ńņš. 16 |

(2Ļ)

d3 k1 d3 k2 (3) ā’iĻP t

(p ā’ k1 ā’ k2 ) aā (k1 ) aā (k2 ) + HC .

(t) = ā’i C

(3)

Hem 3G e

3

(2Ļ) (2Ļ)

(13.43)

ā

The Hamiltonian has the same form in the Heisenberg picture, with a (k1 ) replaced

by aā (k1 , t), etc. Let

N (k1 , t) = aā (k1 , t) a (k1 , t) (13.44)

denote the (Heisenberg-picture) number operator for the k1 -mode, then a straightfor-

ward calculation using eqn (3.26) yields

d3 k2

ā’iĻP t

(p ā’ k1 ā’ k2 ) aā (k1 , t) aā (k2 , t) ā’ HC .

[N (k1 , t) , H] = ā’2ie C

(3)

3G

(2Ļ)

(13.45)

The illuminated volume of the crystal is typically large on the scale of optical wave-

lengths, so the approximation (13.22) can be used to simplify this result to

[N (k1 , t) , H] = ā’2ieā’iĻP t G(3) aā (k1 , t) aā (p ā’ k1 , t) . (13.46)

In this approximation we see that

[N (k1 , t) ā’ N (p ā’ k1 , t) , H] = 0 , (13.47)

i.e. the diļ¬erence between the population operators for signal and idler photons is

a constant of the motion. An experimental test of this prediction is to measure the

expectation values n (k1 , t) = N (k1 , t) and n (p ā’ k1 , t) = N (p ā’ k1 , t) . This can

be done by placing detectors behind each of a pair of stops that select out a particular

signalā“idler pair (k1 , p ā’ k1 ). According to eqn (13.47), the expectation values satisfy

n (k1 , t) ā’ n (p ā’ k1 , t) = N (k1 , t) ā’ N (p ā’ k1 , t)

= N (k1 , 0) ā’ N (p ā’ k1 , 0)

= 0, (13.48)

which provides experimental evidence that the conjugate photons are created at the

same time.

Ā¼ Nonlinear quantum optics

B Entangled state of the signal and idler photons

Even with pump enhancement, the coupling parameter G(3) (k1 , k2 ) is small, so the

interaction-picture state vector, |ĪØ (t) , for the ļ¬eld can be evaluated by ļ¬rst-order per-

turbation theory. These calculations are simpliļ¬ed by returning to the box-quantized

form (13.40). In this notation, the interaction Hamiltonian is

1

G(3) C (p ā’ k1 ā’ k2 ) eā’iāt aā 1 aā 2 + HC ,

Hem (t) = ā’i

(3)

(13.49)

kk

V

k1 ,k2

where we have transformed to the interaction picture by using the rule (4.98), and in-

troduced the detuning, ā = ĻP ā’Ļ2 ā’Ļ1 , for the down-conversion transition. Applying

the perturbation series (4.103) for the state vector leads to

|ĪØ (t) = |0 + ĪØ(1) (t) + Ā· Ā· Ā· ,

(13.50)

2G(3)

1 sin [āt/2] ā ā

C (p ā’ k1 ā’ k2 ) eā’iāt/2

ĪØ(1) (t) = ā’ ak1 ak2 |0 .

V ā

k1 ,k2

According to the discussion in Chapter 6, each term in the k1 , k2 -sum (with the

exception of the degenerate case k1 = k2 ) describes an entangled state of the signal

and idler photons. Combining the limit, V ā’ ā, of inļ¬nite quantization volume with

the large-crystal approximation (13.22) for C yields

d3 k1 d3 k2 2G(3) 3

|ĪØ (t) = |0 ā’ (2Ļ) Ī“ (p ā’ k1 ā’ k2 )

3 3

(2Ļ) (2Ļ)

sin [āt/2] ā

Ć— eā’iāt/2 a (k1 ) aā (k2 ) |0 . (13.51)

ā

The limit t ā’ ā is relevant for cw pumping, so we can use the identity

sin (āt/2) Ļ

lim eā’iāt/2 = Ī“ (ā) , (13.52)

ā 2

tā’ā

which is a special case of eqn (A.102), to ļ¬nd

d3 k1 d3 k2 1 G(3)

|ĪØ (ā) = |0 ā’ 3 3 2

(2Ļ) (2Ļ)

3

Ć— (2Ļ) Ī“ (p ā’ k1 ā’ k2 ) (2Ļ) Ī“ (ĻP ā’ Ļ1 ā’ Ļ2 )

Ć— aā (k1 ) aā (k2 ) |0 , (13.53)

where Ļ1 = Ļk1 and Ļ2 = Ļk2 .

The conclusion is that down-conversion produces a superposition of states that

are dynamically entangled in energy as well as momentum. The entanglement in en-

ergy, which is imposed by the phase-matching condition, Ļ1 + Ļ2 = ĻP , provides

an explanation for the observation that the two photons are created almost simulta-

neously. A strictly correct proof would involve the second-order correlation function

Ā¼

Three-photon interactions

G(2) (r1 , t1 , r1 , t1 ; r2 , t2 , r2 , t2 ), but the same end is served by a simple uncertainty prin-

ciple argument. If we interpret t1 and t2 as the creation times of the two photons, then

the average time, tP = (t1 + t2 ) /2, can be interpreted as the pair creation time, and

the time interval between the two individual photon creation events is Ļ„ = t1 ā’ t2 . The

respective conjugate frequencies are ā„¦ = Ļ1 +Ļ2 and Ī½ = (Ļ1 ā’ Ļ2 ) /2. The uncertainty

in the pair creation time, ātP ā¼ 1/āā„¦, is large by virtue of the tight phase-matching

condition, ā„¦ ĻP . On the other hand, the individual frequencies have large spectral

bandwidths, so that āĪ½ is large and Ļ„ ā¼ 1/āĪ½ is small. Consequently, the absolute

time at which the pair is created is undetermined, but the time interval between the

creations of the two photons is small.

13.3.3 Experimental techniques and results

Spontaneous down-conversion in a lithium niobate crystal was ļ¬rst observed by Harris

et al. (1967). Shortly thereafter, it was observed in an ammonium dihydrogen phos-

phate (ADP) crystal by Magde and Mahr (1967). A sketch of the apparatus used by

Harris et al. is shown in Fig. 13.2. The beam from an argon-ion laser, operating at a

wavelength of 488 nm, impinges on a lithium niobate crystal oriented so that collinear,

type I phase matching is achieved. The laser beam enters the crystal polarized as an

extraordinary ray. Temperature tuning of the index of refraction allows the adjust-

ment of the wavelength of the down-converted, collinear signal and idler beams, which

are ordinary rays produced inside the crystal. These beams are spectrally analyzed

by means of a prism monochromator, and then detected. In the Magde and Mahr ex-

periment, a pulsed 347 nm beam is produced by means of second-harmonic generation

pumped by a pulsed ruby laser beam. The peak pulse power in the ultraviolet beam is

1 MW, with a pulse duration of 20 ns. Spontaneous down-conversion occurs when the

pulsed 347 nm beam of light enters the ADP crystal. Instead of temperature tuning,

angle tuning is used to produce collinearly phase-matched signal and idler beams of

various wavelengths.

Zelā™dovich and Klyshko (1969) were the ļ¬rst to notice that phased-matched, down-

converted photons should be observable in coincidence detection. Burnham and Wein-

berg (1970) performed the ļ¬rst experiment to observe these predicted coincidences,

and in the same experiment they were also the ļ¬rst to produce a pair of non-collinear

signal and idler beams in SDC. Their apparatus, sketched in Fig. 13.3, uses a 9 mW,

LiNbO3

crystal

Polarizer Filter

Oven Analyzer

4880 A

Prism

argon laser

monochromator

Fig. 13.2 Apparatus used to observe spontaneous down-conversion in 1967 by Harris, Osh-

man, and Byer. (Reproduced from Harris et al. (1967).)

Ā¼ Nonlinear quantum optics

Channel 2

2

PM

From

3520 A Ļ2

laser Spike

filters

Ļ1

ADP

Light

UV pass PM 1

trap

filter Iris

Channel 1

Monochromator PM 2

Lens

Fig. 13.3 Apparatus used by Burnham and Weinberg (1970) to observe the simultaneity

of photodetection of the photon pairs generated in spontaneous down-conversion in an am-

monium dihydrogen phosphate (ADP) crystal. Coincidence-counting electronics (not shown)

is used to register coincidences between pulses in the outputs of the two photomultipliers

PM1 and PM2. These detectors are placed at angles Ļ1 and Ļ2 such that phase matching is

satisļ¬ed inside the crystal for the two members (i.e. signal and idler) of a given photon pair.

(Reproduced from Burnham and Weinberg (1970).)

continuous-wave, heliumā“cadmium, ultraviolet laserā”operating at a wavelength of

325 nmā”as the pump beam to produce SDC in an ADP crystal. The crystal is cut so

as to produce conical rainbow emissions of the signal and idler photon pairs around the

pump beam direction. The ultraviolet (UV) laser beam enters an inch-long ADP crys-

tal, and pairs of phase-matched signal (Ī»1 = 633 nm) and idler (Ī»2 = 668 nm) photons

emerge from the crystal at the respective angles of Ļ1 = 52 mrad and Ļ2 = 55 mrad,

with respect to the pump beam. After passing through the crystal, the pump beam

enters a beam dump which eliminates any background due to scattering of the UV

photons. After passing through narrowband ļ¬ltersā”actually a combination of interfer-

ence ļ¬lter and monochromator in the case of the idler photonā”with 4 nm and 1.5 nm

passbands centered on the signal and idler wavelengths respectively, the individual sig-

nal and idler photons are detected by photomultipliers with near-infrared-sensitive S20

photocathodes. Pinholes with eļ¬ective diameters of 2 mm are used to deļ¬ne precisely

the angles of emission of the detected photons around the phase-matching directions.

Most importantly, Burnham and Weinberg were also the ļ¬rst to use coincidence de-

tection to demonstrate that the phase-matched signal and idler photons are produced

Ā¼

Three-photon interactions

essentially simultaneously inside the crystal, within a narrow coincidence window of

Ā±20 ns, that is limited only by the response time of the electronic circuit.

In more modern versions of the Burnhamā“Weinberg experiment, vacuum photomul-

tipliers are replaced by solid-state silicon avalanche photodiodes (single-photon count-

ing modules), which function exactly like a Geiger counter, except thatā”by means

of an internal discriminatorā”the output consists of standardized TTL (transistorā“

transistor logic), ļ¬ve-volt level square pulses with subnanosecond rise times for each

detected photon. This makes the coincidence detection of single photons much easier.

13.3.4 Absolute measurement of the quantum eļ¬ciency of detectors

In Section 13.3.2 we have seen that the process of spontaneous down-conversion pro-

vides a source of entangled pairs of photons. Burnham and Weinberg (1970) used

coincidence-counting techniquesā”originally developed in nuclear and elementary par-

ticle physicsā”to observe the extremely tight correlation between the emission times of

the two photons. As they pointed out, this correlation allows a direct measurement of

the absolute quantum eļ¬ciency of a photon counter. Migdall (2001) subsequently de-

veloped this suggestion into a measurement protocol. The idea behind this technique is

as follows: when a click occurs in one photon counter (the trigger detector), we are then

certain that there must have been another photon emitted in the conjugate direction,

deļ¬ned by momentum and energy conservation. Thus we know precisely the direction

of emission of the conjugate photon, and also its time of arrivalā”within a very nar-

row time window relative to the trigger photonā”at any point along its direction of

propagation.

As shown in Fig. 13.4, the procedure is to place the detector under test (DUT) and

the trigger detector so that the coincidence counter can only be triggered by signals

from a single entangled pair. For a long series of measurements, the respective quantum

eļ¬ciencies Ī·1 and Ī·2 of the trigger detector and the DUT are deļ¬ned by

N1 = Ī·1 N (13.54)

Ī·2

Counter

,76

Ļ2

Coincidence Ī·2

counter

ĻF

Absolute

quantum

efficiency

Parametric Ļ1

crystal Counter

6HECCAH

Fig. 13.4 Scheme for absolute measurement of quantum eļ¬ciency. A pair of entangled

photons originating in the crystal head toward the ā˜triggerā™ detector and the ā˜detector under

testā™ (DUT). The parameter Ī·2 is the quantum eļ¬ciency for the entire path from the point

of emission to the DUT. (Reproduced from Migdall (2001).)

Ā¼ Nonlinear quantum optics

and

N2 = Ī·2 N , (13.55)

where N is the total number of conjugate photon pairs emitted by the crystal into the

directions of the two detectors, N1 is the number of counts registered by the trigger

detector, and N2 is the number of counts registered by the DUT. We may safely

assume that the clicks at the two detectors are uncorrelated, so the probability of a

coincidence count is Ī·coinc = Ī·1 Ī·2 . Thus the number of coincidence counts is

Ncoinc = Ī·1 Ī·2 N , (13.56)

and combing this with eqn (13.54) shows that the absolute quantum eļ¬ciency Ī·2 of

the DUT is the ratio

Ncoinc

Ī·2 = (13.57)

N1

of two measurable quantities. The beauty of this scheme is that this result is indepen-

dent of the quantum eļ¬ciency, Ī·1 , of the trigger detector.

Systematic errors, however, must be carefully taken into account. Any losses along

the optical pathā”from the point of emission of the twin photons inside the crystals

all the way to the point of detection in the DUTā”will contribute to a systematic

error in the measurement. Thus the exit face of the crystal must be carefully anti-

reļ¬‚ection coated, and measured. Care also must be taken to use a large enough iris

in the collection optics for the conjugate photon. This will minimize absorption, by

the iris, of photons which should have impinged on the DUT. Furthermore, this iris

must be carefully aligned, so that it passes all photons propagating in the conjugate

direction determined by phase matching with the trigger photon. This ensures that no

conjugate photons are missed due to misalignment. This alignment error can, however,

be minimized by maximizing the detected signal as a function of small transverse

motions of the test detector.

However, the most serious systematic error arises in the electronic, rather than the

optical, part of the system. The electronic gate window used in the coincidence counter

is usually not a perfectly rectangular pulse shape; typically, it has small tails of lesser

counting eļ¬ciency, due to which some coincidence counts can be missed. These tails

can, however, be calibrated out in separate electronic measurements of the coincidence

circuitry.

13.3.5 Two-crystal source of hyperentangled photon pairs

For many applications of quantum optics, e.g. quantum cryptography, quantum dense

coding, quantum entanglement-swapping, quantum teleportation, and quantum com-

putation, it is very convenientā”and often necessaryā”to employ an intense source of

hyperentangled pairs of photons, i.e. photons that are entangled in two or more

degrees of freedom. A particularly simple, and yet powerful, light source which yields

photon pairs entangled in polarization and momentum was demonstrated by Kwiat

et al. (1999b).

A schematic of the apparatus used for generating hyperentangled photon pairs

with high intensity is shown in Fig. 13.5. The heart of this photon-pair light source

Ā¼

Three-photon interactions

0

8

o

45

(a)

(b)

Fig. 13.5 (a) High-intensity spontaneous down-conversion light source: two identical, thin,

highly nonlinear crystals are stacked in a ā˜crossedā™ conļ¬guration, i.e. the crystal axes lie in

perpendicular planes, as indicated by the diagonal markings on the sides. The crystals are

so thin that it is not possible to tell if a given photon pair emitted by the stack comes from

the ļ¬rst or from the second crystal. Hence the crossed stack produces polarization-entangled

pairs of photons. (b) Schematic of apparatus to produce and to characterize this photon-pair

light source. (Reproduced from Kwiat et al. (1999b).)

consists of two identically cut, thin (0.59 mm), type I down-conversion crystalsā”Ī²

barium borate (BBO)ā”that are stacked in a crossed conļ¬guration, i.e. with their

optic axes lying in perpendicular planes. What we will call the vertical plane is deļ¬ned

by the optic axis of the ļ¬rst crystal and the direction of the pump beam, while the

horizontal plane is deļ¬ned by the optic axis of the second crystal and the pump beam.

The crystals are suļ¬ciently thin so that the waist of the pump beamā”a continuous-

wave, ultraviolet (wavelength 351 nm), argon-ion laserā”overlaps both. Since these

are birefringent (type I) crystals, the ultraviolet pump enters as an extraordinary

ray, and the pair of red, down-converted photon beams leave as ordinary rays. The

two crystals are identically cut with their optic axes oriented at 33.9ā—¦ with respect

to the normal to the input face. The phase-matching conditions guarantee that two

degenerate-frequency photons at 702 nm wavelength are emitted into a cone with a

half-opening angle of 3.0ā—¦ .

Under certain conditions, this arrangement allows one to determine the crystal of

origin of the twin photons. For example, if the pump laser is V -polarized (i.e. linearly-

polarized in the vertical plane), then type I down-conversion would only occur in the

ļ¬rst crystal, which would produce H-polarized (i.e. linearly-polarized in the horizontal

plane) twin photons. Similarly, if the pump laser were H-polarized, then type I down-

conversion would only occur in the second crystal, which would produce V -polarized

twin photons. However, suppose that the pump laser polarization is neither horizontal

nor vertical, but instead makes an angle of 45ā—¦ with respect to the vertical axis. This

state is a coherent superposition, with equal amplitudes, of horizontal and vertical

polarizations. Thus when this 45ā—¦ -polarized pump beam is incident on the two-crystal

Ā½Ā¼ Nonlinear quantum optics

stack, a down-conversion event can occur, with equal probability, either in the ļ¬rst or

in the second crystal. If the photon pair originates in the ļ¬rst crystal, both photons

would be H-polarized, whereas if the photon pair originates in the second crystal, both

photons would be V -polarized.

The thickness of each crystal is much smaller than the Rayleigh range (a few

centimeters) of the pump beam, and diļ¬raction ensures that the spatial modesā”i.e.

the cones of emission in Fig. 13.5(a)ā”from the two crystals overlap in the far ļ¬eld,

where the photons are detected. This situation provides the guiding principle behind

this light source: for a 45ā—¦ -polarized pump beam, it is impossibleā”even in principleā”

to know whether a given photon pair originated in the ļ¬rst or in the second crystal. We

must therefore apply Feynmanā™s superposition rule to obtain the state at the output

of the pair of tandem crystals. If the crystals are identical in thickness and the pump

is normally incident on the crystal face, the result is the entangled state

1 1

Ī¦+ = ā |1k1 H , 1k2 H + ā |1k1 V , 1k2 V . (13.58)

2 2

The notation 1k1 H denotes the horizontal polarization state of one member of the pho-

ton pairā”originating in the ļ¬rst crystalā”and 1k2 H denotes the horizontal polarization

state of the conjugate member, also originating in the ļ¬rst crystal. Similarly, 1k1 V de-

notes the vertical polarization state of one member of the photon pairā”originating in

the second crystalā”and 1k2 V denotes the vertical polarization state of the conjugate

member, also originating in the second crystal. The phase-matching conditions ensure

that the down-converted photon pairs are emitted into azimuthally conjugate direc-

tions along rainbow-like cones, so that they are entangled both in momentum and in

polarization. Hence this light source produces hyperentangled photon pairs.

The entangled state |Ī¦+ is one of the four Bell states deļ¬ned by

1 1

Ī¦+ ā” ā |1k1 H , 1k2 H + ā |1k1 V , 1k2 V , (13.59)

2 2

1 1

Ī¦ā’ ā” ā |1k1 H , 1k2 H ā’ ā |1k1 V , 1k2 V , (13.60)

2 2

1 1

ĪØ+ ā” ā |1k1 H , 1k2 V + ā |1k1 V , 1k2 H , (13.61)

2 2

1 1

ĪØā’ ā” ā |1k1 H , 1k2 V ā’ ā |1k1 V , 1k2 H . (13.62)

2 2

These are maximally entangled states that form a basis set for the polarization states

of pairs of entangled photons with wavevectors k1 and k2 . The states |Ī¦+ and |Ī¦ā’

can be generated by two crossed type I crystals, and the states |ĪØ+ and |ĪØā’ can be

generated by a pair of crossed type II crystals.

More generally, the two crystals could be tilted away from normal incidence around

an axis perpendicular to the direction of the pump laser beam. This would result in

phase changes which lead to the output entangled state

eiĪ¾

1

Ī¦ ; Ī¾ = ā |1k1 H , 1k2 H + ā |1k1 V , 1k2 V ,

+

(13.63)

2 2

Ā½Ā½

Three-photon interactions

where the phase Ī¾ depends on the tilt angle. Instead of tilting the two tandem crystals,

it is more convenient to tilt a quarter-wave plate placed in front of them, so that an

elliptically-polarized pump beam emerges from the quarter-wave plate with the major

axis of the ellipse oriented at 45ā—¦ with respect to the vertical. Then the down-converted

photon pair emerges from the tandem crystals in the entangled state |Ī¦+ ; Ī¾ , with

a nonvanishing phase diļ¬erence Ī¾ between the Hā“H and V ā“V polarization-product

states. The phase of the entanglement parameter Ī¾ can be easily adjusted by changing

the relative phase between the horizontal and vertical polarization components of the

pump light, i.e. by changing the ellipticity of the ultraviolet laser beam polarization.

In the actual experiment, schematically shown Fig. 13.5(b), a combination of a

prism and an iris acts as a ļ¬lter to separate out the ultraviolet laser pump beam from

the unwanted ļ¬‚uorescence of the argon-ion discharge tube. A polarizing beam splitter

(PBS) acts as a preļ¬lter to select a linear polarization of the laser beam. Following

this, a half-wave plate (HWP) allows the selected linear polarization to be rotated

around the laser beam axis. The beam then enters a quarter-wave plate (QWP)ā”

whose tilt angle allows the adjustment of the relative phase Ī¾ of the entangled state

in eqn (13.63)ā”placed in front of the tandem crystals (BBO).

Separate half-wave plates (HWP) and polarizing beam splitters (PBS) provide

polarization analyzers, placed in front of detectors 1 and 2, that allow independent

variations of the two angles of linear polarization, Īø1 and Īø2 , of the photons detected

by Geiger counters 1 and 2, respectively. The irises in front of these detectors were

around 2 mm in size, and the interference ļ¬lters (IF) had typical bandwidths of 5 nm

in wavelength. The iris sizes and interference-ļ¬lter bandwidths were determined by

the criterion that the detection should occur in the far ļ¬eld of the crystals, and by

phase-matching considerations.

Under these conditions, with a 150 mW incident pump beam and a 10% solid-angle

collection eļ¬ciencyā”arising from the ļ¬nite sizes of the irises placed in front of the

detectorsā”the hyperentangled pair production rate was around 20, 000 coincidences

per second. Standard coincidence detection of the correlated photon pairs in this ex-

periment was accomplished by means of solid-state Geiger counters (silicon avalanche

photodiodes with around 70% quantum eļ¬ciency, operated in the Geiger mode), in

conjunction with a time-to-amplitude converter and a single-channel analyzer, with a

coincidence time window of 7 ns. The polarization states of the individual photons were

analyzed by means of rotatable linear polarizers, with the analyzer angle for detector

2 being rotated relative to that of detector 1 (whose analyzer angle was kept ļ¬xed at

ā’45ā—¦ ).

Typical data are shown in Fig. 13.6. The singles rate (the output of an individ-

ual Geiger counter) shows no dependence on the relative angle of the two analyzers,

indicating that the photons were individually unpolarized. On the other hand, coin-

cidence measurements showed that the relative polarization of one photon in a given

entangled pair with respect to the conjugate photon was very high (with a visibility

of 99.6 Ā± 0.3%). This means that an extremely pure two-photon entangled state has

been produced with a high degree of polarization entanglement. Such a high visibility

in the two-photon coincidence fringes indicates a violation of Bellā™s inequalitiesā”see

eqn (19.38)ā”by over 200 standard deviations, for data collected in about 3 minutes.

Ā½Ā¾ Nonlinear quantum optics

s)

s)

ā’

Īø2 (Īø1 =

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

ratesā”the outputs of the individual Geiger countersā”(indicated by squares, with values on

the right axis) versus the relative angle Īø2 ā’Īø1 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 Īø2 with Īø1 kept ļ¬xed at ā’45ā—¦ . (Reproduced from Kwiat et al. (1999b).)

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

Ī¾ continuously over a range from 0 to 5.5Ļ by tilting the quarter-wave plate, placed

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

13.4 Four-photon interactions

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

third-order susceptibility Ļ(3) . The parity argument shows that Ļ(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

photonā“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 ļ¬nal photon. The energy and momentum conservation (phase matching) rules

are then

Ļ0 = Ļ1 + Ļ2 + Ļ3 , (13.64)

k0 = k1 + k2 + k3 , (13.65)

and the Feynman diagram is shown in Fig. 13.7(a). In the degenerate case Ļ1 = Ļ2 =

Ļ3 = Ļ, energy conservation requires Ļ0 = 3Ļ. This eļ¬ect was ļ¬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 ļ¬nal photons are all entangled.

Ā½Āæ

Four-photon interactions

(k0, Ļ0)

(k2, Ļ2)

(k1, Ļ1) (k3, Ļ3)

(k1, Ļ1) (k3, Ļ3)

Fig. 13.7 (a) Sum-frequency generation with

(k2, Ļ2) (k0, Ļ0)

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 Photonā“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

Ļ0 + Ļ1 = Ļ2 + Ļ3 , (13.66)

k0 + k1 = k2 + k3 . (13.67)

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

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

tion of electronā“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 diļ¬erent, since the incident pho-

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

photonā“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, Ļ2) (k3, Ļ3)

(k0, Ļ0) (k1, Ļ1) Fig. 13.8 Photonā“photon scattering medi-

ated by interaction with atoms in the medium.

Ā½ 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 Ī³ (k2 , k3 ; k0 , k1 ) C (k2 + k3 ā’ k1 ā’ k0 ) aā 3 aā 2 ak1 ak0 ,

Hint = kk

4V2

k0 ,k1 ,k2 ,k3

(13.68)

ā—

where the coupling constants satisfy Ī³ (k2 , k3 ; k0 , k1 ) = Ī³ (k0 , k1 ; k2 , k3 ), and C (k)

is deļ¬ned by eqn (13.20).

B Experimental observation of photonā“photon scattering

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

mediated by the atoms in a rubidium vapor cellā”leading to 90ā—¦ 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 ļ¬‚uorescence. The resonant enhancement of the coupling is what makes this

experiment possible, by contrast to the observation of photonā“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) suļ¬ces for a qualitative treat-

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

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

photonsā”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 retroreļ¬‚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 ļ¬‚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 ļ¬ber that

spatially ļ¬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 retroreļ¬‚ecting mirror,

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

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

1.6 Ć— 1010 atoms/cm3 .

The two colliding photonsā”like two hard spheresā”will sometimes scatter oļ¬ 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

Ā½

Four-photon interactions

Diode laser Isolator Isolator

Reference

Rb cell

Reflection

monitor Input fiber

coupler

Output fiber

coupler

SM/PM

fiber

Cell enclosure

Heā’Ne

laser

SPCM SPCM

Lightproof detection enclosure

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

atoms excited oļ¬ 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 heliumā“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 photonā“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.

Ā½ Nonlinear quantum optics

1.1

9coincidence (arb. units)

1.05

1

0.95

ā’2 ā’1 0 1 2 3

Ļ„A ā’ Ļ„B (ns)

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

bidium atoms excited oļ¬ 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 ļ¬t using three measured parameters: the beam shape, the ļ¬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 (ā’iĻ0 t), is given by

2

P (3) = Ļ(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 + Ļ(1) . The nonlinear optical response is conveniently described

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

2

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

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

2

n (E) = n + n2 |E| + Ā· Ā· Ā· ,

(13.71)

1 Ļ(3)

n2 = .

2n

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

n (E) = n + n2 I + Ā· Ā· Ā· ,

(13.72)

Āµ0 Ļ(3)

n2 = .

2

0n

Ā½

Four-photon interactions

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

the intensity of the ļ¬eld is called the optical Kerr eļ¬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 Ļ• = kL = n (E) L/c, and the increment in

phase due to the intensity-dependent term is

Ļ 2Ļn2 I

āĻ• = n2 I L = L. (13.73)

c Ī»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 eļ¬ect, the interaction Hamiltonian is given

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

tions. We consider an experimental conļ¬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 = Ļ 0 aā a + gaā 2 a2 , (13.74)

2

where the coupling constant g is proportional to Ļ(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 ā’ N ,

H = Ļ0 N + (13.75)

2

where N = aā 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 ļ¬xed and only the phase changes.

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

the annihilation operator:

da (t)

= ā’iĻ0 a (t) ā’ igaā (t) a2 (t)

dt

= ā’i (Ļ0 + gN ) a (t) . (13.76)

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

a (t) = eā’i(Ļ0 +gN )t a , (13.77)

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

ā

m |a (t)| m = eā’i(Ļ0 +gm)t m |a| m = Ī“m,m ā’1 m + 1eā’i(Ļ0 +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 āĻ•.

Ā½ Nonlinear quantum optics

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

monochromatic ļ¬elds with diļ¬erent carrier frequencies:

E (t) = E 1 (t) exp (ā’iĻ1 t) + E 2 (t) exp (ā’iĻ2 t) . (13.79)

2 2

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

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

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

the Hamiltonian is

g1 ā 2 2 g2 ā 2 2

H = Ļ 1 aā a1 + Ļ 2 aā a2 + a2 a2 + g12 aā aā 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

Ļ(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 eļ¬ective energy expression (13.18) for

(3)

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

1

gs0 s1 s2 s3 (Ļ1 , Ļ2 , Ļ3 ) [Ī±ā— Ī±1 Ī±2 Ī±3 + CC]

Uem =

(4) (4)

0

V2

k0 s0 ,...,k3 s3

Ć— Ī“Ļ0 ,Ļ1 +Ļ2 +Ļ3 C (k0 ā’ k1 ā’ k2 ā’ k3 )

1

fs0 s1 s2 s3 (Ļ1 , Ļ2 , Ļ3 ) [Ī±0 Ī±1 Ī±ā— Ī±ā— + CC]

(4)

+2 23

V

k0 s0 ,...,k3 s3

Ć— Ī“Ļ0 +Ļ1 ,Ļ2 +Ļ3 C (k0 + k1 ā’ k2 ā’ k3 ) ,

where

gs0 s1 s2 s3 (Ļ1 , Ļ2 , Ļ3 ) = ā’ 0 F0 F1 F2 F3 Ļ(3) 1 s2 s3 (Ļ1 , Ļ2 , Ļ3 ) ,

(4)

s0s

3

fs0 s1 s2 s3 (Ļ1 , Ļ2 , Ļ3 ) = 0 F0 F1 F2 F3 Ļ(3) 1 s2 s3 (Ļ1 , ā’Ļ2 , ā’Ļ3 ) ,

(4)

s0s

4

and

(2)

Ļ(3) 1 s2 s3 (Ļ1 , Ļ2 , Ļ3 ) = (Īµk0 s0 )i (Īµk1 s1 )j (Īµk2 s2 )k (Īµk3 s3 )l Ļijkl (Ļ1 , Ļ2 , Ļ3 ) .

s0 s

13.2 Kerr medium

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

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

eā’igt ā’ 1 |Ī±|

2

Ī± |a (t)| Ī± = exp .

Ā½

Exercises

2

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

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

of Ī± |a (t)| Ī± . 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 Ī± |a (t)| Ī±

as t ā’ ā?

(3) Using the form (13.75) of the Hamiltonian, exhibit the solution of the SchrĀØdinger

o

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

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

of part (2) and to decide if |Ļ (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) = aā (t) a1 (t) and N2 (t) = aā (t) a2 (t) are con-

1 2

stants of the motion.

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

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

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

interaction-picture SchrĀØdinger equation is

o

n1 n2 | ĪØ (0) eā’ig12 n1 n2 t .

|ĪØ (t) =

n1 n2

The cross-Kerr medium as a QNDā—

13.4

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

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

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

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)

input state is |ĪØ (0) = |n1 , Ī±2 , i.e. a number state for the signal and a coherent state

for the meter.

(1) Use the results of Exercise 13.3 to show that |ĪØ (t) = n1 , Ī±2 eā’iĪ³n1 , where

Ī³ = g12 t.

(2) Devise a homodyne measurement scheme that can distinguish between the phase

shifts experienced by the meter beam for diļ¬erent values of n1 , e.g. n1 = 0 and

n1 = 1. For example, measure the quadrature X2 = a2 exp [ā’iĻ•] + aā exp [iĻ•] /2,

2

where Ļ• is the phase of the local oscillator in the homodyne apparatus.

14

Quantum noise and dissipation

In the majority of the applications considered so farā”e.g. photons in an ideal cavity,

photons passing through passive linear media, atoms coupled to the radiation ļ¬eld,

etc.ā”we have neglected all dissipative eļ¬ects, such as absorption and scattering. In

terms of the fundamental microscopic theory, this means that all interactions between

the system under study and the external world have been ignored. When this assump-

tion is in force, the system is said to be closed. The evolution of a closed system

is completely determined by its Hamiltonian. A pure state of a closed system is de-

scribed by a state vector obeying the SchrĀØdinger equation (2.108), and a mixed state

o

is represented by a density operator obeying the quantum Liouville equation (2.119).

With the possible exception of the entire universe, the assumption that a system is

closed is always an approximation. Every experimentally relevant physical system is

unavoidably coupled to other physical systems in its vicinity, and usually very little is

known about the neighboring systems or about the coupling mechanisms. If interac-

tions with the external world cannot be neglected, the system is said to be open. In

this chapter, we begin the study of open systems.

14.1 The world as sample and environment

For the discussion of open systems, we will divide the world into two parts: the

sample1 ā”the physical objects of experimental interestā”and the environmentā”

everything else. Deciding which degrees of freedom should be assigned to the sample

and which to the environment requires some care, as we will shortly see.

In fact, we have already studied three open systems in previous chapters. In the

discussion of blackbody radiation in Section 2.4.2, the radiation ļ¬eld is assumed to be

in thermal equilibrium with the cavity walls. In this case the sample is the radiation

ļ¬eld in the cavity, and some coupling to the cavity walls (the environment) is required

to enforce thermal equilibrium. In line with standard practice in statistical mechanics,

we simply assume the existence of a weak coupling that imposes equilibrium, but

otherwise plays no role. In the discussion of the Weisskopfā“Wigner method in Section

11.2.2 the sample is a two-level atom, and the modes of the radiation ļ¬eld are assigned

to the environment. In this case, an approximate treatment of the coupling to the

environment leads to a derivation of the irreversible decay of the excited atom. A

purely phenomenological treatment of other dissipative terms in the Bloch equation

for the two-level atom can be found in Section 11.3.3.

1 Overuse has leached almost all meaning from the word ā˜systemā™, so we have replaced it with

ā˜sampleā™ for this discussion.

Ā¾Ā½

The world as sample and environment

As an illustration of the choices involved in separating the world into sample and

environment, we begin by revisiting the problem of transmission through a stop. In

Section 8.7 the radiation ļ¬eld is treated as a closed system by assuming that the

screen is a perfect reļ¬‚ector, and by including both the incident and the reļ¬‚ected

modes in the sample. Let us now look at this problem in a diļ¬erent way, by assigning

the reļ¬‚ected modesā”i.e. the modes propagating from right to left in Fig. 8.5ā”to

the environment. The newly deļ¬ned sample consists of the modes propagating from

left to right. It is clearly an open system, since the right-going modes of the sample

scatter into left-going modes that belong to the environment. The loss of photons from

the sample represents dissipation, and the result (8.82) shows that this dissipation is

accompanied by an increase in ļ¬‚uctuations of photon number in the transmitted ļ¬eld.

This is a simple example of a general principle which is often called the ļ¬‚uctuation

dissipation theorem.

14.1.1 Reservoir model for the environment

Our next task is to work out a more systematic way of dealing with open systems.

This eļ¬ort would be doomed from the start if it required a detailed description of the

environment, but there are many experimentally interesting situations for which such

knowledge is not necessary. These favorable cases are characterized by generalizations

of the conditions required for the Weisskopfā“Wigner (WW) treatment of spontaneous

emission.

(1) The modes of the environment (the radiation ļ¬eld for WW) have a continuous

spectrum.

(2) The sample (the two-level atom for WW) hasā”to a good approximationā”the

following properties.

(a) The sample Hamiltonian has a discrete spectrum. This is guaranteed if the

sample (like the atom) has a ļ¬nite number of degrees of freedom. If the sample

has an inļ¬nite number of degrees of freedom (like the radiation ļ¬eld) a discrete

spectrum is guaranteed by conļ¬nement to a ļ¬nite region of space, e.g. a cavity.

(b) The sample is weakly coupled to a broad spectral range of environmental

modes.

In the Weisskopfā“Wigner model these features justify the Markov approximation. Ap-

plying the general rule (11.23) of the resonant wave approximation to the WW model

provides the condition

|ā„¦ks | āK Ļ21 , (14.1)

where |ā„¦ks | is the one-photon Rabi frequency deļ¬ned in eqn (4.153), and āK is the

width of the cut-oļ¬ function for the RWA. This inequality deļ¬nes what is meant by

coupling to a broad spectral range of the radiation ļ¬eld.

Turning now to the general problem, we assume the environmental degrees of free-

dom that couple to the sample have continuous spectra, and that the coupling is weak.

Expressing the characteristic coupling strength as ā„¦S deļ¬nes a characteristic response

frequency ā„¦S , and the condition of weak coupling to a broad range of environmental

excitations is

ā„¦S āE ĻS . (14.2)

Ā¾Ā¾ Quantum noise and dissipation

Here āE is the spectral width of the environmental modes that are coupled to the

sample, and ĻS is a characteristic mode frequency for the unperturbed sample.

In the Weisskopfā“Wigner model, the environment is the radiation ļ¬eld, and we

have a detailed theory for this example. This luxury is missing in the general case,

so we will instead devise a generic model that is based on the assumption of weak

interaction between the sample and the environment. An important consequence of

this assumption is that the sample can only excite low energy modes of the environ-

ment. As we have previously remarked, the low-lying modes of many systems can be

approximated by harmonic oscillators. For example, suppose that the environment

includes some solid material, e.g. the walls of a cavity, and that interaction with the

sample excites vibrations in the crystal lattice of the solid. In the quantum theory

of solids, these lattice vibrations are called phonons (Cohen-Tannoudji et al. 1977a,

Complement JV, p. 586; Kittel 1985, Chap. 2). The Ī½th phonon modeā”which is an

analogue of the ks-mode of the radiation ļ¬eldā”is represented by a harmonic oscil-

lator with fundamental frequency ā„¦Ī½ , analogous to Ļks . For macroscopic bodies, the

discrete index Ī½ becomes eļ¬ectively continuous, so this environment has a continuous

spectrum. Generalizing from this example suggests modeling the environment by one

or more families of harmonic oscillators with continuous spectra. Each family of oscil-

lators is called a reservoir. Weak coupling to the reservoir implies that the amplitudes

of the oscillator displacements and momenta will be small; therefore, we will make the

crucial assumption that the interaction Hamiltonian HSE is linear in the creation and

annihilation operators for the reservoir modes.

Within this schematic model of the worldā”the combined system of sample and

environmentā”the reservoirs can be grouped into two classes, according to their uses. A

reservoir which is not itself subjected to any experimental measurements will be called

a noise reservoir. In this case, the reservoir model simply serves as a useful theoretical

device for describing dissipative eļ¬ects. This is the most common situation, but there

are important applications in which the primary experimental signal is carried by

the modes of one of the reservoirs. In these cases, we will call the reservoir under

observation a signal reservoir. In the optical experiments discussed below, the signal

reservoir excitations areā”naturally enoughā”photons.

For noise reservoirs, the objective is to carry out an approximate elimination of

the reservoir degrees of freedom, in order to arrive at a description of the sample as

an open system. The two principal methods used for this purpose are the quantum

Langevin equations for the ļ¬eld operator and atomic operator (which are formulated

in the Heisenberg picture) and the master equation for the density operator (which

is expressed in the interaction picture). The Langevin approach is, in some ways,

more intuitive and technically simpler. It is particularly useful for problems that have

simple analytical solutions or are amenable to perturbation theory, but it produces

equations of motion for sample operators that do not lend themselves to the numerical

simulations required for more complex problems. For such cases, the approach through

the master equation is essential. We will explain the Langevin method in the present

chapter, and introduce the master equation in Chapter 18.

In the case of a signal reservoirā”which, after all, carries the experimental inform-

ationā”it would evidently be foolish to eliminate the reservoir degrees of freedom.

Ā¾Āæ

The world as sample and environment

Instead, the objective is to determine the eļ¬ect of the sample on the reservoir modes

to be observed. Despite this diļ¬erence in aim, the theoretical techniques developed

for dealing with noise reservoirs can also be applied to signal reservoirs. The principal

reason for this happy outcome is the assumption that both kinds of reservoirs are

coupled to the sample by an interaction Hamiltonian that is linear in the reservoir

operators. This approach to signal reservoirs, which is usually called the inputā“output

method, is described in Section 14.3.

A The world Hamiltonian

The division of the world into sample and environment implies that the Hilbert space

for the world is the tensor product,

HW = HS ā— HE , (14.3)

of the sample and environment spaces. For most applications, it is necessary to model

the environment by means of several independent reservoirs; therefore, the space HE

is itself a tensor product,

HE = H1 ā— H2 ā— Ā· Ā· Ā· ā— HNres , (14.4)

of the Hilbert spaces for the Nres independent reservoirs that deļ¬ne the environment.

Pure states, |Ļ , in HW are linear combinations of product states:

|Ļ = C1 |ĪØ1 |Ī1 + C2 |ĪØ2 |Ī2 + Ā· Ā· Ā· , (14.5)

where |ĪØj and |Īj belong respectively to HS and HE . In most situations, however,

both the sample and the reservoirs must be described by mixed states.

In general, the sample may be acted on by time-dependent external classical ļ¬elds

or currents, and its constituent parts may interact with each other. Thus the total

SchrĀØdinger-picture Hamiltonian for the sample is

o

HS (t) = HS0 + HS1 (t) , (14.6)

where HS0 is the noninteracting part of the sample Hamiltonian. The interaction term

HS1 (t) is

HS1 (t) = HSS + VS (t) , (14.7)

where HSS describes the internal sample interactions and VS (t) represents any inter-

actions with external classical ļ¬elds or currents. The time dependence of the external

ļ¬elds is the source of the explicit time dependence of VS (t) in the SchrĀØdinger picture.

o

In typical cases, VS (t) is a linear function of the sample operators. The Hamiltonian

for the isolated sample is

HS = HS0 + HSS . (14.8)

The total SchrĀØdinger-picture Hamiltonian for the world is then

o

HW = HS (t) + HE + HSE , (14.9)

where

Ā¾ Quantum noise and dissipation

Nres

HE = HJ (14.10)

J=1

is the free Hamiltonian for the environment, HJ is the Hamiltonian for the Jth reser-

voir,

Nres

(J)

HSE = HSE (14.11)

J=1

is the total interaction Hamiltonian between the sample and the environment, and

(J)

HSE is the interaction Hamiltonian of the sample with the Jth reservoir. The world

is, by deļ¬nition, a closed system.

We will initially use a box-quantization description of the reservoir oscillators that

parallels the treatment of the radiation ļ¬eld in Section 3.1.4, i.e. each family of oscil-

lators will be labeled by a discrete index Ī½. The free Hamiltonian for reservoir J is

therefore given by

ā„¦Ī½ bā bJĪ½ ,

HJ = (14.12)

JĪ½

Ī½

where bJĪ½ is the annihilation operator for the Ī½th mode of the Jth reservoir. We have

simpliļ¬ed the model by assuming that each reservoir has the same set of fundamen-

tal frequencies {ā„¦Ī½ }, rather than a diļ¬erent set {ā„¦JĪ½ } for each reservoir. This is not

a serious restriction, since in the continuum limit each ā„¦JĪ½ is replaced by a contin-

uous variable ā„¦. The kinematical independence of the reservoirs is imposed by the

commutation relations

bJĪ½ , bā

[bJĪ½ , bKĀµ ] = 0 , KĀµ = Ī“JK Ī“Ī½Āµ .

In typical applications, the sample is coupled to the environment through sample

operators, OJ , that can be chosen to satisfy

[OJ , HS0 ] ā ĻJ OJ , (14.13)

where ĻJ 0. For ĻJ > 0, this means that OJ is an approximate energy-lowering

operator for the unperturbed sample Hamiltonian HS0 . We will also need the limiting

case ĻJ = 0, which means that OJ is an approximate constant of the motion.

In the resonant wave approximation, the sampleā“environment interaction can be

written as

vJ (ā„¦Ī½ ) OJ bJĪ½ ā’ bā OJ ,

ā

(J)

HSE = i (14.14)

JĪ½

Ī½

where vJ (ā„¦Ī½ ) is a real, positive coupling frequency. This ansatz incorporates the

assumption that each sampleā“reservoir interaction Hamiltonian is a linear function of

the reservoir operators. The restriction to real coupling frequencies is not signiļ¬cant, as

shown in Exercise 14.1. Each coupling frequency is a candidate for the characteristic,

sample-response frequency ā„¦S , so it must satisfy the condition

vJ (ā„¦Ī½ ) āE ĻS . (14.15)

The choice of the sample operator OJ is determined by the physical damping mecha-

nism associated with the Jth reservoir.

Ā¾

The world as sample and environment

B The world density operator

The probability distributions relevant to experiments are determined by the SchrĀØ- o

S

dinger-picture density operator, ĻW (t), that describes the state of the world. We

must, therefore, begin by choosing an initial form, ĻS (t0 ), for the density operator.

W

The natural assumption is that the sample and the reservoirs are uncorrelated for

a suļ¬ciently early time t0 . Since the time-independent, Heisenberg-picture density

operator, ĻH , satisļ¬es ĻH = ĻS (t0 ), this is equivalent to assuming that

W W W

ĻW = ĻS ĻE , (14.16)

where ĻS acts on HS , and ĻE acts on HE . We have dropped the superscript H, since

the remaining argument is conducted entirely in the Heisenberg picture. Furthermore,

it is equally natural to assume that the various reservoirs are mutually uncorrelated

at the initial time, so that

ĻE = Ļ1 Ļ2 . . . ĻNres , (14.17)

where ĻJ acts on HJ for J = 1, 2, . . . , Nres . One or more of the density operators ĻJ is

often assumed to describe a thermal equilibrium state, in which case the corresponding

reservoir is called a heat bath.

The average value of any observable O is given by

O = TrW (ĻW O) , (14.18)

where TrW is deļ¬ned by the sum over a basis set for HW = HS ā— HE . By using the

deļ¬nition of partial traces in Section 6.3.1, it is straightforward to show that

TrW (SR) = (TrS S) (TrE R) , (14.19)

if S acts only on HS and R acts only on HE . The average of an operator product, SR,

with respect to the world density operator ĻW = ĻS ĻE is then

SR = TrW [(ĻS ĻE ) (SR)]

= TrW [ĻS S ĻE R]

= [TrS (ĻS S)] [TrE (ĻE R)]

=S R, (14.20)

where the identities (6.17) and (14.19) were used to get the second and third lines.

Applying this relation to S = 1 (more precisely, S = IS , where IS is the identity

operator for HS ), and R = RJ RK , where RJ acts on HJ , RK acts on HK , and J = K,

yields

RJ RK = RJ RK . (14.21)

In other words, distinct reservoirs are statistically independent.

C Noise statistics

The statistical independence of the various reservoirs allows them to be treated indi-

vidually, so we drop the reservoir index in the present section. For most experimental

Ā¾ Quantum noise and dissipation

arrangements, the reservoir is not subjected to any special preparation; therefore, we

will assume that distinct reservoir modes are uncorrelated, i.e. the reservoir density

operator is factorizable:

Ļ= ĻĪ½ , (14.22)

Ī½

where ĻĪ½ is the density operator for the Ī½th mode. For operators FĪ½ and GĀµ that are

respectively functions of bĪ½ , bā and bĀµ , bā , this assumption implies FĪ½ GĀµ = FĪ½ GĀµ

Ī½ Āµ

for Āµ = Ī½.

For the discussion of quantum noise, only ļ¬‚uctuations around mean values are of

interest. We will say that a factorizable density operator Ļ is a noise distribution if

the natural oscillator variables bĪ½ and bā satisfy

Ī½

bā = bĪ½ = 0 for all Ī½ . (14.23)

Ī½

These conditions can always be achieved by using the ļ¬‚uctuation operator Ī“bĪ½ =

bĪ½ ā’ bĪ½ in place of bĪ½ . By means of suitable choices of the operators FĪ½ and GĀµ ,

the combination of eqns (14.23) and (14.22) can be used to derive restrictions on the

moments of a noise distribution Ļ. For example, the results

bā bĀµ = bĀµ bā = bā bĀµ = 0 and b Ī½ bĀµ = bĪ½ bĀµ = 0 for Āµ = Ī½ (14.24)

Ī½ Ī½ Ī½

lead to the useful rules

bā bĀµ = Ī“Ī½Āµ bā bĪ½ , bĀµ bā = Ī“Ī½Āµ bĪ½ bā , (14.25)

Ī½ Ī½ Ī½ Ī½

bĪ½ bĀµ = Ī“Ī½Āµ b2 (14.26)

Ī½

for the fundamental second-order moments of a noise distribution Ļ. For some appli-

cations it is more convenient to employ symmetrically-ordered moments, e.g.

1ā

b bĀµ + bĀµ bā = NĀµ Ī“ĀµĪ½ , (14.27)

2Ī½ Ī½

where

1

NĀµ = bā bĀµ + (14.28)

Āµ

2

is the noise strength. One virtue of this choice is that the lower bound in the in-

equality NĀµ 1/2 represents the presence of vacuum ļ¬‚uctuations.

If we neglect the weak reservoirā“sample interaction, the time-domain analogue of

these relations can be expressed in terms of the Heisenberg-picture noise operator,

Ī¾ (t), deļ¬ned as a solution of the Heisenberg equation,

d

i Ī¾ (t) = [Ī¾ (t) , Hres ] , (14.29)

dt

where Hres is given by eqn (14.12). The value of Ī¾ (t) at the initial time t = t0 ā”when

the SchrĀØdinger and Heisenberg pictures coincideā”is taken to be

o

Ī¾ (t0 ) = CĪ½ bĪ½ , (14.30)

Ī½

Ā¾

The world as sample and environment

where CĪ½ is a c-number coeļ¬cient. The explicit solution,

CĪ½ bĪ½ eā’iā„¦Ī½ (tā’t0 ) ,

Ī¾ (t) = (14.31)

Ī½

leads to the results

G (t, t ) = Ī¾ ā (t) Ī¾ (t ) = |CĪ½ |2 bā bĪ½ eiā„¦Ī½ (tā’t ) (14.32)

Ī½

Ī½

and

CĪ½ b2 eā’iā„¦Ī½ (t+t ā’2t0 )

2

F (t, t ) = Ī¾ (t) Ī¾ (t ) = (14.33)

Ī½

Ī½

for the second-order correlation functions G (t, t ) and F (t, t ).

The factorizability assumption (14.22) alone is suļ¬cient to show that G (t, t ) is

invariant under the uniform time translation (t ā’ t + Ļ„ , t ā’ t + Ļ„ ) for any set

of coeļ¬cients CĪ½ , but the same cannot be said for F (t, t ). The only way to ensure

time-translation invariance of F (t, t ) is to impose

b2 = 0 , (14.34)

Ī½

which in turn implies F (t, t ) ā” 0. A distribution satisfying eqns (14.27) and (14.34) is

said to represent phase-insensitive noise. It is possible to discuss many noise prop-

erties using only the second-order correlation functions F and G (Caves, 1982), but for

our purposes it is simpler to impose the stronger assumption that the distribution Ļ is

stationary. From the general discussion in Section 4.5, we know that a stationary den-

sity operator commutes with the Hamiltonian. The simple form (14.12) of H in turn

implies that each ĻĪ½ commutes with the mode number operator NĪ½ ; consequently, ĻĪ½

is diagonal in the number-state basis. This very strong feature subsumes eqn (14.34)

in the general result

n n

bā bā

m n

(bĪ½ ) = Ī“nm (bĪ½ ) , (14.35)

Ī½ Ī½

which guarantees time-translation invariance for correlation functions of all orders.

14.1.2 Adiabatic elimination of the reservoir operators

In the SchrĀØdinger picture, the reservoir and sample operators act in diļ¬erent spaces,

o

so [bJĪ½ , O] = 0 for any sample operator, O. Since the SchrĀØdinger and Heisenberg

o

pictures are connected by a time-dependent unitary transformation, the equal-time

commutators vanish at all times,

[O (t) , bJĪ½ (t)] = O (t) , bā (t) = 0 . (14.36)

JĪ½

With this fact in mind, it is straightforward to use the explicit form of HW to ļ¬nd the

Heisenberg equations for the reservoir operators:

ā‚bJĪ½ (t)

= ā’iā„¦Ī½ bJĪ½ (t) ā’ vJ (ā„¦Ī½ ) OJ (t) . (14.37)

ā‚t

Ā¾ Quantum noise and dissipation

Each of these equations has the formal solution

t

ā’iā„¦Ī½ (tā’t0 )

dt eā’iā„¦Ī½ (tā’t ) OJ (t ) ,

ā’ vJ (ā„¦Ī½ )

bJĪ½ (t) = bJĪ½ (t0 ) e (14.38)

t0

where t0 is the initial time at which the SchrĀØdinger and Heisenberg pictures coincide.

o

This convention allows the identiļ¬cation of bJĪ½ (t0 ) with the SchrĀØdinger-picture op-

o

erator bJĪ½ . The ļ¬rst term on the right side of this equation describes free evolution of

the reservoir, and the second term represents radiation reaction, i.e. the emission and

absorption of reservoir excitations by the sample.

The Heisenberg equation for a sample operator OK is

ā‚OK (t) 1 1 1

= [OK (t) , HW (t)] = [OK (t) , HS (t)] + [OK (t) , HSE (t)] . (14.39)

ā‚t i i i

The explicit form (14.14) for HSE (t), together with the equal-time commutation rela-

tions, allow us to express the ļ¬nal term in eqn (14.39) as

1

OK (t) , OJ (t) bJĪ½ (t) ā’ bā (t) [OK (t) , OJ (t)] .

ā

[OK (t) , HSE (t)] = vJ (ā„¦Ī½ ) JĪ½

i Ī½

(14.40)

The equal-time commutation relations (14.36) guarantee that the products of sam-

ple and reservoir operators in this equation can be written in any order without chang-

ing the result, but the individual terms in the formal solution (14.38) for the reservoir

operators do not commute with the sample operators. Consequently, it is essential to

decide on a deļ¬nite ordering before substituting the formal solution for the reservoir

operators into eqn (14.40), and this ordering must be strictly enforced throughout the

subsequent calculation. The ļ¬nal physical predictions are independent of the original

order chosen, but the interpretation of intermediate results may vary. This is another

example of ordering ambiguities like those that allow one to have the zero-point en-

ergy by choosing symmetrical ordering, or to eliminate it by using normal ordering.

We have chosen to write eqn (14.40) in normal order with respect to the reservoir

operators.

Substituting the formal solution (14.38) into eqn (14.40) yields two kinds of terms.

One depends explicitly on the initial reservoir operators bJĪ½ (t0 ) and the other arises

from the radiation-reaction term. We can now proceed to eliminate the reservoir

degrees of freedomā”in parallel with the elimination of the radiation ļ¬eld in the

Weisskopfā“Wigner modelā”but the necessary calculations depend on the details of

the sampleā“environment interaction. Consequently, we will carry out the adiabatic

elimination process in several illustrative examples.

14.2 Photons in a lossy cavity

In this example, the sample consists of the discrete modes of the radiation ļ¬eld in an

ideal physical cavity, and the environment consists of one or more reservoirs which

schematically describe the mechanism for the loss of electromagnetic energy. For an

enclosed cavityā”such as the microcavities discussed in Chapter 12ā”a single reservoir

Ā¾

Photons in a lossy cavity

representing the exchange of energy between the radiation ļ¬eld and the cavity walls

will suļ¬ce. For the commonly encountered four-port devicesā”e.g. a resonant cavity

capped by mirrorsā”it is necessary to invoke two reservoirs representing the vacuum

modes entering and leaving the cavity through each port. In the present section we

will concentrate on the simpler case of the enclosed cavity; the four-port devices will

be discussed in Section 14.3.

In order for the discrete cavity modes to retain their identity, the characteristic

interaction energy, ā„¦S , between the sample and the reservoir must be small compared

to the minimum energy diļ¬erence, āĻ, between adjacent modes, i.e.

ā„¦S āĻ . (14.41)

For example, a rectangular cavity with dimensions L1 , L2 , and L3 satisfying L1

L2 L3 has āĻ = 2Ļc/L3 . When eqn (14.41) is satisļ¬ed the radiation modes are

weakly coupled through their interaction with the reservoir modes, andā”to a good

approximationā”we may treat each radiation mode separately.

We may, therefore, consider a reduced sample consisting of a single mode of the

ļ¬eld, with frequency Ļ0 , and drop the mode index. The unperturbed sample Hamil-

tonian is then

HS0 = Ļ0 aā a , (14.42)

and we will initially allow for the presence of an interaction term HS1 (t). In this case

there is only one sample operator and one reservoir, so the general expression (14.14)

reduces to

v (ā„¦Ī½ ) aā bĪ½ ā’ bā a .

HSE = i (14.43)

Ī½

Ī½

The coupling constant v (ā„¦Ī½ ) is proportional to the RWA cut-oļ¬ function deļ¬ned by

eqn (11.22):

v (ā„¦Ī½ ) = v0 (ā„¦Ī½ ) K (ā„¦Ī½ ā’ Ļ0 ) . (14.44)

This is an explicit realization of the assumption that the sample is coupled to a broad

spectrum of reservoir excitations.

In this connection, we note that the interaction Hamiltonian HSE is similar to

the RWA interaction Hamiltonian Hrwa , in eqn (11.46), that describes spontaneous

emission by a two-level atom. In the present case, the annihilation operator a for the

discrete cavity mode plays the role of the atomic lowering operator Ļā’ and the modes

of the radiation ļ¬eld are replaced by the reservoir excitation modes. The mathematical

similarity between HSE and Hrwa allows similar physical conclusions to be drawn. In

particular, a reservoir excitationā”which carries positive energyā”will never be reab-

sorbed once it is emitted. The implication that the interaction between the sample

and a physically realistic reservoir is inherently dissipative is supported by the explicit

calculations shown below.

This argument apparently rules out any description of an amplifying medium in

terms of coupling to a reservoir. There is a formal way around this diļ¬culty, but it

requires the introduction of an inverted-oscillator reservoir which has distinctly

unphysical properties. In this model, all reservoir excitations have negative energy;

therefore, emitting a reservoir excitation would increase the energy of the sample.

ĀæĀ¼ Quantum noise and dissipation

Since the emission is irreversible, the result would be an ampliļ¬cation of the cavity

mode. For more details, see Gardiner (1991, Chap. 7.2.1) and Exercise 14.5.

14.2.1 The Langevin equation for the ļ¬eld

The Heisenberg equation for a (t) is

d 1

a (t) = ā’iĻ0 a (t) + v (ā„¦Ī½ ) bĪ½ (t) + [a (t) , HS1 (t)] , (14.45)

dt i

Ī½

while the formal solution (14.38) for this case is

t

ā’iā„¦Ī½ (tā’t0 )

dt eā’iā„¦Ī½ (tā’t ) a (t ) .

ā’ v (ā„¦Ī½ )

bĪ½ (t) = bĪ½ (t0 ) e (14.46)

t0

|v (ā„¦Ī½ )|, and we will also assume that HS1 is

The general rule (14.2) requires Ļ0

weak compared to HS0 . Thus the ļ¬rst term on the right side of eqn (14.45) describes

oscillations that are much faster than those due to the remaining terms. This suggests

the introduction of slowly-varying envelope operators,

a (t) = a (t) eiĻ0 t , bĪ½ (t) = bĪ½ (t) eiĻ0 t , (14.47)

that satisfy

d 1

a (t) = v (ā„¦Ī½ ) bĪ½ (t) + [a (t) , HS1 (t)] , (14.48)

dt i

Ī½

and

t

ā’i(ā„¦Ī½ ā’Ļ0 )(tā’t0 )

dt eā’i(ā„¦Ī½ ā’Ļ0 )(tā’t ) a (t ) .

ā’ v (ā„¦Ī½ )

bĪ½ (t) = bĪ½ (t0 ) e (14.49)

t0

The envelope operator a (t) varies on the time scale TS = 1/ā„¦S , so it is the operator

version of the slowly-varying classical envelope introduced in Section 3.3.1.

We are now ready to carry out the elimination of the reservoir degrees of freedom,

by substituting eqn (14.49) into eqn (14.48). The HS1 -term plays no role in this argu-

ment, so we will simplify the intermediate calculation by omitting it. The simpliļ¬ed

equation for a (t) is

t

d

a (t) = ā’ dt K (t ā’ t ) a (t ) + Ī¾ (t) , (14.50)

dt t0

where

|v (ā„¦Ī½ )|2 eā’i(ā„¦Ī½ ā’Ļ0 )(tā’t ) ,

K (t ā’ t ) = (14.51)

Ī½

and

v (ā„¦Ī½ ) bĪ½ (t0 ) eā’i(ā„¦Ī½ ā’Ļ0 )(tā’t0 ) .

Ī¾ (t) = (14.52)

Ī½

At this stage, the passage to the continuum limit is essential; therefore, we change

the sum over the discrete modes to an integral according to the rule

ĀæĀ½

Photons in a lossy cavity

ā

fĪ½ ā’ dā„¦D (ā„¦) f (ā„¦) , (14.53)

0

Ī½

where D (ā„¦) is the density of states for the reservoir modes. The exact form of D (ā„¦)

depends on the particular model chosen for the reservoir. For example, if the reservoir

is deļ¬ned by modes of the radiation ļ¬eld, then D (ā„¦) is given by eqn (4.158). In practice

these details are not important, since they will be absorbed into a phenomenological

decay constant. Applying the rule (14.53) to K (t ā’ t ) and using eqn (14.44) leads to

the useful representation

ā

dā„¦D (ā„¦) |v0 (ā„¦)|2 |K (ā„¦ ā’ Ļ0 )|2 eā’i(ā„¦ā’Ļ0 )(tā’t ) .

K (t ā’ t ) = (14.54)

0

The frequency width of the Fourier transform K (ā„¦) of K (t ā’ t ) is well approxi-

mated by the width āK of the cut-oļ¬ function. According to the uncertainty principle

for Fourier transforms, the temporal width of K (t ā’ t ) is therefore of the order of

1/āK . Since K (t ā’ t ) decays to zero for |t ā’ t | > 1/āK , we use the terminology in-

troduced in Section 11.1.2 to call Tmem = 1/āK the memory interval for the reservoir.

The general rule (14.2) for cut-oļ¬ functions, which in the present case is

ā„¦S = max |v (ā„¦)| āK Ļ0 , (14.55)

imposes the relation Tmem TS . In other words, the assumption of a broad spectral

range for the sampleā“reservoir interaction is equivalent to the statement that the

reservoir has a short memory. This assumption eļ¬ectively restricts the integral in

eqn (14.50) to the interval t ā’ Tmem < t < t, in which a (t ) is essentially constant.

The short memory of the reservoir justiļ¬es the Markov approximation, a (t ) ā a (t),

and this allows us to replace the integro-diļ¬erential equation (14.50) by the ordinary

diļ¬erential equation

ńņš. 16 |