Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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1180 Part F Quantum Optics

is satisﬁed, i. e., the Bloch vector of the atom under-

goes q complete rotations. If the maser happens to reach

such a trapping state |n

, the photon number n

can no

longer increase irrespective of the ﬂux of pump atoms.

With the inclusion of cavity damping at zero tempera-

ture, n

still represents an upper barrier that cannot be

overcome, since damping only causes downward tran-

sitions. Even in the presence of dissipation, generalized

trapping states exist with a photon distribution that van-

ishes for n > n

and has a tail towards smaller photon

numbers n ≤ n

. However, thermal ﬂuctuations at ﬁ-

nite temperatures destabilize the trapping states since

they can momentarily increase the photon number and

allow the distribution to jump over the barrier n = n

Nevertheless, even for n

< 10

−7

, remnants of the trap-

ping behavior persist, and can be seen in the transient

response of the micromaser (Sect. 79.6.4).

79.6.4 Atom Counting Statistics

Direct measurements of the ﬁeld in a single atom maser

resonator are not possible because detector absorption

would drastically degrade its quality. However, the ﬁeld

can be deduced from the statistical signature of the atoms

leaving the resonator.

The probability P(n) of ﬁnding n atoms in a beam

during an observation interval t is given by the classical

Poisson distribution

P(n) = (Rt)

−Rt

/n! . (79.71)

Information on the ﬁeld inside is then revealed by the

conditional probability W(n, |g, m, |e, T) of ﬁnding

n atoms in the ground state and m atoms in the excited

state during a time t. Since there are only two states, it

is sufﬁcient to determine the probability

W(n, |g, t) =

∞



m=0

W(n, |g, m, |e, t) (79.72)

for being in the ground state [79.50]. For n = 0, the

probability of observing no atom in the ground state

during the period t is

W(0, |g, t) = Tr(ρ

stst

) exp{L + R[O

|g

+(1 −η)O

|e

−1]t} , (79.73)

where O

|j

=j|F(τ)|j (79.60)andρ

stst

is the steady

state of the maser ﬁeld. This probability is closely re-

lated to the waiting time statistic P

(0, |g, t) between

two successive ground state atoms, a property which

is easily determined in a start-stop experiment. For an

atom detector with ﬁnite quantum efﬁciency η for state

selective detection, the waiting time probability is

(0, |g, t)



Tr(ρ

stst

|g

×exp[L +R[O

|g

+(1 −η)O

|e

−1]T]O

|g





Tr(ρ

stst

|g



(79.74)

How a speciﬁc ﬁeld state is reﬂected in the atom

counting statistics will be illustrated for two situations:

the region of sub-Poisson statistics and the region where

the trapping condition is satisﬁed. Increased regularity

of the cavity ﬁeld Q

≤ 0 manifests itself in increased

regularity of ground state atoms in the beam. The statisti-

cal behavior exhibits “anti-bunching”, i. e., P

(0, |g, t)

has a maximum at ﬁnite t, indicating “repulsion” be-

tween successive atoms in comparison with a Poissonian

beam. If the transit time τ is chosen in such a way that

gτ  2π, the chance of observing an initially excited

atom in the ground state is negligible. At some point,

however, an unlikely thermal ﬂuctuation occurs, adding

a photon. The rotation angle of the Bloch vector sud-

denly increases to 2π

√

2  3π, and the atoms tend to

leave the cavity in the ground state. After a typical cav-

ity lifetime, the ﬁeld decays and the trapping condition

is restored again. Under this operation condition, the

statistics of ground state atoms is governed by two time

constants:

1. a short interval, in which successive atoms leave the

cavity in the ground state after a thermal ﬂuctuation;

2. a long time interval, in which the trapping condition

is maintained and all atoms leave the resonator in

their excited state until the next ﬂuctuation occurs.

The probability P

(0, |g, t) is plotted in Fig. 79.11.

The plot clearly shows the two time regimes that govern

the imperfect trapping situation.

79.7 Quantum Theory of Measurement

When the object of interest consists of only a few atoms

and a few photons, the puzzling consequences of quan-

tum mechanical measurement become visible. In the

case of the micromaser, the information on the state of

the ﬁeld is imprinted in a subtle way on the atomic beam.

While photon counting is normally a destructive opera-

tion, the dispersive part of the photon-atom interaction

may be used to determine the photon number inside

Part F 79.7

Entangled Atoms and Fields: Cavity QED 79.8 Applications of Cavity QED 1181

–1

–2

–3

–4

–5

–2 –1 0 1 2 3 4 5 6

log(T/T

cav

)

(0, lb>, ;T)

Fig. 79.11 Waiting time probability for atoms in the ground

state while cavity is operated at vacuum trapping-state

condition

a resonator without altering it, on average. Dispersive

effects shift the phase of an oscillating atomic dipole

without changing its state.

The phase shift due to the ﬁeld in the resonator

can be measured in a Ramsey-type experiment [79.51].

Consider an atom with two transitions |g→|e and

|e→|i. The ﬁrst is far from resonance with the cavity

and the second is close to resonance, but with a detuning

=ω −ω

large enough so as not to change the cavity

photon number as the atom passes through. The dynamic

Stark effect of the |g→|e transition frequency due to

state |i is then

∆ω



√

n +1



/δ

. (79.75)

If the resonator is now placed between the two Ram-

sey cavities, which are tuned to ω

≈ ω

, such that

the polarization of the |e→|g transition is rotated by

≈ π/2, then the additional phase shift ∆ω

τ,where

τ is the transit time through the optical resonator,

can be measured, and hence the photon number n.

Since Rydberg states have a large coupling con-

stant g

, the phase shift due to a single atom is

detectable [79.51].

A complete measurement of n requires a sequence

of N atoms because a single Ramsey measurement only

determines whether the atom is in state |e or |g,and

hence ∆ω

τ to within ±π/2. Since each measurement

provides one binary bit of information, a sequence of

N measurements can in principle distinguish 2

pos-

sible Fock states for the photon ﬁeld. However, with

a monoenergetic beam, integral multiples of 2π remain

undetermined. A distribution of velocities, and hence

transit times, is therefore desirable. An entropy reduction

strategy for selecting an optimal velocity distribution,

based on the outcome of previous measurements, is

described in [79.52].

As a consequence of the uncertainty principle,

a measurement of the photon number destroys all in-

formation about the phase of the ﬁeld. In the present

case, the noise in the conjugate variable (the phase)

is prevented from coupling back on the measured one,

and hence the measurement is called a quantum non-

demolition experiment. Many other aspects of phase

diffusion, entangled states, and quantum measurements

in the micromaser are discussed in [79.53].

79.8 Applications of Cavity QED

79.8.1 Detecting and Trapping Atoms

through Strong Coupling

From Fig. 79.8 it is obvious that an atom travelling

through the cavity will modify the transmission prop-

erties of this cavity. Strong coupling thus enables the

experimenter to detect the presence of a single atom

dispersively by monitoring cavity transmission or reﬂec-

tion. Laser cooled atoms have low velocities and spend

sufﬁcient time in the cavity even in free ﬂight to gen-

erate the transmission signal shown in Fig. 79.12.The

signals correspond to individual atom transits, and the

shape depends on the detuning of the probe laser from

the resonantly interacting cavity-atom system.

If an atom absorbs a photon inside the cavity, a strong

dipole force can be exerted due to the inhomogeneous

ﬁeld distribution of the cavity mode. Trapping of atoms

with a single photon was achieved [79.54], and from the

time variation of the cavity transmission a reconstruction

of atomic trajectories became possible.

79.8.2 Generation of Entanglement

In the middle of the 1990s, it was realized that fully

controlled quantum systems could be used to imple-

ment a revolutionary type of information processing now

called quantum computing [79.55]. From the beginning,

cavity QED has conceptually played an important role

for experimental realizations, since it offers a route to

manipulate, in principle, all physical parameters of a co-

herently interacting system. With the well established

microwave-cavity–Rydberg-atom system, it was proven

Part F 79.8

1182 Part F Quantum Optics

0.1

0.01

0 0.2 0.4 0.6 0.8 1

1.5

0.5

0 0.2 0.4 0.6 0.8 1

T(∆)

Time (ms)

T(∆)

Time (ms)

T(∆)

Time (ms)

Fig. 79.12 Transmission of a strongly coupled cavity for

individual atom transits. Caesium atoms and cavity are in

perfect resonance at λ = 852 nm while the probe laser is

increasingly detuned to the red side of the resonance from

top to bottom 79.57

that the generation of correlated and nonlocal, so-called

‘entangled’ quantum states, is possible [79.56].

The ﬁrst ‘application’ of cavity QED was the transfer

of the strong coupling idea to the combined internal and

motional quantum states of trapped ions [79.58]. Here

the harmonic oscillation of the ion replaces the electric

ﬁeld of the conventional cavity-QED system. This quan-

tum gate was realized with a system of two trapped ions

coupled to each other by Coulomb forces [79.59].

Ideas about how to use the strong coupling of atoms

and photons [79.60–62] for the generation of atom–

photon, or atom–atom (by insertion of more than one

atom) entanglement abound, but entanglement genera-

tion by means of cavity-QED with a controlled source

of atoms or ions remains a challenge for the future.

79.8.3 Single Photon Sources

Coherent laser ﬁelds are considered the ultimate source

of classical radiation ﬁelds, and they are characterized

by the random arrival time of photons. Nonclassical

light sources with, for instance, a regularized stream

of photons offer interesting properties for low-noise

measurement applications.

Cavity-QED systems offer an attractive light-matter

process for the generation of such ‘photon-bit-streams’,

or single photon sources [79.63]. In such devices, a sin-

gle photon state can, for instance, be created by Raman

processes involving a classical ﬁeld, which serves as

the control parameter for the process, and the vacuum

ﬁeld of the optical resonator. The Raman process leaves

a single photon in the cavity, which only weakly interacts

with the atom. If the resonator has suitable transmission

properties, this photon will then escape with prede-

termined frequency, shape, and propagation direction.

Deterministic single photon sources have been realized

with quantum dots [79.64,65], single molecules [79.66],

and also with slow [79.67] or trapped [79.68]coldatoms

and ions [79.69] inside optical cavities.

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Part F 79

1184 Part F Quantum Optics

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Part F 79

1185

Quantum Opti

80. Quantum Optical Tests of the Foundations

of Physics

Quantum mechanics began with the solution of

the problem of blackbody radiation by Planck’s

quantum hypothesis: in the interaction of light

with matter, energy can only be exchanged

between the light in a cavity and the atoms in the

walls of the cavity by the discrete amount E = hν,

where h is Planck’s constant and ν is the frequency

of the light. Einstein, in his treatment of the

photoelectric effect, reinterpreted this equation

to mean that a beam of light consists of particles

(“light quanta”) with energy hν. The Compton

effect supported this particle viewpoint of light by

demonstrating that photons carried momentum,

as well as energy. In this way, the wave–particle

duality of quanta made its ﬁrst appearance in

connection with the properties of light.

It might seem that the introduction of the

concept of the photon as a particle would

necessarily also introduce the concept of locality

into the quantum world. However, in view of

observed violations of Bell’s inequalities, exactly

the opposite seems to be true. Here we review

some recent results in quantum optics which

elucidate nonlocality and other fundamental

issues in physics.

In spite of the successes of quantum electro-

dynamics, and of the standard model in particle

physics, there is still considerable resistance to

the concept of the photon as a particle. Many pa-

pers have been written trying to explain all optical

phenomena semiclassically, i. e., with the light

viewed as a classical wave, and the atoms treated

quantum mechanically [80.1–4]. We ﬁrst present

some quantum optics phenomena which exclude

this semiclassical viewpoint.

80.1 The Photon Hypothesis ........................1186

80.2 Quantum Properties of Light.................1186

80.2.1 Vacuum Fluctuations:

Cavity QED ................................ 1186

80.2.2 The Down-Conversion

Two-Photon Light Source ...........1187

80.2.3 Squeezed States of Light ............1187

80.3 Nonclassical Interference .....................1188

80.3.1 Single-Photon

and Matter–Wave Interference ...1188

80.3.2 “Nonlocal” Interference Effects

and Energy–Time Uncertainty .....1189

80.3.3 Two-Photon Interference ...........1190

80.4 Complementarity and Coherence...........1191

80.4.1 Wave–Particle Duality................1191

80.4.2 Quantum Eraser ........................1191

80.4.3 Vacuum-Induced Coherence....... 1192

80.4.4 Suppression of Spontaneous

Down-Conversion ..................... 1192

80.5 Measurements in Quantum Mechanics...1193

80.5.1 Quantum (Anti-)Zeno Effect ........1193

80.5.2 Quantum Nondemolition ...........1193

80.5.3 Quantum Interrogation ..............1194

80.5.4 Weak and “Protected”

Measurements ..........................1195

80.6 The EPR Paradox and Bell’s Inequalities 1195

80.6.1 Generalities..............................1195

80.6.2 Polarization-Based Tests ............ 1196

80.6.3 Nonpolarization Tests ................1196

80.6.4 Bell Inequality Loopholes ...........1198

80.6.5 Nonlocality Without Inequalities . 1199

80.7 Quantum Information ..........................1200

80.7.1 Information Content

of a Quantum: (No) Cloning ........1200

80.7.2 Super-Dense Coding ..................1200

80.7.3 Teleportation ............................1200

80.7.4 Quantum Cryptography ..............1201

80.7.5 Issues in Causality ..................... 1202

80.8 The Single-Photon Tunneling Time .......1202

80.8.1 An Application of EPR Correlations

to Time Measurements...............1202

80.8.2 Superluminal Tunneling Times .... 1203

80.8.3 Tunneling Delay

in a Multilayer Dielectric Mirror...1203

80.8.4 Interpretation

of the Tunneling Time................1204

80.8.5 Other Fast

and Slow Light Schemes.............1205

80.9 Gravity and Quantum Optics ................. 1206

References .................................................. 1207

Part F 80

1186 Part F Quantum Optics

80.1 The Photon Hypothesis

In an early experiment, Taylor reduced the intensity of

a thermal light source in Young’s two-slit experiment,

until, on the average, there was only a single photon

passing through the two slits at a time. He then ob-

served a two-slit interference pattern which was identical

to that for a more intense classical beam of light. In

Dirac’s words, the apparent conclusion is that “each

photon then interferes only with itself” [80.6]. How-

ever, a coherent state, no matter how strongly attenuated,

always remains a coherent state (Sect. 78.2.2); since

a thermal light source can be modeled as a statistical

ensemble of coherent states, a stochastic classical wave

model yields complete agreement with Taylor’s observa-

tions. The one-by-one darkening of grains of ﬁlm can be

explained by treating the matter alone quantum mechan-

ically [80.2]; consequently, the concept of the photon

need not be invoked, and the claim that this experi-

ment demonstrates quantum interference of individual

photons is unwarranted [80.7].

This weakness in Taylor’s experiment can be re-

moved by the use of nonclassical light sources; as

discussed by Glauber [80.8], classical predictions di-

verge from quantum ones only when one considers

counting statistics, or photon correlations. In particular,

two-photon light sources, combined with coincidence

detection, allow the production of single-photon (n = 1

Fock) states with near certainty. In the ﬁrst such experi-

ment [80.9], two photons, produced in an atomic cascade

within nanoseconds of each other, impinged on two

beam splitters, and were then detected in coincidence by

means of four photomultipliers placed at all possible exit

ports. In a simpliﬁed version of this experiment [80.5],

one of the beam splitters and its two detectors are re-

placed with a single detector D

(Fig. 80.1). We deﬁne

Fig. 80.1 Triple-coincidence setup of Grangier et al. [80.5]

the anticorrelation parameter

α ≡ N

123

, (80.1)

where N

123

is the rate of triple-coincidences between

detectors D

, D

and D

; N

is the singles rate

at D

;andN

and N

are double-coincidence

rates. Then from Schwarz’s inequality [80.5, 7, 10],

α ≥ 1 for any classical wave. In essence, since the

wave divides smoothly, the coincidence rate between

and D

is never smaller than the “accidental” coin-

cidence rate, even when measurements are conditioned

on an event at D

. (The Hanbury–Brown and Twiss

experiment [80.11] can be explained classically, be-

cause the thermal ﬂuctuations lead to “bunching,” or

a mean coincidence rate which is greater than the

mean accidental rate; Sect. 78.3.3.) By contrast, the

indivisibility of the photon leads to strong anticorre-

lations between D

and D

,makingα arbitrarily small.

In agreement with this quantum mechanical picture,

Grangier et al. observed a 13-standard-deviation vio-

lation of the inequality [80.5], corroborating the notion

of the “collapse of the wave packet” as proposed by

Heisenberg [80.12].

80.2 Quantum Properties of Light

80.2.1 Vacuum Fluctuations: Cavity QED

The above considerations necessitate the quantization

of the electromagnetic ﬁeld, which in turn leads to

the concept of vacuum ﬂuctuations [80.4] (Sect. 78.1).

Difﬁculties with this idea, such as the implied inﬁ-

nite zero-point energy of the universe, have led some

reseachers to attempt to dispense with this concept

altogether, along with that of the photon, in every ex-

planation of electromagnetic interactions with matter. Of

course, it is impossible to explain all phenomena, such as

spontaneous emission and the Lamb shift, without some

kind of ﬂuctuating electromagnetic ﬁelds (Chapt. 78

and Sect. 79.2.4), but one can go a long way with an

ad hoc ambient classical electromagnetic noise-ﬁeld

ﬁlling all of space, in conjunction with the radiation

reaction [80.1, 4].

In particular, even the Casimir attraction be-

tween two conducting plates (Sect. 79.2.4), which has

now been veriﬁed with high precision [80.13–19]

can be explained semiclassically in terms of dipole

forces between electrons in each plate as they un-

Part F 80.2

Quantum Optical Tests of the Foundations of Physics 80.2 Quantum Properties of Light 1187

dergo zero-point motion that induce image charges

in the other plate. Nevertheless, the effects of cavity

QED (Chapt. 79) [80.20–22], including the inﬂuence

of cavity-induced boundary conditions on energy

levels and spontaneous emission rates, are most eas-

ily uniﬁed via quantization of the electromagnetic

ﬁeld.

By coupling highly excited Rydberg atoms to

photons in a high-ﬁnesse superconducting microwave

cavity, Haroche et al. have observed single-photon

driven Rabi oscillations [80.23],andhaveusedthese

to study decoherence effects [80.24], atom–photon en-

tanglement [80.25], and quantum nondemolition meas-

urements [80.26] (Sect. 80.5.2). Kimble et al. [80.27]

and Rempe et al. [80.28] have performed similar ex-

periments, coupling atoms to small optical cavities, and

even trapping the atoms with light ﬁelds at the single-

photon level [80.29,30]. By monitoring the amplitude of

the light transmitted through the cavity (which depends

on the precise location of the atom inside the cavity

volume), the trajectory of the atom can be determined

with ultrahigh resolution, much smaller than an optical

wavelength [80.31,32].

80.2.2 The Down-Conversion

Two-Photon Light Source

The quantum aspects of electromagnetism are made

more striking with a two-photon light source, in which

two highly correlated photons are produced in spon-

taneous parametric down-conversion, or parametric

ﬂuorescence [80.33–36]. In this process, an ultravio-

let “pump” photon produced in a laser spontaneously

decays inside a crystal with a χ

(2)

nonlinearity into

two highly correlated red photons, conventionally called

the “signal” and the “idler” (Sect. 72.3.4). (The quan-

tum state of the light is more correctly written as

|ψ∝|vacuum+|1

|1

+

|2

+..., but since

the amplitude of the down-conversion process itself is

very weak



 is of order 10

−6



, one often neglects

the terms containing 2 or more pairs. However, recent

experiments have begun to exploit these higher-order

terms, e.g., to investigate 3-, 4-, 5-photon quantum

effects [80.37–40]. Very recently, a stimulated down-

conversion process [80.41] has indicated the presence of

12-photon entanglement [80.42].) As shown in Fig. 80.2,

a rainbow of colored cones is produced around an axis

deﬁned by the direction of the uv beam (for the case

of type-I phase-matching), with the correlated down-

conversion photons always emitted on opposite sides

of the UV beam (Sect. 72.2.2). Their emission times

pump

KDP crystal

Deep red Red

Orange

Fig. 80.2 Conical emissions of down-conversion from

a nonlinear crystal (for type-I phase-matching). Photon en-

ergy depends on the cone opening angle, and conjugate

photons lie on opposite sides of the axis, e.g., the inner

“circle” orange photon is conjugate to the outer “circle”

deep-red photon, etc.

are within femtoseconds of each other, so that detec-

tion of one photon implies with near certainty that

there is exactly one quantum present in the conjugate

mode [80.43]. In type-I phase-matching the correlated

photons share the same polarization, while in type-II

phase-matching they have orthogonal polarizations . We

will see below (Sect. 80.6.2) how both of these can en-

able the production of photons that are entangled in

polarization, as well as in other degrees of freedom.

This production technique allowed for the ﬁrst re-

construction of the Wigner distribution for a single

photon [80.44], which is manifestly non-classical in that

the quasiprobability for both quadratures of the ﬁeld

to vanish is negative. In contrast to an earlier demon-

stration of a negative Wigner function [80.45](using

atoms, not photons), this measurement was possible us-

ing essentially classical measurement techniques with

no quantum assumptions, and is in this sense a direct

demonstration of the non-classical nature of the electro-

magnetic ﬁeld. Later work [80.46] used these techniques

to demonstrate that a single photon forced to choose

between two output ports of a beam splitter exhibits

quantum correlations.

80.2.3 Squeezed States of Light

The creation of correlated photon pairs is closely

related to the process of quadrature-squeezed light

production (Sect. 78.2.2, and the review in [80.47]).

For example, when the gain arising from parametric

ampliﬁcation in a down-conversion crystal becomes

large, there is a transition from spontaneous to stim-

ulated emission of pairs. This gain is dependent on

the phase of ampliﬁed light relative to the phase of

Part F 80.2

1188 Part F Quantum Optics

the pump light. As a result, the vacuum ﬂuctuations

are reduced (“squeezed”) below the standard quan-

tum limit (SQL) in one quadrature, but increased in

theother,insuchawayastopreservetheminimum

uncertainty-principle product [80.48]. This periodicity

of the ﬂuctuations at 2ω is a direct consequence of the

fact that the light is a superposition of states differing

in energy by 2

ω – the quadrature-squeezed vacuum

state



= exp



∗

aa −

ξa

†





(80.2)

represents a vacuum state transformed by the creation

†

) and destruction (aa) of photons two at a time. Es-

sentially any optical processes operating on photon pairs

(e.g., four-wave mixing [80.49, 50]) can also produce

such squeezing.

Amplitude squeezing involves preparation of states

with well-deﬁned photon number, i. e., states lacking the

Poisson ﬂuctuations of the coherent state. The possibility

of producing such states (e.g., via a constant-current-

driven semiconductor laser [80.51]) demonstrates that

“shot noise” in photodetectionshould not be thought of

as merely the result of the probabilistic (à la Fermi’s

Golden Rule, Sect. 69.4) excitation of quantum mech-

anical atoms in a classical ﬁeld, but as representing

real properties of the electromagnetic ﬁeld, accessible

to experimental control.

The highest level of number-squeezing reported

( −5.7 ±0.1) dB, corresponding to a noise reduc-

tion 73% below the SQL) employed an asymmetric

ﬁber loop to squeeze solitons [80.52]. The high-

est level of quadrature-squeezing reported has been

( −7.0 ±0.2) dB (80% below the SQL), using a χ

(2)

crystal optical parametric oscillator [80.53]. This was

a continuous-wave vacuum-squeezed beam – the same

technique was used to produce −5.0 dB of bright

continuous-wave squeezing locked for several hours.

Squeezed light has begun to have impact in

metrology. A few examples sufﬁce: the generation of

audio-band squeezed light [80.54] and the demonstra-

tion of a squeezing-enhanced power-recycled Michelson

interferometer [80.55], both suitable for gravity-wave

interferometry (Sect. 80.9); the use of squeezing to

measure displacement of a light beam below the

standard quantum limit [80.56], suitable for atomic

force microscopy; and demonstrations of squeezed light

spectroscopy [80.57–59].

80.3 Nonclassical Interference

80.3.1 Single-Photon

and Matter–Wave Interference

The ﬁrst truly one-photon interference experiment [80.5]

used the cascade source discussed in Sect. 80.1. One of

the photons was directed to a “trigger” detector, while

the other, thus prepared in an n = 1Fockstate,was

sent through a Mach–Zehnder interferometer. The out-

put photon, detected in coincidence with the trigger

photon, showed fringes with a visibility > 98%. Dirac’s

statement that a single photon interferes with itself is

thus veriﬁed.

Of course, matter can also display interference,

determined by the deBroglie wavelength (Chapt. 77).

There have been signiﬁcant recent advances in atomic

matter–wave interferometry and its applications [80.60],

ever since the early experiments of Pritchard, which used

standing-light gratings and nanofabricated diffraction

gratings to construct Mach–Zehnder-type interferom-

eters for sodium atoms [80.61] and molecules [80.62]

from a supersonic source. Chu and Kasevich introduced

the use of STIRAP (Stimulated Raman Adiabatic fast

Passage; Sect. 69.7) to produce coherent beam splitters

for cold atomic beams, also in a Mach–Zehnder-type

interferometer, but whose source were cesium atoms

cooled in and launched from a magneto-optical trap

(MOT) [80.63]. Matter–wave interferometry has now

been applied to precision measurements of the ac-

celeration g due to Earth’s gravity [80.64], gravity

gradiometry [80.65], and Sagnac matter–wave gyro-

scopes [80.66, 67].

To date some of the largest systems to display quan-

tum interference are large molecules like carbon 60

(“Buckyballs”) and carbon 70 [80.68]. These are sig-

niﬁcant in that the average deBroglie wavelength of

the molecules, emitted from an oven, was 2.8 pm, ac-

tually about 350 times smaller than the molecule itself.

Arndt et al. have also demonstrated multislit diffraction

with the biological molecule porphyrin, and with ﬂuoro-

fullerenes (C

) [80.69]. With a mass of 1632 amu,

the latter are currently the largest single objects to dis-

play interference. Looking ahead, others have suggested

that it may be possible to put a micron-scale mirror



with

≈ 10

atoms



into a superposition of resolvable spatial

Part F 80.3

Quantum Optical Tests of the Foundations of Physics 80.3 Nonclassical Interference 1189

locations [80.70] – the mirror, part of a high-ﬁnesse

optical cavity forming one arm of a Michelson interfer-

ometer, could be mounted on a high-quality mechanical

oscillator, whereby the interaction with a single photon

would change the frequency of the oscillator.

Two other systems, demonstrating Bose–Einstein

condensation (BEC) (see Chapt. 76), have also pro-

duced evidence of macroscopic quantum coherence.

In the experiments of Ketterle et al., atoms from

two different atomic vapor BEC clouds were al-

lowed to fall onto the same detection region, and

display interference fringes [80.71] (more recently in-

terference from an array of 30 independent BECs

has been observed [80.72]). In some ways this is

the matter–wave equivalent of the famous Pﬂeegor–

Mandel experiment [80.73], in which light from two

separate lasers displays interference, even when at-

tenuated to the single-photon level. The explanation

in terms of the indistinguishability of the underly-

ing processes is that one cannot ascertain from which

laser source a given photon originated. However, this

explanation must be applied carefully to the situa-

tion of the two atomic BECs: Unlike the lasers, the

BEC clouds can – at least in principle – be prepared

with a deﬁnite number of atoms, and it would there-

fore seem that one could in principle determine which

cloud emitted a given detected atom. However, this

determinacy is rapidly lost after a few atoms are de-

tected [80.74]. Once the number becomes uncertain,

a well-deﬁned relative phase of the two BECsises-

tablished, according to the number-phase uncertainty

relation ∆(N

−N

)∆(φ

−φ

)  1/2. In fact, discus-

sions have recently arisen over whether or not lasers

should not also be viewed as incoherent number-state

combinations, instead of the usual coherent state |α

(the issue is that in principle there is nothing in

a laser to break the symmetry and select a particular

phase) [80.75–77].

Finally, quantum coherence (though not explicitly

spatial interference as in the previous examples), has

been detected in the operation of a Josephson-junction

linked superconducting loop – the group of Mooij was

able to prepare a superposition of clockwise and counter-

clockwise circulating electrical currents [80.78]. Since

the ≈ 0.5 µA currents corresponded to the motion of

millions of Cooper pairs, this is arguably the largest

system thus far to have displayed quantum coher-

ence. This superconducting system also holds promise

for quantum computing (see Chapt. 81), as Rabi os-

cillations between the different ﬂux states have been

observed [80.79,80].

80.3.2 “Nonlocal” Interference Effects

and Energy–Time Uncertainty

The energy–time uncertainty principle, ∆E∆t ≥ /2

has been tested in a down-conversion interference

experiment [80.81]. The down-conversion process con-

serves energy and momentum:

= ω

+ ω

, (80.3)

≈ k

+ k

, (80.4)

where ω

( k

) is the energy (momentum) of the parent

photon, and

( k

)and ω

( k

) are the energies

(momenta) of the daughter photons; k

and k

sum

to k

to within an uncertainty given by the recipro-

cal of the crystal length [80.82]. Since there are many

ways of partitioning the parent photon’s energy, each

daughter photon may have a broad spectrum, and hence

a wave packet narrow in time. However, ω

+ω

= ω

is extremely well-deﬁned, so that the difference in the

daughter photons’ arrival times, and the sum of their

energies can be simultaneously known to high preci-

sion. Thus, the daughter photons of a parent photon

of sharp energy E

are in an energy-“entangled” state,

a nonfactorizable sum of product states [80.83]:





dEA(E)

|

− E



, (80.5)

where A(E) is the probability amplitude for the pro-

duction of two photons of energies E and E

− E.

A measurement of the energy of one of the photons

to be E

can be interpreted as causing an instantaneous

“collapse” of the system to the state |E

|E

−E

,im-

plying an instantaneous increase of the width of the other

photon’s wave packet. (Of course, the notion of col-

lapse need not be invoked to explain such results. One

can view the detection of the trigger photon as condi-

tionally selecting a particular subensemble of the pairs.

However, as discussed in Sect. 80.6.3, it is not correct

to interpret the down-conversion photons as possessing

a well-deﬁned energy prior to measurement.) In the ex-

periment, one photon was used as a trigger, while the

other was sent into an adjustable Michelson interferom-

eter (Fig. 80.3), used to measure its coherence length.

(The same apparatus was also used to demonstrate that

Berry’s phase in optics has a quantum origin [80.84].)

If the trigger photon passed through an interference ﬁl-

ter F1 of narrow width ∆E and was detected, then the

conjugate photon occupied a broad wavepacket of du-

ration ∆t ≈

/∆E, and displayed interference. When

Part F 80.3