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29. Y. Larionova, W. Stolz, C. O. Weiss, Opt. Lett. 33,321
(2008).
30. A. J. Daley, P. Zoller, B. Trauzettel, Phys. Rev. Lett. 100,
110404 (2008).
31. N. G. Berloff, Turbulence in exciton-polariton
condensates. Preprint available at http://arxiv.org/abs/
1010.5225 (2010).
Acknowledgments: We thank S. Barbay, J. Bloch, R. Kuszelewicz,
W. D. Phillips, L. P. Pitaevskii, and M. Wouters for useful
discussions, and L. Martiradonna for the confocal
masks. This work was supported by the IFRAF,
CLERMONT4, and the Agence Nationale de la
Recherche. A.B. and C.C. are members of the Institut
Universitaire de France.
Supporting Online Material
www.sciencemag.org/cgi/content/full/332/6034/1167/DC1
Materials and Methods
Figs. S1 to S4
28 December 2010; accepted 11 April 2011
10.1126/science.1202307
Observing the Average
Trajectories of Single Photons
in a Two-Slit Interferometer
Sacha Kocsis,
1,2
*Boris Braverman,
1
*Sylvain Ravets,
3
*Martin J. Stevens,
4
Richard P. Mirin,
4
L. Krister Shalm,
1,5
Aephraim M. Steinberg
1
†
A consequence of the quantum mechanical uncertainty principle is that one may not discuss
the path or “trajectory”that a quantum particle takes, because any measurement of position
irrevocably disturbs the momentum, and vice versa. Using weak measurements, however, it is
possible to operationally define a set of trajectories for an ensemble of quantum particles. We sent
single photons emitted by a quantum dot through a double-slit interferometer and reconstructed
these trajectories by performing a weak measurement of the photon momentum, postselected
according to the result of a strong measurement of photon position in a series of planes. The
results provide an observationally grounded description of the propagation of subensembles
of quantum particles in a two-slit interferometer.
In classical physics, the dynamics of a par-
ticle’s evolution are governed by its position
and velocity; to simultaneously know the
particle’s position and velocity is to know its past,
present, and future. However, the Heisenberg
uncertainty principle in quantum mechanics for-
bids simultaneous knowledge of the precise po-
sition and velocity of a particle. This makes it
impossible to determine the trajectory of a single
quantum particle in the same way as one would
that of a classical particle: Any information gained
about the quantum particle’s position irrevocably
alters its momentum (and vice versa) in a way that
is fundamentally uncertain. One consequence is
that in Young’s double-slit experiment one can-
not determine through which slit a particle passes
(position) and still observe interference effects on
a distant detection screen (equivalent to measur-
ing the momentum). Particle-like trajectories and
wavelike interference are “complementary”as-
pects of the behavior of a quantum system, and
an experiment designed to observe one neces-
1
Centre for Quantum Information and Quantum Control and
Institute for Optical Sciences, Department of Physics, University
of Toronto, 60 St. George Street, ON M5S 1A7, Canada.
2
Centre
for Quantum Dynamics, Griffith University, Brisbane 4111,
Australia.
3
Laboratoire Charles Fabry, Institut d'Optique, CNRS,
UniversitéParis-Sud, Campus Polytechnique, 2 avenue Augustin
Fresnel, RD 128, 91127 Palaiseau cedex, France.
4
National In-
stitute of Standards and Technology, 325 Broadway, Boulder,
CO 80305, USA.
5
Institute for Quantum Computing, University
of Waterloo, 200 University Avenue West, Waterloo, ON N2L
3G1, Canada.
*These authors contributed equally to this work.
†To whom correspondence should be addressed. E-mail:
steinberg@physics.utoronto.ca
Fig. 1. Experimental setup for measuring the average photon trajectories.
Single photons from an InGaAs quantum dot are split on a 50:50 beam
splitter and then outcoupled from two collimated fiber couplers that act as
double slits. A polarizer prepares the photons with a diagonal polarization
|D〉=1
ffiffi
2
p(|H〉+|V〉). Quarter waveplates (QWP) and half waveplates (HWP)
before the polarizer allow the number of photons passing through each slit
to be varied. The weak measurement is performed by using a 0.7-mm-thick
piece of calcite with its optic axis at 42° in the x-zplane that rotates the
polarization state to 1
ffiffi
2
p(e−iϕk/2|H〉þeiϕk/2|V〉).AQWPandabeamdis-
placer are used to measure the polarization of the photons in the circular
basis, allowing the weak momentum value k
x
to be extracted. A cooled CCD
measures the final xposition of the photons. Lenses L1, L2, and L3 allow
different imaging planes to be measured. The polarization states of the
photons are represented on the Poincaré sphere, where the six compass points
correspond to the polarization states |H〉,|V〉,|D〉,|A〉=1
ffiffi2
p(|H〉−|V〉),|L〉=1
ffiffi
2
p
(|H〉+i|V〉),and|
R〉¼1
ffiffi
2
p(|H〉−i|V〉).
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sarily gives up the option of observing the other
(1–6). However, it is possible to “weakly”mea-
sure a system, gaining some information about
one property without appreciably disturbing the
future evolution (7); although the information ob-
tained from any individual measurement is lim-
ited, averaging over many trials determines an
accurate mean value for the observable of interest,
even for subensembles defined by some subse-
quent selection (perhaps even on a complementary
observable). It was recently pointed out (8)that
this provides a natural way to operationally de-
fine a set of particle trajectories: One can ascer-
tain the mean momentum of the subensemble of
particles that arrive at any given position, and, by
thus determining the momentum at many posi-
tions in a series of planes, one can experimentally
reconstruct a set of average trajectories. We use
a modified version of this protocol to reconstruct
the “weak-valued trajectories”followed by single
photons as they undergo two-slit interference. In
the case of single-particle quantum mechanics,
the trajectories measured in this fashion repro-
duce those predicted in the Bohm–de Broglie
interpretation of quantum mechanics (9,10).
Weak measurements, first proposed 2 decades
ago (7,11), have recently attracted widespread
attention as a powerful tool for investigating fun-
damental questions in quantum mechanics (12–15)
and have generated excitement for their potential
applications to enhancing precision measurement
(16,17). In a typical von Neumann measure-
ment, an observable of a system is coupled to a
measurement apparatus or “pointer”via its mo-
mentum. This coupling leads to an average shift
in the pointer position that is proportional to the
expectation value of the system observable. In a
“strong”measurement, this shift is large relative
to the initial uncertainty in pointer position, so
that significant informationisacquiredinasingle
shot. However, this implies that the pointer mo-
mentum must be very uncertain, and it is this
uncertainty that creates the uncontrollable, irrevers-
ible disturbance associated with measurement.
In a “weak”measurement, the pointer shift is
small and little information can be gained on a
single shot; but, on the other hand, there may be
arbitrarily little disturbance imparted to the sys-
tem. It is possible to subsequently postselect the
system on a desired final state. Postselecting on
a final state allows a particular subensemble to
be studied, and the mean value obtained from
repeating the weak measurement many times is
known as the weak value. Unlike the results of
strong measurements, weak values are not con-
strained to lie within the eigenvalue spectrum of
the observable being measured (7). This has led
to controversy over the meaning and role of weak
values, but continuing research has made strides
in clarifying their interpretation and demonstrat-
ing a variety of situations in which they are clearly
useful (16–21).
In our experiment, we sent an ensemble of
single photons through a two-slit interferometer
and performed a weak measurement on each pho-
ton to gain a small amount of information about
its momentum, followed by a strong measure-
ment that postselects the subensemble of pho-
tons arriving at a particular position [see (22)for
more details]. We used the polarization degree
of freedom of the photons as a pointer that
weakly couples to and measures the momentum
of the photons. This weak momentum measure-
ment does not appreciably disturb the system,
and interference is still observed. The two mea-
surements must be repeated on a large ensemble
of particles in order to extract a useful amount
of information about the system. From this set
of measurements, we can determine the average
momentum of the photons reaching any partic-
ular position in the image plane, and, by repeat-
ing this procedure in a series of planes, we can
reconstruct trajectories over that range. In this
sense, weak measurement finally allows us to
speak about what happens to an ensemble of
particles inside an interferometer.
Our quantum particles are single photons
emitted by a liquid helium-cooled InGaAs quan-
tum dot (23,24)embeddedinaGaAs/AlAsmi-
cropillar cavity. The dot is optically pumped by a
CW laser at 810 nm and emits single photons at
0
200
400 Z coordinate: 3.2 m
Photon counts
−1
−0.5
0
0.5
1
x 10−3
k
x
/k
A
0
200
400 Z coordinate: 4.5 m
Photon counts
−1
−0.5
0
0.5
1
x 10−3
k
x
/k
B
0
200
400 Z coordinate: 5.6 m
Photon counts
−1
−0.5
0
0.5
1
x 10−3
k
x
/k
C
0
200
400 Z coordinate: 7.7 m
Photon counts
−5 0 5
−1
−0.5
0
0.5
1
x 10−3
Transverse coordinate [mm]
k
x
/k
D
Fig. 2. Measured intensities (photon counts) of
the two circular polarization components of |y〉,
measured on the CCD screen (red and blue curves),
as well as the weak momentum values calculated
from these intensities (black) for imaging planes at
(A)z=3.2m,(B)z=4.5m,(C)z=5.6m,and(D)
z= 7.7 m. The red and blue data points are the
intensity data with constant background sub-
tracted. The errors for the momentum values were
calculated by simulating the effect of Poissonian
noise in the photon counts. The magenta curve
shows momentum values obtained from enforcing
probability density conservation between adjacent
zplanes. Because of the coarse-grained averag-
ing over three imaging planes, the probability-
conserving momentum values are not as sensitive
as the measured weak momentum values to high-
ly localized regions in the pattern with steep mo-
mentum gradients.
Fig. 3. The reconstructed
average trajectories of an
ensemble of single photons
in the double-slit appara-
tus. The trajectories are re-
constructed over the range
2.75 T0.05 to 8.2 T0.1 m
by using the momentum data
(black points in Fig. 2) from
41 imaging planes. Here,
80 trajectories are shown.
Toreconstructasetoftra-
jectories, we determined the
weak momentum values for
the transverse xpositions at
the initial plane. On the basis
of this initial position and
momentum information, the
xposition on the subsequent
imaging plane that each
trajectory lands is calculated, and the measured weak momentum value k
x
at this point found. This
process is repeated until the final imaging plane is reached and the trajectories are traced out. If a
trajectory lands on a point that is not the center of a pixel, then a cubic spline interpolation between
neighboring momentum values is used.
3000 4000 5000 6000 7000 8000
−6
−4
−2
0
2
4
Propagation distance[mm]
Transverse coordinate[mm]
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a wavelength lof 943 nm. A Hanbury Brown-
Twiss interferometer is usedtomeasureasecond-
order correlation function g
(2)
(0) of 0.17 T0.04
(SD), confirming the single-photon nature of the
dot emission (25). The photons are coupled
into single-mode fiber and sent through an in-
fiber 50:50 beam splitter. The outputs of the
beam splitter exit two fiber launchers as Gaussian
beams with their waists at the fiber launchers
and are redirected to be parallel along the zaxis
by mirrored prisms to create the initial “slit func-
tion”(Fig. 1). The two Gaussian beams have a
waist 1/e
2
radius of 0.608 T0.006 mm and a
peak-to-peak separation of 4.69 T0.02 mm.
The polarization of the photons, which serves
as the ancilla system for the weak measurement,
is prepared in the initial state jy〉¼ð1=ffiffiffi
2
pÞ
ðjH〉þjV〉Þ,where|H〉is identified with the x
axis and |V〉with the yaxis.
The weak measurement is accomplished with
a thin piece of birefringent calcite that changes
the polarization of the photons passing through
by introducing a phase shift between the ordinary
and extraordinary components of polarization.
The photons diffract out from the slits and im-
pinge upon the crystal with an incident angle q
that depends on their transverse momentum k
x
(where the momentum of a photon is p=ħk).
By orienting the calcite’s optic axis to lie in the
x-zplane, |H〉becomes the extraordinary polar-
ization that encounters an angle-dependent index
of refraction, n
e
(q), and |V〉becomes the ordinary
polarization that encounters a constant index of
refraction, n
o
. The calcite piece is 0.7 mm thick
with its optic axis in the x-zplane at 42° to the z
axis and imparts a small k
x
-dependent birefringent
phase shift that transforms the incident linear
polarization state of the photons to a slightly el-
liptical polarization state. In this way, we carry
out a measurement of the momentum with the po-
larization serving as a pointer that records the
value of this observable. By arranging for the
magnitude of the polarization rotation to be small
with respect to the uncertainty in the photons’
polarization, we ensure that the measurement is
weak. No single measurement provides unambig-
uous information about the exact propagation
direction, and hence no significant measurement
disturbance is introduced. After averaging the re-
sults over many photons, it becomes possible to
extract the average value of photon momentum.
The birefringent phase shift ϕ(k
x
) that the
photons receive depends on the different paths
and indices of refraction for the two polariza-
tions in the calcite (26). The spread of the angles
of the diffracting photons passing through cal-
cite is small, allowing us to approximate the in-
duced birefringent phase shift ϕ(k
x
) as a linear
function of k
x
:
ϕðkxÞ¼zkx
jkjþϕ0ð1Þ
The coefficient zdesignates the coupling
strength between the phase we are measuring
and the photon momentum, and its value was
found to be 373.5 T3.4 (22). The calcite is tilted
in the x-zplane to tune ϕ
0
= 0 modulo 2p.
A system of three cylindrical lenses, with the
middle lens translatable in the zdirection, allows
the initial slit function to be imaged over an
arbitrary distance. It is important to note that the
thin calcite crystal performing the weak measure-
ment remains fixed in place before the lenses.
This does not affect the outcome of the final
postselection at the various imaging planes along
zas the interaction Hamiltonian between the
polarization pointer and the photon’s transverse
momentum commutes with the free-progagation
Hamiltonian of the system. The trajectories were
reconstructed over the range 2.75 T0.05 to 8.2 T
0.1 m to show the transition from the near-field
to far-field intensity distribution. The polarization
state of each photon is projected into the circular
basis by using a quarter waveplate with its fast
axis set to −45° to x, located in front of the lens
system, and a polarizing beam displacer located
behind the lenses. The beam displacer transmits
the right-hand circularly polarized component of
|y〉undeviated and displaces the left-hand cir-
cularly polarized component of |y〉vertically by
about 2 mm. The photons are then detected on a
cooled charge-coupled device (CCD). The expo-
sure time on the CCD was set to 15 s, allowing
the two vertically separated interference patterns
to accumulate. During each exposure, about
31,000 single photons were detected by the CCD.
By projecting into the circular basis, the mo-
mentum information encoded in polarization is
transformed into an intensity modulation between
the two vertically displaced patterns. The inten-
sity of the top pattern (corresponding to the pro-
jection onto the right-hand circular polarization)
is I
R
º[1 + sinϕ(k
x
)], whereas the intensity of
the bottom pattern (corresponding to the projec-
tion on to the left-hand circular polarization) goes
as I
L
º[1 −sinϕk
x
)]. In the measured inter-
ference patterns at four different imaging planes
(Fig. 2), the pixel on the CCD where each pho-
ton is detected corresponds to the photon’sx
position. The 26-µm pixel width sets the pre-
cision with which the photon’sxposition can be
measured.
By using Eq. 1, we can simultaneously ex-
tract the weak value of the transverse compo-
nent of the photon wave vector k
x
at each pixel
position
kx
jkj¼1
zsin−1IR−IL
IRþIL
ð2Þ
Thus for each value of the photon’s position
x, we are able to calculate the weak value of its
transverse momentum k
x
by taking the difference
in modulated intensity between the two vertically
displaced patterns at the same imaging plane
along the zaxis. The weak momentum values
for four different imaging planes calculated in
this way are shown in Fig. 2. By repeating the
measurement for many imaging planes closely
spaced along z, a vector field is produced from
Fig. 4. The trajectories from Fig. 3 plotted on top of the measured probability density distribution. Even
though the trajectories were reconstructed by using only local knowledge, they reproduce the global
propagation behavior of the interference pattern.
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which the weak-valued photon trajectories are
reconstructed.
For the experimentally reconstructed trajecto-
ries for our double slit (Fig. 3), it is worth stress-
ing that photons are not constrained to follow
these precise trajectories; the exact trajectory of
an individual quantum particle is not a well-
defined concept. Rather, these trajectories rep-
resent the average behavior of the ensemble of
photons when the weakly measured momentum
in each plane is recorded contingent upon the
final position at which a photon is observed. The
trajectories resemble a hydrodynamic flow with a
central line of symmetry clearly visible: Trajec-
tories originating from one slit do not cross the
central line of symmetry into the opposite side of
the interference pattern. Trajectories at the edges
of bright fringes tend to cross over to join more
central bright fringes, thus generating the ob-
served intensity distribution because of interfer-
ence. The trajectories cross over dark fringes at
relatively steep angles; there is a low probability
of finding a photon in these regions that cor-
respond to postselecting on a state nearly or-
thogonal to the initial state of the system. The
separation of the imaging planes sets the scale
over which features in the trajectories can be
observed. The evolution of the interference in
our double-slit apparatus takes place over a scale
that is much longer than the separation between
imaging planes, and our trajectories can accu-
rately track the evolution of this interference. The
one place where the accuracy can suffer is when
a trajectory quickly passes through a dark fringe;
here the fine scale behavior is smaller than the
spacing between imaging planes. By overlaying
the trajectories on top of the measured intensity
distribution (Fig. 4), we observe that the trajec-
tories reproduce the global interference pattern
well. The tendency of the reconstructed trajecto-
ries to “bunch”together within each bright in-
terference fringe is an artifact of measurement
noise with the position error accumulating as the
trajectory reconstruction is carried out further
and further from the initial plane at z=2.75m.
Single-particle trajectories measured in this fash-
ion reproduce those predicted by the Bohm–de
Broglie interpretation of quantum mechanics (8),
although the reconstruction is in no way depen-
dent on a choice of interpretation.
Controversy surrounding the role of mea-
surement in quantum mechanics is as old as the
quantum theory itself, and nowhere have the
paradoxes been thrown into such stark relief as
in the context of the double-slit experiment. Our
experimentally observed trajectories provide an
intuitive picture of the way in which a single par-
ticle interferes with itself. It is of course impos-
sible to rigorously discuss the trajectory of an
individual particle, but in a well-defined opera-
tional sense we gain information about the aver-
age momentum of the particle at each position
within the interferometer, leading to a set of “av-
erage trajectories.”The exact interpretation of
these observed trajectories will require continued
investigation, but these weak-measurement results
can be grounded in experimental measurements
that promise to elucidate a broad range of quan-
tum phenomena (7,11–13,15–17). By using the
power of weak measurements, we are able to pro-
vide a new perspective on the double-slit experi-
ment, which Feynman famously considered to
have in it “the heart of quantum mechanics”(27).
References and Notes
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020404 (2009).
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11, 033011 (2009).
15. G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph,
H. M. Wiseman, Phys. Rev. Lett. 94, 220405 (2005).
16. O. Hosten, P. Kwiat, Science 319, 787 (2008);
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Phys. Rev. Lett. 102, 173601 (2009).
18. A. Peres, Phys. Rev. Lett. 62, 2326 (1989).
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20. Y. Aharonov, L. Vaidman, Phys. Rev. Lett. 62,2327
(1989).
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22. Materials and methods are available as supporting
material on Science Online.
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24. R. H. Hadfield et al., Opt. Express 13, 10846
(2005).
25. R. Loudon, The Quantum Theory of Light (Oxford Univ.
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26. M. Born, E. Wolf, Principles of Optics (Cambridge Univ.
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27. R. Feynman, The Feynman Lectures on Physics
(Addison-Wesley, Boston, 1989).
Acknowledgments: A.M.S. conceived of the experiment and
supervised the work; M.J.S. and R.P.M. designed and
fabricated the single-photon sources; S.K. constructed the
experiment and acquired the data with assistance from
B.B. and S.R. and much guidance from L.K.S.; B.B., S.K.,
and S.R. carried out the data analysis and generated
the figures; S.K. and L.K.S. wrote the text with input
from all the other co-authors. This work was supported
by the Natural Sciences and Engineering Research
Council of Canada, the Canadian Institute for Advanced
Research, and QuantumWorks. S.K. thanks B. Higgins
and L.K.S. thanks K. J. Resch for useful discussions.
The data from the experiment has been archived
and is available at www.physics.utoronto.ca/~aephraim/
data/PhotonTrajectories.
Supporting Online Material
www.sciencemag.org/cgi/content/full/332/6034/1170/DC1
Materials and Methods
27 December 2010; accepted 11 April 2011
10.1126/science.1202218
Spin-Liquid Ground State of the S= 1/2
Kagome Heisenberg Antiferromagnet
Simeng Yan,
1
David A. Huse,
2,3
Steven R. White
1
*
We use the density matrix renormalization group to perform accurate calculations of the ground
state of the nearest-neighbor quantum spin S= 1/2 Heisenberg antiferromagnet on the kagome
lattice. We study this model on numerous long cylinders with circumferences up to 12 lattice
spacings. Through a combination of very-low-energy and small finite-size effects, our results
provide strong evidence that, for the infinite two-dimensional system, the ground state of this
model is a fully gapped spin liquid.
We consider the quantum spin S= 1/2
kagome Heisenberg antiferromagnet
(KHA) with only nearest-neighbor
isotropic exchange interactions (Hamiltonian
H¼S
→
Si⋅
→
Sj,where
→
Siand
→
Sjare the spin
operators for sites i and j, respectively) on a kagome
lattice (Fig. 1A). This frustrated spin system has
long been thought to be an ideal candidate for a
simple, physically realistic model that shows a
spin-liquid ground state (1–3). A spin liquid is a
magnetic system that has “melted”in its ground
state because of quantum fluctuations, so it has
no spontaneously broken symmetries (4). A key
problem in searching for spin liquids in two-
dimensional (2D) models is that there are no ex-
act or nearly exact analytical or computational
methods to solve infinite 2D quantum lattice sys-
tems. For 1D systems, the density matrix renor-
malization group (DMRG) (5,6), the method we
use here, serves in this capacity. In addition to
its interest as an important topic in quantum mag-
netism, the search for spin liquids thus serves
as a test-bed for the development of accurate and
widely applicable computational methods for
2D many-body quantum systems.
1
Department of Physics and Astronomy, University of Cali-
fornia, Irvine, CA 92617, USA.
2
Department of Physics,
Princeton University, Princeton, NJ 08544, USA.
3
Institute
for Advanced Study, Princeton, NJ 08540, USA.
*To whom correspondence should be addressed. E-mail:
srwhite@uci.edu
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