.151)

2

is the dephasing rate for the atomic dipole. We have restored the HS1 (t)-terms and

also imposed ξ11 (t) = ’ξ22 (t) in accord with the conservation of population.

The operators ξ12 (t) and ξ22 (t) represent multiplicative noise, since they involve

products of sample and reservoir operators. This raises a new di¬culty, because there is

no general argument proving that multiplicative noise operators are delta correlated.

Even in the special cases for which a proof can be given”e.g. those considered in

Exercise 14.2”the calculations are quite involved. In this situation, the only general

procedure available is to include the delta-correlation assumption as part of the Markov

approximation. For the problem at hand the ansatz is

†

δ (t ’ t ) .

ξqp (t) ξq p (t ) = Cqp,q (14.152)

p

The coe¬cients Cqp,q p can be evaluated, at least partially, by the general methods

described in Section 14.6.

Quantum noise and dissipation

We will see, in the following section, that the use of atomic transition operators

is a great advantage for the generalization from two-level to N -level atoms, but for

applications to two-level atoms themselves, it is often easier to work in terms of the

familiar Pauli matrices. The relations

1 1

(1 + σz ) , S11 = (1 ’ σz ) , S12 = σ’ , S21 = σ+

S22 = (14.153)

2 2

lead to the equivalent Langevin equations

d 1

σ ’ (t) = ’“12 σ ’ (t) + [σ ’ (t) , HS1 (t)] + ξ’ (t) , (14.154)

dt i

d 1

σ z (t) = ’w21 [1 + σ z (t)] + [σ z (t) , HS1 (t)] + ξz (t) , (14.155)

dt i

where ξ’ (t) = ξ12 (t) and ξz (t) = 2ξ22 (t).

N -level atoms

14.4.2

The derivation of the Langevin equations for atoms with N levels could be carried

out by applying the approach followed for the two-level atom, but this would require

assigning a reservoir for every real decay and another reservoir for each level subjected

to phase-changing perturbations. One can escape burial under this avalanche of reser-

voirs by paying careful attention to the structure of eqns (14.148)“(14.151) for the

two-level atom. If we assume that the dissipative e¬ects involve transitions between

pairs of atomic levels or phase-changing perturbations of single levels, then a little

thought shows that the N -level Langevin equations must have the general form

d 1

Sqp (t) = (iωqp ’ “qp ) Sqp (t) + [Sqp (t) , HS1 (t)] + ξqp (t) for q = p , (14.156)

dt i

d 1

wpq Spp (t) ’

Sqq (t) = wqp Sqq (t) + [Sqq (t) , HS1 (t)] + ξqq (t) . (14.157)

dt i

p p

The envelope operators are de¬ned by generalizing eqn (14.135) to

Sqp (t) = S qp (t) eiωqp t ei[θq (t)’θp (t)] , (14.158)

where each θq (t) is a real function. The reason for including the θq s in this de¬nition is

that”in favorable cases”they can be chosen to eliminate explicit time dependencies

due to S qp (t) , HS1 (t) . Substituting eqn (14.158) into eqns (14.156) and (14.157)

leads to the envelope equations

d 1

™ ™

S qp (t) = ’i θq ’ θp ’ “qp S qp (t) + S qp (t) , HS1 (t) + ξqp (t) for q = p ,

dt i

(14.159)

d 1

wpq S pp (t) ’

S qq (t) = wqp S qq (t) + S qq (t) , HS1 (t) + ξqq (t) , (14.160)

dt i

p p

where

Incoherent pumping

§

⎪ transition rate for p ’ q if µp > µq ,

⎨

= 0 if µp < µq ,

wpq (14.161)

⎪

©

the phase-changing rate for the qth level when q = p .

1

For q = p, “qp = (wqr + wpr ) is the dephasing rate for S qp (t) . (14.162)

2 r

Strictly speaking, one should also de¬ne envelope noise operators,

ξ qp (t) = e’iωqp t e’i[θq (t)’θp (t)] ξqp (t) , (14.163)

but the assumption that the original operators ξqp (t) are delta correlated implies that

the envelope noise operators would have the same correlation functions. Since the

correlation functions are all that matters for noise operators, it is safe to ignore the

distinction between ξqp (t) and ξ qp (t).

14.5 Incoherent pumping

Incoherent pumping processes”which raise rather than lower the energy of an atom”

are used to produce population inversion; consequently, they play a central role in

laser physics. As we have seen in Section 14.4, the interaction of an atom with a

short-memory reservoir is necessarily dissipative. This raises the following question:

Can incoherent pumping be described by a reservoir model? This feat has been ac-

complished, but only at the cost of introducing an unphysical reservoir (Gardiner,

1991, Sec. 7.2.1). The idea is to describe pumping by coupling the atom to a reservoir

composed of oscillators with an inverted energy spectrum, µ„¦ = ’ „¦, as in Exercise

14.5. Emitting an excitation into this reservoir lowers the reservoir energy and there-

fore raises the energy of the atom. We have previously mentioned the formal use of

unphysical negative-energy modes in the discussion of the input“output method in

Section 14.3, but in that situation the probability for exciting the unphysical modes

is negligible. This cannot be the case for the inverted-oscillator reservoir; otherwise,

there would be no pumping. Since this model violates Einstein™s rule, we must accept

some added complexity.

The interaction between an atom and a classical ¬eld, with rapid ¬‚uctuations in

phase, provides a physically acceptable model for incoherent pumping. Unfortunately,

building such a model for the simplest case of a two-level atom is pointless, since the

discussion in Section 11.3.3 shows that no pumping scheme for a two-level atom can

produce an inverted population. We will, therefore, grudgingly admit that real atoms

have more than two levels and add a third. The added complexity will be o¬set by

ignoring phase-changing perturbations.

The sample is a collection of three-level atoms, with the energy-level diagram shown

in Fig. 14.2. The 3 ” 1 transition is driven by a strong, classical pump ¬eld

E P (t) = eP EP 0 ei‘P (t) e’iωP t , (14.164)

where ωP ≈ ω31 and ‘P (t) is a rapidly ¬‚uctuating phase. Since there is no coupling

between the atoms, we can restrict our attention to a single atom located at r = 0.

For this reduced sample, the interaction Hamiltonian is HS1 = VP (t) + HS1 , where

Quantum noise and dissipation

!

Fig. 14.2 A three-level atom with dipole al-

M!

lowed transitions 1 ” 3 and 1 ” 2. The spon-

taneous emission rates are w31 and w21 respec-

„¦2

tively. The 1 ” 3 transition is also driven by

M!

M

a classical ¬eld with Rabi frequency „¦P . A

non-radiative decay 3 ’ 2, with rate w32 , is

indicated by the dashed arrow. The wavy ar-

rows denote the spontaneous emissions.

„¦P ei‘P (t) e’iωP t eiω31 t ei[θ3 (t)’θ1 (t)] S 31 + HC ,

VP (t) = (14.165)

S 31 is the envelope operator de¬ned by eqn (14.158), „¦P is the Rabi frequency associ-

ated with the constant amplitude EP 0 , and HS1 includes any other interactions with

external ¬elds as well as any sample“sample interactions. The remaining interaction

term HS1 in¬‚uences some of the choices to be made, but the terms contributed by

HS1 to the equations of motion play no direct role in the following argument. We

will therefore omit them from the intermediate steps and restore them at the end. In

addition to the spontaneous emissions, 2 ’ 1 and 3 ’ 1, we assume that there is a

non-radiative decay channel: 3 ’ 2.

The Langevin equations for this problem are derived in Exercise 14.6 by dropping

the phase-changing terms from the (N = 3)-case of eqns (14.159) and (14.160). It is

also useful to impose θ3 (t) ’ θ1 (t) = ∆P t ’ ‘P (t)”where ∆P = ωP ’ ω31 is the pump

detuning”in order to eliminate the explicit time dependence of VP (t). The remaining

phase di¬erences θ1 ’ θ2 and θ2 ’ θ3 are related by

θ2 ’ θ3 = (θ1 ’ θ3 ) ’ (θ1 ’ θ2 )

= ∆P t ’ ‘P (t) ’ (θ1 ’ θ2 ) , (14.166)

so we can only impose one more condition on the phases. The choice of this condition

depends on HS1 . In the problem at hand, we have assumed that the transition 2 ” 1