s)

s)

’

θ2 (θ1 =

Fig. 13.6 Coincidence rates (indicated by circles, with values on the left axis) and singles

rates”the outputs of the individual Geiger counters”(indicated by squares, with values on

the right axis) versus the relative angle θ2 ’θ1 between the two linear analyzers (i.e. polarizing

beam splitters, PBSs) placed in front of detectors 1 and 2 in Fig. 13.5(b). These data were

taken by varying θ2 with θ1 kept ¬xed at ’45—¦ . (Reproduced from Kwiat et al. (1999b).)

A further experiment demonstrated that it is possible to tune the entanglement phase

ξ continuously over a range from 0 to 5.5π by tilting the quarter-wave plate, placed

in front of the tandem crystals, from 0—¦ to 30—¦ .

13.4 Four-photon interactions

Four-photon processes correspond to classical four-wave mixing, so they involve the

third-order susceptibility χ(3) . The parity argument shows that χ(3) can be nonzero

for an isolated atom, therefore four-photon processes can take place in any medium,

including a vapor. In Section 13.4.2-B we will describe experimental observation of

photon“photon scattering in a rubidium vapor cell.

13.4.1 Frequency tripling and down-conversion

The four-photon analogue of sum-frequency generation is frequency tripling or

third harmonic generation in which three photons are absorbed to produce a

single ¬nal photon. The energy and momentum conservation (phase matching) rules

are then

ω0 = ω1 + ω2 + ω3 , (13.64)

k0 = k1 + k2 + k3 , (13.65)

and the Feynman diagram is shown in Fig. 13.7(a). In the degenerate case ω1 = ω2 =

ω3 = ω, energy conservation requires ω0 = 3ω. This e¬ect was ¬rst observed in the

early 1960s by Maker et al. (1963).

The time-reversed process, which describes down-conversion of one photon into

three, is shown in Fig. 13.7(b). In the photon indivisibility experiment described in

Section 1.4, one of the two entangled photons is used to trigger the counters. This

guaranteed that a genuine one-photon state would be incident on the beam splitter. In

nondegenerate three-photon down-conversion, the three ¬nal photons are all entangled.

½¿

Four-photon interactions

(k0, ω0)

(k2, ω2)

(k1, ω1) (k3, ω3)

(k1, ω1) (k3, ω3)

Fig. 13.7 (a) Sum-frequency generation with

(k2, ω2) (k0, ω0)

three photons. (b) Down-conversion of one

(a) (b) photon into three.

It would therefore be possible to use one photon to trigger the counters, and thus

guarantee that a genuine entangled state of two photons is incident on another part

of the apparatus.

13.4.2 Photon“photon scattering

In three-photon coupling, the phase-matching conditions (13.23) are the only possi-

bility, but with four photons there are two arrangements for conserving energy and

momentum: namely, eqns (13.64) and (13.65), and

ω0 + ω1 = ω2 + ω3 , (13.66)

k0 + k1 = k2 + k3 . (13.67)

The corresponding Feynman diagram, shown in Fig. 13.8, describes photon“photon

scattering. In quantum electrodynamics, this process depends on the virtual produc-

tion of electron“positron pairs in the vacuum. This scattering cross-section is so small

that it cannot be observed with currently available techniques (Schweber, 1961, Chap.

16a). The situation in a nonlinear medium is quite di¬erent, since the incident pho-

tons can excite an atom near resonance and thus produce an enormously enhanced

photon“photon cross-section.

A The phenomenological Hamiltonian

We will restrict our attention to a vapor, since this is the simplest medium allowing

four-photon processes. In this case there are no preferred directions, so the coupling

(k2, ω2) (k3, ω3)

(k0, ω0) (k1, ω1) Fig. 13.8 Photon“photon scattering medi-

ated by interaction with atoms in the medium.

½ Nonlinear quantum optics

between modes can only depend on the inner products of the polarization basis vec-

tors. These geometrical factors are readily calculated for any given process, so we will

simplify the notation by suppressing the polarization indices. From this point on, the

argument parallels the one used for the three-photon Hamiltonian, so the simplest

interaction Hamiltonian that yields Fig. 13.8 in lowest order is

11

nat γ (k2 , k3 ; k0 , k1 ) C (k2 + k3 ’ k1 ’ k0 ) a† 3 a† 2 ak1 ak0 ,

Hint = kk

4V2

k0 ,k1 ,k2 ,k3

(13.68)

—

where the coupling constants satisfy γ (k2 , k3 ; k0 , k1 ) = γ (k0 , k1 ; k2 , k3 ), and C (k)

is de¬ned by eqn (13.20).

B Experimental observation of photon“photon scattering

An experiment has been performed to observe head-on photon“photon collisions”

mediated by the atoms in a rubidium vapor cell”leading to 90—¦ scattering. In the

experiment the rubidium atoms are excited close enough to resonance to get resonant

enhancement, but far enough from resonance to eliminate photon absorption and res-

onance ¬‚uorescence. The resonant enhancement of the coupling is what makes this

experiment possible, by contrast to the observation of photon“photon scattering in

the vacuum.

The detailed theoretical analysis of this experiment is rather complicated (Mitchell

et al., 2000), but the model Hamiltonian of eqn (13.68) su¬ces for a qualitative treat-

ment. In particular, one would expect coincidence detections for pairs of photons

scattered in opposite directions”in the center-of-mass frame of a pair of incident

photons”as if the two incident photons had undergone an elastic hard-sphere scatter-

ing in a head-on collision.

As shown in Fig. 13.9, a diode laser beam at 780 nm wavelength passes through

two isolators (this prevents the retrore¬‚ected beam from a mirror placed behind the

cell from re-entering the laser, and thus interfering with its operation). In order to

minimize absorption and resonance ¬‚uorescence, the frequency of the laser beam is

detuned from the nearest rubidium-atom absorption line by 1.3 GHz, which is some-

what larger than the atomic Doppler line width at room temperature. The incident

diode laser beam passes through a single-mode, polarization-maintaining ¬ber that

spatially ¬lters it. This produces a single-transverse (TEM00 ) mode beam that is inci-

dent onto a square, glass rubidium vapor cell. This cell is identical in shape and size to

the standard cuvettes used in Beckmann spectrophotometers. Two vertically-polarized

photons, one from the incident beam direction, and one from the retrore¬‚ecting mirror,

thus could collide head-on”inside a beam waist of area (0.026 cm)2 ”in the interior

of the vapor cell. The atomic density of rubidium atoms inside the cell is around

1.6 — 1010 atoms/cm3 .

The two colliding photons”like two hard spheres”will sometimes scatter o¬ each

other at right angles to the incident laser beam direction. The scattered photons

would be produced simultaneously, much like the twin photons in spontaneous down-

conversion. They could therefore be detected by means of coincidence counters, e.g.

two silicon avalanche photodiode Geiger counters, or single-photon counting modules

½

Four-photon interactions

Diode laser Isolator Isolator

Reference

Rb cell

Reflection

monitor Input fiber

coupler

Output fiber

coupler

SM/PM

fiber

Cell enclosure

He’Ne

laser

SPCM SPCM

Lightproof detection enclosure

Fig. 13.9 The apparatus used for observing photon“photon scattering mediated by rubidium

atoms excited o¬ resonance. (Reproduced from Mitchell et al. (2000).)

(SPCM). The reference rubidium cell is used to monitor how close to atomic resonance

the diode laser is tuned, and an auxiliary helium“neon laser is used to align the optics

of the scattered-light detection system.

In Fig. 13.10, we show experimental data for the coincidence-counting signal as

a function of the time delay between coincidence-counting pulses. The coincidence-

counting electronic circuitry was used to scan the time delay from negative to positive

values. By inspection, there is a peak in coincidence counts around zero time delay,

which is consistent with the coincidence-detection window of 1 ns. This is evidence

for photon“photon collisions mediated by the atoms. As a control experiment, the

same scan of coincidence counts was made after a deliberate misalignment of the

two detectors by 0.14 rad with respect to the exact back-to-back scattering direction.

This misalignment was large enough to violate the momentum-conservation condition

(13.67). As expected, the coincidence peak disappeared.

½ Nonlinear quantum optics

1.1

9coincidence (arb. units)

1.05

1

0.95

’2 ’1 0 1 2 3

„A ’ „B (ns)

Fig. 13.10 Observed coincidence rates for 90—¦ photon“photon scattering mediated by ru-

bidium atoms excited o¬ resonance. See Fig. 13.9 for the setup of the apparatus. Error bars

indicate statistical errors on the data acquired with detectors aligned to collect back-to-back

scattering products. The observed maximum in the coincidence rate disappears when the

two detectors are deliberately misaligned from the back-to-back scattering direction. The

solid curve is a theoretical ¬t using three measured parameters: the beam shape, the ¬nite

detection time, and the detector area. (Reproduced from Mitchell et al. (2000).)

13.4.3 Kerr media

For vapors and liquids the second-order susceptibility vanishes, and the absence of

any preferred direction implies that the third-order polarization envelope for a single

monochromatic wave, E (t) exp (’iω0 t), is given by

2

P (3) = χ(3) |E| E . (13.69)

This is also valid for some centrosymmetric crystals, e.g. those with cubic symmetry. In

these cases the lowest-order optical response of the medium is given by the linear index

of refraction n = 1 + χ(1) . The nonlinear optical response is conveniently described

in terms of a ¬eld-dependent index n (E) de¬ned as

2

n2 (E) = 1 + χ = n2 + χ(3) |E| + · · · . (13.70)

Since χ(3) |E|2 is small, this can be approximated by

2

n (E) = n + n2 |E| + · · · ,

(13.71)

1 χ(3)

n2 = .

2n

This is more often expressed in terms of the intensity I as

n (E) = n + n2 I + · · · ,

(13.72)

µ0 χ(3)

n2 = .

2

0n

½

Four-photon interactions

The dependence of the atomic polarization, or equivalently the index of refraction, on

the intensity of the ¬eld is called the optical Kerr e¬ect. Media with non-negligible

values of n2 /n are called Kerr media. In a Kerr medium, the phase of a classical plane

wave traversing a distance L increases by • = kL = n (E) L/c, and the increment in

phase due to the intensity-dependent term is

ω 2πn2 I

∆• = n2 I L = L. (13.73)

c »0

This dependence of the phase on the intensity is called self-phase modulation. The

intensity dependence of the index of refraction also leads to the phenomenon of self-

focussing (Saleh and Teich, 1991, Sec. 19.3).

In the quantum description of the Kerr e¬ect, the interaction Hamiltonian is given

by the general expression (13.68); but substantial simpli¬cations occur in real applica-

tions. We consider an experimental con¬guration in which the Kerr medium is enclosed

in a resonant cavity with discrete modes. In this case, one mode is typically dominant.

In principle, the quantization scheme should be carried out from the beginning using

the cavity modes as a basis, but the result would have the same form as obtained from

the degenerate case k0 = k1 = k2 = k3 of Fig. 13.8. The model Hamiltonian is then

1

H = ω 0 a† a + ga†2 a2 , (13.74)

2

where the coupling constant g is proportional to χ(3) and a is the annihilation op-

erator for the favored mode. By means of the canonical commutation relations, the

Hamiltonian can be expressed as

1

g N2 ’ N ,

H = ω0 N + (13.75)

2

where N = a† a. In the Heisenberg picture, this form makes it clear that N (t) is a

constant of the motion: N (t) = N (0) = N. This corresponds to the classical result

that the intensity is ¬xed and only the phase changes.

The evolution of the quantum amplitude is given by the Heisenberg equation for

the annihilation operator:

da (t)

= ’iω0 a (t) ’ iga† (t) a2 (t)

dt

= ’i (ω0 + gN ) a (t) . (13.76)

Since the number operator is independent of time, the solution is

a (t) = e’i(ω0 +gN )t a , (13.77)

and the matrix elements of the annihilation operator in the number-state basis are

√

m |a (t)| m = e’i(ω0 +gm)t m |a| m = δm,m ’1 m + 1e’i(ω0 +gm)t . (13.78)

Thus the modulus of the matrix element is constant, and the term mgt in the phase

represents the quantum analogue of the classical phase shift ∆•.

½ Nonlinear quantum optics

It is also useful to consider situations in which the classical ¬eld is the sum of two

monochromatic ¬elds with di¬erent carrier frequencies:

E (t) = E 1 (t) exp (’iω1 t) + E 2 (t) exp (’iω2 t) . (13.79)

2 2

The polarization will then have contributions of the form |E 1 | E 1 and |E 2 | E 2 ”

describing self-phase modulation”and also terms proportional to |E 1 |2 E 2 and |E 2 |2 E 1

”describing cross-phase modulation. This is called a cross-Kerr medium, and

the Hamiltonian is

g1 †2 2 g2 †2 2

H = ω 1 a† a1 + ω 2 a† a2 + a2 a2 + g12 a† a† a1 a2 .

a1 a1 + (13.80)

1 2 12

2 2

The coupling frequencies g1 , g2 , and g12 are all proportional to components of the

χ(3) -tensor. For isotropic media, the three coupling frequencies are identical; but for

crystals it is possible to have g1 = g2 = 0, while g12 = 0. This situation represents

pure cross-phase modulation.

13.5 Exercises

13.1 The fourth-order classical energy

Apply the line of argument used to derive the e¬ective energy expression (13.18) for

(3)

Uem to show that the fourth-order e¬ective energy is

1

gs0 s1 s2 s3 (ω1 , ω2 , ω3 ) [±— ±1 ±2 ±3 + CC]

Uem =

(4) (4)

0

V2

k0 s0 ,...,k3 s3

— δω0 ,ω1 +ω2 +ω3 C (k0 ’ k1 ’ k2 ’ k3 )

1

fs0 s1 s2 s3 (ω1 , ω2 , ω3 ) [±0 ±1 ±— ±— + CC]

(4)

+2 23

V

k0 s0 ,...,k3 s3

— δω0 +ω1 ,ω2 +ω3 C (k0 + k1 ’ k2 ’ k3 ) ,

where

gs0 s1 s2 s3 (ω1 , ω2 , ω3 ) = ’ 0 F0 F1 F2 F3 χ(3) 1 s2 s3 (ω1 , ω2 , ω3 ) ,

(4)

s0s

3

fs0 s1 s2 s3 (ω1 , ω2 , ω3 ) = 0 F0 F1 F2 F3 χ(3) 1 s2 s3 (ω1 , ’ω2 , ’ω3 ) ,

(4)

s0s

4

and

(2)

χ(3) 1 s2 s3 (ω1 , ω2 , ω3 ) = (µk0 s0 )i (µk1 s1 )j (µk2 s2 )k (µk3 s3 )l χijkl (ω1 , ω2 , ω3 ) .

s0 s

13.2 Kerr medium

Consider a Kerr medium with the Hamiltonian given by eqn (13.74).

(1) For a coherent state |± , use the result of part (2) of Exercise 5.2 to show that

e’igt ’ 1 |±|

2

± |a (t)| ± = exp .

½

Exercises

2

(2) For a nearly classical state, i.e. |±| 1, one might intuitively expect that the

number operator N in eqn (13.77) could be replaced by |±|2 in the evaluation

of ± |a (t)| ± . Write down the resulting expression and compare it to the exact

result given above to determine the range of values of t for which the conjectured

expression is valid. What is the behavior of the correct expression for ± |a (t)| ±

as t ’ ∞?

(3) Using the form (13.75) of the Hamiltonian, exhibit the solution of the Schr¨dinger

o

equation i ‚/‚t |ψ (t) = H |ψ (t) ”with initial condition |ψ (0) = |± ”as an ex-

pansion in number states. Use this solution to explain the counterintuitive results

of part (2) and to decide if |ψ (t) remains a nearly coherent state for all times t.

13.3 Cross-Kerr medium

Consider a cross-Kerr medium described by the Hamiltonian in eqn (13.80).

(1) Derive the Heisenberg equations of motion for the annihilation operators and show

that the number operators N1 (t) = a† (t) a1 (t) and N2 (t) = a† (t) a2 (t) are con-

1 2

stants of the motion.

(2) For the two-mode coherent state |±1 , ±2 , evaluate ±1 , ±2 |a1 (t)| ±1 , ±2 .

(3) For a pure cross-Kerr medium, expand the interaction-picture state vector |Ψ (t)

in the number-state basis {|n1 , n2 } and show that the exact solution of the

interaction-picture Schr¨dinger equation is

o

n1 n2 | Ψ (0) e’ig12 n1 n2 t .

|Ψ (t) =

n1 n2

The cross-Kerr medium as a QND—

13.4

In a quantum nondemolition (QND) measurement (Braginsky and Khalili, 1996; Grang-

ier et al., 1998) the quantum back actions of normal measurements”e.g. the random-

ization of the momentum of a free particle induced by a measurement of its position”

are partially avoided by forming an entangled state of the signal with a second system,

called the meter. For the pure cross-Kerr medium in Section 13.4.3, identify a1 and a2

as the signal and meter operators respectively. Assume that the (interaction-picture)