2

Chapter 2
Quantisation of the Electromagnetic Field












Abstract The study of the quantum features of light requires the quantisation of the electromagnetic field. In this chapter we quantise the field and introduce three possible sets of basis states, namely, the Fock or number states, the coherent states and the squeezed states. The properties of these states are discussed. The phase operator and the associated phase states are also introduced.



2.1 Field Quantisation


The major emphasis of this text is concerned with the uniquely quantum-mechanical properties of the electromagnetic field, which are not present in a classical treatment. As such we shall begin immediately by quantizing the electromagnetic field. We shall make use of an expansion of the vector potential for the electromagnetic field in terms of cavity modes. The problem then reduces to the quantization of the harmonic oscillator corresponding to each individual cavity mode.
We shall also introduce states of the electromagnetic field appropriate to the de-
scription of optical fields. The first set of states we introduce are the number states corresponding to having a definite number of photons in the field. It turns out that it is extremely difficult to create experimentally a number state of the field, though fields containing a very small number of photons have been generated. A more typ- ical optical field will involve a superposition of number states. One such field is the coherent state of the field which has the minimum uncertainty in amplitude and phase allowed by the uncertainty principle, and hence is the closest possible quan- tum mechanical state to a classical field. It also possesses a high degree of optical coherence as will be discussed in Chap. 3, hence the name coherent state. The coher- ent state plays a fundamental role in quantum optics and has a practical significance in that a highly stabilized laser operating well above threshold generates a coher- ent state.
A rather more exotic set of states of the electromagnetic field are the squeezed states. These are also minimum-uncertainty states but unlike the coherent states the


7


quantum noise is not uniformly distributed in phase. Squeezed states may have less noise in one quadrature than the vacuum. As a consequence the noise in the other quadrature is increased. We introduce the basic properties of squeezed states in this chapter. In Chap. 8 we describe ways to generate squeezed states and their applications.
While states of definite photon number are readily defined as eigenstates of the
number operator a corresponding description of states of definite phase is more diffi- cult. This is due to the problems involved in constructing a Hermitian phase operator to describe a bounded physical quantity like phase. How this problem may be re- solved together with the properties of phase states is discussed in the final section of this chapter.
A convenient starting point for the quantisation of the electromagnetic field is
the classical field equations. The free electromagnetic field obeys the source free Maxwell equations.

? • B = 0 , (2.1a)
? B
? × E = ? ? t , (2.1b)
? • D = 0 , (2.1c)
? D

? × H =

? t , (2.1d)


where B = ?0H, D = ?0E, ?0 and ?0 being the magnetic permeability and electric permittivity of free space, and ?0?0 = c?2. Maxwell’s equations are gauge invariant
when no sources are present. A convenient choice of gauge for problems in quan- tum optics is the Coulomb gauge. In the Coulomb gauge both B and E may be determined from a vector potential A(r, t) as follows
B = ? × A , (2.2a)
E = ? A , (2.2b)
? t


with the Coulomb gauge condition



? • A = 0 . (2.3)


Substituting (2.2a) into (2.1d) we find that A(r, t) satisfies the wave equation


?2A(r,t)= 1 ?
c2

2A(r,t)
. (2.4)
? t2


We separate the vector potential into two complex terms

A (r,t)= A(+) (r,t)+ A(?) (r,t) , (2.5)

where A(+)(r, t) contains all amplitudes which vary as e?i?t for ? > 0 and
A(?)(r, t) contains all amplitudes which vary as ei?t and A(?) = (A(+))?.

2.1 Field Quantisation 9
It is more convenient to deal with a discrete set of variables rather than the whole continuum. We shall therefore describe the field restricted to a certain volume of space and expand the vector potential in terms of a discrete set of orthogonal mode functions:
A(+) (r,t)= ?ckuk (r)e?i?kt , (2.6)
k

where the Fourier coefficients ck are constant for a free field. The set of vector mode functions uk(r) which correspond to the frequency ?k will satisfy the wave equation

. 2 .

?2 + ?k
c2


uk (r)= 0 (2.7)


provided the volume contains no refracting material. The mode functions are also required to satisfy the transversality condition,
? • uk (r)= 0 . (2.8)

The mode functions form a complete orthonormal set
¸
k (r) uk? (r)dr = ?kk? . (2.9)
V

The mode functions depend on the boundary conditions of the physical volume under consideration, e.g., periodic boundary conditions corresponding to travelling- wave modes or conditions appropriate to reflecting walls which lead to standing waves. For example, the plane wave mode functions appropriate to a cubical volume of side L may be written as
uk (r)= L?3/2eˆ(? ) exp (ik • r) (2.10) where eˆ(? ) is the unit polarization vector. The mode index k describes several dis-
crete variables, the polarisation index (? = 1, 2) and the three Cartesian components
of the propagation vector k. Each component of the wave vector k takes the values



kx =

2?nx
L , ky =

2?ny
L , kz =

2?nz
L , nx, ny, nz = 0, ±1, ±2,... (2.11)

The polarization vector eˆ(? ) is required to be perpendicular to k by the transversality condition (2.8).
The vector potential may now be written in the form


. k .1/2 .


i?kt † ?

i?kt .

A (r,t)= ?
k

.
2?k?0

akuk(r)e?

+ ak uk (r) e

. (2.12)


The corresponding form for the electric field is

. k?k .1/2 .


i?kt


† ? i?kt .

E(r,t)= i ?
k

2?0

akuk (r) e?

? ak uk (r)e

. (2.13)


The normalization factors have been chosen such that the amplitudes ak and a† are
dimensionless.
In classical electromagnetic theory these Fourier amplitudes are complex num- bers. Quantisation of the electromagnetic field is accomplished by choosing ak and k to be mutually adjoint operators. Since photons are bosons the appropriate com- mutation relations to choose for the operators ak and a† are the boson commutation

relations

. † † .

. † .

[ak, ak? ]= ak , ak?

= 0,

ak, ak?

= ?kk? . (2.14)


The dynamical behaviour of the electric-field amplitudes may then be described by an ensemble of independent harmonic oscillators obeying the above commutation relations. The quantum states of each mode may now be discussed independently of
one another. The state in each mode may be described by a state vector |? )k of the
Hilbert space appropriate to that mode. The states of the entire field are then defined
in the tensor product space of the Hilbert spaces for all of the modes.
The Hamiltonian for the electromagnetic field is given by

H = 1 ¸ .? E2 + ? H2.
2


dr . (2.15)


Substituting (2.13) for E and the equivalent expression for H and making use of the conditions (2.8) and (2.9), the Hamiltonian may be reduced to the form


H = ?k?k k

. .
a†ak +
2



. (2.16)


This represents the sum of the number of photons in each mode multiplied by the energy of a photon in that mode, plus 1 h¯ ?k representing the energy of the vacuum
fluctuations in each mode. We shall now consider three possible representations of the electromagnetic field.



2.2 Fock or Number States

The Hamiltonian (2.15) has the eigenvalues h?k(nk + 1 ) where nk is an integer
(nk = 0, 1, 2, ..., ?). The eigenstates are written as |nk) and are known as number
or Fock states. They are eigenstates of the number operator Nk = a†ak

k ak|nk) = nk|nk) . (2.17)

The ground state of the oscillator (or vacuum state of the field mode) is defined by

2.2 Fock or Number States 11

ak|0) = 0 . (2.18)

From (2.16 and 2.18) we see that the energy of the ground state is given by

1
(0|H|0 = ?k?k . (2.19)
k

Since there is no upper bound to the frequencies in the sum over electromagnetic field modes, the energy of the ground state is infinite, a conceptual difficulty of quan- tized radiation field theory. However, since practical experiments measure a change in the total energy of the electromagnetic field the infinite zero-point energy does not lead to any divergence in practice. Further discussions on this point may be found
in [1]. ak and a† are raising and lowering operators for the harmonic oscillator ladder
of eigenstates. In terms of photons they represent the annihilation and creation of a
photon with the wave vector k and a polarisation eˆk . Hence the terminology, annihi- lation and creation operators. Application of the creation and annihilation operators to the number states yield


1/2 †


1/2

ak|nk) = nk |nk ? 1), ak |nk) = (nk + 1)

|nk + 1) . (2.20)


The state vectors for the higher excited states may be obtained from the vacuum by successive application of the creation operator
. †.nk
k

|nk) =


(nk!)1/2

|0), nk = 0, 1, 2 ... . (2.21)


The number states are orthogonal



and complete



(nk|mk) = ?mn , (2.22)


?

? |nk)(nk| = 1 . (2.23)
nk =0
Since the norm of these eigenvectors is finite, they form a complete set of basis vectors for a Hilbert space.
While the number states form a useful representation for high-energy photons,
e.g. ? rays where the number of photons is very small, they are not the most suitable
representation for optical fields where the total number of photons is large. Experi- mental difficulties have prevented the generation of photon number states with more than a small number of photons (but see 16.4.2). Most optical fields are either a su- perposition of number states (pure state) or a mixture of number states (mixed state). Despite this the number states of the electromagnetic field have been used as a basis for several problems in quantum optics including some laser theories.


2.3 Coherent States


A more appropriate basis for many optical fields are the coherent states [2]. The coherent states have an indefinite number of photons which allows them to have a more precisely defined phase than a number state where the phase is completely random. The product of the uncertainty in amplitude and phase for a coherent state is the minimum allowed by the uncertainty principle. In this sense they are the closest quantum mechanical states to a classical description of the field. We shall outline the basic properties of the coherent states below. These states are most easily generated using the unitary displacement operator
D (?)= exp .?a† ? ??a. , (2.24) where ? is an arbitrary complex number.
Using the operator theorem [2]

eA+B = eAeBe?[A,B]/2 , (2.25)


which holds when

we can write D(?) as



[A, [A, B]] = [B, [A, B]] = 0,

2 † ?
D (?)= e?|?| /2e?a e?? a . (2.26)
The displacement operator D(?) has the following properties
D† (?)= D?1 (?)= D (??) , D† (?) aD (?)= a + ?,
D† (?) a†D (?)= a† + ?? . (2.27)
The coherent state |?) is generated by operating with D(?) on the vacuum state
|?) = D (?) |0) . (2.28)

The coherent states are eigenstates of the annihilation operator a. This may be proved as follows:
D† (?) a|?) = D† (?) aD (?) |0) = (a + ?) |0) = ?|0) . (2.29)
Multiplying both sides by D(?) we arrive at the eigenvalue equation
a|?) = ?|?) . (2.30)
Since a is a non-Hermitian operator its eigenvalues ? are complex.
Another useful property which follows using (2.25) is
D (? + ? )= D (?) D (? ) exp (?i Im {?? ?}) . (2.31)

2.3 Coherent States 13

The coherent states contain an indefinite number of photons. This may be made ap- parent by considering an expansion of the coherent states in the number states basis.
Taking the scalar product of both sides of (2.30) with (n| we find the recursion
relation



It follows that

(n + 1)1/2 (n + 1|?) = ?(n|?) . (2.32)

?n

(n|?) =


(n!)1/2

(0|?) . (2.33)

We may expand |?) in terms of the number states |n) with expansion coefficients
(n|?) as follows
?n
|?) = ?|n)(n|?) = (0|?) ? (n!)1/2 |n) . (2.34)


The squared length of the vector |?) is thus

2n
|(?|?)|2 = |(0|?)|2 ? | |




= |(0|?)|2e|?|




. (2.35)



It is easily seen that

n n!


(0|?) = (0|D (?) |0)
= e?|?| /2 . (2.36)

Thus |(?|?)|2 = 1 and the coherent states are normalized.
The coherent state may then be expanded in terms of the number states as


|?) = e?|?| /2 ?

?n
(n!)1/2 |n) . (2.37)


We note that the probability distribution of photons in a coherent state is a Poisson distribution

P (n)= |(n|?)|2 = |?|

2

2ne?|? 2
n!
†


, (2.38)

2

where |?|

is the mean number of photons (n¯ = (?|a

a|?) = |?| ).

The scalar product of two coherent states is
(? |?) = (0|D† (? ) D (?) |0) . (2.39)


Using (2.26) this becomes



. 1 2 2. ?.

(? |?) = exp

? 2 |?|

+ |? |

+ ??

. (2.40)


The absolute magnitude of the scalar product is


|(? |?)|2 = e?|??? |


. (2.41)


Thus the coherent states are not orthogonal although two states |?) and |? ) become approximately orthogonal in the limit |? ? ? |± 1. The coherent states form a two-
dimensional continuum of states and are, in fact, overcomplete. The completeness
relation





may be proved as follows.

1 ¸
? |?)(?|d ? = 1 , (2.42)

We use the expansion (2.37) to give


¸ d2?

? ? |n)(m| ¸


?|?| ?


?m?nd2? .

|?)(?|

? n=0 m=0 ??n!m!

(2.43)


Changing to polar coordinates this becomes







Using

¸
|?)(?|


d2 ?
=
?


?
?
n,m=0

|n)(m|
??n!m!


2?

?
rdre?r2

0


rn+m

2?
¸
d? e

0


i(n?m)?



. (2.44)




we have

d? ei(n?m)? = 2??nm , (2.45)
0

?

¸ d2?

? |n)(n| ¸ ? n

|?)(?|

=
? n=0 n!

d? e? ?

0

, (2.46)

where we let ? = r2. The integral equals n!. Hence we have

¸ d2? ?

|?)(?|

= ? |n)(n = 1 , (2.47)
n=0


following from the completeness relation for the number states.
An alternative proof of the completeness of the coherent states may be given as follows. Using the relation [3]


e? BAe?? B = A + ? [B, A]+ ?


[B, [B, A]] + ••• , (2.48)

2!

it is easy to see that all the operators A such that

D† (?) AD (?)= A (2.49)

are proportional to the identity.



We consider


then


A = d2?|?)(?|

D† (? ) ¸ d2? ?


? D (? )= d2? ?


? ? ?


= d2? ?


? . (2.50)

| )( |

| ? )( ? |

| )( |


Then using the above result we conclude that

d2?|?)(?| ? I . (2.51)

The constant of proportionality is easily seen to be ?.
The coherent states have a physical significance in that the field generated by a highly stabilized laser operating well above threshold is a coherent state. They form a useful basis for expanding the optical field in problems in laser physics and nonlinear optics. The coherence properties of light fields and the significance of the coherent states will be discussed in Chap. 3.




Squeezed States


A general class of minimum-uncertainty states are known as squeezed states. In general, a squeezed state may have less noise in one quadrature than a coherent state. To satisfy the requirements of a minimum-uncertainty state the noise in the other quadrature is greater than that of a coherent state. The coherent states are a particular member of this more general class of minimum uncertainty states with equal noise in both quadratures. We shall begin our discussion by defining a family of minimum-uncertainty states. Let us calculate the variances for the position and momentum operators for the harmonic oscillator

. k
q =
2?

.
a + a† , p = i
2


.a ? a†. . (2.52)


The variances are defined by
V (A)= (?A)2 = (A2)? (A)




. (2.53)


In a coherent state we obtain

(?q)2




= k , (?p)2



= k? . (2.54)

coh 2?

coh 2

Thus the product of the uncertainties is a minimum


(?p ?q) = k . (2.55)
coh 2

Thus, there exists a sense in which the description of the state of an oscillator by a coherent state represents as close an approach to classical localisation as possible. We shall consider the properties of a single-mode field. We may write the annihila- tion operator a as a linear combination of two Hermitian operators

X1 + iX2
a = 2 . (2.56)

X1 and X2, the real and imaginary parts of the complex amplitude, give dimension- less amplitudes for the modes’ two quadrature phases. They obey the following commutation relation
[X1, X2]= 2i (2.57)

The corresponding uncertainty principle is
?X1 ?X2 ? 1 . (2.58)

This relation with the equals sign defines a family of minimum-uncertainty states. The coherent states are a particular minimum-uncertainty state with

?X1 = ?X2 = 1 . (2.59)
The coherent state |?) has the mean complex amplitude ? and it is a minimum- uncertainty state for X1 and X2, with equal uncertainties in the two quadrature phases. A coherent state may be represented by an “error circle” in a complex am- plitude plane whose axes are X1 and X2 (Fig. 2.1a). The center of the error circle lies
at 1 (X1 + iX2) = ? and the radius ?X1 = ?X2 = 1 accounts for the uncertainties in
X1 and X2.




(a)


Y2 X2



e–r

(b)


er


Y1

?

X1

Fig. 2.1 Phase space representation showing contours of constant uncertainty for (a) coherent state and (b) squeezed state |? , ?)


There is obviously a whole family of minimum-uncertainty states defined by
?X1?X2 = 1. If we plot ?X1 against ?X2 the minimum-uncertainty states lie on a
hyperbola (Fig. 2.2). Only points lying to the right of this hyperbola correspond to physical states. The coherent state with ?X1 = ?X2 is a special case of a more
general class of states which may have reduced uncertainty in one quadrature at the expense of increased uncertainty in the other (?X1 < 1 < ?X2). These states
correspond to the shaded region in Fig. 2.2. Such states we shall call squeezed states
[4]. They may be generated by using the unitary squeeze operator [5]
S (?)= exp .1/2??a2 ? 1/2?a†2. . (2.60)

where ? = re2i? .
Note the squeeze operator obeys the relations
S† (?)= S?1 (?)= S (??) , (2.61)

and has the following useful transformation properties

S† (?) aS (?)= a cosh r ? a†e?2i? sinh r, S† (?) a†S (?)= a† cosh r ? ae?2i? sinh r ,
S† (?) (Y1 + iY2)S (?)= Y1e?r + iY2er , (2.62)


where


Y1 + iY2 = (X1 + iX2) e?i? (2.63)


is a rotated complex amplitude. The squeeze operator attenuates one component of the (rotated) complex amplitude, and it amplifies the other component. The degree
of attenuation and amplification is determined by r = |?|, which will be called the
squeeze factor. The squeezed state |?, ?) is obtained by first squeezing the vacuum
and then displacing it












Fig. 2.2 Plot of ?X1 ver- sus ?X2 for the minimum-
uncertainty states. The dot marks a coherent state while the shaded region corresponds to the squeezed states


|?, ?) = D (?) S (?) |0) . (2.64)
A squeezed state has the following expectation values and variances
(X1 + iX2) = (Y1 + iY2)ei? = 2?,
?Y1 = e?r , ?Y2 = er ,
(N) = |?2| + sinh2 r,
(?N)2 = |? cosh r ? ??e2i? sinh r|2 + 2 cosh2 r sinh2 r . (2.65)

Thus the squeezed state has unequal uncertainties for Y1 and Y2 as seen in the error ellipse shown in Fig. 2.1b. The principal axes of the ellipse lie along the Y1 and Y2
axes, and the principal radii are ?Y1 and ?Y2. A more rigorous definition of these
error ellipses as contours of the Wigner function is given in Chap. 3.



Two-Photon Coherent States


We may define squeezed states in an alternative but equivalent way [6]. As this definition is sometimes used in the literature we include it for completeness.
Consider the operator



where

b = ?a + ?a† (2.66)

|?|2 ? |?|2 = 1 .

Then b obeys the commutation relation
.b, b†. = 1 . (2.67)


We may write (2.66) as


b = U aU † (2.68)


where U is a unitary operator. The eigenstates of b have been called two-photon coherent states and are closely related to the squeezed states.
The eigenvalue equation may be written as
b|? )g = ? |? )g . (2.69)


From (2.68) it follows that

where |? ) are the eigenstates of a.


|? )g = U |? ) (2.70)

The properties of |? )g may be proved to parallel those of the coherent states. The
state |? )g may be obtained by operating on the vacuum
|? )g = Dg (? ) |0)g (2.71)

2.5 Two-Photon Coherent States 19
with the displacement operator
† ?
Dg (? )= e? b ?? b (2.72)
and |0)g = U |0). The two-photon coherent states are complete

and their scalar product is

¸
|? )g g(? |

d2?
= 1 (2.73)
?

. 1 2 1 2.

g(? |? ?)g = exp

? ?? ? ? 2 |? |

? 2 ? ?

. (2.74)

. .

We now consider the relation between the two-photon coherent states and the squeezed states as previously defined. We first note that
U ? S (?)
with ? = cosh r and ? = e2i? sinh r. Thus
|0)g ? |0, ?) (2.75)

with the above relations between (?, ?) and (r, ? ). Using this result in (2.71) and rewriting the displacement operator, Dg(? ), in terms of a and a† we find
|? )g = D (?) S (?) |0) = |?, ?| (2.76)


where


? = ?? ? ?? ? .

Thus we have found the equivalent squeezed state for the given two-photon coher- ent state.
Finally, we note that the two-photon coherent state |? )g may be written as
|? )g = S (?) D (? ) |0) .

Thus the two-photon coherent state is generated by first displacing the vacuum state, then squeezing. This is the opposite procedure to that which defines the squeezed
state |?, ?). The two procedures yield the same state if the displacement parameters
? and ? are related as discussed above.
The completeness relation for the two-photon coherent states may be employed to derive the completeness relation for the squeezed states. Using the above results we have

¸ d2?
? |? cosh r ? ? ?e


2i?


sinh r, ?)(? cosh r ? ? ?e


2i?


sinh r, ?| = 1 . (2.77)



The change of variable



? = ? cosh r ? ? ?e2i? sinh r (2.78)

leaves the measure invariant, that is d2? = d2? . Thus

¸ d2?
? |?, ?)(?, ?| = 1 . (2.79)



2.6 Variance in the Electric Field


The electric field for a single mode may be written in terms of the operators X1 and
X2 as
1 . k? .1/2

E (r,t)= ?
L

2?0

[X1 sin (?t ? k • r) ? X2 cos (?t ? k • r)] . (2.80)


The variance in the electric field is given by
V (E (r,t)) = K .V (X1) sin2 (?t ? k • r)+ V (X2) cos2 (?t ? k • r)
? sin [2 (?t ? k • r)]V (X1, X2)} (2.81)


where



K = 1 . 2k? . ,

L3 ?0

V (X , X )= ((X1 X2 )+(X2 X1 ))
2


? (X1)(X2).


For a minimum-uncertainty state

V (X1, X2)= 0 . (2.82)

Hence (2.81) reduces to
V (E (r,t)) = K .V (X1) sin2 (?t ? k • r)+ V (X2) cos2 (?t ? k • r). . (2.83) The mean and uncertainty of the electric field is exhibited in Figs. 2.3a–c where the
line is thickened about a mean sinusoidal curve to represent the uncertainty in the electric field.
The variance of the electric field for a coherent state is a constant with time (Fig. 2.3a). This is due to the fact that while the coherent-state-error circle rotates
about the origin at frequency ?, it has a constant projection on the axis defining
the electric field. Whereas for a squeezed state the rotation of the error ellipse leads to a variance that oscillates with frequency 2?. In Fig. 2.3b the coherent excitation

2.6 Variance in the Electric Field 21

Fig. 2.3 Plot of the electric field versus time showing schematically the uncertainty in phase and amplitude for
(a) a coherent state, (b) a squeezed state with reduced amplitude fluctuations, and
(c) a squeezed state with reduced phase fluctuations

























appears in the quadrature that has reduced noise. In Fig. 2.3c the coherent excitation appears in the quadrature with increased noise. This situation corresponds to the phase states discussed in [7] and in the final section of this chapter.
The squeezed state |?, r) has the photon number distribution [6]

n

p (n)= (n! cosh r)?1 . 1 tanh r.

.
exp ? 2

tanh r . ? ?)2 ei? + ? 2e?i? .. H (z) 2

2 ?|

| ? 2 (

| n |


where



z = ? + ??e



tanh r
.

(2.84)

2ei? tanh r
The photon number distribution for a squeezed state may be broader or narrower than a Poissonian depending on whether the reduced fluctuations occur in the phase (X2) or amplitude (X1) component of the field. This is illustrated in Fig. 2.4a where
we plot P(n) for r = 0, r > 0, and r < 0. Note, a squeezed vacuum (? = 0) contains
only even numbers of photons since Hn(0)= 0 for n odd.



Fig. 2.4 Photon number distribution for a squeezed state |? , r): (a) ? = 3, r = 0, 0.5, ?0.5, (b) ? = 3, r = 1.0


For larger values of the squeeze parameter r, the photon number distribution ex- hibits oscillations, as depicted in Fig. 2.4b. These oscillations have been interpreted as interference in phase space [8].



Multimode Squeezed States


Multimode squeezed states are important since several devices produce light which is correlated at the two frequencies ?+ and ??. Usually these frequencies are sym-
metrically placed either side of a carrier frequency. The squeezing exists not in the
single modes but in the correlated state formed by the two modes.
A two-mode squeezed state may be defined by [9]
|?+, ?? ) = D+ (?+ ) D? (?? ) S (G) |0) (2.85)

where the displacement operator is
. † .

D± (?)= exp

?a ? ??a±

, (2.86)


and the unitary two-mode squeeze operator is

S (G)= exp .G?a+a

? Ga† a† . . (2.87)

? + ?

The squeezing operator transforms the annihilation operators as




where G = rei? .


S†(G)a


S(G)= a±


cosh r a† ei? sinh r , (2.88)
?

This gives for the following expectation values
(a±) = ?±
2

(a±a±) = ?
(a+a?) = ?+?? ? e



sinh r cosh r


(2.89)

a† a
± ±

) = |?±

| + sinh r.


The quadrature operator X is generalized in the two-mode case to
1 . † † .

X = ?2

a+ + a+ + a? + a?

. (2.90)


As will be seen in Chap. 5, this definition is a particular case of a more general definition. It corresponds to the degenerate situation in which the frequencies of the two modes are equal.
The mean and variance of X in a two-mode squeezed state is
(X ) = 2(Re {?+ } + Re {??}),


V (X )=

e?2r cos2 ? + e2r sin2 ? .


. (2.91)

2 2

These results for two-mode squeezed states will be used in the analyses of nonde- generate parametric oscillation given in Chaps. 4 and 6.



Phase Properties of the Field


The definition of an Hermitian phase operator corresponding to the physical phase of the field has long been a problem. Initial attempts by P. Dirac led to a non-Hermitian operator with incorrect commutation relations. Many of these difficulties were made quite explicit in the work of Susskind and Glogower [10]. Pegg and Barnett [11] showed how to construct an Hermitian phase operator, the eigenstates of which, in an appropriate limit, generate the correct phase statistics for arbitrary states. We will first discuss the Susskind–Glogower (SG) phase operator.
Let a be the annihilation operator for a harmonic oscillator, representing a single field mode. In analogy with the classical polar decomposition of a complex ampli- tude we define the SG phase operator,

ei? = .aa†.?1/2 a . (2.92)

The operator ei? has the number state expansion





and eigenstates |ei? ) like


?
ei? = ? |n)(n + 1| (2.93)
n=1



?

|ei? ) = ? ein? |n) for ? ? < ? ? ? . (2.94)
n=1

It is easy to see from (2.93) that ei? is not unitary,

.ei? , .ei? .†.


= |0


)(0| . (2.95)



An equivalent statement is that the SG phase operator is not Hermitian. As an im- mediate consequence the eigenstates |ei? ) are not orthogonal. In many ways this
is similar to the non-orthogonal eigenstates of the annihilation operator a, i.e. the coherent states. None-the-less these states do provide a resolution of identity

?
¸ . .
d? .ei? )(ei? . = 2? . (2.96)
. .
??
The phase distribution over the window ?? < ? ? ? for any state |?) is then de- fined by




The normalisation integral is

1 i?
P (? )= 2? |(e

|?)|2


. (2.97)


?
¸
P (? ) d? = 1 . (2.98)
??

The question arises; does this distribution correspond to the statistics of any physical phase measurement? At the present time there does not appear to be an answer.
However, there are theoretical grounds [12] for believing that P(? ) is the correct
distribution for optimal phase measurements. If this is accepted then the fact that the SG phase operator is not Hermitian is nothing to be concerned about. However, as we now show, one can define an Hermitian phase operator, the measurement statistics of which converge, in an appropriate limit, to the phase distribution of (2.97) [13].
Consider the state |?0 ) defined on a finite subspace of the oscillator Hilbert
space by


s
|?0) = (s + 1)?1/2 ? ein?0 |n) . (2.99)
n=1
It is easy to demonstrate that the states |? ) with the values of ? differing from ?0 by integer multiples of 2?/(s + 1) are orthogonal. Explicitly, these states are
. a†a m 2? .

|?m) = exp i


s + 1

|?0); m = 0, 1,..., s , (2.100)



with


2?m
?m = ?0 + s + 1 .

Thus ?0 ? ?m < ?0 + 2? . In fact, these states form a complete orthonormal set on
the truncated (s + 1) dimensional Hilbert space. We now construct the Pegg–Barnett
(PB) Hermitian phase operator

s
? = ? ?m|?m)(?m| . (2.101)
m=1
For states restricted to the truncated Hilbert space the measurement statistics of ?
are given by the discrete distribution
Pm = |(?m|?)s |2 (2.102)
where |?)s is any vector of the truncated space.
It would seem natural now to take the limit s ? ? and recover an Hermitian phase
operator on the full Hilbert space. However, in this limit the PB phase operator does
not converge to an Hermitian phase operator, but the distribution in (2.102) does converge to the SG phase distribution in (2.97). To see this, choose ?0 = 0.
Then

. s .

nm2? . .2

. .
. i s + 1 n.
.

(2.103)


where ?n = (n|?)s.

.n=0

As ?m are uniformly distributed over 2? we define the probability density by

.. 2?

.?1 .

1 . ? .2

P (? )= lim

Pm =

. ? ein? ?n.

(2.104)




where

s??


s + 1




2?m

. .
.n=1 .

? = lim
s?? s + 1

, (2.105)

and ?n is the number state coefficient for any Hilbert space state. This convergence
in distribution ensures that the moments of the PB Hermitian phase operator con- verge, as s ? ?, to the moments of the phase probability density.



The phase distribution provides a useful insight into the structure of fluctuations in quantum states. For example, in the number state |n), the mean and variance of
the phase distribution are given by





and

(? ) = ?0 + ? , (2.106)


2
V (? )= 3 ? , (2.107)


respectively. These results are characteristic of a state with random phase. In the case of a coherent state |rei? ) with r ± 1, we find
(? ) = ? , (2.108)
1
V (? )= 4n¯ , (2.109)
where n¯ = (a†a) = r2 is the mean photon number. Not surprisingly a coherent state has well defined phase in the limit of large amplitude.


Exercises


If |X1) is an eigenstate for the operator X1 find (X1|?) in the cases (a) |?) = |?); (b) |?) = |?, r).
Prove that if |?) is a minimum-uncertainty state for the operators X1 and X2,
then V (X1, X2)= 0.
Show that the squeeze operator

S (r, ? )= exp . r .e?2i? a2
2

may be put in the normally ordered form

? e2i? a†2..

S (r, ? )= (cosh r)?1/2 exp .

? a†2. exp .

. ? ? .
ln (cosh r) a†a. exp a2

? 2 ? 2

where ? = e2i? tanh r.
Evaluate the mean and variance for the phase operator in the squeezed state
|?, r) with ? real. Show that for |r|± |?| this state has either enhanced or
diminished phase uncertainty compared to a coherent state.