Direct measurement of superluminal group velocity and of signal velocity in an optical fiber

0.19 MB·English

by  Nicolas Brunner

Direct measurement of superluminal group velocity and of signal velocity
in an optical fiber
Nicolas Brunner, Valerio Scarani, Mark Wegmu¨ller, Matthieu Legr´e and Nicolas Gisin
Group of Applied Physics, University of Geneva,
20 rue de l’Ecole-de-M´edecine, CH-1211 Geneva 4, Switzerland
(February 1, 2008)
We present an easy way of observing superluminal group velocities using a birefringent optical
fiberandotherstandard devices. Inthetheoretical analysis, weshow thattheoptical propertiesof
thesetupcanbedescribedusingthenotionof”weak value”. Theexperimentshowsthatthegroup
velocity can indeed exceed c in the fiber; and we report the first direct observation of the so-called
5 ”signal velocity”, thespeed at which information propagates and that cannot exceed c.
0
0 The physics of lightpropagationis a verytimely topic mationofthepulseisweak,thegroupvelocityisstillthe
2 because of its relevance for both classical [1] and quan- speed of the center-of-mass, now given by

n tum [2] communication. Two kind of velocities are usu-
L
a ally introduced to describe the propagation of a wave v = . (1)
J g t +hti
in a medium with dispersion ω(k): the phase velocity f

0 v = ω and the group velocity v = ∂ω. Both of these
1 ph k g ∂k This can become either very large and even negative
velocities can exceed the speed of light in vacuum c in
(hti → −∞) or very small (hti → ∞) — although in
2 suitable cases [3]; hence, neither can describe the speed theselimitingsituationsthepulseisusuallystronglydis-
v atwhichtheinformationcarriedbyapulsepropagatesin
torted, so that our reasoning breaks down.
5 the medium. Indeed, since the seminal workof Sommer-
We can now move to our setup, sketched out in Fig.
5 feld,extendedandcompletedbyBrillouin[4],itisknown
1 1. ThemediumisabirefringentopticalfiberoflengthL,
7 that informationtravelsat the signal velocity, defined as whose refraction index is n ≃ 3 for the telecom wave-
the speed of the front of a square pulse. This velocity f 2
0 length 1550 nm. The fiber is sandwiched between two
4 cannotexceedc [5]. The factthatno modificationofthe
polarizers. This setup is also known in classical optics
0 group velocity can increase the speed at which informa-
as a Lyot filter [8]. As shown in [9], this simple situ-
h/ tion is transmitted has been directly demonstrated in a ation can be described with the quantum formalism of
p recent experiment [6]. Superluminal (or even negative)
weak values [10]. In fact, the three essential ingredients
- and, on the other extreme, exceedingly small group ve-
t of weak values are present here: (I) The input polarizer
n locities, have been observed in several media [7]. In this
allows the pre-selection of a pure polarizationstate |ψ i.
a letter we report observation of both superluminal and 0
u (II)Thebirefringentfiberperformsapre-measurementof
delayed pulse propagationin a tabletop experiment that
q polarizationbyspatiallyseparatingthetwofiber’seigen-
: involvesonlyahighlybirefringentopticalfiberandother modes, supposed to be described by the eigenstates |Hi
v
standard telecom devices.
Xi and |Vi of σz — that is, we choose z to be the bire-
fringence axis. This pre-measurement,knownin telecom
r Before describing our setup, it is useful to under- physics as polarization-mode dispersion (PMD), is weak
a standinsomemore detailthe mechanismthroughwhich
wheneverthe temporalshift δτ (the so-calleddifferential
anomalous group velocities can be obtained. For a light
group delay, DGD) between the two eigenmodes of the
pulse sharply peaked in frequency, the speed of the
fiber is much smaller than the coherence time t of the
center-of-mass is the group velocity v of the medium c
g optical pulse. (III) The output polarizer performs the
for the central frequency [3]. In the absence of anoma-
post-selection on another chosen polarization state |φi.
lous light propagation, the local refractive index of the
The mean time of arrival of the optical pulse can be ex-
mediumisn ,supposedindependentonfrequencyforthe
f pressed as a function of a weak value W [9,1]:
region of interest. The free propagation simply yields
vg = L/tf where L is the length of the medium and δτ δτ hφ|σz|ψi
hti= ReW = Re (2)
t = n L/c is the free propagation time. One way to
f f 2 2 (cid:20) hφ|ψi (cid:21)
allowfast- andslow-lightamountsto modify the proper-
tiesofthemediuminsuchawaythatitbecomesopaque where|ψiisthepolarizationobtainedbythefreerotation
for all but the fastest (slowest) frequency components. of |ψ i in the fiber. Thus, hti is in principle unbounded:
0
The center-of-mass of the outgoing pulse appears then by carefully choosing |ψ i and |φi, i.e the direction of
0
at a time t = t +hti, with hti the mean time of arrival each polarizer, one can tune the group velocity (1) to
f
oncethefreepropagationhasbeensubtracted;obviously anydesiredvalue. In summary,by appending twopolar-
hti < 0 for fast-light, hti > 0 for slow-light. If the defor- izers at the ends of a birefringent fiber, one creates an
1
effective medium whose dispersion and absorption prop- G(ω)=einfωL/c(A+B)F(ω,W ) (6)
0
erties depend on the pre- and post-selection; in this ef-
fective medium, the group velocity can be increased or where we have written A ≡ a∗a and B ≡ b∗b ,
1 0 1 0
decreasedbyanarbitraryamountbymerelychoosingthe F(ω,W )=cos(ωδτ/2)+iW sin(ωδτ/2) and where
0 0
orientations of two polarizers.
The rest of the paper is organized in three parts: first A−B hφ|σz|ψ0i
W = = ≡ W +iW (7)
0 R I
we derive the optical properties (dispersion and absorp- A+B hφ|ψ i
0
tion) of our effective medium, stressing the role of the
weak value W. Then we describe the experiment and is the weak value involving the orientations of the two
present our main results in Figs 4 and 5. We shall con- polarizers[13]. Note that this is not the weak value that
clude bycomparingourexperimentto arelatedone[11], entersthemeantimeofarrival(2): thatoneiscomputed
andbystressingthedifferenceswithotherfast-andslow- using |ψ(ω)i instead of |ψ0i; using (5), one obtains
light techniques.
W +i W cosωδτ + 1(1−|W |2)sinωδτ
W = R I 2 0 . (8)
t (cid:0) |F(ω,W0)|2 (cid:1)
f
From (3) and (6) it is straightforward to derive expres-
PBS PMD fibers PBS sions for the absorption coefficient κ(ω) and the index
z
of refraction n(ω). For simplicity, the global phases of
<t>
bothpolarizationstates|ψ iand|φiarechosensuchthat
. 0
FIG.1. Basic scheme. Bycarefully adjusting thedirection (A+B) is real and contributes thus only to the absorp-
of both polarizing beam splitters (PBS), one can modify the tion. The results are:
group velocity of thepulse.
1
κ(ω)=− [ln(A+B)+ln|F(ω,W )|] (9)
0
L
c W
R
Theoretical analysis. For any linear medium in which n(ω)=nf + arctan . (10)
Lω (cid:20)cot(ωδτ/2)−W (cid:21)
I
light propagates,the absorptioncoefficient κ(ω) and the
index of refraction n(ω) are defined through
From n(ω), one can derive the group index n (ω) =
g
n+ωdn. Using (8) and (2) one finds
G(ω)=e−κ(ω)Lein(ωc)ωL (3) dω
δτ c
where L is the length of the fiber, and where G(ω) is n (ω)=n +c ReW = n + hti. (11)
g f f
2L L
the linear response function which characterizesthe evo-
lution of a plane wave of frequency ω in the medium:
Thisgivesexactlythebehavior(1)forthegroupvelocity
e−iωt → G(ω)e−iωt. The calculation of G(ω) for the
— indeed, both derivations are based onthe assumption
effective medium sketchedinFig. 1 hasto takepolariza-
that the coherence time of the pulse is large compared
tion into account. The first polarizer prepares the state
to δτ. When ReW = WR is small, the second
|F(ω,W0)|2
|ψ0i ⊗ e−iωt =(a0|Hi+b0|Vi)⊗e−iωt (4) term in (11) disappears and ng ≈ nf: this is normal
propagation, obtained for most choices of the pre- and
where |Hi and |Vi are the eigenmodes of the fiber, the post-selection. The conditions ReW = ±1 are obtained
eigenstatesofthePaulimatrixσz fortheeigenvalues±1. for the slow, resp. the fast, polarization mode of the
As discussedinRef. [9],the evolutionofthe polarization fiber, |ψ i = |Hi, resp |Vi. When ReW > 1, we obtain
0
oftheplanewavebecauseofbirefringenceisgivenbythe the slow-light regime; when ReW < −1, the fast-light
unitary operator eiωδτσz/2, which describes a global ro- regime. In particular, since n ≃ 3, superluminal group
f 2
tation around the z axis of the Poincar´e sphere — with velocity (n < 1) requires ReW < − L , negative group
g cδτ
ourconventions,|Hiistheslowmode and|Viisthefast velocity (n <0) requires ReW <−3 L .
mode. So, the state that reaches the second polarizer is g cδτ
In fig. 2, the absorption coefficient (9) and the index
|ψ(ω)i ⊗ einfωL/ce−iωt where one recognizes the phase
of refraction (10) are plotted for different weak values.
acquired during free propagation through the fiber and
When W increases, the slope of the index of refraction
R
where
becomessteeper: alargerpositive(negative)groupdelay
|ψ(ω)i=a0eiδ2τω|Hi + b0e−iδ2τω|Vi. (5) corresponds to a slower(faster) group velocity. It is also
clear from fig. 2 that a strong absorption occurs when-
The second polarizer is represented by the projector on ever the change in the group velocity is important. Al-
a polarization state |φi = a |Hi+b |Vi. So the output thoughthesefeaturesareusualintheirqualitativeformu-
1 1
statereads|φi⊗G(ω)e−iωt witharesponsefunction[12] lation, we note (as the authors of Ref. [11] did) that the
2
standard Kramers-Kronig relations do not apply: these post-selectedpolarizationstates. Furthermoreamicrom-
relations predict that the absorption coefficient (”imagi- eter step motor permits to slightly change the length of
narypartoftheindex”)determinesuniquelytheindexof thefiber. Thisallowsalignmentofthepost-selectionalso
refraction(”realpart”),and viceversa;while in ourcase, in the x−y plane of the Poincar´esphere, since polariza-
thesameabsorptionisassociatedtodifferentdispersions. tion is rotating around the z axis in the fiber.
Modified Kramers-Kronig relations have been discussed The crucial part of the experiment is the alignment of
in [14]. However,a full study ofthe questionlies beyond bothPBScubes. Theinputpolarizerhastobealignedat
the approachpresentedhere,becauseourformula(6)for 45orelativetothefiber’saxis[17]. Thisisdonebyinject-
G(ω) is valid only in a limited frequency range — e.g., ing incoherent light (from a LED) into the system and
the index of the fiber is not constant over all possible minimizing the degree of polarization at the output of
frequencies. the PM fiber. For the post-selection, the position of the
secondPBScubeandthelengthofthefiberareadjusted
such that transmission through the system is minimum.
A first set of results is presented in fig. 4. Figure
w) n (a) shows the data of a sequence of successive measure-
n( f
w w ments on a dB scale. The largest curve is the refer-
ence pulse, i.e. the pulse in the normal regime, with-
out any specific alignment of the PBS cubes. Between
w)
k( each measurement the post-selection is slightly changed,
in order to decrease transmission: we observe that the
w w
lower the transmission, the higher the group velocity, in
FIG.2. Index of refraction and absorption coefficient ver- agreementwiththetheory. Inaddition,figure(a)clearly
sus frequency for W0 =WR =±60. illustrates the difference between group and signal ve-
locities [4]. In fact, even though the group velocity is
higher for each successive curve, the signal velocity re-
mains constant and equal to c/n , since the front parts
f
of all pulses are strictly identical. Figures (b) and (c)
Photon counting OTDR
show,respectively,afast- anda slow-lightexample. The
Pulse PBS 1 PBS 2
generator PM fiber measured data are plotted together with the reference
Pulse
Odopmticaainl tainmaelyzePrulse Step motor pulse (dashedcurve)anda fit (smoothsolidcurve). The
fit is obtained by applying the following transformation
to the observed intensity of the reference pulse I (t):
in
FIG.3. Experimental setup FFT
I (t) → A (t) = I (t) → A (ω) → A (ω) =
in in in in out
G(ω)A (ω) F→FT Ap(t) → I (t) = |A (t)|2, where
in out out out
FFT is a discrete Fourier transformprocedure. The first
Experiment. The experimental setup is sketched out in stepofthis fitting procedureassumesthe referencepulse
fig. 3. As a source and for detection we use an optical to be Fourier-transform limited; the slight discrepancies
time domain reflectometer (OTDR). This is a telecom found between the data and the fits (in the front and
instrument designed to measure loss profiles of fibers: back parts of the pulses) are probably due to this ap-
it sends short laser pulses and analyzes the amount of proximation. The fitting parameter is the weak value
back-scatteredlight as a function of time. Here we use a W, supposedto be real. Note that in (b), |W| lies below
commercial prototype OTDR working in photon count- L ≃ 2000, the value needed in order to demonstrate
cδτ
ing mode at the telecom wavelength λ = 1.55µm [15]. superluminalgroupvelocity. Toreachthatbound, much
It contains a pulsed DFB laser (∆ν ≃ 500MHz , pulse longer input pulses are needed. We replace the OTDR
duration 2 ns) and a gated peltier-cooled InGaAs pho- sourcebyanexternalsource: aDFBlaser(∆ν ≃2MHz)
ton counter (gate duration 2 ns). The photon counting working in CW mode, modulated by an external electro
OTDR is well adapted to this experiment since it al- opticmodulator(EOM).Thissourcecreatesnearlygaus-
lows to monitor the optical pulse, even in the presence sian pulses with a coherence time of about 50ns. The
ofstrongabsorption. Thebirefringentfiberisapolariza- OTDR is still used for detection and triggers the mod-
tion maintaining (PM) fiber of length L = 1.5 m, with ulator. Results are presented in fig. 5. The pulse on
a DGD δτ = 2.66 ps, measured with an interferometric the right shows clearly superluminal group velocity. By
low-coherence method [16]. The fiber is placed between fitting the position of the maximum of the output pulse
two polarizing beam splitter (PBS) cubes with specified [18], we find W ≃ −3500, consistent with superluminal
extinction ratios of 50 dB. Both cubes are mounted on but non-negative group velocity, as expected.
rotationalstages,allowingaprecisealignmentofpre-and Comparison with other experiments. Solli and co-
3
workers reported on a similar, but indirect, experiment [12] We have found a scalar response function G(ω) be-
using polarized microwaves and a photonic crystal as causewehavepre-andpost-selectedonpurepolarization
birefringent medium, with identical conclusions on the states. In general, the response function Gij(ω) will be
group velocity [11]. Their conclusions were extracted tensorial, relating thepolarization components[11].
[13] A more complex link between response functions and
fromphasemeasurementsonly,soinparticularthey had
weak values in a similar situation was pointed out in:
no result on the signal velocity.
D. R. Solli, C.F. McCormick, R.Y. Chiao, S. Popescu,
Most experiments on slow light have been performed J.M. Hickmann,Phys.Rev.Lett. 92, 043601 (2004).
using electromagnetically induced transparency (EIT) [14] J.S. Toll, Phys.Rev. 104, 1760 (1956)
[7,19]. In EIT, the mode that carries information can [15] M.Wegmu¨ller, F.Scholder,N.Gisin,J.LightwaveTech.
be transmitted without losses, which is not the case in 22, 390 (2004)
our experiment. [16] N.Gisin,J.P.Pellaux,J.P.vonderWeid,IEEEJ.Light-
wave Technology 9, 821 (1991)
Inconclusion,wepresentedaneasywaytocreateslow
[17] It can be easily shown from (2) that this pre-selection
andfast lightusing a birefringentopticalfiber andother
maximizes themean time of arrival hti.
standard telecom components. The theoretical part of
[18] Wewerenotabletocorrectlyfittheshapeofthesecurves.
this work was mainly devoted to the study of the linear ThisisprobablyduetoasaturationoftheOTDRdetec-
responsefunctionofoursystemG(ω),thatisdetermined tionortosomephasenoisethatinvalidatesthefirststep
byaweakvalue,thusextending thepreviousworkof[9]. of thefitting procedure.
In the experiment, we obtained clear evidence for super- [19] S. Harris, Physics Today, July 1997; M.D. Lukin, A.
luminal group velocities (fig. 5); we also gave the first Imamo˘glu, Nature 413, 273 (2001).
direct measurement of the ”signal velocity” [fig. 4 (a)],
showing that the increase of the group velocity does not
increase the speed at which information travels.
WethankC.Barreirofortechnicalsupport. Thiswork
was supported by the Swiss NCCR ”Quantum photon-
ics”.
(a)
-10 1
B
d
unts-20
o
c
#
[1] B. Huttner, C. Geiser and N. Gisin, IEEE J. Sel. Top.
QuantumElectron. 6, 317 (2000).
[2] D. Bouwmeester, A. Ekert and A. Zeilinger (eds), The -30
physics ofquantum information(Springer,Berlin,2000).
[3] J. D. Jackson, Classical Electrodynamics, 2nd edition
(Wiley,New-York,1975), Sect.7-8.
2 Time of arrival 0 (TOA) ns -2
[4] L. Brillouin, Wave propagation and group velocity (Aca- 1.2 1.2
(b) (c)
demic Press, NewYork,1960). units 1 W = -241 units W = +59
[5] Actually,someweakexcitationscalled”forerunners”may arb.0.8 0.8 arb.
ac.rrTivheeiervsepneebdefohraes tbheeenmcaailnledfr,osnotm, behuotwnoutnffoarsttuenratthealyn, counts0.4 0.4 counts
# #
”front velocity”.
[6] M. D. Stenner, D.J. Gauthier and M. Neifeld, Nature 0.0 0.0
425, 695 (2003) 2 1 TOA0ns -1 -2 2 1 TOA0ns -1 -2
[7] R.W. Boyd, D.J. Gauthier, Progress in Optics Vol. 43, FIG. 4. In the three figures, the reference pulse (dashed)
edited by E. Wolf (Elsevier, Amsterdam, 2002) has been normalized for convenience. (a) Sequence of mea-
[8] W. Demtr¨oder, Laser spectroscopy (Springer, Berlin, surementsobtained by varyingtheorientation of apolarizer,
2003) in a log scale. The stronger the absorption, the larger the
[9] N. Brunner, A. Ac´ın, D. Collins, N. Gisin, V. Scarani, group velocity, as expected. However, the pulse is distorted
Phys.Rev.Lett. 91, 180402 (2003) in such a way that its front travels with a constant speed,
[10] For a review see: Y. Aharonov and L. Vaidman, quant- the signal velocity c/nf. (b),(c) Examples of measurements
ph/0105101; published in: J.G. Muga, R. Sala Mayato, of fast-light and of slow-light, in a linear scale, together with
I.L.Egusquiza(eds),TimeinQuantumMechanics,Lec- theoretical fits (smooth solid curve). The real weak value W
tureNotes in Physics (Springer, Berlin, 2002). obtained from thefit is given.
[11] D.R.Solli, C.F. McCormick, C. Ropers, J.J. Morehead,
R.Y.Chiao,J.M.Hickmann,Phys.Rev.Lett.91,143906
(2003).
4
FIG. 5. Fast- and slow-light curves compared to the ref-
erence pulse, whose center is marked by the dashed vertical
2000
line. The full vertical line marks the position of the summit
ofapulsethatwouldhavetravelledatspeedc. Thefastpulse
s
nit1500 shows superluminal group velocity.
u
b.
ar
1000
s
nt
u
o
c
# 500
0
40 20 0 -20 -40
Time of arrival ns
5