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3
(2π)
d3 k1 d3 k2 (3) ’iωP t
(p ’ k1 ’ k2 ) a† (k1 ) a† (k2 ) + HC .
(t) = ’i C
(3)
Hem 3G e
3
(2π) (2π)
(13.43)

The Hamiltonian has the same form in the Heisenberg picture, with a (k1 ) replaced
by a† (k1 , t), etc. Let
N (k1 , t) = a† (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
’iωP t
(p ’ k1 ’ k2 ) a† (k1 , t) a† (k2 , t) ’ HC .
[N (k1 , t) , H] = ’2ie C
(3)
3G
(2π)
(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] = ’2ie’iωP t G(3) a† (k1 , t) a† (p ’ k1 , t) . (13.46)

In this approximation we see that

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

i.e. the di¬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 ’ k1 , t) = N (p ’ k1 , t) . This can
be done by placing detectors behind each of a pair of stops that select out a particular
signal“idler pair (k1 , p ’ k1 ). According to eqn (13.47), the expectation values satisfy

n (k1 , t) ’ n (p ’ k1 , t) = N (k1 , t) ’ N (p ’ k1 , t)
= N (k1 , 0) ’ N (p ’ k1 , 0)
= 0, (13.48)

which provides experimental evidence that the conjugate photons are created at the
same time.
¼ 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, |Ψ (t) , for the ¬eld can be evaluated by ¬rst-order per-
turbation theory. These calculations are simpli¬ed by returning to the box-quantized
form (13.40). In this notation, the interaction Hamiltonian is
1
G(3) C (p ’ k1 ’ k2 ) e’i∆t a† 1 a† 2 + HC ,
Hem (t) = ’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, ∆ = ωP ’ω2 ’ω1 , for the down-conversion transition. Applying
the perturbation series (4.103) for the state vector leads to

|Ψ (t) = |0 + Ψ(1) (t) + · · · ,
(13.50)
2G(3)
1 sin [∆t/2] † †
C (p ’ k1 ’ k2 ) e’i∆t/2
Ψ(1) (t) = ’ ak1 ak2 |0 .
V ∆
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 ’ ∞, of in¬nite quantization volume with
the large-crystal approximation (13.22) for C yields

d3 k1 d3 k2 2G(3) 3
|Ψ (t) = |0 ’ (2π) δ (p ’ k1 ’ k2 )
3 3
(2π) (2π)
sin [∆t/2] †
— e’i∆t/2 a (k1 ) a† (k2 ) |0 . (13.51)

The limit t ’ ∞ is relevant for cw pumping, so we can use the identity
sin (∆t/2) π
lim e’i∆t/2 = δ (∆) , (13.52)
∆ 2
t’∞

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

d3 k1 d3 k2 1 G(3)
|Ψ (∞) = |0 ’ 3 3 2
(2π) (2π)
3
— (2π) δ (p ’ k1 ’ k2 ) (2π) δ (ωP ’ ω1 ’ ω2 )
— a† (k1 ) a† (k2 ) |0 , (13.53)

where ω1 = ωk1 and ω2 = ωk2 .
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, ω1 + ω2 = ωP , 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
¼
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 „ = t1 ’ t2 . The
respective conjugate frequencies are „¦ = ω1 +ω2 and ν = (ω1 ’ ω2 ) /2. The uncertainty
in the pair creation time, ∆tP ∼ 1/∆„¦, is large by virtue of the tight phase-matching
condition, „¦ ωP . On the other hand, the individual frequencies have large spectral
bandwidths, so that ∆ν is large and „ ∼ 1/∆ν 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 ¬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.
Zel™dovich and Klyshko (1969) were the ¬rst to notice that phased-matched, down-
converted photons should be observable in coincidence detection. Burnham and Wein-
berg (1970) performed the ¬rst experiment to observe these predicted coincidences,
and in the same experiment they were also the ¬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).)
¼ Nonlinear quantum optics


Channel 2

2
PM
From
3520 A φ2
laser Spike
filters
φ1
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 φ1 and φ2 such that phase matching is
satis¬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, helium“cadmium, ultraviolet laser”operating at a wavelength of
325 nm”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 (»1 = 633 nm) and idler (»2 = 668 nm) photons
emerge from the crystal at the respective angles of φ1 = 52 mrad and φ2 = 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 ¬lters”actually a combination of interfer-
ence ¬lter and monochromator in the case of the idler photon”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 e¬ective diameters of 2 mm are used to de¬ne precisely
the angles of emission of the detected photons around the phase-matching directions.
Most importantly, Burnham and Weinberg were also the ¬rst to use coincidence de-
tection to demonstrate that the phase-matched signal and idler photons are produced
¼
Three-photon interactions

essentially simultaneously inside the crystal, within a narrow coincidence window of
±20 ns, that is limited only by the response time of the electronic circuit.
In more modern versions of the Burnham“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 that”by means
of an internal discriminator”the output consists of standardized TTL (transistor“
transistor logic), ¬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 e¬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 techniques”originally developed in nuclear and elementary par-
ticle physics”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 e¬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,
de¬ned by momentum and energy conservation. Thus we know precisely the direction
of emission of the conjugate photon, and also its time of arrival”within a very nar-
row time window relative to the trigger photon”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
e¬ciencies ·1 and ·2 of the trigger detector and the DUT are de¬ned by

N1 = ·1 N (13.54)

·2
Counter
,76
ω2
Coincidence ·2
counter
ωF
Absolute
quantum
efficiency
Parametric ω1
crystal Counter
6HECCAH

Fig. 13.4 Scheme for absolute measurement of quantum e¬ciency. A pair of entangled
photons originating in the crystal head toward the ˜trigger™ detector and the ˜detector under
test™ (DUT). The parameter ·2 is the quantum e¬ciency for the entire path from the point
of emission to the DUT. (Reproduced from Migdall (2001).)
¼ Nonlinear quantum optics

and
N2 = ·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 ·coinc = ·1 ·2 . Thus the number of coincidence counts is

Ncoinc = ·1 ·2 N , (13.56)

and combing this with eqn (13.54) shows that the absolute quantum e¬ciency ·2 of
the DUT is the ratio
Ncoinc
·2 = (13.57)
N1
of two measurable quantities. The beauty of this scheme is that this result is indepen-
dent of the quantum e¬ciency, ·1 , of the trigger detector.
Systematic errors, however, must be carefully taken into account. Any losses along
the optical path”from the point of emission of the twin photons inside the crystals
all the way to the point of detection in the DUT”will contribute to a systematic
error in the measurement. Thus the exit face of the crystal must be carefully anti-
re¬‚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 e¬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 convenient”and often necessary”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
¼
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 ˜crossed™ con¬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 ¬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 crystals”β
barium borate (BBO)”that are stacked in a crossed con¬guration, i.e. with their
optic axes lying in perpendicular planes. What we will call the vertical plane is de¬ned
by the optic axis of the ¬rst crystal and the direction of the pump beam, while the
horizontal plane is de¬ned by the optic axis of the second crystal and the pump beam.
The crystals are su¬ciently thin so that the waist of the pump beam”a continuous-
wave, ultraviolet (wavelength 351 nm), argon-ion laser”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 33.9—¦ 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 3.0—¦ .
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
¬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 45—¦ with respect to the vertical axis. This
state is a coherent superposition, with equal amplitudes, of horizontal and vertical
polarizations. Thus when this 45—¦ -polarized pump beam is incident on the two-crystal
½¼ Nonlinear quantum optics

stack, a down-conversion event can occur, with equal probability, either in the ¬rst or
in the second crystal. If the photon pair originates in the ¬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 di¬raction ensures that the spatial modes”i.e.
the cones of emission in Fig. 13.5(a)”from the two crystals overlap in the far ¬eld,
where the photons are detected. This situation provides the guiding principle behind
this light source: for a 45—¦ -polarized pump beam, it is impossible”even in principle”
to know whether a given photon pair originated in the ¬rst or in the second crystal. We
must therefore apply Feynman™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
¦+ = √ |1k1 H , 1k2 H + √ |1k1 V , 1k2 V . (13.58)
2 2
The notation 1k1 H denotes the horizontal polarization state of one member of the pho-
ton pair”originating in the ¬rst crystal”and 1k2 H denotes the horizontal polarization
state of the conjugate member, also originating in the ¬rst crystal. Similarly, 1k1 V de-
notes the vertical polarization state of one member of the photon pair”originating in
the second crystal”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 |¦+ is one of the four Bell states de¬ned by
1 1
¦+ ≡ √ |1k1 H , 1k2 H + √ |1k1 V , 1k2 V , (13.59)
2 2
1 1
¦’ ≡ √ |1k1 H , 1k2 H ’ √ |1k1 V , 1k2 V , (13.60)
2 2
1 1
Ψ+ ≡ √ |1k1 H , 1k2 V + √ |1k1 V , 1k2 H , (13.61)
2 2
1 1
Ψ’ ≡ √ |1k1 H , 1k2 V ’ √ |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 |¦+ and |¦’
can be generated by two crossed type I crystals, and the states |Ψ+ and |Ψ’ 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
eiξ
1
¦ ; ξ = √ |1k1 H , 1k2 H + √ |1k1 V , 1k2 V ,
+
(13.63)
2 2
½½
Three-photon interactions

where the phase ξ 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 45—¦ with respect to the vertical. Then the down-converted
photon pair emerges from the tandem crystals in the entangled state |¦+ ; ξ , with
a nonvanishing phase di¬erence ξ between the H“H and V “V polarization-product
states. The phase of the entanglement parameter ξ 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 ¬lter to separate out the ultraviolet laser pump beam from
the unwanted ¬‚uorescence of the argon-ion discharge tube. A polarizing beam splitter
(PBS) acts as a pre¬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)”
whose tilt angle allows the adjustment of the relative phase ξ of the entangled state
in eqn (13.63)”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, θ1 and θ2 , 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 ¬lters (IF) had typical bandwidths of 5 nm
in wavelength. The iris sizes and interference-¬lter bandwidths were determined by
the criterion that the detection should occur in the far ¬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 e¬ciency”arising from the ¬nite sizes of the irises placed in front of the
detectors”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 e¬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 ¬xed at
’45—¦ ).
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 ± 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 Bell™s inequalities”see
eqn (19.38)”by over 200 standard deviations, for data collected in about 3 minutes.
½¾ Nonlinear quantum optics



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
ƥ = 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 ƥ.
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)
input state is |Ψ (0) = |n1 , ±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 |Ψ (t) = n1 , ±2 e’iγn1 , where
γ = g12 t.
(2) Devise a homodyne measurement scheme that can distinguish between the phase
shifts experienced by the meter beam for di¬erent values of n1 , e.g. n1 = 0 and
n1 = 1. For example, measure the quadrature X2 = a2 exp [’i•] + a† exp [i•] /2,
2
where • is the phase of the local oscillator in the homodyne apparatus.
14
Quantum noise and dissipation

In the majority of the applications considered so far”e.g. photons in an ideal cavity,
photons passing through passive linear media, atoms coupled to the radiation ¬eld,
etc.”we have neglected all dissipative e¬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 Schr¨dinger 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 ”the physical objects of experimental interest”and the environment”
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 ¬eld is assumed to be
in thermal equilibrium with the cavity walls. In this case the sample is the radiation
¬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 Weisskopf“Wigner method in Section
11.2.2 the sample is a two-level atom, and the modes of the radiation ¬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 ˜system™, so we have replaced it with
˜sample™ for this discussion.
¾½
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 ¬eld is treated as a closed system by assuming that the
screen is a perfect re¬‚ector, and by including both the incident and the re¬‚ected
modes in the sample. Let us now look at this problem in a di¬erent way, by assigning
the re¬‚ected modes”i.e. the modes propagating from right to left in Fig. 8.5”to
the environment. The newly de¬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 ¬‚uctuations of photon number in the transmitted ¬eld.
This is a simple example of a general principle which is often called the ¬‚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 e¬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 Weisskopf“Wigner (WW) treatment of spontaneous
emission.
(1) The modes of the environment (the radiation ¬eld for WW) have a continuous
spectrum.
(2) The sample (the two-level atom for WW) has”to a good approximation”the
following properties.
(a) The sample Hamiltonian has a discrete spectrum. This is guaranteed if the
sample (like the atom) has a ¬nite number of degrees of freedom. If the sample
has an in¬nite number of degrees of freedom (like the radiation ¬eld) a discrete
spectrum is guaranteed by con¬nement to a ¬nite region of space, e.g. a cavity.
(b) The sample is weakly coupled to a broad spectral range of environmental
modes.
In the Weisskopf“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
|„¦ks | ∆K ω21 , (14.1)
where |„¦ks | is the one-photon Rabi frequency de¬ned in eqn (4.153), and ∆K is the
width of the cut-o¬ function for the RWA. This inequality de¬nes what is meant by
coupling to a broad spectral range of the radiation ¬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 „¦S de¬nes a characteristic response
frequency „¦S , and the condition of weak coupling to a broad range of environmental
excitations is
„¦S ∆E ωS . (14.2)
¾¾ Quantum noise and dissipation

Here ∆E is the spectral width of the environmental modes that are coupled to the
sample, and ωS is a characteristic mode frequency for the unperturbed sample.
In the Weisskopf“Wigner model, the environment is the radiation ¬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 νth phonon mode”which is an
analogue of the ks-mode of the radiation ¬eld”is represented by a harmonic oscil-
lator with fundamental frequency „¦ν , analogous to ωks . For macroscopic bodies, the
discrete index ν becomes e¬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 world”the combined system of sample and
environment”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 e¬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 are”naturally enough”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 ¬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 reservoir”which, after all, carries the experimental inform-
ation”it would evidently be foolish to eliminate the reservoir degrees of freedom.
¾¿
The world as sample and environment

Instead, the objective is to determine the e¬ect of the sample on the reservoir modes
to be observed. Despite this di¬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 input“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 — 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 — H2 — · · · — HNres , (14.4)

of the Hilbert spaces for the Nres independent reservoirs that de¬ne the environment.
Pure states, |χ , in HW are linear combinations of product states:

|χ = C1 |Ψ1 |Λ1 + C2 |Ψ2 |Λ2 + · · · , (14.5)

where |Ψj and |Λj 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 ¬elds
or currents, and its constituent parts may interact with each other. Thus the total
Schr¨dinger-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 ¬elds or currents. The time dependence of the external
¬elds is the source of the explicit time dependence of VS (t) in the Schr¨dinger 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 Schr¨dinger-picture Hamiltonian for the world is then
o

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

where
¾ 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 de¬nition, a closed system.
We will initially use a box-quantization description of the reservoir oscillators that
parallels the treatment of the radiation ¬eld in Section 3.1.4, i.e. each family of oscil-
lators will be labeled by a discrete index ν. The free Hamiltonian for reservoir J is
therefore given by
„¦ν b† bJν ,
HJ = (14.12)

ν
where bJν is the annihilation operator for the νth mode of the Jth reservoir. We have
simpli¬ed the model by assuming that each reservoir has the same set of fundamen-
tal frequencies {„¦ν }, rather than a di¬erent set {„¦Jν } for each reservoir. This is not
a serious restriction, since in the continuum limit each „¦Jν is replaced by a contin-
uous variable „¦. The kinematical independence of the reservoirs is imposed by the
commutation relations
bJν , b†
[bJν , bKµ ] = 0 , Kµ = δJK δνµ .

In typical applications, the sample is coupled to the environment through sample
operators, OJ , that can be chosen to satisfy
[OJ , HS0 ] ≈ ωJ OJ , (14.13)
where ωJ 0. For ωJ > 0, this means that OJ is an approximate energy-lowering
operator for the unperturbed sample Hamiltonian HS0 . We will also need the limiting
case ωJ = 0, which means that OJ is an approximate constant of the motion.
In the resonant wave approximation, the sample“environment interaction can be
written as
vJ („¦ν ) OJ bJν ’ b† OJ ,

(J)
HSE = i (14.14)

ν
where vJ („¦ν ) is a real, positive coupling frequency. This ansatz incorporates the
assumption that each sample“reservoir interaction Hamiltonian is a linear function of
the reservoir operators. The restriction to real coupling frequencies is not signi¬cant, as
shown in Exercise 14.1. Each coupling frequency is a candidate for the characteristic,
sample-response frequency „¦S , so it must satisfy the condition
vJ („¦ν ) ∆E ωS . (14.15)
The choice of the sample operator OJ is determined by the physical damping mecha-
nism associated with the Jth reservoir.
¾
The world as sample and environment

B The world density operator
The probability distributions relevant to experiments are determined by the Schr¨- o
S
dinger-picture density operator, ρW (t), that describes the state of the world. We
must, therefore, begin by choosing an initial form, ρS (t0 ), for the density operator.
W
The natural assumption is that the sample and the reservoirs are uncorrelated for
a su¬ciently early time t0 . Since the time-independent, Heisenberg-picture density
operator, ρH , satis¬es ρH = ρS (t0 ), this is equivalent to assuming that
W W W

ρW = ρS ρE , (14.16)

where ρS acts on HS , and ρE 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
ρE = ρ1 ρ2 . . . ρNres , (14.17)
where ρJ acts on HJ for J = 1, 2, . . . , Nres . One or more of the density operators ρJ 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 (ρW O) , (14.18)

where TrW is de¬ned by the sum over a basis set for HW = HS — HE . By using the
de¬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 ρW = ρS ρE is then

SR = TrW [(ρS ρE ) (SR)]
= TrW [ρS S ρE R]
= [TrS (ρS S)] [TrE (ρE 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
¾ 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:
ρ= ρν , (14.22)
ν

where ρν is the density operator for the νth mode. For operators Fν and Gµ that are
respectively functions of bν , b† and bµ , b† , this assumption implies Fν Gµ = Fν Gµ
ν µ
for µ = ν.
For the discussion of quantum noise, only ¬‚uctuations around mean values are of
interest. We will say that a factorizable density operator ρ is a noise distribution if
the natural oscillator variables bν and b† satisfy
ν

b† = bν = 0 for all ν . (14.23)
ν

These conditions can always be achieved by using the ¬‚uctuation operator δbν =
bν ’ bν in place of bν . By means of suitable choices of the operators Fν and Gµ ,
the combination of eqns (14.23) and (14.22) can be used to derive restrictions on the
moments of a noise distribution ρ. For example, the results

b† bµ = bµ b† = b† bµ = 0 and b ν bµ = bν bµ = 0 for µ = ν (14.24)
ν ν ν

lead to the useful rules

b† bµ = δνµ b† bν , bµ b† = δνµ bν b† , (14.25)
ν ν ν ν

bν bµ = δνµ b2 (14.26)
ν

for the fundamental second-order moments of a noise distribution ρ. For some appli-
cations it is more convenient to employ symmetrically-ordered moments, e.g.
1†
b bµ + bµ b† = Nµ δµν , (14.27)
2ν ν

where
1
Nµ = b† bµ + (14.28)
µ
2
is the noise strength. One virtue of this choice is that the lower bound in the in-
equality Nµ 1/2 represents the presence of vacuum ¬‚uctuations.
If we neglect the weak reservoir“sample interaction, the time-domain analogue of
these relations can be expressed in terms of the Heisenberg-picture noise operator,
ξ (t), de¬ned as a solution of the Heisenberg equation,
d
i ξ (t) = [ξ (t) , Hres ] , (14.29)
dt
where Hres is given by eqn (14.12). The value of ξ (t) at the initial time t = t0 ”when
the Schr¨dinger and Heisenberg pictures coincide”is taken to be
o

ξ (t0 ) = Cν bν , (14.30)
ν
¾
The world as sample and environment

where Cν is a c-number coe¬cient. The explicit solution,

Cν bν e’i„¦ν (t’t0 ) ,
ξ (t) = (14.31)
ν

leads to the results

G (t, t ) = ξ † (t) ξ (t ) = |Cν |2 b† bν ei„¦ν (t’t ) (14.32)
ν
ν

and
Cν b2 e’i„¦ν (t+t ’2t0 )
2
F (t, t ) = ξ (t) ξ (t ) = (14.33)
ν
ν

for the second-order correlation functions G (t, t ) and F (t, t ).
The factorizability assumption (14.22) alone is su¬cient to show that G (t, t ) is
invariant under the uniform time translation (t ’ t + „ , t ’ t + „ ) for any set
of coe¬cients Cν , 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)
ν

which in turn implies F (t, t ) ≡ 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 ρ 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 ρν commutes with the mode number operator Nν ; consequently, ρν
is diagonal in the number-state basis. This very strong feature subsumes eqn (14.34)
in the general result
n n
b† b†
m n
(bν ) = δnm (bν ) , (14.35)
ν ν


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

14.1.2 Adiabatic elimination of the reservoir operators
In the Schr¨dinger picture, the reservoir and sample operators act in di¬erent spaces,
o
so [bJν , O] = 0 for any sample operator, O. Since the Schr¨dinger and Heisenberg
o
pictures are connected by a time-dependent unitary transformation, the equal-time
commutators vanish at all times,

[O (t) , bJν (t)] = O (t) , b† (t) = 0 . (14.36)


With this fact in mind, it is straightforward to use the explicit form of HW to ¬nd the
Heisenberg equations for the reservoir operators:
‚bJν (t)
= ’i„¦ν bJν (t) ’ vJ („¦ν ) OJ (t) . (14.37)
‚t
¾ Quantum noise and dissipation

Each of these equations has the formal solution
t
’i„¦ν (t’t0 )
dt e’i„¦ν (t’t ) OJ (t ) ,
’ vJ („¦ν )
bJν (t) = bJν (t0 ) e (14.38)
t0

where t0 is the initial time at which the Schr¨dinger and Heisenberg pictures coincide.
o
This convention allows the identi¬cation of bJν (t0 ) with the Schr¨dinger-picture op-
o
erator bJν . The ¬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

‚OK (t) 1 1 1
= [OK (t) , HW (t)] = [OK (t) , HS (t)] + [OK (t) , HSE (t)] . (14.39)
‚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 ¬nal term in eqn (14.39) as
1
OK (t) , OJ (t) bJν (t) ’ b† (t) [OK (t) , OJ (t)] .

[OK (t) , HSE (t)] = vJ („¦ν ) Jν
i ν
(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 de¬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 ¬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 bJν (t0 ) and the other arises
from the radiation-reaction term. We can now proceed to eliminate the reservoir
degrees of freedom”in parallel with the elimination of the radiation ¬eld in the
Weisskopf“Wigner model”but the necessary calculations depend on the details of
the sample“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 ¬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 cavity”such as the microcavities discussed in Chapter 12”a single reservoir
¾
Photons in a lossy cavity

representing the exchange of energy between the radiation ¬eld and the cavity walls
will su¬ce. For the commonly encountered four-port devices”e.g. a resonant cavity
capped by mirrors”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, „¦S , between the sample and the reservoir must be small compared
to the minimum energy di¬erence, ∆ω, between adjacent modes, i.e.

„¦S ∆ω . (14.41)

For example, a rectangular cavity with dimensions L1 , L2 , and L3 satisfying L1
L2 L3 has ∆ω = 2πc/L3 . When eqn (14.41) is satis¬ed the radiation modes are
weakly coupled through their interaction with the reservoir modes, and”to a good
approximation”we may treat each radiation mode separately.
We may, therefore, consider a reduced sample consisting of a single mode of the
¬eld, with frequency ω0 , and drop the mode index. The unperturbed sample Hamil-
tonian is then
HS0 = ω0 a† 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 („¦ν ) a† bν ’ b† a .
HSE = i (14.43)
ν
ν

The coupling constant v („¦ν ) is proportional to the RWA cut-o¬ function de¬ned by
eqn (11.22):
v („¦ν ) = v0 („¦ν ) K („¦ν ’ ω0 ) . (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 σ’ and the modes
of the radiation ¬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 excitation”which carries positive energy”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 di¬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.
¿¼ Quantum noise and dissipation

Since the emission is irreversible, the result would be an ampli¬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 ¬eld
The Heisenberg equation for a (t) is
d 1
a (t) = ’iω0 a (t) + v („¦ν ) bν (t) + [a (t) , HS1 (t)] , (14.45)
dt i
ν

while the formal solution (14.38) for this case is
t
’i„¦ν (t’t0 )
dt e’i„¦ν (t’t ) a (t ) .
’ v („¦ν )
bν (t) = bν (t0 ) e (14.46)
t0

|v („¦ν )|, and we will also assume that HS1 is
The general rule (14.2) requires ω0
weak compared to HS0 . Thus the ¬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) eiω0 t , bν (t) = bν (t) eiω0 t , (14.47)

that satisfy
d 1
a (t) = v („¦ν ) bν (t) + [a (t) , HS1 (t)] , (14.48)
dt i
ν

and
t
’i(„¦ν ’ω0 )(t’t0 )
dt e’i(„¦ν ’ω0 )(t’t ) a (t ) .
’ v („¦ν )
bν (t) = bν (t0 ) e (14.49)
t0

The envelope operator a (t) varies on the time scale TS = 1/„¦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 simpli¬ed
equation for a (t) is
t
d
a (t) = ’ dt K (t ’ t ) a (t ) + ξ (t) , (14.50)
dt t0

where
|v („¦ν )|2 e’i(„¦ν ’ω0 )(t’t ) ,
K (t ’ t ) = (14.51)
ν

and
v („¦ν ) bν (t0 ) e’i(„¦ν ’ω0 )(t’t0 ) .
ξ (t) = (14.52)
ν

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
¿½
Photons in a lossy cavity


fν ’ d„¦D („¦) f („¦) , (14.53)
0
ν

where D („¦) is the density of states for the reservoir modes. The exact form of D („¦)
depends on the particular model chosen for the reservoir. For example, if the reservoir
is de¬ned by modes of the radiation ¬eld, then D („¦) 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 ’ t ) and using eqn (14.44) leads to
the useful representation

d„¦D („¦) |v0 („¦)|2 |K („¦ ’ ω0 )|2 e’i(„¦’ω0 )(t’t ) .
K (t ’ t ) = (14.54)
0

The frequency width of the Fourier transform K („¦) of K (t ’ t ) is well approxi-
mated by the width ∆K of the cut-o¬ function. According to the uncertainty principle
for Fourier transforms, the temporal width of K (t ’ t ) is therefore of the order of
1/∆K . Since K (t ’ t ) decays to zero for |t ’ t | > 1/∆K , we use the terminology in-
troduced in Section 11.1.2 to call Tmem = 1/∆K the memory interval for the reservoir.
The general rule (14.2) for cut-o¬ functions, which in the present case is
„¦S = max |v („¦)| ∆K ω0 , (14.55)
imposes the relation Tmem TS . In other words, the assumption of a broad spectral
range for the sample“reservoir interaction is equivalent to the statement that the
reservoir has a short memory. This assumption e¬ectively restricts the integral in
eqn (14.50) to the interval t ’ Tmem < t < t, in which a (t ) is essentially constant.
The short memory of the reservoir justi¬es the Markov approximation, a (t ) ≈ a (t),
and this allows us to replace the integro-di¬erential equation (14.50) by the ordinary
di¬erential equation

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