|Ψ = √ (6.83)

ks k

2 ks,k s

where the normalization condition Ψ |Ψ = 1 is

2

|Cks,k s | = 1 , (6.84)

ks,k s

and the expansion (6.83) can be inverted to give

1ks , 1k s |Ψ

√

Cks,k s = . (6.85)

2

By comparing eqns (6.83) and (6.75), we can see that a two-photon state is sepa-

rable if the coe¬cients in eqn (6.83) factorize:

Cks,k s = γks γk s , (6.86)

where the γks s are c-number coe¬cients. In this case, |Ψ can be expressed as

¾½¾ Entangled states

1 2

|Ψ = √ “† |0 , (6.87)

2

where

γks a† ,

“† = (6.88)

ks

ks

and the normalization condition (6.84) becomes

|γks |2 = 1 . (6.89)

ks

The normalization of the γks s in turn implies “, “† = 1; therefore, “† can be inter-

preted as a creation operator for a photon in the classical wave packet:

E (r) = γks Fk eks eik·r , (6.90)

ks

where

ωk

Fk = i . (6.91)

2 0V

Thus the bosonic character of photons implies that a separable state necessarily con-

tains two photons in the same classical wave packet, in agreement with the de¬nition

(6.74) for massive bosons.

A two-photon state that is not separable is said to be entangled. This leads in

particular to the useful rule

|1ks , 1k s is entangled if ks = k s . (6.92)

The factorization condition (6.86) provides a de¬nition of separable states and entan-

gled states that works in the absence of position-space wave functions for photons, but

the physical meaning of entanglement is not as intuitively clear as it is in ordinary

quantum mechanics. The best remedy is to ¬nd a substitute for the missing wave

function.

6.6.2 The detection amplitude

Let us pretend, for the moment, that the operator Es (r) = e— · E(’) (r) creates a

(’)

s

photon, with polarization es , at the point r. If this were true, then the state vector

(’)

|r, s = Es (r) |0 would describe a situation in which one photon is located at r with

polarization es . For a one-photon state |Ψ , this suggests de¬ning a single-photon ˜wave

function™ by

Ψ (r, s) = r, s |Ψ

(+)

= 0 Es (r) Ψ

= e— 0 Ej

(+)

(r) Ψ . (6.93)

sj

Now that our attention has been directed to the appropriate quantity, we can discard

this very dubious plausibility argument, and directly investigate the physical signi¬-

cance of Ψ (r, s). One way to do this is to use eqn (4.74) to evaluate the ¬rst-order

¾½¿

Entanglement for photons

¬eld correlation function for the one-photon state |Ψ . For equal time arguments, the

result is

(1) (’) (+)

Gij (r ; r) = Ψ Ei (r ) Ej (r) Ψ

(’) (+)

Ψ Ei (r ) n n Ej (r) Ψ

=

n

(’) (+)

= Ψ Ei (r ) 0 0 Ej (r) Ψ , (6.94)

where the last line follows from the observation that the vacuum state alone can

contribute to the sum over the number states |n . By combining these two equations,

one ¬nds that

G(1) (r s ; rs) = es i e— Gij (r ; r)

(1)

sj

= Ψ (r, s) Ψ— (r , s ) . (6.95)

This result for G(1) (rs; r s ) is quite suggestive, since it has the form of the density

matrix for a pure state with wave function Ψ (r, s). On the other hand, the usual Born

interpretation does not apply to Ψ (r, s), since there is no photon position operator. An

important clue pointing to the correct physical interpretation of Ψ (r, s) is provided by

the theory of photon detection. In Section 9.1.2-A it is shown that the counting rate for

a photon detector”located at r and equipped with a ¬lter transmitting polarization

2

es ”is proportional to G(1) (rs; rs). According to eqn (6.95), this means that |Ψ (r, s)|

is the probability that a photon is detected at r, the position of the detector. In view

of this fact, we will refer to Ψ (r, s) as the one-photon detection amplitude. The

important point to keep in mind is that the detector is a classical object which”unlike

the photon”has a well-de¬ned location in space. This is what makes the detection

amplitude a useful replacement for the missing photon wave function.

We extend this approach to two photons by pretending that |r1 , s1 ; r2 , s2 =

(’) (’)

Es1 (r1 ) Es2 (r2 ) |0 is a state with one photon at r1 (with polarization es1 ) and

another at r2 (with polarization es2 ). For a two-photon state |Ψ this suggests the

e¬ective wave function

Ψ (r1 , s1 ; r2 , s2 ) = r1 , s1 ; r2 , s2 |Ψ

(+) (+)

= 0 Es1 (r1 ) Es2 (r2 ) Ψ

= e—1 i e—2 j Ψij (r1 , r2 ) , (6.96)

s s

where

(+) (+)

Ψij (r1 , r2 ) = 0 Ei (r1 ) Ej (r2 ) Ψ . (6.97)

Applying the method used for G(1) to the evaluation of eqn (4.75) for the second-order

correlation function (with all time arguments equal) yields

(2) (’) (’) (+) (+)

Gklij (r1 , r2 ; r1 , r2 ) = Ek (r1 ) El (r2 ) Ei (r1 ) Ej (r2 )

= Ψij (r1 , r2 ) Ψ— (r1 , r2 ) , (6.98)

kl

¾½ Entangled states

which has the form of the two-particle density matrix corresponding to the pure two-

particle wave function Ψij (r1 , r2 ).

The physical interpretation of Ψij (r1 , r2 ) follows from the discussion of coincidence

counting in Section 9.2.4, which shows that the coincidence-counting rate for two fast

detectors placed at equal distances from the source of the ¬eld is proportional to

(es1 )k (es2 )l e—1 e—2

(2) 2

Gklij (r1 , r2 ; r1 , r2 ) = |Ψ (r1 , s1 ; r2 , s2 )| , (6.99)

s s

i j

where es1 and es2 are the polarizations admitted by the ¬lters associated with the

2

detectors. Since |Ψ (r1 , s1 ; r2 , s2 )| determines the two-photon counting rate, we will

refer to Ψ (r1 , s1 ; r2 , s2 )”or Ψij (r1 , r2 )”as the two-photon detection amplitude.

6.6.3 Pure state entanglement de¬ned by detection amplitudes

We are now ready to formulate an alternative de¬nition of entanglement, for pure

states of photons, that is directly related to observable counting rates. The detection

amplitude for the two-photon state |Ψ , de¬ned by eqn (6.83), can be evaluated by

using eqns (3.69) and (6.85) in eqn (6.97), with the result:

√

Cks,k s Fk (eks )i eik·r1 Fk (ek s )j eik ·r2 .

Ψij (r1 , r2 ) = 2 (6.100)

ks,k s

This expansion for the detection amplitude can be inverted, by Fourier transforming

with respect to r1 and r2 and projecting on the polarization basis, to get

(2 0 / )2

√

=’

Cks,k s Ψks,k s , (6.101)

2ωk ωk

where

1

d3 r2 e’ik·r1 e’ik ·r2 (e— )i (e— s )j Ψij (r1 , r2 ) .

d3 r1

Ψks,k s = (6.102)

ks k

V

According to eqns (6.100) and eqn (6.101), the two-photon detection amplitude and

the expansion coe¬cients Cks,k s provide equivalent descriptions of the two-photon

state. From eqn (6.100) we see that factorization of the expansion coe¬cients, accord-

ing to eqn (6.86), implies factorization of the detection amplitude, i.e.

Ψij (r1 , r2 ) = φi (r1 ) φj (r2 ) , (6.103)

where

γks Fk (eks )i eik·r .

φi (r) = 21/4 (6.104)

ks

In other words, the detection amplitude for a separable state factorizes, just as a two-

particle wave function does in nonrelativistic quantum mechanics. On the other hand,

eqn (6.101) shows that factorization of the detection amplitude implies factorization of

the expansion coe¬cients. Thus we are at liberty to use eqn (6.103) as a de¬nition of

a separable state that agrees with the de¬nition (6.86). This approach has the decided

¾½

Entanglement for photons

advantage that the detection amplitude is closely related to directly observable events,

e.g. current pulses emitted by the coincidence counter. The coincidence-counting rate

is proportional to the square of the amplitude, so for separable states the coincidence

rate is proportional to the product of the singles rates at the two detectors. This means

that the random counting events at the two detectors are stochastically independent,

i.e. the quantum ¬‚uctuations of the electromagnetic ¬eld at any pair of detectors are

uncorrelated. This is the analogue of Theorem 6.3, which states that a separable state

of two distinguishable particles yields uncorrelated quantum ¬‚uctuations for any pair

of observables.

For ks = k s the state |Ψ = |1ks , 1k s is entangled”according to the traditional

de¬nition”and evaluating eqn (6.100) in this case gives

(eks )i eik·r1 (ek s )j eik ·r2 + (eks )j eik·r2 (ek s )i eik ·r1 .

Ψij (r1 , r2 ) = Fk Fk

(6.105)

The de¬nition (6.96) in turn yields

Ψ (r1 , s1 ; r2 , s2 ) = φks (r1 , s1 ) φk s (r2 , s2 ) + φks (r2 , s2 ) φk s (r1 , s1 ) , (6.106)

where

φks (r, s1 ) = Fk e—1 · eks eik·r . (6.107)

s

This has the structure of an entangled-state wave function for two bosons”as shown in

eqn (6.80)”with similar physical consequences. In particular, if one photon is detected

in the mode ks, then a subsequent detection of the remaining photon is guaranteed

to ¬nd it in the mode k s . More generally, quantum ¬‚uctuations in the electromag-

netic ¬eld at the two detectors are correlated. According to the general de¬nition in

Section 6.5.3, an entangled two-photon state is dynamically entangled if the detection

amplitude cannot be expressed in the minimal form (6.106) required by Bose statistics.

We saw in Section 6.4.1 that reduced density operators, de¬ned by partial traces,

are quite useful in the discussion of distinguishable particles, but systems of identical

particles”such as photons”cannot be divided into distinguishable subsystems. The

key to overcoming this di¬culty is found in eqn (6.98) which shows that the second-

order correlation function has the form of a density matrix corresponding to the two-

photon detection amplitude Ψij (r1 , r2 ). This suggests that the analogue of the reduced

(1)

density matrix is the ¬rst-order correlation function Gij (r ; r), evaluated for the two-

photon state |Ψ .

The ¬rst evidence supporting this proposal is provided by considering a separable

state de¬ned by eqn (6.87). In this case

(1) (’) (+)

Gij (r ; r) = Ψ Ei (r ) Ej (r) Ψ

1

0 “2 Ei (r ) Ej (r) “†2 0

(’) (+)

=

2

1

0 “2 , Ei (r ) Ej (r) , “†2

(’) (+)

= 0, (6.108)

2

¾½ Entangled states

(+)

where the last line follows from the identity Ej (r) |0 = 0 and its adjoint. The ¬eld

operators and the operators “ and “† are both linear functions of the creation and

annihilation operators, so

(r) , “†2 = 2 Ej (r) , “† “† .

(+) (+)

Ej (6.109)

The remaining commutator is a c-number which is evaluated by using the expansions

(3.69) and (6.88) to get

Ej (r) , “† = 2’1/4 φj (r) ,

(+)

(6.110)

where φi (r) is de¬ned by eqn (6.104). Substituting this result, and the corresponding

(’)

expression for “, Ei (r ) , into eqn (6.108) yields

√

Gij (r ; r) = 2φj (r) φ— (r ) .

(1)

(6.111)

i

The conclusion is that the ¬rst-order correlation function for a separable state factor-

izes. This is the analogue of Theorem 6.1 for distinguishable particles.

Next let us consider a generic entangled state de¬ned by |Ψ = “† ˜† |0 , where

θks a†

˜† = (6.112)

ks

ks

and

|θks |2 = 1 . (6.113)

ks

For this argument, we can con¬ne attention to operators satisfying “, ˜† = 0, which

is equivalent to the orthogonality of the classical wave packets:

—

(θ, γ) ≡ θks γks = 0 . (6.114)

ks

The ¬rst-order correlation function for this state is

(1) (’) (+)

Gij (r ; r) = Ψ Ei (r ) Ej (r) Ψ

1

= √ {φj (r) φ— (r ) + ·j (r) ·i (r )} ,

—

(6.115)

i

2

where ·j (r) is de¬ned by replacing γks with θks in eqn (6.104). Thus for the entangled,

two-photon state |Ψ , the ¬rst-order correlation function (reduced density matrix) has

the standard form of the density matrix for a one-particle mixed state. This is the

analogue of Theorem 6.2 for distinguishable particles.

6.7 Exercises

6.1 Proof of Theorem 6.1

(1) To prove assertion (a), use the expression for the density operator resulting from

eqns (6.40) and (2.81) to evaluate the reduced density operators.

(2) To prove assertion (b), assume that |Ψ is entangled”so that it has Schmidt rank

r > 1”and derive a contradiction.

¾½

Exercises

6.2 Proof of Theorem 6.3

(1) For a separable state |Ψ show that Ψ |δA δB| Ψ = 0.

(2) Assume that Ψ |δA δB| Ψ = 0 for all A and B. Apply this to operators that are

diagonal in the Schmidt basis for |Ψ and thus show that |Ψ must be separable.

6.3 Singlet spin state

(1) Use the standard treatments of the Pauli matrices, given in texts on quantum

mechanics, to express the eigenstates of n · σ in the usual basis of eigenstates of

σz .

(2) Show that the singlet state |S = 0 , given by eqn (6.37), has the same form for all

choices of the quantization axis n.

2

|S = 0 = 0.

(3) Show that SA + SB

6.4 Correlations in a separable mixed state

Consider a system of two distinguishable spin-1/2 particles described by the ensemble

{|Ψ1 = |‘ |“ , |Ψ2 = |“ |‘ B}

A B A

of separable states, where the spin states are eigenstates of sA and sB .

z z

(1) Show that the density operator can be written as

ρ = p |Ψ1 Ψ1 | + (1 ’ p) |Ψ2 Ψ2 | ,

where 0 p 1.

(2) Evaluate the correlation function δsA δsB and use the result to show that the

z z

spins are only uncorrelated for the extreme values p = 0, 1.

(3) For intermediate values of p, argue that the correlation is exactly what would be

found for a pair of classical stochastic variables taking on the values ±1/2 with

the same assignment of probabilities.

7

Paraxial quantum optics

The generation and manipulation of paraxial beams of light forms the core of exper-

imental practice in quantum optics; therefore, it is important to extend the classical

treatment of paraxial optics to situations involving only a few photons, such as the

photon pairs produced by spontaneous down-conversion. In addition to the interac-

tion of quantized ¬elds with standard optical elements, the theory of quantum paraxial

propagation has applications to fundamental issues such as the generation and control

of orbital angular momentum and the meaning of localization for photons.

In geometric optics a beam of light is a bundle of rays making small angles with a

central ray directed along a unit vector u0 . The constituent rays of the bundle are said

to be paraxial. In wave optics, the bundle of rays is replaced by a bundle of unit vectors

normal to the wavefront; so a paraxial wave is de¬ned by a wavefront that is nearly

¬‚at. In this situation it is natural to describe the classical ¬eld amplitude, E (r, t), as a

function of the propagation variable ζ = r·u0 , the transverse coordinates r tangent to

the wavefront, and the time t. Paraxial wave optics is more complicated than paraxial

ray optics because of di¬raction, which couples the r -, ζ-, and t-dependencies of the

¬eld. For the most part, we will only consider a single paraxial wave; therefore, we can

choose the z-axis along u0 and set ζ = z.

The de¬nite wavevector associated with the plane wave created by a† (k) makes it

s

possible to recast the geometric-optics picture in terms of photons in plane-wave states.

This way of thinking about paraxial optics is useful but”as always”it must be treated

with caution. As explained in Section 3.6.1, there is no physically acceptable way to

de¬ne the position of a photon. This means that the natural tendency to visualize the

photons as beads sliding along the rays at speed c must be strictly suppressed. The

beads in this naive picture must be replaced by wave packets containing energy ω

and momentum k, where k is directed along the normal to the paraxial wavefront.

In the following section, we begin with a very brief review of classical paraxial

wave optics. In succeeding sections we will de¬ne a set of paraxial quantum states,

and then use them to obtain approximate expressions for the energy, momentum,

and photon number operators. This will be followed by the de¬nition of a slowly-

varying envelope operator that replaces the classical envelope ¬eld E (r, t). Some more

advanced topics”including the general paraxial expansion, angular momentum, and

an approximate notion of photon localizability”will be presented in the remaining

sections.

¾½

Paraxial states

7.1 Classical paraxial optics

As explained above, each photon is distributed over a wave packet, with energy ω and

momentum k, that propagates along the normal to the wavefront. However, this wave

optics description must be approached with equal caution. The standard approach in

classical, paraxial wave optics (Saleh and Teich, 1991, Sec. 2.2C) is to set

E (r, t) = E (r, t) ei(k0 ·r’ω0 t) , (7.1)

where ω0 and k0 = u0 n (ω0 ) ω0 /c are respectively the carrier frequency and the carrier

wavevector. The four-dimensional Fourier transform, E (k, ω), of the slowly-varying

envelope is assumed to be concentrated in a neighborhood of k = 0, ω = 0. The

equivalent conditions in the space“time domain are