quant- ph/ 9501013 16 Jan 95
FTL Faster-Than-Light Quantum Tunneling (time travel)
Sub-femtosecond Determination of Transmission Delay Times For A Dielectric Mirror (photonic bandgap) As A Function Of Angle Of Incidence
Aephraim M. Steinberg and Raymond Y. Chiao
Department of Physics, U.C. Berkeley, Berkeley, CA 94720
Internet: aephraim@physics.berkeley.edu
(Preprint quant-ph/9501013; Received Phys. Rev. 17 August, 1994)
Using a two-photon interference technique, we measure the delay for single-photon wavepackets to be transmitted through a multilayer dielectric mirror, which functions as a \photonic bandgap" medium. By varying the angle of incidence, we are able to conrm the behavior predicted by the group delay (stationary phase approximation), including a variation of the delay time from superluminal to subluminal as the band edge is tuned towards to the wavelength of our photons.
The agreement with theory is better than 0.5 femtoseconds (less than one quarter of an optical period) except at large angles of incidence. The source of the remaining discrepancy is not yet fully understood.
PACS numbers: 03.65.Bz, 42.50.Wm, 73.40.Gk
In recent years, there has been a great deal of interest in two related topics: tunneling times [1, 2, 3, 4] and
photonic bandgaps [5, 6, 7]. A standard quarter-wave-stack dielectric mirror is in fact the simplest example of a
one-dimensional photonic bandgap, and consequently may be thought of as a tunnel barrier for photons within its
\stop band." The periodic modulation of the refractive index is analogous to a periodic Kronig-Penney potential in
solid-state physics, and leads to an imaginary value for the quasimomentum in certain frequency ranges{ that is, to
an exponentially decaying eld envelope within the medium, and high re ectivity due to constructive interference
(Bragg re ection). We have exploited this analogy to perform the rst measurement of the single-photon tunneling
delay time [8, 9, 10, 11], using as our barrier an 11-layer mirror of alternating high (n=2.22) and low (n=1.41)
index quarter-wave layers, with minimum transmission of about 1% at the center of the bandgap. We conrmed the
striking prediction which drives the tunneling time controversy: in certain limits, a transmitted wave packet peak may
appear on the far side of the barrier faster than if the peak had traversed the barrier at the vacuum speed of light c.
Meanwhile, several microwave experiments have investigated other instances of superluminal propagation including
electromagnetic analogies to tunneling [12, 13, 14, 15, 16, 17].
While in itself, this anomalous peak propagation does not constitute a violation of Einstein causality [18, 19, 20,
21, 22, 23, 24, 25, 26, 27], it certainly leads one to ask whether there may exist another, longer timescale in tunneling,
with more physical signicance than the group (i.e., peak) delay. After all, in a certain sense, the bulk of the
transmitted wave originates in the leading edge of the incident wave packet, not near the incident peak [28, 23, 29].
Many theories have been propounded to describe the duration of the tunneling interaction, and the leading contenders
involve studying oscillating barriers [1, 30] or Larmor precession of a tunneling electron in a barrier with a conned
magnetic eld [31, 32, 33]. It should be stressed that these theories are not intended to describe the propagation of
wave packets, but rather the dynamical timescale of the tunneling process; several experiments have supported their
predictions [34].
Nevertheless, there is a popular misconception that these times (and in particular the B uttiker-Landauer time in
its \semiclassical" or WKB limit{ md= h, where represents the evanescent decay constant inside the barrier, i.e.,
the magnitude of the imaginary wavevector) predict the arrival time of wave packets. In [8], we were able to exclude
the semiclassical time, but not B uttiker's version of the Larmor time, as describing peak propagation. Furthermore,
some workers have expressed concern about the paucity of data supporting the superluminality of the group delay, in
spite of our nding of a seven-standard-deviation eect. Microwave experiments have also traditionally been met with
skepticism (see, for example, [35].) In light of these objections, we have extended the earlier experiment to study the
delay time as a function of angle-of-incidence. As the angle is changed, the frequency and the width of the bandgap
change as well, so this is essentially a way to study the energy-dependence of the tunneling time.
Our apparatus is shown in Fig. 1. As the technique and the sample have both been described at length elsewhere
[36, 37, 8, 38, 39], we will content ourselves with an abbreviated sketch of the method. A crystal with an optical
(2) nonlinearity is pumped by a cw ultraviolet laser, and in the process of spontaneous parametric down-conversion
emits simultaneous pairs of horizontally-polarized infrared photons. The two photons in each pair leave the crystal on
As of 1995, at National Institute of Standards and Technology, Phys A167, Gaithersburg, MD 20899
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opposite sides of the ultraviolet pump, conserving momentum. They are correlated in time to within their reciprocal
bandwidth of about 15 fs. They are also correlated in energy, their frequencies summing to that of the (narrow-
band) 351 nm pump. When the two photons arrive simultaneously at a beam splitter, there is no way to distinguish
the two Feynman paths leading to coincidences between detectors placed at the beam splitter's two output ports:
both photons being transmitted, and both photons being re ected. This leads to an interference eect in which the
coincidence rate is suppressed (the two photons tending to head o to the same detector). By contrast, if the photons
arrive at the beam splitter at dierent times (on the scale of their 15 fs correlation time), coincidence counts occur
half the time. Thus by placing a dielectric mirror in one arm of the interferometer and adjusting the external path
length to minimize the coincidence rate, we can measure the delay experienced by the photon wavepackets which are
transmitted through this barrier. We nd that near the transmission minimum, the photons travel through the mirror
faster than they travel through an equivalent length of air, whereas when the mirror is angled to bring the band-edge
closer to the photons' wavelength, they travel slower than through air, as one would expect. Figure 2 shows sample
data for these two situations, where the sign change can be clearly seen.
In [8], our results were consistent with the group delay predictions, and also with B uttiker's proposed Larmor time
[33], but not with the \semiclassical" time. The measured times exceeded the predictions by approximately 0.5 fs, but
this result was at the borderline of statistical signicance, and not discussed. Since then, further data taken at various
angles of incidence have continued to show a discrepancy, ranging from an excess of 0.5 fs near normal incidence to a
decit of over 1 fs at large angles of incidence. At the same time, the data oer close agreement with the group delay,
and appear to rule out identication of the Larmor theory with a peak propagation time. Our attempts to eliminate
systematic eects and characterize those which remain were described in [8]. Since then, unable to nd any other
sources of error to explain the discrepancy, we are convinced that it is a property of the sample under study, and not of
the interferometer used for the measurements. We therefore obtained a second dielectric mirror of design parameters
identical to the rst, to see whether the errors could be attributed to deviations from the ideal quarter-wave-stack
structure. As can be seen from Figure 3, both mirrors show quite similar behavior. Both are 11-layer quarter-wave
stacks as described above. Mirror 1 shows a minimum transmission at 692 nm, while mirror 2's minimum is at 688 nm;
this dierence is insignicant on the scale of the bandgap, which extends from 600 nm to 800 nm. We conclude that
some real eect is at work, modifying the stationary-phase prediction. In principle, frequency-dependent transmission
could lead to such an eect, as does second-order group-velocity dispersion; both eects are much too small to explain
the present discrepancy. As discussed in [40], attempts to numerically model dielectric mirrors with small, random
uctuations in layer thicknesses were able to produce deviations on the right order, but in general they did not lead
to deviations of the form we observed experimentally. It is conceivable that loss or scattering in the dielectrics could
also help explain the eect, and we are beginning to investigate this possibility [41]; see also [42, 43].
Theoretical curves are plotted along with the data in gures 3 and 4. The group delay is calculated by the method
of stationary phase. The transmission phase of the 11-layer structure is calculated numerically, and dierentiated
rst with respect to angle of incidence to give the transverse shift and then with respect to incident frequency
to give the time delay, according to the formulas y = @T =@ky = @T =@(k sin ) = (k cos )@T =@ and
g = @T =@! + (y=c) sin , where T is the transmission phase [44]. B uttiker's Larmor time [33] is equal to the
magnitude of the complex time [45, 46] c = i@(ln t)=@
L, where t is the complex transmission amplitude, and
L
the Larmor frequency. For our optical structure, an eective Larmor frequency
L corresponds to a uniform (over
the barrier region) scaling of the local index of refraction by a factor of 1 +
L=!. Since in the limit of interest, the
\in-plane portion" of the Larmor time (i.e., the real part of the complex time) diers little from the group delay, we
take them to be equal in order to include the eects of the transverse shift in the Larmor theory. The \out-of-plane
portion" (or imaginary part) is calculated numerically, and added in quadrature to the group delay in order to generate
the Larmor time. Since our measurements compare the transit time through the barrier with that through air, we
subtract the time parallel wavefronts propagating at c would take to reach a point on the far side of the barrier (with
a transverse shift of y) from both the group delay and the Larmor time, so as to facilitate comparison with the
experimental data.
At the moment, more work (both experimental and theoretical) is needed to understand the discrepancy. We are
therefore planning to repeat this experiment with s-polarized light (by introducing half-wave plates before and after
the sample being studied), which has very dierent transmission characteristics as a function of angle (also leading
to a larger dierence between the group delay and the Larmor theories). As shown in Figure 4, our preliminary data
are again consistent with the group delay and not with the Larmor time, but due to the lower transmission for this
polarization, we need to improve our signal-to-noise ratio before reaching any denitive conclusions.
The superluminality of the barrier traversal near mid-gap is now well supported by the data, and the group delay
(stationary-phase) theory can be seen to be relatively accurate for a variety of angles of incidence, but there is a
residual discrepancy on the order of 0.5 femtoseconds, which is not yet fully understood.
This work was supported by the U.S. Oce of Naval Research under grant N00014-90-J-1259. We would like to
thank P. G. Kwiat, M. Mitchell, B. Johnson, G. McKinney, and J. Holden for their assistance; W. Heitmann for
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discussions about data analysis; and R. Landauer, J. D. Jackson, and E. Yablonovitch for useful disagreements.
Notes added during revision: Since the submission of this manuscript, a paper has appeared [47] extending our
previous experimental results to barriers of varying thicknesses (and transmission as low as 0:01%) near normal
incidence, using classical femtosecond pulses. It reports general agreement with the group delay theory, aside from a
discrepancy on the order of 1.5 fs. Two papers have also appeared discussing the eects of dissipation on tunneling
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