Path Distinguishability in Double Scattering of Light by Atoms
Path Distinguishability in Double Scattering of Light by Atoms
Christian Miniatura,1, 2 Cord A. Müller,3 Yin Lu,1 Guangquan Wang,1 and Berthold-Georg Englert1
Department of Physics, Faculty of Science, National University of Singapore, Singapore 117542, Singapore
Institut Non Linéaire de Nice, UMR 6618 du CNRS, UNSA, 1361 route des Lucioles, 06560 Valbonne, France
Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
(Dated: 15 April 2007)
Wave-particle duality finds a natural application for electrons or light propagating in disordered
media where coherent corrections to transport are given by two-wave interference. For scatterers
with internal degrees of freedom, these corrections are observed to be much smaller than would be
expected for structureless scatterers. By examining the basic example of the scattering of one photon
by two spin- 1
atoms—a case-study for coherent backscattering—we demonstrate that the loss of
interference strength is associated with which-path information stored by the scattering atoms.
PACS numbers: 42.25.Hz, 03.65.Nk
Einstein’s and de Broglie’s wave-particle duality
(WPD)—the ability of a quantum system to display the
seemingly contradictory attributes that one would have
regarded as wave-like and particle-like and, therefore, as
mutually exclusive in pre-quantum physics—is arguably
the most important phenomenological consequence of
Bohr’s principle of complementarity. The quantitative
aspects of WPD are particularly well understood [1] in
the context of two-paths interferometers where a definite
path is particle-like and the interference between the am-
plitudes of the two paths is wave-like.
The wave-like interference strength is quantified by the
familiar Michelson’s fringe visibility V , and the particle-
like path knowledge is measured by the path distinguisha-
bility D, which is perhaps less familiar and has this op-
erational meaning: The odds for guessing the path right
are (1+D)/2. The extreme situations of no path knowl-
edge and high fringe visibility (ideally D = 0, V = 1)
or full path knowledge and no fringes (D = 1, V = 0)
are standard textbook fare. A rather recent experiment
with a bearing on the matter discussed below is the one
carried out by Eichmann et al. in 1993 [2].
The compromises allowed by the laws of physics in in-
termediary situations are restricted by the duality rela-
tion [3, 4]
D2 + V2 ≤ 1 . (1)
It is worth noting that well pronounced wave-like and
particle-like aspects can coexist: With odds for guessing
the path right of 99% (D = 0.98), we can have well visible
fringes of 20% visibility (V = 0.2).
The two pioneering experiments that tested the dual-
ity relation employed two-path interferometers for atoms
[5] and photons [6], whereby internal degrees of freedom
of the interfering objects themselves were used to pro-
vide the path information. By contrast, in the situation
that we examine here—coherent double scattering—the
which-path information is stored in the deflecting ele-
ments of the interferometer (atoms) and not carried by
the interfering objects (photons).
We wish to show how these notions of path knowledge
and interference strength are naturally applied to coher-
ent wave transport. In the semi-classical regime of weak
localization, most of the coherent effects in wave trans-
port can be explained by two-wave interference between
amplitudes propagating in opposite direction along loop-
like scattering paths [7, 8]. These interference corrections
to transport can dramatically alter the diffusion process
and even suppress it [9, 10]. They are sensitive to sev-
eral “dephasing” processes but most of them are circum-
vented at sufficiently low temperatures [11, 12, 13].
Here we address an intrinsic dephasing mechanism
which survives at zero temperature: the path knowledge
stored in the internal degrees of freedom of the scatterers.
Indeed, when the scatterers have an internal structure,
the interference corrections to transport are observed to
be rather small, for example in the scattering of electrons
by magnetic impurities at very low temperatures [12].
The same effect has been observed in the coherent
backscattering (CBS) of light by cold rubidium atoms
[14, 15, 16]. This coherent multiple scattering effect
arises when an optically thick sample of scatterers is il-
luminated by coherent light. It, too, results from inter-
ference of light amplitudes, here of the two amplitudes
associated with traversing the same path in opposite di-
rection. The endpoints of each scattering path thus play
the role of Young slits and give rise to an angular fringe
pattern in the far-field. Owing to the varying separa-
tion between the endpoints, these patterns have differ-
ent fringe spacings but they all display a bright fringe
at backscattering. Thus, the sum of all fringe patterns
displays an angular peak around the backscattering di-
rection [17].
We quantify the strength of this interference in a nat-
ural manner by the relative excess of the peak intensity
over the background, the analog of the fringe visibility V
in this context. For atoms with a spin-0 ground state,
this CBS peak-to-background ratio reaches its maximal
possible value of 2 in the helicity-preserving polarization
channel [18], corresponding to V = 1, whereas it is very
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FIG. 1: (a) The two paths in coherent backscattering. Along
path A the photon is first scattered by atom 1, then by atom 2;
along path B the order is reversed. (b) Level scheme of the
transition. Both the ground state and the excited
state are doublets with total angular momentum j = 1
, and
the magnetic quantum numbers m = ± 1
label the sublevels.
small for atoms with a degenerate ground state [14], as is
the situation with a Zeeman fine structure or a hyperfine
structure.
Our point is that these results can be recast and
understood, both qualitatively and quantitatively, in
terms of wave-particle duality. Indeed, when scatter-
ing the photon the atoms may undergo a change in
their ground state—a circumstance equally crucial in the
single-scattering situation of the Eichmann et al. exper-
iment [2]. This is to say that the atoms can store which-
path information so that the experimenter can find out,
in principle if not in practice, which of the two atoms
scattered first and which second. As a consequence,
the strength of the coherent corrections to transport is
bounded by the distinguishability of the paths inside the
sample, and the height of the CBS peak is limited by the
amount of path knowledge available.
We consider the simplest possible scenario that ex-
hibits the effect: double scattering off two identical spin-
atoms (atom 1 and atom 2), with the photon resonant
with a 1
dipole transition. In fact, this is the situ-
ation of the Eichmann et al. experiment where the scat-
terers are Hg+ ions. This geometry is simplest for mul-
tiple scattering to set in and is at the heart of the CBS
phenomenon. Since our focus is on the influence of the
internal atomic structure, we assume that the atoms are
so stiffly trapped that there is no relevant contribution
from the atomic recoil (the storage of CBS which-path
information in the center-of-mass degrees of freedom of
mobile atoms is studied in Ref. [19]). Put differently, we
take for granted that the atomic center-of-mass degrees
of freedom do not store which-path information.
To simplify the problem further, we assume that the
distance between the atoms is sufficiently large for the
double scattering contribution to dominate over all other
multiple scattering processes (triple, quadruple, . . . ). As
illustrated in Fig. 1, path A is the case when atom 1
scatters first and atom 2 second (sequence 1 → 2); path
B is the sequence 2 → 1. The paths are geometrically
identical but traversed in opposite directions. The two
atoms together compose the path detector: the change
of their internal states bears witness of the actual path.
Since the ground states are spin- 1
states, the path de-
tector is a qubit pair, which is a 4-state system. During
the scattering process, however, the excited states of the
atoms are involved as well, and the details of the scatter-
ing interaction determine the over-all effect on the atoms.
This net before-to-after change in the combined ground
states of both atoms is given by a completely positive
two-qubit map.
We establish this map by first recalling that the atom-
photon interaction is described by quasi-resonant point-
dipole elastic scattering. The corresponding transition
operator is proportional to T = (dd) ⊗ |r〉〈r| where r
is the atom’s position vector and d is the dipole vector
operator of the atomic transition. The omitted propor-
tionality factor depends on the oscillator strength of the
transition. It determines the probability of the double
scattering event and is, therefore, crucial for an actual
experiment. But in the present context this probability is
irrelevant because the final two-atom state is conditioned
on successful double scattering. Bearing this conditioning
in mind, we consistently leave all further proportionality
factors implicit.
For an incoming photon with wave vector k and trans-
verse polarization ǫ, the matrix elements of T are
〈m′,k′ǫ′|T |m,kǫ〉 = 〈m′, ǫ′|(dd)|m, ǫ〉 ei(k−k
′)·r , (2)
where k′ and ǫ′ are the wave vector and polarization of
the outgoing photon, and m and m′ are the magnetic
quantum numbers of the initial and final ground state,
respectively [16]. Since the scattering is elastic, we have
|k| = |k′|.
The dyadic operator T = (dd) acts on the internal
degrees of freedom of the photon (the polarization states)
and of the atom (the magnetic sublevels of the angular
momentum multiplets). Its matrix elements read
〈m′, ǫ′|(dd)|m, ǫ〉 = 〈m′|(ǫ′∗ · d)(d · ǫ)|m〉 . (3)
The matrix elements of the vector operator d are the
Clebsch–Gordan coefficients that characterize the cou-
pling of spin-1 (photon) with spin- 1
(ground state) to
give spin- 1
(excited state); all the coefficients have equal
magnitude for such a 1
transition. As a conse-
quence, we have effectively T = (σ σ) for initial and final
ground states, where σ is the Pauli vector operator for
the spin- 1
ground state. [20]
We consider the exact backscattering geometry where
k = −k′, which we choose parallel to the z axis of the
coordinate system; the magnetic quantum numbers ±1/2
in Fig. 1(b) also refer to the z direction. For path A, the
double scattering operator that acts on the two atomic
ground-state qubits is
TA = ǫ
′∗ · (σ2σ2) · (1− nn) · (σ1σ1) · ǫ
= −ǫ′∗ · σ2 (σ2 × n) · (n× σ1)σ1 · ǫ , (4)
where 1− nn is the dyadic projector onto the plane or-
thogonal to the unit vector n that points from one scat-
terer to the other. The double-scattering operator TB for
path B is obtained by interchanging 1 ↔ 2 in (4).
With ρin denoting the initial two-qubit state of the two
scattering atoms, the final states are then given by
ρA,B =
TA,B ρin TA,B
with wA,B = tr
TA,B ρin TA,B
, (5)
where the normalizing denominators take care of all the
proportionality factors that we left implicit. The weights
of the two paths are wA/(wA +wB) and wB/(wA +wB),
respectively. In addition to the initial two-atom state
ρin, these weights and the final states depend on the pre-
selected polarization ǫ of the incoming photon and the
post-selected polarization ǫ′ of the outgoing photon, on
which the ensemble of events is conditioned.
Since the final states of the atoms are different for the
two paths, there is which-path information stored in the
atoms, which—in principle—can be extracted by a suit-
able measurement, although in practice it could be very
difficult to implement such a measurement. The optimal
measurement would provide as much path knowledge as
is available, quantified by the distinguishability of the
paths, which is given by [1, 3, 4]
∣wAρA − wBρB
wA + wB
∣TA ρin TA
† − TB ρin TB
wA + wB
. (6)
This is supplemented by the visibility
TA ρin TB
wA + wB
, (7)
the quantitative measure for the interference strength of
the two paths. Irrespective of the detailed form of ρin and
the operators TA, TB, the duality relation (1) is obeyed
by this distinguishability and visibility [1].
We now restrict the discussion to symmetric initial
two-qubit states of the form ρin =
(11 − pσ1 · σ2)
with − 1
≤ p ≤ 1 as required by the positivity of ρin.
This one-parameter family of initial states encompasses
some cases of particular physical interest: the completely
mixed state (p = 0); the projector on the singlet state of
vanishing total angular momentum (p = 1); the projec-
tor on the triplet sector of unit total angular momentum
(p = − 1
). It is worth recalling that two-qubit states of
this form are separable for p ≤ 1
and entangled for p > 1
but, as illustrated by Eqs. (9) below, nothing remarkable
happens to D and V at the transition from p < 1
p > 1
For all values of p, the initial state ρin is invariant under
the interchange 1 ↔ 2 and, therefore, the interferometer
is symmetric in the sense that both paths occur with
equal a priori probability (wA = wB). As a consequence,
the difference of operators in (6) is antisymmetric under
1 ↔ 2 and thus of the form
TA ρin TA
† − TB ρin TB
wA + wB
= a·(σ1−σ2)+b·(σ1×σ2) (8)
with two numerical vectors a and b that depend on the
photon polarizations ǫ, ǫ′, the unit vector n, and the
initial-state parameter p. The right-hand side in (8) is
a rank-2 operator with its nonzero eigenvalues given by
a2 + b2, and so we get D = 4
a2 + b2 for the dis-
tinguishability of the paths.
In this manner we arrive at explicit expressions for D
and V [21, 22]. We will report the full technical details
elsewhere and focus here on the particular situation in
which the line connecting the two atoms in Fig. 1(a) is
perpendicular to the incoming and outgoing propagation
directions, that is: we choose the unit vector n along the
x axis. For this perpendicular geometry, one has
1 + p+ 2pu
2(1 + pu)
1− u′2 ,
∣(1 + p)(1 + uu′)− 2p(1− u′)
2(1 + pu)
, (9)
where u and u′ are the x components of the Stokes vec-
tors associated with the incoming and outgoing photon
polarizations [23]. These are such that
(1− V)V for p ≤ 0 ,
(1− V)
2p/(1 + p) + V
for p ≥ 0 .
Clearly, the duality relation (1) is obeyed for all p values.
The relation is only saturated for p = 1, in which case the
initial atomic state is pure and the equal sign is expected
to hold in (1) on general grounds [4].
For the completely-mixed initial state (p = 0) we have
1− u′2 , V =
(1 + uu′) . (11)
Here, the distinguishability does not depend at all on the
initial polarization—a surprising feature that is particu-
lar to the 1
transition in the perpendicular geometry
and is not generic. This observation about the perpen-
dicular geometry can be understood as follows.
Since all Clebsch–Gordan coefficients are of equal size,
the first scatterer has uniform a priori probability of
reaching either one of its ground states, irrespective of
the polarization of the incoming photon. Yet, when con-
ditioned on the direction into which the photon is re-
emitted, the probability is not uniform as a rule, but it
is for the perpendicular geometry. Therefore, there is no
which-path information stored in the final completely-
mixed state of the atom that scatters first.
When the observed polarization of the outgoing photon
is an equal-weight superposition of the in-plane and out-
of-plane linear polarizations (u′ = 0), the final state of
the second scatterer is a corresponding pure state. So,
when finding only one atom in this pure state, we can
infer the path with certainty, but if both atoms are found
in this state, we know nothing about the path and will
guess wrong half of the time. Both cases are equally
probable, so that our betting odds are 75%, which is
consistent with D = 1
for u′ = 0 in (11), as it should
be. In this case the visibility is V = 1
irrespective of the
incoming polarization.
The distinguishability is zero for an outgoing photon
linearly polarized in the plane of the drawing in Fig. 1(a),
when u′ = 1, or perpendicular to this plane (u′ = −1).
The corresponding visibility takes on any value between
0 and 1. The case V = 0 happens for photons with per-
pendicular polarizations, one with in-plane polarization,
the other perpendicular (uu′ = −1). The case V = 1 oc-
curs for photons with parallel linear polarizations, both
in the plane or both perpendicular to it (uu′ = 1).
Let us now turn to the situation of an initial singlet
state (p = 1). As noted above, the duality relation (1) is
then saturated and we have
1− u′2 , V =
∣ . (12)
The fact that the distinguishability does not depend on
the initial polarization can be understood by an argu-
ment similar to the one given above for the p = 0 case.
Irrespective of the incoming photon polarization and for,
say, a left-circular outgoing photon, the final states of
the atoms have (m1,m2) = (
) for path A and
) for path B whereas the reversed situation occurs
for a right-circular outgoing photon. This means that
if the outgoing light is analyzed in the circular channels
(u′ = 0), perfect path knowledge is available (D = 1) and
no interference will be observed (V = 0). Conversely, if
the outgoing light is analyzed in the channels of linear in-
plane and out-of-plane polarization (
∣ = 1), no path
knowledge is available (D = 0) and one recovers full in-
terference strength (V = 1).
Finally, one can think of mimicking the physics of the
CBS phenomenon by an angular average over the direc-
tion n. We first calculate the average of the difference
operator in (8) and then compute the resulting distin-
guishability as the trace of its modulus. The correspond-
ing visibility is obtained as the angular average of the n
dependent visibility (7). For p = 0, which applies to most
of the available experimental data, the largest averaged
distinguishability is D = 1
; it is found in the helicity-
preserving polarization channel. The smallest average
visibility is also found in this channel, its value is V = 2
Even if a direct quantitative comparison with the real
CBS situation cannot be made at this stage, this result
is nevertheless consistent with the experimental observa-
tion that the lowest CBS peaks are actually found in this
detection channel [14].
In summary, we have demonstrated that the concept
of wave-particle duality proves relevant and useful for our
understanding of some aspects of the interference effects
in multiple scattering. To make solid quantitative con-
tact with actual CBS experiments, the analysis must be
extended to account for scattering by three and more
atoms. The stage for this future research is now set.
Ch. M. and C. M. wish to thank the Science Faculty
and the Physics Department of NUS for their kind hospi-
tality and financial support. This work was supported by
by A∗Star Grant No. 012-104-0040, and by NUS Grant
WBS: R-144-000-179-112.
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[20] In the case of a 1
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http://arxiv.org/abs/cond-mat/0202044
