Photons as quasi-charged particles
Photons as quasi-charged particles
K.-P. Marzlin, Jürgen Appel, A. I. Lvovsky
Institute for Quantum Information Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada
(Dated: November 3, 2018)
The Schrödinger motion of a charged quantum particle in an electromagnetic potential can be
simulated by the paraxial dynamics of photons propagating through a spatially inhomogeneous
medium. The inhomogeneity induces geometric effects that generate an artificial vector potential to
which signal photons are coupled. This phenomenon can be implemented with slow light propagating
through an a gas of double-Λ atoms in an electromagnetically-induced transparency setting with
spatially varied control fields. It can lead to a reduced dispersion of signal photons and a topological
phase shift of Aharonov-Bohm type.
Introduction.— It is known since the ground-breaking
work of Berry on geometric phases [1] that artificial gauge
potentials can be induced if the spatial dynamics of a sys-
tem that obeys a wave equation is confined in a certain
way. For instance, if the internal Hamiltonian of neu-
tral atoms contains an energy barrier but the spin eigen-
states are spatially varying, gauge field dynamics can be
induced [2]. In the limit of ray optics, moving atomic en-
sembles could simulate the propagation of light around
a black hole or generate topological phase factors of the
Aharonov-Bohm type [3], and inhomogeneous dielectric
media could generally exhibit geometric effects such as
an optical spin-Hall effect and the optical Magnus force
In this paper, we propose to use electromagnetically in-
duced transparency (EIT) to generate an artificial vector
potential for the paraxial dynamics of signal photons that
simulates quantum dynamics of charged particles in a
static electromagnetic field. Not only the ray of light but
also its mode structure is affected, resulting in a paraxial
wave equation that is equivalent to the Schrödinger equa-
tion for charged particles. Furthermore, the form of the
artificial vector potential can be easily controlled through
spatial variations in the control fields. We suggest con-
figurations that generate homogeneous quasi-electric and
magnetic fields as well as a vector potential of Aharonov-
Bohm type.
Although the treatment in this paper is based on EIT,
the effect presented here is more general: it will occur in
any medium that supports a set of discrete eigenmodes
for a propagating signal fields with different indices of
refraction. If the parameters governing these eigenmodes
vary in space, the signal modes will adiabatically follow,
acquiring geometric phases that affect their paraxial dy-
namics.
Review of EIT with multi-Λ atoms.— The effect takes
place in an atomic multi-Λ system, in which two ground
states are coupled to Q excited states by Q pairs of con-
trol (Ωq) and signal (âq) fields (Fig. 1). An experimen-
tally relevant example of such system is the fundamental
D1 transition in atomic rubidium, where both the ground
and excited levels are split into two hyperfine sublevels
[5]. We assume that the detunings are small so each sig-
nal field âq interacts only with the respective transition
|B〉 ↔ |Aq〉 with the associated atomic operator σ̂B,Aq
and vacuum Rabi frequency gq. In this case, the parax-
ial wave equation for each signal mode can be cast into
the form
âq = iNgqσ̂B,Aq , (1)
where the wave propagates along the z axis, ∆⊥ = ∂
∂2y , N is the number of atoms and k is the wavevector
which we assume approximately independent of q. In
Ref. [6] we have constructed a unitary transformation
âq =
Wqs b̂s (2)
that maps the original field modes aq to a new set of
modes b̂q, such that one and only one of the new modes,
b̂Q =
R∗q âq, (3)
(where Rq ≡ Ωq/(gqΩ⊥) and Ω⊥ ≡
q=1 |Ωq/gq|2 de-
pend on the control fields) couples only to an atomic dark
state and experiences EIT [6, 7, 8]. All other superposi-
tions of field modes are absorbed. This transformation is
given explicitly by Wqq′ = γwqw
q′−δqq′ , with γ = RQ+1
and wq = γ
−1(δQq +Rq).
The EIT mode b̂Q interacts with the multi-Λ atoms
in the same fashion as does the signal field in a regu-
lar 3-level system. While propagating through the EIT
medium, it gives rise to a dark-state polariton associ-
ated with zero interaction energy [9]. All other modes
couple to atomic states whose energy levels are Stark-
shifted by the interaction with either the pump field or
the other signal modes b̂q (q 6= Q). The resulting energy
gap guarantees that, if the amplitudes and phases of the
control fields are slowly changed, the composition of the
dark-state polariton, and hence the EIT mode b̂Q, will
adiabatically follow. It has been proposed [6] and ex-
perimentally demonstrated [5] that a variation in time of
the control fields can therefore be used to adiabatically
http://arxiv.org/abs/0704.0814v2
FIG. 1: Multi Λ-system: Q excited states |Aq〉 are each cou-
pled by a classical control field Ωq to the ground state |C〉
and by a quantized field âq with detuning δ to state |B〉.
transfer optical states between signal modes. In this pa-
per, we focus on spatial propagation of the EIT mode
under control fields that are constant in time, but varied
in space.
Derivation of the gauge potential.— We proceed by ex-
pressing Eq. (1) in terms of the new signal modes b̂q.
Employing the vector notation ~a = {â1, · · · , âQ} and
~σB,A = {g1σ̂B,A1 , · · · , gQσ̂B,AQ} we get
b = iN~σB,A. (4)
Throughout the paper, the double arrow denotes a Q ×
Q matrix. Because
W depends on space and time, the
differential operators have to be applied to both
W and
b. As a result, transformation (2) brings about additional
terms into the equation of motion, that can be written
in form of a minimal coupling scheme by introducing the
Hermitian gauge field
i ≡ i
W †∂i
W, (5)
where i = t, x, y, z. We multiply both sides of Eq. (4) by
W † and exploit the unitarity of
W to show that ∂i
W † =
W †(∂i
W † from which it follows that −
W †∂2i
i + i∂iA
i. The dynamic equation for the b̂ modes can
then be written as
i∂t +A
b = −
ic∂z + cA
b (6)
(−i∇⊥ −A
2~̂b−
W †N~σBA
with ∇⊥ = (∂x, ∂y). This equation has the structure of
a 2+2 dimensional field theory with minimal coupling.
Under the assumption that the control fields do not
depend on t and z we can make a temporal Fourier trans-
formation of the slowly varying amplitudes, which results
in the paraxial wave equation
b(δ) =
(−i∇⊥ −A
~σBA(δ). (7)
The gauge potential is given explicitly by
⊥ = i
R∗q(∇⊥Rq)~w~w† − iγ(∇⊥ ~w)~w† + iγ∗ ~w∇⊥ ~w† .
The full matrix A
⊥ is a pure gauge: it has emerged solely
as a consequence of the unitary transformation (2), which
reflects our choice to describe the system in terms of the
new modes b̂q rather than the original modes âq. How-
ever, this choice is motivated by the fact that the EIT
mode b̂Q is the only mode that is not absorbed. Ab-
sorption of other modes b̂q (with q 6= Q) means that
the index of refraction for these modes has a significant
imaginary part. This separates the EIT mode b̂Q from
other b-modes and ensures that it will adiabatically follow
variations of the control fields. Therefore, when analyz-
ing the evolution of b̂Q, we can neglect the off-diagonal
terms in the matrix (−i∇⊥ −A
2 in Eq. (7) and write
i∂z b̂Q(δ) = −(
~σBA)Q(δ)−
b̂Q(δ) (9)
(−i∇⊥ −A⊥)Qq(−i∇⊥ −A⊥)qQb̂Q(δ).
This equation does not include the whole matrix A
Consequently, this potential no longer acts like a pure
gauge but attains physical significance in determining the
spatial dynamics of the EIT mode.
The first term on the right-hand side of Eq. (9), re-
sponsible for the interaction of the light field with the
EIT medium, takes the same form as the susceptibility
of EIT in a single Λ-system. Neglecting decoherence, we
can write it as [6] (
W † N
~σBA)Q(δ) =
b̂Q, with the
EIT group velocity vEIT = cΩ
2/(Ng2). Note that vEIT
depends on the spatial position because Ω does. This
transforms Eq. (9) to
i∂z b̂Q =
(−i∇⊥ −AQQ)2 −
b̂Q (10)
AQQ = i
R∗q∇⊥Rq = −
|Rq|2∇⊥Arg(Rq),
|(A⊥)Qq|2 = −A2QQ +
|∇⊥Rq|2 (11)
being, respectively, the “quasi-vector” and “quasi-scalar”
potentials.
We see that the paraxial spatial evolution of the EIT
signal mode is governed by the equation that is identi-
cal (up to coefficients) to the Schrödinger equation of a
charged particle in an electromagnetic field. This is the
main result of this work. By arranging the control field in
a certain configuration, one can control the spatial prop-
agation of the signal mode through the EIT medium.
Some steering of the EIT mode is possible even in a
single-Λ system by affecting the term δ/vEIT in Eq. (10),
which results in nonuniform refraction for this mode
[10, 11]. The action of quasi-gauge fields (11) is fun-
damentally different: deflection of the signal field occurs
not due to refraction (the refraction index on resonance
is 1), but due to adiabatic following.
The case of two control fields: homogeneous electric
and magnetic quasi-fields.— Of particular practical im-
portance is the simplest non-trivial case with Q = 2.
We parametrize the control fields by writing R1,2 =
1/2±Rei(φ±θ). The corresponding Rabi frequencies
are then Ωi = h(x, y) giRi, with h(x, y) being an arbi-
trary common prefactor. This parametrization yields the
gauge potentials
AQQ = −∇⊥φ− 2R∇⊥θ; (12)
(∇⊥R)2
1− 4R2 + (∇⊥θ)
2(1 − 4R2).
Similarly to usual electrodynamics we can use a gauge
transformation [12], A′QQ = AQQ + ∇⊥f , to eliminate
the term ∇⊥φ from Eq. (12). The common phase φ of
the control fields therefore does not contribute and can
be set to zero.
A simple way to generate a term that corresponds to a
one-dimensional scalar potential V (x) for a Schrödinger
particle is to choose R = 0 and θ =
2kV (x′)
This choice of control fields leads to AQQ = 0 and Φ =
2kV (x).
For the special case of a constant electric quasi-field
along the x axis, V (x) = −Fx and subsequently
θ = −
4kF |x|3/3, (13)
where x < 0 is assumed for the region of interest. A res-
onant (δ = 0) Gaussian solution to Eq. (10) is displayed
in Fig. 2(a). The center of the Gaussian beam is shifted
by an amount xctr = Fz
2/2k, which is equivalent to the
motion of a charged particle in a constant electric field.
The control field phase profile (13) can be implemented
using, for example, a phase plate. The assumption that
the control fields do not depend on z implies that the
Fresnel number for these fields must be above 1, i.e. that
the characteristic transverse distance over which these
fields significantly change must be larger than ∼
where L is the EIT cell length. This imposes a limi-
tation on the magnitude of the electric quasi-field: from
Eq. (13) we find F <∼ λ−1/2L−3/2 and thus xctr <∼
Assuming that the signal field also has a Fresnel number
of at least 1, and thus satisfies 2zR >∼ L (with Rayleigh
length zR = kw
2/2, w being the signal beam width at
the cell entrance), we find that in a realistic experiment,
the maximum possible signal beam displacement due to
the quasi-electric field is on the order of the signal beam
width w.
To generate a homogeneous magnetic quasi-field along
the z-axis the quantity B = ∇ × AQQ = 2∇⊥θ × ∇⊥R
should be constant. However, it seems difficult to si-
multaneously achieve a vanishing electric quasi-field E =
−∇⊥Φ. A choice that minimizes the electric quasi-
field around the origin is given by θ =
B/2x and
B/2 y. The quasi-potentials then become AQQ =
−B y ex, which corresponds to the Landau gauge in stan-
dard electrodynamics, and Φ = B + 2B3y4 + O(y6). If
Φ is neglected, a Gaussian solution to the paraxial wave
equation is given by
bQ = N cscu(z) exp
cotu(z)∆x2 +∆x · pc
∆x∆y − 1
pc,xpc,y +
ycpc,y
where we have set ∆x ≡ (x − xc, y − yc), u(z) ≡
Bz/(2k) − i tanh−1(2η) and η ≡ Bw2/4. Here xc =
x0 + (k/B)(x
0 sin(Bz/k) + x̃
0(1 − cos(Bz/k)) denotes
the classical spiral trajectory of a charged particle in a
magnetic field, with x′c = dxc/dz, initial position x0 and
initial velocity x′0. For convenience we also have defined
x̃′0 = (y
0,−x′0) and the classical canonical momentum
pc. We remark that pc,x is a constant of motion. The
evolution of the signal mode is displayed in Fig. 2(b).
A surprising feature of solution (14) is that the diffrac-
tive divergence of the signal beam is reduced: the width
squared of the Gaussian,
Re(iB cotu)
1 + 4η2 − (1 − 4η2) cos(Bz
varies periodically with z instead of monotonically in-
creasing. This effect is known for electron wavepackets
[13] and can be understood as a consequence of the cir-
cular motion of particles in a magnetic field: instead of
dispersing, two-dimensional particles in a magnetic field
will simply move on circles of different size (depending on
their velocity), but with the same angular velocity. The
particle cloud will therefore not spread but “breathe”.
It remains to show that non-adiabatic coupling to other
modes can be suppressed for realistic experimental pa-
rameters. This is the case if the strength of the gauge
field terms coupling bQ to other modes, which for the
quasi-magnetic field are of the order B/(2k), are much
smaller than the difference in the respective linear sus-
ceptibilities χ1. For the EIT mode bQ, χ1 = δ/vEIT
with vEIT defined above Eq. (10); for the other modes
it can be approximated by the susceptibility of a two-
level medium, χ1 = −4Ng2(δ − iγ/2)/(cγ2). Evalu-
ating this relation at resonance leads to the condition
η ≪ (kw)2n3π/2, with n ≡ N/(V k3) being the number
of atoms in the volume k−3, which can easily be fulfilled
in an experiment.
Aharonov-Bohm potential for photons.— One of the
most intriguing phenomena of charged quantum particles
FIG. 2: Paraxial propagation of a signal beam over twice the
Rayleigh length in the presence (solid) and absence (grey) of
a constant (a) electric field along the x axis and (b) magnetic
field along z. The dashed line represents the center of the
grey beam. The effect of the fields is somewhat exaggerated.
in electromagnetic fields is the Aharonov-Bohm (AB) ef-
fect [14]. Its two astonishing features are (i) a phase shift
induced by the vector potential in a region in which elec-
tric and magnetic fields are absent, and (ii) its topological
nature: the phase shift does not depend on the particle
trajectory as long as it encloses a magnetic flux. Because
(unlike genuine electromagnetism) the potential (5) is a
differential function of the control fields, it is impossible
to simulate feature (i) with quasi-charged photons. How-
ever, we will show here that a mathematically equivalent
topological phase shift does exist for the optical case.
To generate an AB potential for photons we propose
to use two counter-rotating Laguerre-Gaussian control
fields, i.e., fields that possess an orbital angular momen-
tum. If these control fields are spatially wider than the
signal fields, the corresponding Rabi frequencies can be
approximated in cylindrical coordinates (r, ϕ) by Ω1 =
g1s1re
iϕ and Ω2 = g2s2re
−iϕ. The gauge potentials (12)
then become AQQ = −2R/r ~eϕ and Φ = (1 − 4R2)/r2,
with R = 1
(|s1|2 − |s2|2)/(|s1|2 + |s2|2). The potential
AQQ corresponds exactly to an Aharonov-Bohm poten-
tial for charged particles as it is created by a solenoid.
Solutions of the paraxial wave equation (10) can be
found in cylindrical coordinates by expanding the field
mode as bQ = r
m∈ZZ Bm(z, r) exp(imϕ). Because
of Ω ∼ r, the EIT group velocity can be written as vEIT =
ṽ r2 with ṽ ≡ c
|s1|2 + |s2|2/N . Exact solutions are
given by Bessel functions,
Bm = e
−iκ2z/(2k)
κrJν(κr) + βm
κrYν(κr)
with ν =
1 +m2 + 4Rm− 2kṽδ. For monochromatic
signal fields this corresponds to a rotation of the trans-
verse mode structure. For R = ±1/2 the potential trans-
fers a unit amount of angular momentum to the signal
light, but generally the amount can vary continuously be-
tween −h̄ and h̄. Signal photons in the EIT mode there-
fore form a two-dimensional bosonic quantum system in
an Aharonov-Bohm potential.
Conclusion.— We showed that EIT in a multi-Λ sys-
tem can be used to generate a variety of geometric ef-
fects on propagating signal pulses that mimic the be-
havior of a charged particle in an electromagnetic field.
We found specific arrangements of two spatially inhomo-
geneous pump fields in a double-Λ system which gener-
ate quasi-gauge potentials which correspond to constant
electric and magnetic fields. Furthermore topological ef-
fects like the Aharonov-Bohm phase shift can be induced.
The latter is significantly different from the proposal of
Ref. [3] in that it is based on spatially inhomogeneous
pump fields rather than the Doppler effect in moving me-
This paper investigated EIT in systems with two
ground levels. In such a system, there is only one EIT
mode, which results in an Abelian U(1) gauge theory,
making the physics analogous to electromagnetism. By
extending to multiple ground levels, it may be possible to
obtain multiple EIT modes and model non-Abelian gauge
potentials. This will be explored in a future publication.
We thank David Feder and Alexis Morris for fruit-
ful discussions. This work was supported by iCORE,
NSERC, CIAR, QuantumWorks and CFI.
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http://arxiv.org/abs/cond-mat/0602501
http://arxiv.org/abs/quant-ph/0611181
http://arxiv.org/abs/quant-ph/0403171
