away from the barrier in order to recover the HOM dip. This is contrary to the normal

expectation that all such delays should be positive in sign. For example, one would ex-

pect a positive sign if the tunnel barrier were an ordinary piece of glass, in which case

the mirror would have to be moved towards the barrier to recover the HOM dip. Thus

the sign of the necessary displacement of mirror M1 determines whether tunneling is

superluminal or subluminal in character.

The tunnel barrier used in this experiment”which was ¬rst performed at Berke-

ley in 1993 (Steinberg et al., 1993; Steinberg and Chiao, 1995)”is a dielectric mir-

ror formed by an alternating stack of high- and low-index coatings, each a quarter

wavelength thick. The multiple Bragg re¬‚ections from the successive interfaces of the

dielectric coatings give rise to constructive interference in the backwards direction of

propagation for the photon and destructive interference in the forward direction. The

result is an exponential decay in the envelope of the electric ¬eld amplitude as a func-

tion of propagation distance into the periodic structure, i.e. an evanescent wave. This

constitutes a photonic bandgap, that is, a range of classical wavelengths”equivalent

to energies for photons”for which propagation is forbidden. This is similar to the ex-

M1

Tunnel Geiger

barrier counter

γ1

UV laser

Beam

γ0

splitter

Down- γ2

conversion

crystal Geiger

M2 counter

Fig. 10.11 Schematic of a realistic tunneling-time experiment, such as that performed in

Berkeley (Steinberg et al., 1993; Steinberg and Chiao, 1995), to measure the tunneling time of

a photon by means of Hong“Ou“Mandel two-photon interference. The double-headed arrow to

the right of mirror M1 indicates that it can be displaced so as to compensate for the tunneling

time delay introduced by the tunnel barrier. The sign of this displacement indicates whether

the tunneling time is superluminal or subluminal.

¿¼ Experiments in linear optics

ponential decay of the electron wave function inside the classically forbidden region of

a tunnel barrier.

In this experiment, the photonic bandgap stretched from a wavelength of 600 nm

to 800 nm, with a center at 700 nm, the wavelength of the photon pairs used in the

Hong“Ou“Mandel interferometer. The exponential decay of the photon probability

amplitude with propagation distance is completely analogous to the exponential decay

of the probability amplitude of an electron inside a periodic crystal lattice, when its

energy lies at the center of the electronic bandgap. The tunneling probability of the

photon through the photonic tunnel barrier was measured to be around 1%, and

was spectrally ¬‚at over the typical 10 nm-wide bandwidths of the down-conversion

photon wave packets. This is much narrower than the 200 nm total spectral width of

the photonic bandgap. The carrier wavelength of the single-photon wave packets was

chosen to coincide with the center of the bandgap. After the tunneling process was

completed, the transmitted photon wave packets su¬ered a 99% reduction in intensity,

but the distortion from the initial Gaussian shape was observed to be completely

negligible.

In Fig. 10.12, the data for the tunneling time obtained using the Hong“Ou“Mandel

8 100%

Transmission

6 80%

Larmor time

Delay time (fs)

4 60%

Transmission

Group delay time

2 40%

Subluminal

0 20%

Superluminal

0%

’2

0o 10o 20o 30o 40o 50o 60o 70o 80o 90o

Angle

Fig. 10.12 Summary of tunneling time data taken using the Hong“Ou“Mandel interferom-

eter, shown schematically in Fig. 10.11, as the tunnel barrier sample was tilted: starting from

normal incidence at 0—¦ towards 60—¦ for p-polarized down-converted photons. As the sample

was tilted towards Brewster™s angle (around 60—¦ ), the tunneling time changed sign from a

negative relative delay, indicating a superluminal tunneling time, to a positive relative delay,

indicating a subluminal tunneling time. Note that the sign reversal occurs at a tilt angle of

40—¦ . Two di¬erent samples used as barriers are represented respectively by the circles and

the squares. (Reproduced from Steinberg and Chiao (1995).)

Tunneling time measurements— ¿½

interferometer are shown as a function of the tilt angle of the tunnel barrier sample

relative to normal incidence, with the plane of polarization of the incident photon lying

in the plane of incidence (this is called p-polarization). As the tilt angle is increased

towards Brewster™s angle (around 60—¦ ), the re¬‚ectivity of the successive interfaces

between the dielectric layers tends to zero. In this limit the destructive interference in

the forward direction disappears, so the photonic bandgap, along with its associated

tunnel barrier, is eliminated.

Thus as one tilts the tunnel barrier towards Brewster™s angle, it e¬ectively behaves

more and more like an ordinary glass sample. One then expects to obtain a positive

delay for the passage of the photon γ1 through the barrier, corresponding to a sublu-

minal tunneling delay time. Indeed, for the three data points taken at the large tilt

angles of 45—¦ , 50—¦ , and 55—¦ (near Brewster™s angle) the mirror M1 had to be moved

towards the sample, as one would normally expect for the compensation of positive

delays. However, for the three data points at the small tilt angles of 0—¦ , 22—¦ , and 35—¦ ,

the data show that the tunneling delay of photon γ1 is negative relative to photon γ2 .

In other words, for incidence angles near normal the mirror M1 had to be moved in

the counterintuitive direction, away from the tunnel barrier. The change in sign of the

e¬ect implies a superluminal tunneling time for these small angles of incidence. The

displacement of mirror M1 required to recover the HOM dip changed from positive to

negative at 40—¦ , corresponding to a smooth transition from subluminal to superluminal

tunneling times. From these data, one concludes that, near normal incidence, the tun-

neling wave packet γ1 passes through the barrier superluminally (i.e. e¬ectively faster

than c) relative to wave packet γ2 . The interpretation of this seemingly paradoxical

result evidently requires some care.

We ¬rst note that the existence of apparently superluminal propagation of classi-

cal electromagnetic waves is well understood. An example, that shares many features

with tunneling, is propagation of a Gaussian pulse with carrier frequency in a region

of anomalous dispersion. The fact that this would lead to superluminal propagation

of a greatly reduced pulse was ¬rst predicted by Garrett and McCumber (1969) and

later experimentally demonstrated by Chu and Wong (1982). The classical explanation

of this phenomenon is that the pulse is reshaped during its propagation through the

medium. The locus of maximum constructive interference”the pulse peak”is shifted

forward toward the leading edge of the pulse, so that the peak of a small replica of

the original pulse arrives before the peak of a similar pulse propagating through vac-

uum. Another way of saying this is that the trailing edge of the pulse is more strongly

absorbed than the leading edge. The resulting movement of the peak is described by

the group velocity, which can be greater than c or even negative. These phenomena

are actually quite general; in particular, they will also occur in an amplifying medium

(Bolda et al., 1993). In this case it is possible for a Gaussian pulse with carrier fre-

quency detuned from a gain line to propagate”with little change in amplitude and

shape”with a group velocity greater than c or negative (Chiao, 1993; Steinberg and

Chiao, 1994).

The method used above to explain classical superluminal propagation is mathemat-

ically similar to Wigner™s theory of tunneling in quantum mechanics (Wigner, 1955).

This theory of the tunneling time was based on the idea, roughly speaking, that the

¿¾ Experiments in linear optics

peak of the tunneling wave packet would be delayed with respect to the peak of a

nontunneling wave packet by an amount determined by the maximum constructive

interference of di¬erent energy components, which de¬nes the peak of the tunneling

wave packet. The method of stationary phase then leads to the expression

d arg T (E)

„Wigner = (10.104)

dE E0

for the group-delay tunneling time, where E0 is the most probable energy of the

tunneling particle™s wave packet, and T (E) is the particle™s tunneling probability am-

plitude as a function of its energy E. Wigner™s theory predicts that the tunneling

delay becomes superluminal because”for su¬ciently thick barriers”the time „Wigner

depends only on the tunneling particle™s energy, and not on the thickness of the bar-

rier. Since the Wigner tunneling time saturates at a ¬nite value for thick barriers, this

produces a seeming violation of relativistic causality when „Wigner < d/c, where d is

the thickness of the barrier.

Wigner™s theory was not originally intended to apply to photons, but we have

already seen in Section 7.8 that a classical envelope satisfying the paraxial approxima-

tion can be regarded as an e¬ective probability amplitude for the photon. This allows

us to use the classical wave calculations to apply Wigner™s result to photons. From

this point of view, the rare occasions when a tunneling photon penetrates through the

barrier”approximately 1% of the photons appear on the far side”is a result of the

small probability amplitude that is transmitted. This in turn corresponds to the 1%

transmission coe¬cient of the sample at 0—¦ tilt. It is only for these lucky photons that

the click of the upper Geiger counter occurs earlier than a click of the lower Geiger

counter announcing the arrival of the nontunneling photon γ2 . The average of all data

runs at normal incidence shows that the peak of the tunneling wave packet γ1 arrived

1.47 ± 0.21 fs earlier than the peak of the wave packet γ2 that traveled through the air.

This is in reasonable agreement (within two standard deviations) with the prediction

of 1.9 fs based on eqn (10.104).

Some caveats need to be made here, however. The ¬rst is this: the observation of

a superluminal tunneling time does not imply the possibility of sending a true signal

faster than the vacuum speed of light, in violation of special relativity. By ˜true signal™

we mean a signal which connects a cause to its e¬ect; for example, a signal sent by

closing a switch at one end of a transmission-wire circuit that causes an explosion to

occur at the other end. Such causal signals are characterized by discontinuous fronts”

produced by the closing of the switch, for example”and these fronts are prohibited

by relativity from ever traveling faster than c. However, it should be stressed that

it is perfectly permissible, and indeed, under certain circumstances”arising from the

principle of relativistic causality itself”absolutely necessary, for the group velocity of

a wave packet to exceed the vacuum speed of light (Bolda et al., 1993; Chiao and

Steinberg, 1997). From a quantum mechanical point of view, this kind of superluminal

behavior is not surprising in the case of the tunneling phenomenon considered here.

Since this phenomenon is fundamentally probabilistic in nature, there is no determin-

istic way of controlling whether any given tunneling event will occur or not. Hence

The meaning of causality in quantum optics— ¿¿

there is no possibility of sending a controllable signal faster than c by means of any

tunneling particle, including the photon.

It may seem paradoxical that a particle of light can, in some sense, travel faster

than light, but we must remember that it is not logically impossible for a particle of

light in a medium to travel faster than a particle of light in the vacuum. Nevertheless,

it behooves us to discuss the fundamental questions raised by these kinds of coun-

terintuitive superluminal phenomena concerning the meaning of causality in quantum

optics. This will be done in more detail below.

The second caveat is this: it would seem that the above data would rule out all

theories of the tunneling time other than Wigner™s, but this is not so. One can only

say that for the speci¬c operational method used to obtain the data shown in Fig.

10.12, Wigner™s theory is singled out as the closest to being correct. However, by

using a di¬erent operational method which employs di¬erent experimental conditions

to measure a physical quantity”such as the time of interaction of a tunneling particle

with a modulated barrier, as was suggested by B¨ttiker and Landauer (1982)”one will

u

obtain a di¬erent result from Wigner™s. One striking di¬erence between the predictions

of these two particular theories of tunneling times is that in Wigner™s theory, the

group-delay tunneling time is predicted to be independent of barrier thickness in the

case of thick barriers, whereas in B¨ ttiker and Landauer™s theory, their interaction

u

tunneling time is predicted to be linearly dependent upon barrier thickness. A linear

dependence upon the thickness of a tunnel barrier has indeed been measured for one of

the two tunneling times observed by Balcou and Dutriaux (1997), who used a 2D tunnel

barrier based on the phenomenon of frustrated total internal re¬‚ection between two

closely spaced glass prisms. Thus in Balcou and Dutriaux™s experiment, the existence

of B¨ ttiker and Landauer™s interaction tunneling time has in fact been established.

u

For a more detailed review of these and yet other tunneling times, wave propagation

speeds, and superluminal e¬ects, see Chiao and Steinberg (1997).

The con¬‚icts between the predictions of the various tunneling-time theories dis-

cussed above illustrate the fact that the interpretation of measurements in quantum

theory may depend sensitively upon the exact operational conditions used in a given

experiment, as was emphasized early on by Bohr. Hence it should not surprise us that

the operationalism principle introduced at the beginning of this chapter must always

be carefully taken into account in any treatment of these problems. More concretely,

the phrase ˜the tunneling time™ is meaningless unless it is accompanied by a precise

operational description of the measurement to be performed.

The meaning of causality in quantum optics—

10.5

The appearance of counterintuitive, superluminal tunneling times in the above ex-

periments necessitates a careful re-examination of what is meant by causality in the

context of quantum optics. We begin by reviewing the notion of causality in classical

electromagnetic theory. In Section 8.1, we have seen that the interaction of a classical

electromagnetic wave with any linear optical device”including a tunnel barrier”can

be described by a scattering matrix. We will simplify the discussion by only considering

planar waves, e.g. superpositions of plane waves with all propagation vectors directed

along the z-axis. An incident classical, planar wave Ein (z, t) propagating in vacuum

¿ Experiments in linear optics

is a function of the retarded time tr = t ’ z/c only; therefore we replace Ein (z, t) by

Ein (tr ). This allows the incident ¬eld to be expressed as a one-dimensional Fourier

integral transform:

∞

dω

Ein (ω) e’iωtr .

Ein (tr ) = (10.105)

’∞ 2π

The output wave, also propagating in vacuum, is described in the same way by a

function Eout (ω) that is related to Ein (ω) by

Eout (ω) = S(ω)Ein (ω) , (10.106)

where S(ω) is the scattering matrix”or transfer function”for the device in question.

The transfer function S(ω) describes the reshaping of the input wave packet to produce

the output wave packet. By means of the convolution theorem, we can transform the

frequency-domain relation (10.106) into the time-domain relation

+∞

Eout (tr ) = S(„ )Ein (tr ’ „ )d„ , (10.107)

’∞

where ∞

dω

S(ω)e’iω„ .

S(„ ) = (10.108)

2π

’∞

The fundamental principle of causality states that no e¬ect can ever precede its

cause. This implies that the transfer function must strictly vanish for all negative

delays, i.e.

S(„ ) = 0 for all „ < 0 . (10.109)

Therefore, the range of integration in eqn (10.107) is restricted to positive values, so

that ∞

Eout (tr ) = S(„ )Ein (tr ’ „ )d„ . (10.110)

0

Thus we reach the intuitively appealing conclusion that the output ¬eld at time tr can

only depend on values of the input ¬eld in the past. In particular, if the input signal

has a front at tr = 0, that is

Ein (tr ) = 0 for all tr < 0 (or equivalently z > ct) , (10.111)

then it follows from eqn (10.110) that

Eout (tr ) = 0 for all tr < 0 . (10.112)

Thus the classical meaning of causality for linear optical systems is that the reshaping,

by whatever mechanism, of the input wave packet to produce the output wave packet

cannot produce a nonvanishing output signal before the arrival of the input signal

front at the output face.

In the quantum theory, one replaces the classical electric ¬eld amplitudes by time-

dependent, positive-frequency electric ¬eld operators in the Heisenberg picture. By

Interaction-free measurements— ¿

virtue of the correspondence principle, the linear relation between the classical input

and output ¬elds must also hold for the ¬eld operators, so that

+∞

(+) (+)

S(„ )Ein (tr ’ „ )d„ .

Eout (tr ) = (10.113)

0

One new feature in the quantum version is that the frequency ω in S(ω) is now

interpreted in terms of the Einstein relation E = ω for the photon energy. Another

important change is in the de¬nition of a signal front. We have already learnt that

¬eld operators cannot be set to zero; consequently, the statement that the input signal

has a front must be reinterpreted as an assumption about the quantum state of the

¬eld. The quantum version of eqn (10.111) is, therefore,

(+)

Ein (tr )ρ = 0 for all tr < 0 , (10.114)

where ρ is the time-independent density operator describing the state of the system

in the Heisenberg picture. It therefore follows from eqn (10.113) that

(+)

Eout (tr )ρ = 0 for all tr < 0 . (10.115)

The physics behind this statement is that if the system starts o¬ in the vacuum state

at t = 0 at the input, nothing that the optical system can do to it can promote it out

of the vacuum state at the output, before the arrival of the front. Therefore, causality

has essentially similar meanings at the classical and the quantum levels of description

of linear optical systems.

Interaction-free measurements—

10.6

A familiar procedure for determining if an object is present in a given location is to

illuminate the region with a beam of light. By observing scattering or absorption of the

light by the object, one can detect its presence or determine its absence; consequently,

the ¬rst step in locating an object in a dark room is to turn on the light. Thus in

classical optics, the interaction of light with the object would seem to be necessary for

its observation. One of the strange features of quantum optics is that it is sometimes

possible to determine an object™s presence or absence without interacting with the

object. The idea of interaction-free measurements was ¬rst suggested by Elitzur and

Vaidman (1993), and it was later dubbed ˜quantum seeing in the dark™ (Kwiat et al.,

1996). A useful way to think about this phenomenon is to realize that null events”

e.g. a detector does not click during a given time window”can convey information

just as much as the positive events in which a click does occur.

When it is certain that there is one and only one photon inside an interferom-

eter, some very counterintuitive nonlocal quantum e¬ects”including interaction-free

measurements”are possible. In an experiment performed in 1995 (Kwiat et al., 1995a),

this aim was achieved by pumping a lithium-iodate crystal with a 351 nm wavelength

ultraviolet laser, in order to produce entangled photon pairs by spontaneous down-

conversion. As shown in Fig. 10.13, one member of the pair, the gate photon, is di-

rected to a silicon avalanche photodiode T , and the signal from this detector is used to

¿ Experiments in linear optics

Dark

T

UV

Fig. 10.13 Schematic of an experiment using

a down-conversion source to demonstrate one

LiIO3

form of interaction-free measurement. The ob-

ject to be detected is represented by a trans-

latable 100% mirror, with translation denoted

Obj

by the double-arrow symbol ”. (Reproduced

from Kwiat et al. (1995a).)

open the gate for the other detectors. The other member of the pair, the test photon,

is injected into a Michelson interferometer, which is prepared in a dark fringe near the

equal-path length, white-light fringe condition; see Exercise 10.6. Thus the detector

Dark at the output port of the Michelson is a dark fringe detector. It will never reg-

ister any counts at all, if both arms of the interferometer are unblocked. However, the

presence of an absorbing or nontransmitting object in the lower arm of the Michelson

completely changes the possible outcomes by destroying the destructive interference

leading to the dark fringe.

In the real experimental protocol, the unknown object is represented by a translat-

able, 100% re¬‚ectivity mirror. In the original Elitzur“Vaidman thought experiment,

this role is played by a 100%-sensitivity detector that triggers a bomb. This raises the

stakes,2 but does not alter the physical principles involved. When the mirror blocks

the lower arm of the interferometer in the real experiment, it completely de¬‚ects the

test photon to the detector Obj. A click in Obj is the signal that the blocking ob-

ject is present. When the mirror is translated out of the lower arm, the destructive

interference condition is restored, and the test photon never shows up at the Dark

detector.

For a central Michelson beam splitter with (intensity) re¬‚ectivity R and transmis-

sivity T = 1 ’ R (neglecting losses), an incident test photon will be sent into the lower

arm with probability R. If the translatable mirror is present in the lower arm, the pho-

ton is de¬‚ected into the detector Obj with unit probability; therefore, the probability

of absorption is

P (absorption) = P (failure) = R . (10.116)

This is not as catastrophic as the exploding bomb, but it still represents an unsuc-

cessful outcome of the interaction-free measurement attempt. However, there is also a

mutually exclusive possibility that the test photon will be transmitted by the central

beam splitter, with probability T , and”upon its return”re¬‚ected by the beam split-

ter, with probability R, to the Dark detector. Thus clicks at the Dark port occur with

probability RT . When a Dark click occurs there is no possibility that the test photon

was absorbed by the object”the bomb did not go o¬”since there was only a single

photon in the system at the time. Hence, the probability of a successful interaction-free

measurement of the presence of the object is

2 Oneof the virtues of thought experiments is that they are not subject to health and safety

inspections.

Interaction-free measurements— ¿

P (detection) = P (success) = RT . (10.117)

For a lossless Michelson interferometer, the fraction · of successful interaction-free

measurements is therefore

P (detection)

P (success)

·≡ =

P (success) + P (failure) P (detection) + P (absorption)

1’R