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

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1200 Part F Quantum Optics

deal with real experiments, inequalities once again

result; these have also been experimentally vio-

lated, using non-maximally entangled states from

down-conversion [80.215, 216], further underscor-

ing the inconsistency between quantum theory and

locality.

80.7 Quantum Information

80.7.1 Information Content

of a Quantum: (No) Cloning

The inherent nonlocality of particles in an entangled

state cannot be used to transmit superluminal messages.

For example, if A and B receive a polarization-entangled

pair of photons, which A then collapses in a certain ba-

sis by performing a polarization measurement, B can

only extract one bit of information from a measurement

on his photon – this bit corresponds not to A’s choice

of basis, but to the (random) outcome of A’s measure-

ment. However, instantaneous communication would be

possible if one could make copies (“clones”) of a single

photon in an unknown polarization state: by performing

measurements on n copies of his photon, B could de-

termine its polarization to a resolution of n bits, thereby

accurately determining A’s choice of basis. Taking into

account quantum ﬂuctuations [80.217], one ﬁnds that

no physically allowed ampliﬁer can make a sufﬁciently

faithful copy for such a scheme to work – it is impossible

to clone an unknown quantum state [80.218]. A simple

proof is as follows. Consider an ideal cloner, initially in

the state |i

, which would take |0|i

→|0|0

and

|1|i

→|1|1

. For an accurate copier, one would

expect (|0+|1)|i

→ (|0+|1)(|0

+|1

), but the

linearity of quantum transformations instead yields

|0|0

+|1|1

, i. e., an entangled state.

Although perfect cloning is impossible, it is never-

theless possible to create copies which are “pretty good”.

Speciﬁcally, using an optimal cloning strategy, one can

in principle create a copy with a ﬁdelity of 5/6 with the

original state [80.219, 220]. Such a cloning procedure

has been experimentally realized by several groups, e.g.,

relying on stimulated emission, or sending the photon to

be cloned through a low noise optical ampliﬁer, with re-

sults matching the theoretical predictions [80.221–223].

80.7.2 Super-Dense Coding

The previous considerations make the work by Ben-

nett et al. on “quantum teleportation” and related effects

all the more remarkable. In the quantum dense-coding

protocol, a single photon can be used to transfer

two bits of information, when it is part of an en-

tangled EPR pair [80.224]. Again consider A and B,

each possessing one photon of such a pair. By ma-

nipulating only her photon (via a polarization rotator

and a phase shifter), A can convert the initial joint

state |ψ

−

 (80.9) into any of the four two-particle

“Bell states” [|ψ

=

(

, V

±|V

, H



)

√

|φ

=

(

, H

±|V

, V



)

√

2], and then send

her photon to B. By making a suitable measurement on

both photons, B can then in principle determine which of

the four states A produced [80.225]: A’s single photon

carried two bits. This protocol has been experimen-

tally realized using down-conversion photons [80.226],

though only two of the four Bell states could be reliably

distinguished. (Standard polarization Bell state analysis

is implemented by combining the two photons on a non-

polarizing 50–50 beam splitter. For any of the triplet

states |ψ

, |φ

, the photons will both travel to the

same output port due to the Hong–Ou–Mandel interfer-

ence discussed in Sect. 80.3.3; only for the state |ψ

−



will the photons travel to different outputs, resulting in

a coincidence detection.)

80.7.3 Teleportation

In the even more striking quantum teleportation ef-

fect [80.227], an unknown polarization state f (with

its in-principle inﬁnite amount of information) can be

“teleported” from A to B, if each already possesses one

photon of an EPR pair (e.g., in the singlet state |ψ

−

).

First, A jointly measures her EPR photon and the pho-

ton F (whose state f is to be teleported) in the basis

deﬁned by the Bell states of these two photons. Via

a mere two bits of classical information, A then in-

forms B which of the four (equally probable) Bell states

she actually measured. With this information, B can

transform the state of his EPR particle into f .Forex-

ample, if A found the singlet state |ψ

−

, then the

polarization of her EPR photon must have been orthog-

onal to that of F (because the polarizations of particles

in a singlet state are always perfectly anticorrelated, re-

gardless of the quantization basis). But because the two

EPR photons were initially also in a singlet state, their

polarizations must also be orthogonal, so B’s EPR pho-

ton is already in state f . If instead A found |φ

−

,

Part F 80.7

Quantum Optical Tests of the Foundations of Physics 80.7 Quantum Information 1201

for example, then B simply makes the same transfor-

mation (with a polarization rotator) that would have

changed |ψ

−

 into |φ

−

, again leaving his photon

in the state f . Thus, although one may only extract one

bit of (normally useless) information from an EPR par-

ticle, the perfect correlations may be used to transfer

an inﬁnite amount of information, i. e., precise speciﬁ-

cation of a point in the state space of the particle (the

Poincaré sphere for a photon or the Bloch sphere for

an electron). The “no-cloning” theorem (Sect. 80.7.1)is

not violated, since A irrevocably alters the state of F by

the measurement she performs, leaving only one particle

in f .

A number of experiments have now experi-

mentally realized quantum teleportation. The ﬁrst

of these used polarization-entangled down-conversion

photons [80.181], but was limited by the impossibil-

ity to resolve all four Bell states using only linear

optics. Teleportation of continous-variable states has

also been observed [80.228], using twin-beam squeezed

states (Sect. 80.2.3). Recently, the ﬁrst teleportation in

a matter system has been achieved: the groups of Blatt

et al. [80.229]andWineland et al. [80.230]havesuc-

cessfully teleported the (energy) quantum state of an ion

to a separate ion. Although the overall distance was less

than 1 mm, these challenging experiments are signiﬁ-

cant because they incorporate most of the techniques

necessary for scalable quantum information processing

in an ion-trap system (see Sect. 81.7.2).

In an interesting extension of the original teleporta-

tion protocol, one can ask what happens if the photon

to be teleported is itself entangled to a 4th photon G.

In this case, known as “entanglement swapping”, a suc-

cessful teleportation will lead to the entanglement of

photons G and B, even though these have never di-

rectly interacted. Entangled down-conversion photons

have been used to demonstrate such “entanglement

swapping” achieving a violation of Bell’s inequalities

between the two noninteracting photons [80.231]; simi-

lar results have been achieved with continous variables

as well [80.232]. Such procedures may one day enable

construction of a quantum “repeater” which could en-

able the transmission of quantum information over long

distances [80.233, 234].

A fundamental connection between quantum infor-

mation and black holes has recently been suggested

by Lloyd and Ng [80.235] and others. The basic idea

is that every physical object, including a black hole,

can be thought of as a quantum computer that uni-

tarily transforms input states to output states; i. e.,

“in” quantum bits (“qubits”) can always be reversibly

interconverted into “out” qubits, thus obeying time-

reversal symmetry. To resolve the paradox of the

apparent loss of information of matter falling into

a black hole, Lloyd and Ng propose that pairs of

entangled photons can materialize at the event hori-

zon of a black hole. One member of the photon pair

ﬂies outward to become the Hawking blackbody radi-

ation; the other falls into the black hole and hits the

singularity together with the matter that formed the

hole. The annihilation of the infalling photon acts as

a measurement on the infalling matter in a quantum

teleportation-like process, transporting the informa-

tion contained in the infalling matter to the outgoing

Hawking radiation, using the Horowitz–Maldacena

mechanism [80.236].

80.7.4 Quantum Cryptography

Although EPR schemes cannot send signals super-

luminally, they have other potential applications in

cryptography. In the “one-time pad” of classical cryptog-

raphy [80.237], two collaborators share a secret “key”

(a random string of binary digits) in order to encode and

decode a message. Such a key may provide an abso-

lutely unbreakable code, provided that it is unknown to

an eavesdropper. The problem arises in key distribution:

any classical distribution scheme is subject to noninva-

sive eavesdropping, e.g., using a ﬁber-coupler to tap the

line, without disturbing the transmitted classical signal.

In quantum cryptography proposals, security is guaran-

teed by using single-photon states [80.238, 239], some

of the schemes employing particles prepared in an EPR-

entangled state. Each collaborator receives one member

of each correlated pair, and measures the polarization

(or whatever degree of freedom is carrying the informa-

tion) in a random basis. After repeating the process many

times, the two then discuss publicly which bases were

used for each measurement, but not the actual meas-

urement results. The cases where different bases were

chosen are not used for conveying the key, and may

be discarded, along with instances where one party de-

tected no photon. In cases where the same bases were

used, however, the participants will now have corre-

lated information, from which a random, shared key can

be generated. As long as single photons are used, any

attempt at eavesdropping, even one relying on QND,

will necessarily introduce errors due to the uncertainty

principle. If the eavesdropper uses the wrong basis to

study a photon before sending it on to the real recipi-

ent, the very act of measuring will disturb the original

state.

Part F 80.7

1202 Part F Quantum Optics

Although one can perform quantum cryptography

with single-photon states or even weak coherent states

(Sect. 81.2.1), there are a number of advantages to us-

ing entanglement: there is inherent randomness; photons

from different pairs show no correlations (unlike mul-

tiple photons in an attenuated coherent state, which

may be used by an eavesdropper to gain informa-

tion), and any information leaked to other degrees of

freedom automatically shows up as increased error

rate [80.240]. This last feature means that the sources

are “self-checking” in a sense, and one could even let

an adversary have control over the source, secure in

the knowledge that any tampering would become evi-

dent in the nonlocal correlations. A number of quantum

cryptography implementations using down-conversion

photon pairs have been realized, either using polarization

entanglement [80.179, 180, 241, 242], or energy–time

entanglement [80.243, 244]. One main experimental

challenge will be to increase the rates of entangled

pair production (the typical source emission rate for

these cryptography experiments is only 10 000 /s), to

make them competitive with current weak coherent pulse

schemes (which can easily operate at over 10 MHz).

80.7.5 Issues in Causality

Outstanding causal paradoxes in optics include the para-

dox of Barton and Scharnhorst [80.245], closely related

to the Casimir effect (Sect. 79.2.4). In this case, the am-

plitude for light-by-light scattering is modiﬁed by the

presence of closely-spaced parallel conducting plates. It

appears that this can lead to propagation of light in vac-

uum (albeit a vacuum “colored” by the presence of the

Casimir plates) faster than c. Hegerfeldt has pointed out

similar paradoxes in connection with localization of any

particle in a quantum ﬁeld theory [80.246] and with the

Glauber theory of photodetection (Sect. 78.4) [80.247].

However, at least for the simplest example – the inter-

action between two widely separated atoms – as long

as one considers only probabilities that depend on the

separation r, the second atom cannot be excited by light

from the ﬁrst until after a time r/c [80.248].

80.8 The Single-Photon Tunneling Time

80.8.1 An Application of EPR Correlations

to Time Measurements

In this section we discuss experiments involving the

quantum propagation of light in matter. Due to the

sharp time correlations of the paired photons from

spontaneous down-conversion, one can use the HOM

interferometer (Sect. 80.3.3, and Fig. 80.4) to measure

very short relative propagation delay times for the

signal and idler photons. One early application was

therefore to conﬁrm that single photons in glass travel

at the group velocity [80.91]. At least until recently,

the only quantum theory of light in dispersive me-

dia was an ad hoc one [80.249–256]. The shift of the

interference dip resulting from a medium introduced

into one of the interferometer arms can be accurately

measured by determining how much the path lengths

must be changed to compensate the shift and recover

the dip. This result suggests that when looking for

a microscopic description of dielectrics, it is unnec-

essary to consider the medium as being polarized by

an essentially classical electric ﬁeld due to the col-

lective action of all photons present, and reradiating

accordingly. Linear dielectric response is not a col-

lective effect in this sense – each photon interferes

only with itself (as per Dirac’s dictum) as it is par-

tially scattered from the atoms in the medium. The

single-photon group velocity thus demonstrates “wave–

particle unity.”

The standard limitation for measurements of short-

time phenomena is that to have high time-resolution,

one needs short pulses (or at least short coherence

lengths), but these in turn require broad bandwidths

and are therefore very susceptible to dispersive broad-

ening. It is a remarkable consequence of the EPR

energy correlations of the down-conversion photons

(Sect. 80.6.3) that time measurements made with the

HOM interferometer are essentially immune to such

broadening [80.91, 257, 258]. In effect, the measure-

ment is sensitive to the difference in emission times

while the broadening is sensitive to the sum of the

frequencies. While frequency and time cannot both be

speciﬁed for a given pulse, the crucial feature of EPR

correlations such as those exhibited by down-converted

photons is that this difference and this sum correspond

to commuting observables, and both may be arbitrar-

ily well deﬁned. The photon which reaches detector

1 could either have traversed the dispersive medium

and been transmitted, or traversed the empty path and

been reﬂected, leaving its twin to traverse the medium.

The medium thus samples both of the (anticorrelated)

frequencies, leading to an automatic cancellation of

Part F 80.8

Quantum Optical Tests of the Foundations of Physics 80.8 The Single-Photon Tunneling Time 1203

any ﬁrst-order (and in fact, all odd-order) dispersive

broadening. Measurements can be more than 5 orders

of magnitude more precise than would be possible

via electronic timing of direct detection events, and

in principle better than those performed with nonlin-

ear autocorrelators (which rely on the same nonlinear

physics as down-conversion, but do not beneﬁt from

a cancellation of dispersive broadening).

80.8.2 Superluminal Tunneling Times

Another well-known problem in the theory of quantum

propagation is the delay experienced by a particle as it

tunnels. There are difﬁculties associated with calculating

the “duration” of the tunneling process, since evanescent

waves do not accumulate any phase [80.259–261]. First,

the kinetic energy in the barrier region is negative, so

the momentum is imaginary. Second, the transit time of

a wavepacket peak through the barrier, deﬁned in the

stationary phase approximation by

(φ)

≡ ∂



arg t(ω) e

ikd



/∂ω ,

(80.14)

tends to a constant as the barrier thickness diverges, in

seeming violation of relativistic causality. (Actually, it

is shown in [80.88] that such saturation of the delay

time is a natural consequence of time-reversal sym-

metry,andin[80.262] that one can deduce from the

principle of causality itself that every system possesses

a superluminal group delay, at least at the frequency

where its transmission is a minimum.) For example, the

transmission function for a rectangular barrier,

t(k,κ)=

−ikd

cosh κd +i

−k

2kκ

sinh κd

(80.15)

leads in the opaque limit (κd  1) to a traversal time of

2m/

kκ, independent of the barrier width d.Thesame

result applies to photons undergoing frustrated total in-

ternal reﬂection [80.88], when the mass m is replaced

by n

ω/c

, and similar results apply to other forms of

tunneling.

Some researchers have therefore searched for some

more meaningful “interaction time” for tunneling, which

might accord better with relativistic intuitions and per-

haps have implications for the ultimate speed of devices

relying on tunneling [80.263]. The “semiclassical time”

corresponds to treating the magnitude of the (imagi-

nary) momentum as a real momentum. This time is of

interest mainly because it also arises in Büttiker and

Landauer’s calculation of the critical timescale in prob-

lems involving oscillating barriers, which they take to

imply that it is a better measure of the duration of the

interaction than is the group delay [80.259]. The Lar-

mor time [80.264] is one of the early efforts to attach

a “clock” to a tunneling particle, in the form of a spin

aligned perpendicular to a small magnetic ﬁeld conﬁned

to the barrier region. The basic idea is that the amount

of Larmor precession experienced by a transmitted par-

ticle is a measure of the time spent by that particle in

the barrier. This clock turns out to contain components

corresponding both to the distance-independent “dwell

time” and the linear-in-distance semiclassical time. Cu-

riously, the most common theories for tunneling times

become superluminal in certain cases anyway, whether

or not they deal with the motion of wave packets.

Here, we will restrict ourselves to discussing the

time of appearance of a peak of a single-photon wave

packet. While other tunneling-time experiments have

been performed in the past [80.265], optical tests offer

certain unique advantages [80.266], including the ease

of construction of a barrier with no dissipation, very lit-

tle energy-dependence, and a superluminal group delay.

The transmitted wave packets suffer little distortion, and

are essentially indistinguishable from the incident wave

packets. At a theoretical level, the fact that photons are

described by Maxwell’s (fully relativistic) equations is

an important argument against interpreting superluminal

tunneling predictions as a mere artifact of the nonrela-

tivistic Schrödinger equation. Also, one is denied the

recourse suggested by some workers [80.267]ofin-

terpreting the superluminal appearance of transmitted

peaks to mean that only the high-energy components

(which, for matter waves, traveled faster even before

reaching the barrier [80.268]) were transmitted.

80.8.3 Tunneling Delay

in a Multilayer Dielectric Mirror

A suitable optical tunnel barrier can be a standard mul-

tilayer dielectic mirror. The alternating layers of low

and high index material, each one quarter-wave thick at

the design frequency of the mirror, lead to a photonic

bandgap [80.269] analogous to that in the Kronig–

Penney model of solid state physics (Sect. 79.2.6). The

gap represents a forbidden range of energies, in which

the multiple reﬂections will interfere constructively so

as to exponentially damp any incident wave. The anal-

ogy with tunneling in nonrelativistic quantum mechanics

arises because of the exponential decay of the ﬁeld en-

velope within the periodic structure, i. e., the imaginary

value of the quasimomentum. The same qualitative fea-

tures arise for the transmission time: for thick barriers,

Part F 80.8

1204 Part F Quantum Optics

it should saturate at a constant value, as was veri-

ﬁed in a recent experiment employing short classical

pulses [80.270]. (A more direct analogy, that of waveg-

uides beyond cutoff, yielded similar results in a classical

microwave experiment [80.271], while another paper

has reported superluminal effects related to the penetra-

tion of diffracted or “leaky” microwaves into a shadow

region [80.272]. All these experiments involve very

small detection probabilities, just as in Chu and Wong’s

pioneering experiment on propagation within an ab-

sorption line [80.273]. However, it has been predicted

that superluminal propagation could occur without high

loss or reﬂection [80.274–276], by operating outside

the resonance line of an inverted medium (Sect. 70.1).

One can understand the effect as off-resonance “virtual

ampliﬁcation” of the leading edge of a pulse.)

The phenomenon was investigated at the single-

photon level by using the high time-resolution

techniques discussed in Sect. 80.8.1 to measure the

relative delay experienced by down-conversion pho-

tons [80.277] when such a tunnel barrier (consisting

of 11 layers) was introduced into one arm of a HOM

interferometer. The transmissivity of the barrier was

relatively ﬂat throughout the bandgap (extending from

600 nm to 800 nm; Fig. 80.11), with a value of 1% at

the gap center (700 nm), where the experiment was per-

formed. The HOM coincidence dip was measured both

with the barrier (and its substrate) and with the substrate

100

450 550 650 750 850 950 1050

Transmission probability (%)

Wavelength (nm)

Time (fs)

Fig. 80.11 Transmission probability for the tunnel barrier used

in [80.277](heavy dotted curve); heavy black, dashed brown,and

solid grey curves show group delay, Larmor time, and semiclassical

time. Also shown for comparison is the “causality limit” d/c =3.6fs

(horizontal line)

Coincidence counts (normalized)

Time

With

barrier

Without

barrier

∆t ≈ –2 fs

Fig. 80.12 Coincidence proﬁles with and without the tunnel

barrier map out the single-photon wave packets. The lower

proﬁle shows the coincidences with the barrier; this proﬁle

is shifted by ≈ 2 fs to negative times relative to the one with

no barrier (upper curve): the average particle which tunnels

arrives earlier than the one which travels the same distance

in air

alone (Fig. 80.12). Each dip was subsequently ﬁtted to

a Gaussian, and the difference between their centers was

calculated. When several such runs were combined, it

was found that the tunneling peak arrived 1.47±0.21 fs

earlier than the one traveling through air, in reasonable

agreement with the theoretical prediction of approxi-

mately 1.9 fs. Taking into account the 1.1 µm thickness

of the barrier, this implies an effective photon tunneling

velocity of 1.7 c. The results exclude the “semiclassical”

time but are consistent with the group delay. An inves-

tigation of the energy-dependence of the tunneling time

was also performed, by angling the dielectric mirror,

thus shifting its bandgap [80.278]. The data conﬁrm the

group delay in this limit as well, and rule out identiﬁca-

tion of Büttiker’s Larmor time with a peak propagation

time, at least in this optical system.

80.8.4 Interpretation

of the Tunneling Time

Even though a wave packet peak may appear on the far

side of a barrier sooner than it would under allowed prop-

agation, it is important to stress that no information is

transmitted faster than c, nor on average is any energy.

These effects occur in the limit of low transmission,

where the transmitted wavepacket can be considered as

a “reshaped” version of the leading edge of the inci-

dent pulse [80.273, 279, 280]. At a physical level, the

Part F 80.8

Quantum Optical Tests of the Foundations of Physics 80.8 The Single-Photon Tunneling Time 1205

reﬂection from a multilayer dielectric is due to destruc-

tive interference among coherent multiple reﬂections

between the different layers. At times before the ﬁeld

inside the structure reaches its steady-state value, there

is little interference, and a non-negligible fraction of the

wave is transmitted. This preferential treatment of the

leading ramp-up engenders a sort of “optical illusion,”

shifting the transmitted peak earlier in time. A signal,

such as a front with a sharp onset, relies on arbitrarily

high-frequency components, which would not beneﬁt

from this illusion, but instead travel arbitrarily close to

the vacuum speed of light c. Even for a smooth wave

packet, no energy travels faster than light; most is sim-

ply reﬂected by the barrier. Only if one considers the

Copenhagen interpretation of quantum mechanics, with

its instantaneous collapse, does one ﬁnd superluminal

propagation of those particles which happen to be trans-

mitted. This leads to the question of whether it is possible

to ask which part of a wave packet a given particle comes

from.

One paper argued that transmitted particles do in

fact stem only from the leading ramp-up of the wave

packet [80.281]. While it is true that the transmission

only depends on causally connected portions of the

incident wave packet, further analysis revealed that si-

multaneous discussion of such particle-like questions

and the wave nature of tunneling ran afoul of the com-

plementarity principle [80.282]. In essence, labeling the

initial positions of a tunneling particle destroys the care-

ful interference by which the reshaping occurs (as in

the quantum eraser, Sect. 80.4.1). However, one picture

in which the transmitted particles really do originate

earlier is the Bohm–de Broglie model of quantum mech-

anics [80.283, 284]. This theory considers Ψ to be

a real ﬁeld (residing however in conﬁguration space,

thus incorporating nonlocality) which guides pointlike

particles in a deterministic manner. It reproduces all the

predictions of quantum mechanics without incorporat-

ing any randomness; the probabilistic predictions of QM

arise from a range of initial conditions. Bohm’s equa-

tion of motion has the form of a ﬂuid-ﬂow equation,

v(x) =

∇ arg Ψ(x)/m, implying that particle trajecto-

ries may never cross, as velocity is a single-valued

function of position. Consequently, all transmitted par-

ticles originate earlier in the ensemble than all reﬂected

particles [80.282, 285]. This approach yields trajecto-

ries with well-deﬁned (and generally subluminal) dwell

times in the barrier region. However, the fact that the

mean tunneling delay of Bohm particles diverges as the

incident bandwidth becomes small, along with other

interpretational issues [80.286, 287], leaves open the

question of whether time scales as deﬁned by the Bohm

model have any physical meaning.

The “weak measurement” approach of Aharonov

et al. [80.142, 143] or equivalently, complex condi-

tional probabilitiy amplitudes obeying Bayes’s theo-

rem [80.288, 289], can be used to address the question

of tunneling interaction times in an experimentally

unambiguous way. The real part of the resulting com-

plex times determine the effect a tunneling particle

would have on a “clock” to which it coupled, while

the imaginary part indicates the clock’s back-action on

the particle. They unify various approaches such as

the Larmor and Büttiker–Landauer times, as well as

Feynman-path methods. In addition, they allow one to

discuss separately the histories of particles which have

been transmitted or reﬂected by a barrier, rather than

discussing only the wave function as a whole. Interest-

ingly, these calculations do not support the assertion that

transmitted particles originate in the leading ramp-up of

a wave packet.

80.8.5 Other Fast

and Slow Light Schemes

In addition to the case of tunneling described already,

apparently superluminal propagation was observed in

a Bessel-beam geometry [80.290], and in the case of

an inverted medium [80.291], as described in [80.274–

276]. While the former case may be explained geo-

metrically, the latter – in which superluminal group

velocities occur without signiﬁcant gain, loss, or dis-

tortion – raises difﬁcult questions about the speeds

of propagation of both energy and information. For

two contrasting perspectives, see [80.292] and [80.293].

Much more work has followed, including theoretical

treatments of the role of quantum noise in prevent-

ing superluminal information transfer [80.294], and

attempts to experimentally compare the velocity of in-

formation transfer with the group velocity [80.295].

The latter work seemed to verify the claim, previously

tested only in an electronic analog [80.296], that even

in the regime of superluminal propagation, new in-

formation was limited to causal speeds. Some dispute

has persisted [80.297], and it seems clear that a more

rigorous deﬁnition of information velocity is required;

somewhere between the idealized extremes of inﬁnitely

sharp signal fronts and strictly ﬁnite signal bandwidths

lies the real world, and neither the front velocity nor

the group velocity should be expected to completely

describe the behavior of actual information-carrying

pulses. At the same time, the deﬁnition of the energy

Part F 80.8

1206 Part F Quantum Optics

velocity in active media cannot be resolved without re-

viving long-standing conundra about how to apportion

energy between the propagating ﬁeld and energy stored

in the medium [80.298].

Since 1999, there has actually been much more

excitement over so-called “slow light” than over

“fast light” [80.299]. Building on the concepts

of electromagnetically-induced transparency [80.300]

(Sect. 69.7), two groups succeeded in that year in

utilizing the steep dispersion curves which can be

generated by extremely narrow holes in absorption

lines to slow light by a remarkable factor, with

group velocities as low as 17 m/s [80.301, 302]. Two

years later, two experiments succeeded in bringing

light to a standstill [80.303, 304]. This can be un-

derstood in terms of a “dark-state polariton” model,

in which the propagating photon is adiabatically con-

verted into a (stationary) metastable atomic excitation.

In addition to the obvious possibilities for storage of

optical pulses, particularly tantalizing at the quantum

level [80.305], there have been several proposals for

generating extremely strong optical nonlinearities in

this system, perhaps directly applicable to quantum

information processing [80.306, 307]. Both slow light

and fast light have now been observed in solid-state

systems [80.308, 309], as well. In both cases, the exten-

sion to the single-photon level remains a major goal,

for both practical and theoretical reasons. For a re-

cent review on anomalous optical propagation velocities,

see [80.310].

80.9 Gravity and Quantum Optics

According to general relativity, gravitational radiation

can be produced and detected by moving mass distri-

butions [80.311–313]. However, gravitational radiation

is coupled only to time-varying mass quadrupole mo-

ments in lowest order, since the mass dipole moment is



= M R

and R

for a closed system can only

exhibit uniform rectilinear motion. Current efforts focus

on detecting gravitational waves (typically at 100 Hz to

1 kHz) from astrophysical sources, such as supernovae

or collapsing binary stellar systems. For example, it is

expected that in the nearby Virgo cluster of galaxies,

several such events should occur per year, each yielding

a fractional strain (∆L/L)of10

−21

on Earth. However,

there are large uncertainties in this estimate.

Two main efforts have been pursued toward gravi-

tational wave detection. The ﬁrst type of detector, the

resonant-mass detector (sometimes known as a “We-

ber bar,” after its inventor), utilizes a large cylindrical

mass whose fundamental mode of acoustical oscilla-

tion is resonantly excited by time-varying tidal forces

produced by the passage of a gravitational wave. The

induced motions are typically detected by piezoelec-

tric crystals, or by SQUIDs (superconducting quantum

interference devices) [80.314], yielding strain sensitivi-

ties better than 10

−18

√

Hz. Such detectors were ﬁrst

constructed in the 1960’s [80.313], and are still in

use (e.g., such as the 2.3-ton bar at Louisiana State

University, ALLEGRO [80.315]),buttodatenoin-

controvertible detections have been reported. (Attempts

to improve the signal-to-noise ratio in resonant-mass

GW detectors led to the consideration of back-action-

evading sensors (a special case of QND measurements,

Sect. 80.5.2) to circumvent the standard quantum noise

limit [80.316].)

More recently, a large amount of research has been

devoted to using optical interferometry to detect grav-

itational radiation. A passing gravitational wave alters

the relative path length in the arms of a Michelson inter-

ferometer, thereby slightly shifting the output fringes.

Although the effective gravitational mass of the light

is much smaller than that of the Weber bar, very long

interferometer arms (2–4 km, with a Fabry–Perot cav-

ity in each arm to increase the effective length) more

than make up for this disadvantage. The signal-to-

noise ratio for the detection of a fringe shift depends

on the power of the light. The US initiative, called

LIGO (Laser Interferometer Gravitational-Wave Obser-

vatory [80.317]) uses 10 W from a Nd:YAG laser, and

an additional external mirror to recirculate the unmeas-

ured light, thus increasing the stored light power up

to 10 kW.

There are three LIGO interferometers, located re-

spectively in Hanford, Washington (with both a 2 km

and a 4 km version) and Livingston, Louisiana (4 km

version). The registration of coincident events at the sep-

arated interferometers allows one to rule out terrestial

artifacts, but many problems involving seismic and ther-

mal isolation, absorption and heating, intrinsic thermal

noise and optical quality had to be addressed. The LIGO

interferometers have had several preliminary science

runs since Fall 2002; the ﬁrst true “search run” is sched-

uled for 2005 [80.318]. The present sensitivities, which

range from 10

−21

to 10

−22

√

Hz (at ≈ 200–300 Hz) are

approaching the initial design goal of 3 × 10

−23

√

Hz.

Part F 80.9

Quantum Optical Tests of the Foundations of Physics References 1207

Other similar detectors nearing, or currently in, op-

eration are VIRGO (in Pisa, Italy), GEO 600 (near

Hannover, Germany), and TAMA (in Tokyo, Japan);

planning for an instrument (ACIGA) in Australia is un-

derway, as is planning for the NASA/ESA collaborative

Laser Interferometer Space Antenna (LISA), currently

aiming for launch in 2012 [80.319]); this space-based

interferometer, with arms up to 5 × 10

km long, would

probe frequencies from 10

−4

to 10

−1

Hz, inaccessible

to terrestial experiments due to seismic and atmospheric

disturbances.

Because the standard quantum noise limit of

these detectors is ultimately determined by the

vacuum ﬂuctuations incident on the unused input

ports of the interferometers, it is in principle pos-

sible to achieve reduced noise levels by using

squeezed vacuum instead [80.313, 320] (Sect. 80.2.3).

This has been demonstrated experimentally in table-

top experiments [80.55, 321], though there are no

plans to incorporate it into the current version

of LIGO.

It has also been suggested that matter waves which

interact with gravity waves inside a matter–wave in-

terferometer (Sect. 80.3.1) could lead to a sensitive

method to detect gravitation radiation [80.322, 323].

Such a “Matter–wave Interferometric Gravitational-

wave Observatory” (MIGO) may allow the detection

of primordial gigahertz gravity waves arising from

the Big Bang [80.324]. Moreover, quantum mechan-

ical detectors based on the use of macroscopically

coherent entangled states may enable quantum trans-

ducers which can interconvert between electromagnetic

and gravitational radiations, based on time-reversal

symmetry [80.325, 326].

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