EXCERPTS, NOTES, DETAILS, AND REFERENCES:
our experiments at Berkeley: the Franson experiment [2,3], the \quantum eraser" [4], the \dispersion-cancellation"
eect [5], and tunnelling-time measurements [6,7]. Let us begin by stating that we consider the EPR phenomenon to
be an \eect," not a \paradox": EPR's experimental predictions are internally consistent, and a contradiction is only
reached if one assumes both EPR's notion of locality and the completeness of quantum mechanics (QM). The three
central elements that constitute the EPR argument are 1) a belief in some of the quantum-mechanical predictions
concerning two separated particles, 2) a very reasonable denition of an \element of reality" [namely, that \if, without
in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a
physical quantity, then there exists an element of physical reality corresponding to this physical quantity"], and 3)
a belief that nature is local, i.e., that no expectation values at a spacetime point x2 can depend on an event at a
spacelike-separated point x1 (this denition of locality is now seen to be more stringent than Einsteinian causality, and
is inconsistent with QM).
--Quantum Nonlocality in Two-Photon Raymond Y. Chiao y , Paul G. Kwiat z and Aephraim M. Steinberg x y Department of Physics, University of California, Berkeley, CA 94720-7300, U.S.A. Institut fur Experimental physik, Universit· at Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria. National Institute of Standards and Technology, Phys A167, Gaithersburg, MD 20899, U.S.A. (Preprint quant-ph/9501016; This version was produced on December 21, 1994). Pp 1
Erwin Schr· odinger [14], in response to the EPR paper, pointed out that at the heart of these nonlocal eects is
what he called \entangled states" in quantum mechanics, i.e., nonfactorizable superpositions of product states. For if a two-particle wavefunction were factorizable,
(x1; x2) ="(x1)"(x2) (2)
then the probability of joint detection would also factorize,
j (x1; x2) j
2 = j "(x1) j
2
j "(x2) j
2 (3)
so that the outcomes of two spatially separated measurements would be independent of one another. In cases where the
two-particle state cannot be factorized as above, this means that the quantum-mechanical prediction implies nonlocal
correlations in the behavior of remote particles. The Bohm singlet state (1) is such an entangled state. It predicts
correlations between spin measurements made on the two particles. But these correlations persist even if the particles
and their analyzers are separated by space-like intervals, implying the existence of non-local in uences. Though each
particle considered individually is unpolarized, the two particles will always have opposite spin projections when
measured along the same quantization axis. Einstein et al. would conclude that each spin component is an \element
of reality" in that it would be possible to predict its value with 100% certainty without disturbing the particle, simply
by measuring the corresponding spin component of the particle's twin (a measurement which according to EPR's
locality hypothesis cannot disturb the particle in question). As discussed above, this reasoning led EPR to conclude
that quantum mechanics was incomplete; if one instead considers QM to be a complete theory, one must then admit
the existence of nonlocal eects. As we shall see below, experiment supports this latter interpretation.
--Opt Sit. Pp 2.
Using a \fair-sampling" assumption and the
symmetry properties of the interferometer (in particular, that the coincidence rate of the unused ports in Fig. 7
is equal to that of the used ports, an assumption supported by tests done with a third detector not shown in the
gure), we can directly obtain the value of the Bell-parameter S from the coincidence rates obtained at two values
each of °1 and °2. S is a measure of the strength of the correlations between the two particles, evaluated for the
four combinations of the two values of °1 and °2, and according to the Clauser-Horne-Shimony-Holt form of Bell's
inequality [18], satisfy j S j 2 for any local realistic model. For appropriate choices of °1 (45 and 135) and°2 (0
and 90), we obtain S = 2:63 0:08, clearly displaying quantum nonlocality.
--Opt Sit. Pp 5
We know that the peak of a classical electromagnetic wavepacket
propagating through a piece of glass will travel at the group velocity, but it is not entirely clear that one can interpret
this classical wavepacket as if it were the wavefunction of the single photon and then use the Born interpretation for
this wavefunction. If this interpretation were correct, then the photon would simply travel at the group velocity in
this medium. However, as Sommerfeld and Brillouin have pointed out [30], at the classical level there are at least ve
kinds of propagation velocities in a dispersive medium: the phase, group, energy, \signal," and front velocities, all of
which dier from one another in the vicinity of an absorption line, where there is a region of anomalous dispersion. In
particular, the group velocity can become \superluminal," i.e., faster than the vacuum speed of light, in these regions.
If the photon were to travel at the group velocity in this medium, would it also travel \superluminally"? If not, then
at which of these velocities does the photon travel in dispersive media? (These questions become especially acute in
media with inverted populations, where o-resonance wavepackets can travel superluminally without attenuation and
with little dispersion [31]; see also the accompanying article by Chiao et al.)
Motivated by the above questions, we did the following experiment. We removed the HWP and the polarizers from
the quantum eraser setup and inserted a piece of glass in the path of one of the photons; see Fig. 17. The glass
slows down the photon which traverses it, and in order to observe the coincidence dip, it is necessary to introduce an
equal, compensating delay " by adjusting the trombone prism. We measured the magnitude of this delay for various
samples of glass and were able to determine traversal times on the order of 35 ps, with 1 fs accuracy. In this way,
we were able to conrm that single photons travel through glass at the group velocity in transparent spectral regions,
an interesting example of particle-wave unity.
Clearly, the interest of measuring optical delays is greatest for media with dispersion. Consider the limiting time-
resolution of this interferometer. For a short wavepacket or pulse, a broad spectrum is necessary. In dispersive media,
however, the broad spectrum required for an ultrafast pulse (or single-photon wavepacket) can lead to a great deal
of dispersion. One might expect that this broadening of the wavepacket would also broaden the coincidence dip in
the HOM interferometer, since the physical explanation of the dip (in terms of which-path information carried by the
photons' arrival times) seems to imply that the width of the dip should be the size of the wavepackets which impinge
on the beam splitter. Thus the tradeo between pulse width and dispersive broadening would place an ultimate
limit on the resolution of a measurement made on a given sample. For example, a 15 fs wavepacket propagating
through half an inch of SF11 glass (one of the samples we studied) would classically broaden to about 60 fs due to
the dispersion in this glass. The nature of the broadening is that of a chirp, i.e., the local frequency sweeps from low
to high values (for normal dispersion, in which redder wavelengths travel faster than bluer wavelengths). Hence the
earlier part of the broadened pulse consists of redder wavelengths, and the later part of this pulse consists of bluer
----opt cit pp 9
wavelengths; see Fig. 18.
In our experiment, however, we found that the combination of the time-correlations and energy-correlations exhib-
ited by our entangled photons led to a cancellation of these dispersive eects. While the individual wavepacket which
travels through the glass does broaden according to classical optics, it is impossible to know whether this photon was
re ected or transmitted at the beam splitter (recall Fig. 12). This means that when an individual photon arrives
at a detector, it is unknowable whether it travelled through the glass or whether its conjugate (with anticorrelated
frequency) did so; due to the chirp, the delay in these two cases is opposite, relative to the peak of the wavepacket.
An exact cancellation occurs for the (greatly dominant) linear group-velocity dispersion term, and no appreciable
broadening of the 15 fs interference dip occurs. This is a direct consequence of the nature of the EPR state, in that
it relies on the simultaneous correlations of energy and time. A detailed theoretical analysis predicted these results,
in agreement with the simple argument presented here Coincidence rate (sec -1 )
Coincidence rate (sec -1 )
--- Opt Sir pp 10
tunnelling is one of the most striking consequences of quantum mechanics. The Josephson eect in solid state
physics, fusion in nuclear physics, and instantons in high energy physics are all manifestations of this phenomenon.
Every quantum mechanics text treats the calculation of the tunnelling probability. And yet, the issue of how much
time it takes a particle to tunnel through a barrier, a problem rst addressed in the 1930s, remains controversial to
the present day. The question arises because the momentum in the barrier region is imaginary. The rst answer, the
group delay (also known as the \phase time" because it describes the time of appearance of a wavepacket peak by
using the stationary phase approximation), can in certain limits be paradoxically small, implying barrier traversal at
a speed greater than that of light in vacuum [33,34]. This apparent violation of Einstein causality does not arise from
the use of the nonrelativistic Schr· odinger equation, since it also arises in solutions of Maxwell's equations, which are
fully relativistic. It has generally been assumed that such superluminal velocities cannot be physical [30], but in the
case of tunnelling, no resolution has been universally accepted.
As a result of developments in solid state physics, such as tunnelling in heterostructure devices, the issue has
acquired a new sense of urgency since the 1980s, leading to much con icting theoretical work [35{37]. Several
experimental papers presenting more or less indirect measurements of barrier traversal times have appeared. Some
seem to agree with the \semiclassical time" of B· uttiker and Landauer [35,38], while others [39,40] seem to agree with
the group delay (\phase time"). We presented the rst direct time measurement conrming that the time delay in
tunnelling can be superluminal, studying single photons traversing a dielectric mirror [6]. Since then, several microwave
experiments have conrmed that the eective group velocity of classical evanescent waves in various congurations
may be superluminal [41{43]. Also, recently a femtosecond laser experiment has conrmed our earlier ndings of
superluminal tunnelling in dielectric mirrors [44], using classical pulses.
As a result of developments in solid state physics, such as tunnelling in heterostructure devices, the issue has
acquired a new sense of urgency since the 1980s, leading to much con icting theoretical work [35{37]. Several
experimental papers presenting more or less indirect measurements of barrier traversal times have appeared. Some
seem to agree with the \semiclassical time" of B· uttiker and Landauer [35,38], while others [39,40] seem to agree with
the group delay (\phase time"). We presented the rst direct time measurement conrming that the time delay in
tunnelling can be superluminal, studying single photons traversing a dielectric mirror [6]. Since then, several microwave
experiments have conrmed that the eective group velocity of classical evanescent waves in various congurations
may be superluminal [41{43]. Also, recently a femtosecond laser experiment has conrmed our earlier ndings of
superluminal tunnelling in dielectric mirrors [44], using classical pulses.
Our experiment again employs the down-conversion source in a HOM interferometer arrangement. The advantage
of using these conjugate particles is that after one particle traverses a tunnel barrier its time of arrival can be compared
with that of its twin (which encounters no barrier), thus oering a clear operational denition and direct measurement
of the time delay in tunnelling. Since this technique relies on coincidence detection, the particle aspect of tunnelling
can be clearly observed: Each coincidence detection corresponds to a single tunnelling event.
In our apparatus, the tunnel barrier is a multilayer dielectric mirror. Such mirrors are composed of quarter-wave
layers of alternating high- and low-index materials, and hence possess a one-dimensional \photonic band gap" [45],
i.e., a range of frequencies which correspond to pure imaginary values of the wavevector. They are optical realizations
of the Kronig-Penney model of solid state physics, and thus analogous to crystalline solids possessing band gaps, as
well as to superlattices. Our mirrors have an (HL) 5 H structure, where H represents titanium oxide (with an index
of 2.22) and L represents fused silica (with an index of 1.41). Their total thickness d is 1.1 m, implying a traversal
time of d=c = 3:6 fs if a particle were to travel at c. Their band gaps extend approximately from 600 to 800 nm, and
their transmission amplitudes reach a minimum of 1% at 692 nm.
Transmission prob. (%)
Time (fs)
The semiclassical time is calculated from the group velocity which would hold inside an innite periodic medium
(i.e., neglecting re ections at the extremities of the barrier). As the wavevector becomes pure imaginary for frequencies
---Opt Cit PP 11
within the band gap, so does the semiclassical time; in order to extend it into the band-gap region, we simply drop
the factor of i, in analogy with the interaction time of B· uttiker and Landauer [35]. The \Larmor time" is a measure
of the amount of Larmor precession a tunnelling electron would experience in an innitesimal magnetic eld conned
to the barrier region. B· uttiker has suggested a Larmor time which takes into account the tendency of the transmitted
electrons to align their spins along the magnetic eld as well as the precession about the eld [46]. The group delay
is the derivative of the barrier's transmission phase with respect to the angular frequency of the light, according to
the method of stationary phase. All three times dip below d=c = 3:6 fs and are thus superluminal, although their
detailed behaviors are quite dierent; see Fig. 20. For example, the group delay remains relatively constant near 1.7 fs
over most of the band gap. The semiclassical time, on the other hand, dips below 3.6 fs only over a narrower range
of frequencies, and actually reaches zero at the center of the gap. B· uttiker's Larmor time approaches the group delay
far from the band gap as well as at its center, but diers from it at intermediate points.
Our apparatus is shown in Fig. 21. As before, a KDP crystal is pumped by a cw uv laser at 351 nm, producing pairs
of down-conversion photons, directed by mirrors to impinge simultaneously on the surface of a 50/50 beam splitter.
One photon of each pair travels through air, while the conjugate photon impinges on our sample, consisting of an
etalon substrate of fused silica, which is coated over half of one face with the 1.1 m coating described above, and
uncoated on the other half of that face. (The entire opposite face is antire ection coated.) This sample is mounted
on two stacked stages. The rst is a precision translation stage, which can place the sample in either of two positions
transverse to the beam path. In one of these positions, the photon must tunnel through the 1.1 m coating in order
to be transmitted, while in the other position, it travels through 1.1 m of air. In both positions, it traverses the same
thickness of substrate. The second stage allows the sample to be tilted with respect to normal incidence.
If the two photons' wavepackets are made to overlap in time at the beam splitter, the destructive interference
eect described above leads to a theoretical null in the coincidence detection rate. Thus as the path-length dierence
is changed by translating a \trombone" prism with a Burleigh Inchworm system (see Fig. 21) the coincidence rate
exhibits a dip with an rms width of approximately 20 fs, which is the correlation time of the two photons (determined
by their 6 nm bandwidths) [24,32,47]. As explained above, the rate reaches a minimum when the two wavepackets
overlap perfectly at the beam splitter. For this reason, if an extra delay is inserted in one arm of this interferometer
(i.e., by sliding the 1.1m coating into the beam), the prism will need to be translated in order to compensate for
this delay and restore the coincidence minimum. In order to eliminate so far as possible any systematic errors, we
conducted each of our data runs by slowly scanning the prism across the dip, while sliding the coating in and out of
the beam periodically, so that at each prism position we had directly comparable data with and without the barrier.
--Opt Cit Pp 12
We found that inserting the barrier at normal incidence (for which it was designed) did in fact cause the dip to be
shifted to a position in which the prism was located farther from the barrier. This determines the sign of the eect:
The external delay had to be lengthened, implying that the mean delay time experienced by the photon inside the
barrier was less than the delay time for propagating through the same distance in air. As we rotated the mirror about
the vertical axis, the bandgap shifted to lower wavelengths according to Bragg's law, and for the p-polarized photons
we studied, the width of the bandgap also diminished due to the decreased re ectivity of the dielectric interfaces at
non-normal incidence (cf. Brewster's angle). Thus at 0, our 702 nm photons are near the center of the bandgap,
while at 55, they are near the band edge, where the transmission is over 40%. As can be seen clearly from Fig. 22, the
delay time changes from a superluminal value to a subluminal one as the angle of incidence is scanned, in agreement
with theory.
Our normal-incidence data [6] demonstrated that the semiclassical time was inadequate for describing these prop-
---Opt cit Pp 13
We have thus conrmed that the peak of a tunnelling wave packet may indeed far side of a barrier
sooner than if it had been travelling at the vacuum speed of light. No signal can be sent with these smooth wavepackets,
however; only a small portion of the leading edge of the incident Gaussian is actually transmitted, and whether the
photon \collapses" into this portion or into the re ected portion is not under experimental control.
The superluminality can be understood by thinking of the low transmission
through our barrier as arising from destructive interference between waves which have spent dierent lengths of time
in the barrier. While the incident wavepacket is rising, multiple re ections can be neglected, since their intensities are
small relative to the partial wave which makes a single pass; thus the destructive interference is not very eective. At
later times, when the elds stored in the barrier have had time to reach a steady state, the interference reduces the
transmission to its steady-state value. Thus the leading edge of the packet is transmitted preferentially with respect
to the rest of the packet, shifting the transmitted peak earlier in time.
Recent work based on \weak measurement" theory [48] and the idea of conditional probability distributions for the
position of a quantum particle suggests that this superluminal eect is related to the fact that a tunnelling particle
spends very little time in the barrier region, except within an evanescent decay length of the two barrier edges [49{51].
It is as though the particle \skipped" the bulk of the barrier. Furthermore, the nonlocality is underscored by the fact
that this approach allows one to describe conditional probability distributions for a particle which is rst prepared
incident on the left and later detected emerging on the right. These probability distributions describe in-principle
measurable eects, and do indeed traverse the barrier faster than the vacuum speed of light. They suggest that a
single tunnelling particle could aect the expectation values of two dierent measuring devices located at spacelike
separated positions, so long as the coupling to the devices was too weak to disturb the tunnelling process, and hence
too weak to shift either measuring device by an amount comparable to its intrinsic uncertainty.
--Opt cit Pp 14
The experiments which we have described in this paper demonstrate some of the stranger nonlocal features of
quantum mechanics. The rst three of these experiments explore them in connection with the Einstein-Podolsky-
Rosen eect. In the Franson experiment, the behaviors of the two space-like separated particles at the nal beam
splitters (i.e., which exit port they choose) are correlated or anticorrelated with each other, depending on the settings
of the phase shifters in the interferometer. Likewise, in the quantum eraser, whether interference or the complementary
which-path information is observed can be controlled by the experimenter's choice of the settings of polarizers placed
after the nal beam splitter of the interferometer. In the dispersion cancellation experiment, one cannot know, even
in principle, which of two photons propagated through a piece of the glass. This in turn leads to a cancellation
of the eect of dispersive broadening on the measurement. The fourth of these experiments shows that, even at
the one-particle level, there exist nonlocal eects in quantum mechanics: in tunnelling there exist superluminal time
delays of the tunnelling particle.
--Opt cit Pp 15
OPT CIT REFERENCES
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[7] Steinberg A M and Chiao R Y 1994 Phys. Rev. Submitted (quant-ph/9501013) Sub-femtosecond determination of trans-
mission delay times for a dielectric mirror (photonic bandgap) as a function of angle of incidence
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D Han, Y S Kim and W W Zachary (NASA Conference Publication 3135) p 61
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[49] Steinberg A M 1994 Phys. Rev. Lett. submitted (quant-ph/9501015) How much time does a tunnelling particle spend in
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[51] Steinberg A M 1994 When Can Light Go Faster Than Light? The tunnelling time and its sub-femtosecond measurement
via quantum interference PhD thesis (U C Berkeley)
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In 1962 Josephson predicted the existence of a tunnelling supercurrent which traversed a gap separating two superconductors. This superconducting tunnel effect was conrmed experimentally by Giaever and others in Josephson junctions consisting of superconducting thin lms separated by a thin oxide barrier.
----tunnelling Times and Superluminality: a Tutorial. Raymond Y. Chiao. Dept. of Physics, Univ. of California. Berkeley, CA 94720-7300, U. S. A. November 6, 1998
Does the click of the detector which registers the
arrival of the photon which traversed the tunnel barrier go o earl ier or later
(on the average) than the click of the detector which registers the arrival the
photon which traversed the vacuum? If the tunnel barrier had simply been a
thin piece of transparent glass, then the answer would obviously be \later,"
since the group velocity for a photon inside the glass would be less than the
speed of light, and the group delay for the photon traversing the glass relative
to that of the vacuum would be positive. However, if, as some tunnelling-time
theories predict, the tunnelling process is superluminal, then the counterintuitive answer would be \earlier," since the eective group velocity for a photon inside the tunnel barrier would be greater than the speed of light, and the group delay for the photon traversing the barrier relative to that of the vacuum would be negative. Hence it is the sign of the relative time between the clicks in the two
detectors which determines whether tunnelling is subluminal or superluminal.
The reader may ask why relativistic causality is not violated by the super-
luminality of the tunnelling process, if it should indeed be superluminal. It has
been shown [8] that special relativity does not forbid the group velocity to be
faster than c; only Sommerfeld's front velocity must not exceed c. Alsore-
member that due to the uncertainty principle the time of emission of the signal
photon is not under the experimenter's control.
Presently, the best detectors for photons have picosecond-scale response
times, which are still not fast enough to detect the femtosecond-scale time dif-
ferences expected in our tunnelling-time experiment. Hence it was necessary to
utilize a Hong-Ou-Mandel interferometer, which has a femtosecond-scale tem-
poral resolution for measuring the time dierence between the travel times of
the two photons traversing the two arms of the interferometer. By placing the
tunnel barrier in one of these arms, a precise measurement of the delay due to
tunnelling could then be performed.
The tunnel barrier used in our experiments was a dielectric mirror in which
periodic layers of alternately high and low index media produce a photonic
band gap at the rst Brillouin zone edge. The problem of photon propagation
in this periodic structure is analogous to that of the Kronig-Penney model for
electrons propagating inside a crystal. In particular, near the midgap point
on the rst Brillouin zone edge, there exists due to Bragg re ection inside the
periodic structure an evanescent (i.e., exponential) decay of the transmitted
wave amplitude, which is equivalent to tunnelling. Note that this Bragg re ection
eect is completely analogous to the one occurring in the Esaki tunnel diode
mentioned above. One important feature of this kind of tunnel barrier is the
fact that it is nondispersive near midgap, and therefore there is little distortion
of the tunnelling wave packet.
tunnelling Time Theories
Another strong motivation for performing experiments to measure the tunnelling
time was the fact that there were many con icting theories for this time (see
the reviews by Hauge and St"vneng [9], by Landauer and Martin [10], and by
Chiao and Steinberg [8]). It suces here to list the three main contenders:
(1) The Wigner time (i.e., \phase time" or \group delay").
(2) The B· uttiker-Landauer time (i.e., \semiclassical time").
(3) The Larmor time (with B· uttiker's modication).
The Wigner time calculates how long it takes for the peak of a wave packet
to emerge from the exit face of the tunnel barrier relative to the time the peak
of the incident wave packet arrives at the entrance face. Since the peak of the
wave packet in the Born interpretation is the point of highest probability for
a click to occur(seetheaboveGedankenexperiment), it is natural to expect
this to be the relevant time for our experiments. This calculation is based on
an asymptotic treatment of tunnelling as a scattering problem, and utilizes the
method of stationary phase to calculate the position of the peak of a wave
packet. The result is simple: this tunnelling time is the derivative of the phase
of the tunnelling amplitude with respect to the energy of the particle.
The B· uttiker-Landauer time is based on a dierent Gedankenexperiment.
Suppose that the height of the tunnel barrier is perturbed sinusoidally in time.
If the frequency of the perturbation is very low, the tunnelling particle will see
the instantaneous height of the barrier, and the transmission probability will
adiabatically follow the perturbation. However, as one increases the frequency
of the perturbation, at some characteristic frequency the tunnelling probability
will no longer be able to adiabatically follow the rapidly varying perturbation.
It is natural to dene the tunnelling time as the inverse of this characteristic
frequency. The result is again simple: for opaque barriers, this tunnelling time
is the distance traversed by the particle (i.e., the barrier width d) divided by
the absolute value of the velocity of the particle j v j . (In the classically forbidden
region of the barrier, this velocity is imaginary, but its characteristic size is given
by the absolute value).
The Larmor time is based on yet another Gedankenexperiment. Suppose that
the tunnelling particle had a spin magnetic moment (e.g., the electron). Suppose
further that a magnetic eld were applied to region of the barrier, but only to
that region. Then the angle of precession of the spin of the tunnelling particle
is a natural measure of the tunnelling time. However, B· uttiker noticed that in
addition to this Larmor precession eect, there is a considerable tendency for
the spin to align itself either along or against the direction of the magnetic eld
during tunnelling, since the energy for these two spin orientations is dierent.
The total angular change of the tunnelling particle's spin divided by the Larmor
precession frequency is B· uttiker's Larmor time.
One consequence of the Wigner time is the Hartman eect: The tunnelling
time saturates for opaque barriers, and approaches for large d a limiting value given by the uncertainty principle, h=(V0 E). The apparent superluminality
of tunnelling is a consequence of this eect, since as d is increased, there is a
point beyond which the saturated value of the tunnelling time is exceeded by
the vacuum traversal time d=c, and the particle appears to have tunneled faster
than light.
By contrast, the B· uttiker-Landauer theory predicts a tunnelling time which
increases linearly with d for opaque barriers, as one would expect classically.
For a rectangular barrier with a height V0 << mc 2 , the eective velocity j v j is
always less than c. However, for the periodic structure which we used in our
experiment, the effective velocity j v j at midgap is infinite, which is a behavior
even more superluminal than that predicted by the Wigner time. This fact
makes it easy to distinguish experimentally between these two theories of the
tunnelling time. However, we hasten to add that the B· uttiker-Landauer time
may not apply to our experimental situation, as the Gedankenexperiment on
which it is based is quite dierent from the one relevant to our experiment.
B·uttiker's Larmor time predicts a tunnelling time which is independent of d
for thin barriers, but which asymptotically approaches a linear dependence on d
in the opaque barrier limit, where it coincides with the B·uttiker-Landauer time.
In our rst experiment it was impossible to distinguish experimentally between
this time and the Wigner time. Only in our second experiment could these two
theories be clearly distinguished from one another.
Details of the Berkeley Experiments
Spontaneous parametric down-conversion was the light source used in our ex-
periments [11, 12]. An ultraviolet (UV) beam from an argon laser operating at
a wavelength of 351 nm was incident on a crystal of potassium dihydrogen phos-
phate (KDP), which has a " (2) nonlinearity. During the process of parametric
down-conversion inside the crystal, a rainbow of many colors was generated in
conical emissions around the ultraviolet laser beam, in which one parent UV
photon broke up into two daughter photons, conserving energy and momentum.
The KDP crystal was cut with an optic axis oriented so that the two degener-
ate (i.e., equal energy) daughter photons at a wavelength of 702 nm emerged
at a small angle relative to each other. We used two pinholes to select out
these two degenerate photons. The size of these pinholes determined the band-
width of the light which passed through them, and the resulting single-photon
wavepackets had temporal widths around 20 fs and a bandwidth of around 6
nm in wavelength.
The tunnel barrier consisted of a dielectric mirror with eleven quarter-
wavelength layers of alternately high index material (titanium oxide with n =
2:22) and low index material (fused silica with n = 1:45). The total thickness
of the eleven layers was 1.1 m. This implied an in vacuo traversal time across
the structure of 3.6 fs. Viewed as a photonic bandgap medium, this periodic structure had a lower band edge located at a wavelength of 800 nm and an upper band edge at 600 nm. The transmission coecient of the two photons which
were tuned near midgap (700 nm) was 1%. Since the transmission had a broad
minimum at midgap compared to the wave packet bandwidth, there was little
pulse distortion. The Wigner theory predicted at midgap a tunnelling delay time
of around 2 fs, or an effective tunnelling velocity of 1:8 c. The B·uttiker-Landauer
theory predicted at midgap an infinite eective tunnelling velocity, which implies
a zero tunnelling time.
To achieve the femtosecond-scale temporal resolutions necessary for measur-
ing the tiny time delays associated with tunnelling, we brought together these
two photons by means of two mirrors, so that they impinged simultaneously at
a beam splitter before they were detected in coincidence by two Geiger-mode
silicon avalanche photodiodes. There resulted a narrow null in the coincidence
count rate as a function of the relative delay between the two photons, a de-
structive interference effect first observed by Hong, Ou, and Mandel [13]. The
narrowness of this coincidence minimum, combined with a good signal-to-noise
ratio, allowed a measurement of the relative delay between the two photons to
a precision of 0:2 fs.
A simple way to understand this two-photon interference is to apply Feyn-
man's rules for the interference of indistinguishable processes. Consider two
photons impinging simultaneously on a 50/50 beam splitter followed by two de-
tectors in coincidence detection. When two simultaneous clicks occur at the two
detectors, it is impossible even in principle to tell whether both photons were re-
flected by the beam splitter or whether both photons were transmitted through
the beam splitter. In this case, Feynman's rules tell us to add the probability
amplitudes for these two indistinguishable process, and then take the absolute
square to find the probability. Thus the probability of a coincidence count to
occur is given by j r 2 + t 2j 2 , wherer is the complex reflection amplitude for one
photon to be reflected, and t is the complex transmission amplitude for one
photon to be transmitted. For a lossless beam splitter, time-reversal symmetry
leads to the relation t = ± ir. Substituting this into the expression for the coin-
cidence probability, and using the fact that j r j = j t j for a 50/50 beam splitter,
we find that this probability vanishes. Thus the two photons must always pair
off in the same (random) direction towards only one of the two detectors, an
effect which arises from the bosonic nature of the photons.
A schematic of the apparatus we used to measure the tunnelling time is given
in Fig. 2. The delay between the two daughter photons was adjustable by means
of the \trombone prism" mounted on a Burleigh inchworm system, and was
measured by means of a Heidenhein encoder with a 0.1 m resolution. A positive
sign of the delay due to a piece of glass was determined as corresponding to a
motion of the prism towards the glass. The multilayer coating of the dielectric
mirror (i.e., the tunnel barrier) was evaporated on only half of the glass mirror
substrate. This allowed us to translate the mirror so that the beam path passed
either through the tunnel barrier in an actual measurement of the tunnelling time, or through the uncoated half of the substrate in a control experiment. In
this way, one could obtain data with and without the barrier in the beam, i.e.,
a direct comparison between the delay through the tunnelling barrier and the
delay for traversing an equal distance in air. The normalized data obtained in
this fashion is shown in Fig. 3(a), with the barrier oriented at normal incidence
( = 0). Note that the coincidence minimum with the tunnel barrier in the
beam is shifted to a negative value of delay relative to that without the barrier in
the beam. This negative shift indicates that the tunnelling delay is superluminal.
To double-check the sign of this shift, which is crucial for the interpretation of
superluminality, we tilted the mirror towards Brewster's angle for the substrate
( = 56), where there is a very broad minimum in the re ection coecient as
a function of angle. Near Brewster's angle this minimum is so broad that it
is not very sensitive to the dierence between the high and low indices of the
successive layers of dielectrics. Thus to a good approximation, the re ections
from all layers vanish simultaneously near this angle. Hence the Bragg re ection
responsible for the band gap disappears, and the evanescent wave behavior and
the tunnelling behavior seen near normal incidence disappears. The dielectric
mirror should then behave like a thin piece of transparent glass with a positive
delay time relative to that of the vacuum. Detailed calculations not using the
above approximations also show that at = 55, the sign of the shift should
indeed revert to its normal positive value.
The data taken in p-polarization at = 55 is shown in Fig. 3(b). The
reversal of the sign of the shift is clearly seen. Therefore one is confronted witha choice of the data either in Fig. 3(a) as showing a superluminal
shift. Since we know that the delay in normal dielectrics as represented by
Fig. 3(b) should be subluminal, this implies that the tunnelling delay in Fig. 3(a) or in Fig. 3(b)
should be superluminal. Therefore the data in Fig. 3(a) implies that after
traversing the tunnel barrier, the peak of a photon wave packet arrived 1:47 0:21 fs earl ier than it would had it traversed only vacuum.
Mirror (normalized)
Another reason for tilting the mirror is that one can thereby distinguish
between the Wigner time and B·uttiker's Larmor time, as they dier consid-
erably in the region near the band edge, which occurs near Brewster's angle.
This can be seen in Fig. 4, where there is a considerable divergence as the
band edge is approached between the solid line representing the theoretical
prediction of the Wigner time, and the long-dashed line representing that of
B·uttiker's Larmor time. The data points in Fig. 4 seem to rule out B· uttiker's
Larmor time (although again we hasten to add that this theory may not apply
to our experiment). The agreement with Wigner's theory is better, but there
are discrepancies which are not understood.
Other experiments conrming the superluminality of tunnelling have been
performed in Cologne, Florence, and Vienna [14, 15, 16]. The Cologne and
Florence groups performed microwave experiments, and the Vienna group per-
formed a femtosecond laser experiment. All these groups have conrmed the
Hartman eect. One of these groups [17] has claimed to have sent Mozart's
40th symphony at a speed of 4:7c through a microwave tunnel barrier 114 mm
long consisting of a periodic dielectric structure similar to our dielectric mirror.
However, the further implication that their experiment represents a violation of
causality is in our opinion unfounded [8].
Recently, an experiment indicating the simultaneous existence of two different tunnelling times was performed in Rennes [18]. In frustrated total re ection
(FTIR), the tunnelling of photons through an air gap occurs between two glass
prisms when a light beam is incident upon this gap beyond the critical angle.
The Rennes group observed in FTIR both a lateral displacement of the tunnelling
beam of light and an angular de ection of this beam. These two eects could
be interpreted as evidence for two dierent tunnelling times that simultaneously
occurred in the same tunnelling barrier. The lateral displacement is related to
the Wigner time, and the angular de ection is related to the B· uttiker-Landauer
time. As evidence for this, they cited the saturation of the beam displacement
(the Hartman eect), and the linear increase of the beam de ection, as the gap
was increased.
Conclusions
The experiments at Berkeley and elsewhere thus indicate that the tunnelling pro-
cess is superluminal. In our opinion, this does not imply that one can communi-
cate faster than c, despite claims to the contrary by Heitmann and Nimtz [17].
The group velocity cannot be identied as the signal velocity of special relativ-
ity, by which a cause is connected to its eect. Rather, it is Sommerfeld's front
velocity which exclusively plays this role. However, even if one were to dene
the group velocity as a \signal" velocity, no causal loop paradoxes can arise [19].
Although the controversies amongst the various tunnelling theories have not
yet been fully resolved by experiment, a good beginning has been made in this direction.
In particular, it is now clear that one cannot rule out the Wigner
time simply on the grounds that it yields a superluminal tunnelling time. It
also appears that there may exist more than one tunnelling time. Hopefully,
the mysterious role of time in quantum mechanics will be elucidated by these
studies.
