∇2 = ’4πδ (r ’ r0 ) .

|r ’ r0 |

(c) Combine the last two results to derive the Coulomb potential term in eqn

(4.31).

4.2 Maxwell™s equations from the Heisenberg equations of motion

Derive Maxwell™s equations and Lorentz equations of motion as given by eqns (4.33)“

(4.37), and eqn (4.42), using Heisenberg™s equations of motions and the relevant equal-

time commutators.

Spatial inversion and time reversal—

4.3

(1) Use eqn (4.55) to evaluate UP |n for a general number state, and explain how to

extend this to all states of the ¬eld.

(2) Verify eqn (4.61) and ¬ll in the details needed to get eqn (4.64).

(3) Evaluate ΛT |n for a general number state, and explain how to extend this to all

states of the ¬eld. Watch out for antilinearity.

4.4 Stationary density operators

Use eqns (3.83), (4.67), and U (’t) = U † (t), together with cyclic invariance of the

trace, to derive eqns (4.69) and (4.71).

4.5 Spin-¬‚ip transitions

The neutron is a spin-1/2 particle with zero charge, but it has a nonvanishing magnetic

moment MN = ’ |gN | µN σ, where gN is the neutron gyromagnetic ratio, µN is the

nuclear magneton, and σ = (σx , σy , σz ) is the vector of Pauli matrices. Since the

neutron is a massive particle, it is a good approximation to treat its center-of-mass

½ Interaction of light with matter

motion classically. All of the following calculations can, therefore, be done assuming

that the neutron is at rest at the origin.

(1) In the presence of a static, uniform, classical magnetic ¬eld B0 the Schr¨dinger-

o

picture Hamiltonian”neglecting the radiation ¬eld”is H0 = ’MN · B0 . Take the

z-axis along B0 , and solve the time-independent Schr¨dinger equation, H0 |ψ =

o

µ |ψ , for the ground state |µ1 , the excited state |µ2 , and the corresponding en-

ergies µ1 and µ2 .

(2) Include the e¬ects of the radiation ¬eld by using the Hamiltonian H = H0 + Hint,

where Hint = ’MN · B and B is given by eqn (3.70), evaluated at r = 0.

(a) Evaluate the interaction-picture operators aks (t) and σ± (t) in terms of the

Schr¨dinger-picture operators aks and σ± = (σx ± iσy ) /2 (see Appendix

o

C.3.1). Use the results to ¬nd the time dependence of the Cartesian com-

ponents σx (t), σy (t), σz (t).

(b) Find the condition on the ¬eld strength |B0 | that guarantees that the zero-

order energy splitting is large compared to the strength of Hint , i.e.

µ2 ’ µ1 | µ1 , 1ks |Hint | µ2 , 0 | ,

where |µ1 , 1ks = |µ1 |1ks , |µ2 , 0 = |µ2 |0 , and |1ks = a† |0 . Explain the

ks

physical signi¬cance of this condition.

(c) Using Section 4.9.3 as a guide, calculate the spontaneous emission rate (Ein-

stein A coe¬cient) for a spin-¬‚ip transition. Look up the numerical values

of |gN | and µN and use them to estimate the transition rate for magnetic

¬eld strengths comparable to those at the surface of a neutron star, i.e.

|B0 | ∼ 1012 G.

4.6 The quantum top

Replace the unperturbed Hamiltonian in Exercise 4.5 by H0 = ’MN · B0 (t), where

B0 (t) changes direction as a function of time. Use this Hamiltonian to derive the

Heisenberg equations of motion for σ (t) and show that they can be written in the

same form as the equations for a precessing classical top.

4.7 Transition probabilities for a neutron in combined static and

radio-frequency ¬elds—

Solve the Schr¨dinger equation for a neutron in a combined static and radio-frequency

o

magnetic ¬eld. A static ¬eld of strength B0 is applied along the z-axis, and a circularly-

polarized, radio-frequency ¬eld of classical amplitude B1 and frequency ω is applied

in the (x, y)-plane, so that the total Hamiltonian is H = H0 + Hint , where

H0 = ’Mz B0 ,

Hint = ’MxB1 cos ωt + My B1 sin ωt ,

Mx = 1 µσx , My = 1 µσy , Mz = 1 µσz , µ is the magnetic moment of the neutron,

2 2 2

and the σs are Pauli matrices. Show that the probability for a spin ¬‚ip of the neutron

initially prepared (at t = 0) in the ms = + 1 state to the ms = ’ 1 state is given by

2 2

½

Exercises

1

P 1 ’’ 1 (t) = sin2 ˜ sin2 at ,

2

2 2

where

2

ω1

sin2 ˜ = ,

(ω0 ’ ω)2 + ω1

2

2

(ω0 ’ ω) + ω1 ,

2

a=

ω0 = µB0 / , and ω1 = µB1 / . Interpret this result geometrically (Rabi et al., 1954).

5

Coherent states

In the preceding chapters, we have frequently called upon the correspondence principle

to justify various conjectures, but we have not carefully investigated the behavior of

quantum states in the correspondence-principle limit. The di¬culties arising in this

investigation appear in the simplest case of the excitation of a single cavity mode

E κ (r). In classical electromagnetic theory”as described in Section 2.1”the state of

a single mode is completely described by the two real numbers (Qκ0 , Pκ0 ) specifying

the initial displacement and momentum of the corresponding radiation oscillator. The

subsequent motion of the oscillator is determined by Hamilton™s equations of motion.

The set of classical ¬elds representing excitation of the mode κ is therefore represented

by the two-dimensional phase space {(Qκ , Pκ )}.

In striking contrast, the quantum states for a single mode belong to the in¬nite-

dimensional Hilbert space spanned by the family of number states, {|n , n = 0, 1, . . .}.

In order for a state |Ψ to possess a meaningful correspondence-principle limit, each

member of the in¬nite set, {cn = n |Ψ , n = 0, 1, . . .}, of expansion coe¬cients must

be expressible as a function of the two classical degrees of freedom (Qκ0 , Pκ0 ). This

observation makes it clear that the number-state basis is not well suited to demonstrat-

ing the correspondence-principle limit. In addition to this fundamental issue, there are

many applications for which a description resembling the classical phase space would

be an advantage.

These considerations suggest that we should search for quantum states of light that

are quasiclassical; that is, they approach the classical description as closely as possi-

ble. To this end, we ¬rst review the solution of the corresponding problem in ordinary

quantum mechanics, and then apply the lessons learnt there to the electromagnetic

¬eld. After establishing the basic form of the quasiclassical states, we will investigate

possible physical sources for them and the experimental evidence for their existence.

The ¬nal sections contain a review of the mathematical properties of quasiclassical

states, and their use as a basis for representations of general quantum states.

5.1 Quasiclassical states for radiation oscillators

In order to simplify the following discussion, we will at ¬rst only consider situations

in which a single mode of the electromagnetic ¬eld is excited. For example, excitation

of the mode E κ (r) in an ideal cavity corresponds to the classical ¬elds

½

Quasiclassical states for radiation oscillators

1

A (r, t) = √ Qκ (t) E κ (r) ,

0

(5.1)

1

E (r, t) = ’ √ Pκ (t) E κ (r) .

0

5.1.1 The mechanical oscillator

In Section 2.1 we guessed the form of the quantum theory of radiation by using the

mathematical identity between a radiation oscillator and a mechanical oscillator of

unit mass. The real Q and P variables of the classical oscillator can be simultaneously

speci¬ed; therefore, the trajectory (Q (t) , P (t)) of the oscillator is completely described

by the time-dependent, complex amplitude

ωQ (t) + iP (t)

√

A (t) = , (5.2)

2ω

where the is introduced for dimensional convenience only. Hamilton™s equations of

motion for the real variables Q and P are equivalent to the complex equation of motion

™

A = ’iωA , (5.3)

with the general solution given by the phasor (a complex number of ¬xed modulus)

A (t) = ± exp (’iωt) . (5.4)

The initial complex amplitude of the oscillator is related to ± by

ωQ0 + iP0

√

A (t = 0) = = ±, (5.5)

2ω

and the conserved classical energy is

1 22

ω Q0 + P0 = ω±— ± .

Ecl = (5.6)

2

Taking the real and imaginary parts of A (t), as given in eqn (5.4), shows that the

solution traces out an ellipse in the (Q, P ) phase space. An equivalent representation

is the circle traced out by the tip of the phasor A (t) in the complex (Re A, Im A)

space.

For the quantum oscillator, the classical amplitude A (0) and the energy ω |±|2

are respectively replaced by the lowering operator

ωq + ip

a= √ (5.7)

2ω

and the Hamiltonian operator Hosc = ωa† a. The Heisenberg equation of motion for

a(t),

da i

= ’ [a, Hosc ] = ’iωa , (5.8)

dt

has the same form as the classical equation of motion (5.3).

½¼ Coherent states

We can now use an argument from quantum mechanics (Cohen-Tannoudji et al.,

1977a, Chap. V, Complement G) to construct the quasiclassical state. According to

the correspondence principle, the classical quantities ± and Ecl must be identi¬ed with

the expectation values of the corresponding operators, so the quasiclassical state |φ

corresponding to the classical value ± should satisfy φ |a| φ = ± and φ Hosc φ =

Ecl = ω |±|2 . Inserting Hosc = ωa† a into the latter condition and using the former

2

condition to evaluate |±| produces

φ a † a φ = φ a† φ φ |a| φ . (5.9)

The joint variance of two operators X and Y , de¬ned by

V (X, Y ) = (X ’ X ) (Y ’ Y ) = XY ’ X Y, (5.10)

reduces to the ordinary variance V (X) for X = Y . In this language, the meaning of

eqn (5.9) is that the joint variance of a and a† vanishes,

V a† , a = 0 , (5.11)

i.e. the operators a and a† are statistically independent for a quasiclassical state. In

its present form it is not obvious that V a† , a refers to measurable quantities, but

this concern can be addressed by using eqn (5.7) to get the equivalent form

ω 1 1

V a† , a =

2 2

(q ’ q ) (p ’ p ) ’ .

+ (5.12)

2 2ω 2

The condition (5.11) is the fundamental property de¬ning quasiclassical states, and

it determines |φ up to a phase factor. To see this, we de¬ne a new operator b = a ’ ±

and a new state |χ = b |φ , to get

χ| χ = φ b† b φ = V a† , a = 0 . (5.13)

The squared norm χ |χ only vanishes if |χ = 0; consequently, a |φ = ± |φ . Thus

the quasiclassical state |φ is an eigenstate of the lowering operator a with eigenvalue

±. For this reason it is customary to rename |φ as |± , so that

a |± = ± |± . (5.14)

For non-hermitian operators, there is no general theorem guaranteeing the existence

of eigenstates, so we need to ¬nd an explicit solution of eqn (5.14). In this section,

we will do this in the usual coordinate representation, in order to gain an intuitive

understanding of the physical signi¬cance of |± . In the following section, we will ¬nd

an equivalent form by using the number-state basis. This is useful for understanding

the statistical properties of |± .

The coordinate-space wave function for |± is φ± (Q) = Q| ± , where q |Q =

Q |Q . In this representation, the action of q is qφ± (Q) = Qφ± (Q), and the action of

½½

Quasiclassical states for radiation oscillators

the momentum operator is pφ± (Q) = ’i (d/dQ) φ± (Q). After inserting this into eqn

(5.7), the eigenvalue problem (5.14) is represented by the di¬erential equation

1 d

√ ωQ + φ± (Q) = ±φ± (Q) , (5.15)

dQ

2ω

which has the normalizable solution

2

(Q ’ Q0 )

1/4

ω P0 Q

exp ’

φ± (Q) = exp i (5.16)

2

π 4∆q0

for any value of the complex parameter ±. The parameters Q0 and P0 are given by

√

Q0 = 2 /ω Re ±, P0 = 2 ω Im ±, and the width of the Gaussian is ∆q0 = /2ω.

We have chosen the prefactor so that φ± (Q) is normalized to unity. For Q0 = P0 =

0, φ0 (Q) is the ground-state wave function of the oscillator; therefore, the general

quasiclassical state, φ± (Q), represents the ground state of an oscillator which has been

displaced from the origin of phase space to the point (Q0 , P0 ). For the Q dependence

this is shown explicitly by the probability density |φ± (Q)|2 , which is a Gaussian in

Q centered on Q0 . An alternative representation using the momentum-space wave

function, φ± (P ) = P | ± , can be derived in the same way”or obtained from φ± (Q)

by Fourier transform”with the result

2

(P ’ P0 ) Q0 P

’1/4

exp ’ exp ’i

φ± (P ) = (π ω) , (5.17)

4∆p20

ω/2. The product ∆p0 ∆q0 = /2, so |± is a minimum-uncertainty

where ∆p0 =

state; it is the closest we can come to the classical description. The special values

∆q0 = /2ω and ∆p0 = ω/2 de¬ne the standard quantum limit for the

harmonic oscillator.

5.1.2 The radiation oscillator

Applying these results to the radiation oscillator for a particular mode E κ involves a

change of terminology and, more importantly, a change in physical interpretation. For

the radiation oscillator corresponding to the mode E κ , the de¬ning equation (5.14) for

a quasiclassical state is replaced by

aκ |±κ = δκ κ ±κ |±κ ; (5.18)

in other words, the quasiclassical state for this mode is the vacuum state for all other

modes. This is possible because the annihilation operators for di¬erent modes commute

with each other. A simple argument using eqn (5.18) shows that the averages of all

normal-ordered products completely factorize:

m

a† ±κ = (±— ) (±κ )

n m n

±κ (aκ )

κ κ

m

±κ a† ±κ

n

( ±κ |aκ | ±κ ) ;

= (5.19)

κ

consequently, |±κ is called a coherent state. The de¬nition (5.18) shows that |±κ

belongs to the single-mode subspace Hκ ‚ HF that is spanned by the number states

for the mode E κ .

½¾ Coherent states

The new physical interpretation is clearest for the radiation modes of a physical

cavity. In the momentum-space representation, the operator pκ is just multiplication

by the eigenvalue Pκ , and the expansion (2.99) shows that the electric ¬eld oper-

√

ator is a function of the pκ s, so that E (r) φ± (Pκ ) = Eκ V E κ (r) φ± (Pκ ) , where

√

Eκ = Pκ / √ is the electric ¬eld strength associated with Pκ . The dimension-

0V

less function V E κ (r) is of order unity and describes the shape of the mode func-

tion. The corresponding result in the coordinate representation is B (r) φ± (Qκ ) =

√ √

Bκ V Bκ (r) φ± (Qκ ) with Bκ = kκ Qκ / 0 V = µ0 /V ωκ Qκ . Eliminating Pκ in

favor of Eκ allows the Gaussian factor in φ± (P ) to be expressed as

(Eκ ’ µκ )2

0V 2

exp ’ (Eκ ’ µκ ) = exp ’ , (5.20)

4e2

2 ωκ κ

where eκ is the vacuum ¬‚uctuation strength de¬ned by eqn (2.188). Thus a coherent

state displays a Gaussian probability density in the electric ¬eld amplitude Eκ with

average µκ , and variance V (Eκ ) = 2e2 . Similarly the coordinate-space wave function

κ

is a Gaussian in Bκ with average βκ and variance 2b2 . The classical limit corresponds

κ

to |Eκ | eκ and |Bκ | bκ , which are both guaranteed by |±κ | 1. As an example,

15 ’1

(»κ ≈ 2 µm) and V = 1 cm , then the vacuum ¬‚uctuation

3

consider ωκ = 10 s

strength for the electric ¬eld is eκ 0.08 V/m.

The fact that ±κ is a phasor provides the useful pictorial representation shown in

Fig. 5.1. This is equivalent to a plot in the phase plane (Qκ , Pκ ). The result (5.17) for

the wave function and the phase plot Fig. 5.1 are expressed in terms of the excitation

of a single radiation oscillator in a physical cavity, but the idea of coherent states is not

restricted to this case. The annihilation operator a can refer to a cavity mode (aκ ), a

(box-quantized) plane wave (aks ), or a general wave packet operator (a [w]), as de¬ned

in Section 3.5.2, depending on the physical situation under study. In the interests of

simplicity, we will initially consider situations in which only one annihilation operator a

(one electromagnetic degree of freedom) is involved. This is su¬cient for a large variety

Im (±)

±

±0

Fig. 5.1 The coherent state (displaced ground

state) |±0 is pictured as an arrow joining the

origin to the point ±0 in the complex plane.

The quantum uncertainties of the ground state

(at the origin) and the displaced ground state Re (±)

are each represented by an error circle (quan-

tum fuzzball ).

½¿

Sources of coherent states

of applications, but the physical justi¬cation for isolating the single-mode subspace

associated with a is that coupling between modes is weak. This fact should always be

kept in mind, since a more complete calculation may involve taking the weak coupling

into account, e.g. when considering dissipative or nonlinear e¬ects.

5.1.3 Coherent states in the number-state basis

We now consider a single mode and represent |± by the number-state expansion

∞

|± = bn |n . (5.21)

n=0

According to eqn (2.78) the eigenvalue equation (5.14) can then be written as

∞ ∞

√

nbn |n ’ 1 = ± bn |n . (5.22)

n=0 n=0

Equating the coe¬cients of the number states yields the recursion relation, bn+1 =

√

√

±/ n + 1 bn , which has the solution bn = b0 ±n / n!. Thus each coe¬cient bn is a

function of the complex parameter ±, in agreement with the discussion at the beginning

of the chapter. The vacuum coe¬cient b0 is chosen to get a normalized state, with the

result ∞

±n

’|±|2 /2

√ |n .

|± = e (5.23)

n!

n=0

This construction works for any complex number ±, so the spectrum of the operator a

is the entire complex plane. A similar calculation for a† fails to ¬nd any normalizable