Transverse multimode effects on the performance of photon-photon gates

Abstract

The multimode character of quantum fields imposes constraints on the implementation of high-fidelity quantum gates between individual photons. So far this has only been studied for the longitudinal degree of freedom. Here we show that effects due to the transverse degrees of freedom significantly affect quantum gate performance. We also discuss potential solutions, in particular separating the two photons in the transverse direction.

Full text

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PHYSICAL REVIEW A 83, 022312 (2011)
Transverse multimode effects on the performance of photon-photon gates
Bing He,1Andrew MacRae,1Yang Han,1,2A. I. Lvovsky,1and Christoph Simon1
1Institute for Quantum Information Science and Department of Physics and Astronomy, University of Calgary,
Calgary T2N 1N4, Alberta, Canada
2College of Science, National University of Defense Technology, Changsha 410073, China
(Received 17 June 2010; published 14 February 2011)
Themultimodecharacterofquantumfieldsimposesconstraintsontheimplementationofhigh-fidelityquantum
gates between individual photons. So far this has only been studied for the longitudinal degree of freedom. Here
we show that effects due to the transverse degrees of freedom significantly affect quantum gate performance. We
also discuss potential solutions, in particular separating the two photons in the transverse direction.
DOI: 10.1103/PhysRevA.83.022312PACS number(s): 03.67.Lx, 42.50.Ex, 03.67.Hk
I. INTRODUCTION
Photons are attractive as carriers of quantum informa-
tion because they propagate quickly over long distances
and interact weakly with their environment. Their utility
for quantum information processing applications, such as
quantum repeaters [1] or quantum computing [2], would be
further enhanced if it were possible to efficiently implement
two-qubit gates between individual photons. Such two-qubit
gates can be implemented probabilistically using just linear
optics and photon detection [3], but strong photon-photon
interactions would allow much more direct and deterministic
implementations. Several approaches to the implementation
of interaction-based photon-photon gates have been proposed,
including those based on Kerr nonlinearities in fibers or
crystals [4], on electromagnetically induced transparency
(EIT) in atomic ensembles [5], and on the interaction of
both photons with an individual quantum system [6]. See
Refs. [7–11] for recent experimental progress.
In the simplest case, an ideal controlled-phase gate
performsthetransformations
|0?|1?,|1?|0? → |1?|0?,|1?|1? → eiφ|1?|1?, where |0? and |1?
are zero- and one-photon states, respectively. In the present
context, the phase φ acquired by the state |1?|1? is due
to the interaction of the two input photons. For quantum
information processing applications it is desirable to achieve
φ = π. In early theoretical papers the main focus was
determining how to achieve interactions that are sufficiently
strong to allow large phase shifts. The photonic pulses were
typically idealized as single mode. Later it was realized that
the (longitudinal) multimode character of the pulses impose
important constraints on the implementation of high-fidelity
quantum gates [12–19]. The phase shifts due to the interaction
depend on the relative position of the two photons, and take
different values over the pulses because the photons have
to be described as extended wave packets rather than point
particles [20]. As a consequence, an initial product state of the
two photons is mapped by the interaction onto an output state
that exhibits unwanted entanglement in the photons’ external
degrees of freedom. For large phase shifts this leads to low
fidelities for the simplest quantum gate proposals [18].
It has been suggested that this difficulty can be overcome
by more sophisticated quantum gate designs where the two
photons pass through each other. This can be achieved by
trapping one [8,13] or both [14] of the two photons, by having
|0?|0? → |0?|0?,|0?|1? →
a counterpropagation geometry [15,16], or by considering two
photons with different group velocities [19].
Here we show that having the photons pass through each
other is not sufficient on its own, due to the presence of
the transverse degrees of freedom. In short, the interaction-
induced phases also depend on the relative transverse position
of the two photons, which leads to fidelity limitations that
cannot be mitigated by counterpropagation. In the following
we describe these limitations in detail. We also discuss
potential solutions, in particular separating the two wave
packets in the transverse direction, which is possible for
nonlinearities based on long-range interactions.
II. INTERACTING PULSE EVOLUTION
The general picture for the photon-photon gate is the
interaction between two fields,ˆ?1(x,t) andˆ?2(x,t), in three
spatial dimensions. The field operators at t = 0 are defined as
ˆ?i(x) =
They might describe photons in a Kerr medium or polaritons
in an EIT medium. The field operators satisfy the equal-time
commutation relation [ˆ?i(x,t),ˆ?†
means that photon absorption is negligible (e.g., in the EIT
case the spectra of the pulses are well inside the transparency
window), so the evolution of the interacting fields can be
regarded as unitary. Making the standard slowly varying
envelope and paraxial approximations, we have the effective
Hamiltonian [21] (where ¯ h ≡ 1 is adopted hereafter)
?
to describe their free evolution. Here, v is the group velocity
in the positive or negative z direction, k =2π
wave vector, and ∇2
interaction for the two fields is [22]
?
׈?j(x2,t)ˆ?i(x1,t)],
where the terms i = j and i ?= j in the sum correspond to
self-phase and cross-phase modulation effects, respectively.
Given the total Hamiltonian ˆ H =ˆ K +ˆV, the equations of
1
√V
?
kˆ ai,keik·x,whereV isthequantizationvolume.
j(x?,t)] = δijδ(x − x?). This
ˆ K =
j
?
d3xˆ?†
j(x,t)
?
v1
i∇z−v∇2
T
2k
?
ˆ?j(x,t),
(1)
λis the carrier
Tis the transverse Laplace operator. The
ˆV =
i,j
1
2
?
d3x1
?
d3x2ˆ [?†
i(x1,t)ˆ?†
j(x2,t)?(x1− x2)
(2)
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HE, MACRAE, HAN, LVOVSKY, AND SIMONPHYSICAL REVIEW A 83, 022312 (2011)
motion i∂tˆ?i(x,t) = [ˆ?i(x,t),ˆ H] for the counterpropagating
fields read as
?
∂z1
2k
?
∂z2
2k
∂
∂t+ v
∂
− iv∇2
T,1
?
?
ˆ?1(x1,t) = −iˆ α(x1,t)ˆ?1(x1,t),
(3)
∂
∂t− v
∂
− iv∇2
T,2
ˆ?2(x2,t) = −iˆ α(x2,t)ˆ?2(x2,t),
with the interaction potential
ˆ α(xi,t) =
?
d3x??(xi− x?)[ˆI1(x?,t) +ˆI2(x?,t)],
(4)
whereˆIn(x) =ˆ?†
A photon-photon
evolving an
?d3x1f1(x1)ˆ?†
ˆU(t) = Te−i?t
whereT denotesthetime-orderingoperation,andtheoperator
is factorized using the Baker-Campbell-Hausdorff (BCH)
formula.ThefirstfactorinEq.(5)describesthefreeevolution,
including pulse propagation and pulse diffraction. The second
factor is from the interaction between pulses. The third factor
contains all commutators between the exponents of the first
two terms; they reflect the interplay between pulse motion and
pulse interaction, which generally changes the pulse profiles.
The ideal output state under a gate operation would
be eiφe−i?t
trolled phase, where we take into account the free evo-
lution of the photons. The actual output state, however,
will be ˆU(t)|?? =?d3x1
?0|ˆ?1(x1,t)ˆ?2(x2,t)|??, which is generally nonfactorizable
with respect to x1and x2. The fidelity F and the controlled
phase φ of a gate operation are determined via the overlap
between the actual output state and the freely evolved state
|?R? = e−i?t
√Feiφ= ??R|ˆU(t)|?? = ??|e−i?t
=
n(x)ˆ?n(x).
gate is
state
2(x2)|0?,
implemented
|?? = |1?1|1?2=
wherefi(x) =
by
input
1(x1)?d3x2f2(x2)ˆ?†
0dt? ˆ H(t?)= e−i?t
biphoton
?0|ˆ?i(x)|1? are the pulse profiles, under unitary time evolution
0dt?ˆ Ke−i?t
0dt?ˆVe−iˆC,
(5)
0dt?ˆ K|??, with φ being a homogeneous con-
?d3x2ψ(x1,x2,t)ˆ?†
1(x1)ˆ?†
2(x2)|0?
[23], giving the two-particle wave function ψ(x1,x2,t) ≡
0dt?ˆ K|??:
0dt?ˆVe−iˆC|??
?
d3x1d3x2ψ∗
0(x1,x2,t)ψ(x1,x2,t),
(6)
where ψ0(x1,x2,t) is the corresponding two-particle function
for the freely evolved state |?R?.
The field equations (3) allow one to obtain the evolution
of ψ(x1,x2,t) by multiplyingˆ?2(x2,t) to the right of the first
equation of (3) andˆ?1(x1,t) to the left of the second. Then,
the product with ?0| and |?? takes on the addition of the
equations, yielding the following linear equation for the two-
particle function ?0|ˆ?1(x1,t)ˆ?2(x2,t)|??:
?∂
?∂
∂t+ v
∂
∂z1
?
∂
?0|ˆ?1(x1,t)ˆ?2(x2,t)|??
?
+
∂t− v
∂z2
?0|ˆ?1(x1,t)ˆ?2(x2,t)|??
−
?
iv∇2
T,1
2k0
+ iv∇2
T,2
2k0
?
?0|ˆ?1(x1,t)ˆ?2(x2,t)|??
= −i?(x2− x1)?0|ˆ?1(x1,t)ˆ?2(x2,t)|??.
Here we have used ?i(x)|0? = 0, ?i(x,t)?†
1
V
coordinate X =x1+x2
for ψ(x1,x2,t) = R(X,t)ξ(x,t) can be reduced to
?
2k
(7)
i(x?,t)|0? =
?
keik·(x−x?)|0? = δ(3)(x − x?)|0?. With the center-of-mass
2
and x = x1− x2, the derived equation
∂
∂t+ 2v∂
∂z+ iv∇2
T,x
?
ξ(x,t) = −i?(x)ξ(x,t),
(8)
with R(X,t) being trivially evolved under the diffraction term
e−ivt
The linear equation noted above greatly simplifies the
determination of the evolution of interacting pulses. This
simplification is possible because we consider the interaction
between two single photons (as opposed to multiphoton
pulses).Inthecomovingcoordinate,whicheliminatestheterm
2v∂zξ(x,t) in (8), the three factors in Eq. (5) are translated
into e−ivt
ξ(x,0). Now, the third factor contains the exponentials of the
commutators betweenvt
?
2
?
3
?vt
The commutators in the exponentials are of order l/r, where
l = vt is the medium length and r = kσ2is the Rayleigh
length, with σ the transverse size of the pulses at t = 0.
It is not difficult to achieve l/r ? 1 in practice, making
the effects from the third factor insignificant. For simplicity
we will not include the first and third factors in the two-
particle functions derived in the following, but the first-order
contribution from the third factor will be considered in our
numerical calculations. The second factor, e−iϕ(x,t), where
ϕ(x,t) =?t
examples.
k∇2
T,X.
2k∇2
T,x, e−iϕ(x,t), and e−iˆC?, respectively, to evolve
2k∇2
T,xand ϕ(x,t) as follows:
?
?
?
e−iˆC?= exp
−1
ϕ,vt
2k∇2
?
T,x
??
×exp
i
ϕ,ϕ,vt
2k∇2
T,x
??
???
+i
62k∇2
T,x,ϕ,vt
2k∇2
T,x
···.
(9)
0dt??[xT,z − 2v(t − t?)], is the main concern of
the present paper. We will explain its effects with two
III. CONTACT POTENTIAL
Our first example is a highly local interaction described by
a ?-function potential, ?(x1− x2) = V0δ(3)(x1− x2). This is
a good model for nonlinearities that are due to the interaction
of both photons with the same atom in an atomic ensemble or
crystal, or to short-range atomic collisions [4,5,13–15]. This
interaction gives an output two-particle function of
ψ(x1,x2,t) = f1(z1− vt,xT,1)f2(z2+ vt,xT,2)
×exp
?
iV0
2v[H(z−2vt)−H(z)]δ(2)(xT)
?
, (10)
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TRANSVERSE MULTIMODE EFFECTS ON THE ...
PHYSICAL REVIEW A 83, 022312 (2011)
1x
2x
12
( ,
x x
)
ψ
11
( )
f x
22
()
f x
FIG. 1. (Color online) Qualitative plot showing the two-particle
wave function as a function of the transverse coordinates x1and x2
of the two photons. The photons are distributed over a multitude of
transverse positions. For a contact interaction the interaction induces
a nonzero phase only if the two photons happen to be at exactly
the same position (see the diagonal line x1= x2); see Eq. (10). The
probability of this occurring is infinitesimal. As a consequence, the
effective output phase defined by Eq. (6) is always zero, independent
of the strength of the interaction.
where H(z) is the Heaviside step function. From Eq. (10) one
sees that the interaction-induced phase is nonzero only if the
transverse coordinates of the two photons coincide, i.e., on a
subset of configuration space, xT = 0, which has a measure
of zero (see Fig. 1). As a consequence, one has F = 1 and
φ = 0. In the case of an ideal three-dimensional ?-function
potential the effective output phase is exactly zero, no matter
how strong the interaction between the two photons. This is
closely related to the results of Refs. [12,17] for the one-
dimensional, but copropagating, case. One essentially finds
equivalent results for any interaction whose range is much
shorter than the transverse size of the wave packets. Note that
this result is consistent with the nonzero conditional phase for
photon-photon interactions obtained in Refs. [24,25], where
the evolution is nonunitary, as manifested by a different field
operator commutator.
IV. DIPOLE-DIPOLE INTERACTION
Related difficulties also arise for more long-range inter-
actions. This can be seen from our second example, which is
motivatedbyRefs.[11,16].Itconcernstheinteractionbetween
polaritons whose atomic component is in a highly excited
Rydberg state in an external electric field (cf. Fig. 2). This
induces a dipole-dipole interaction between the polaritons,
?(x1− x2) = C(1 − 3cos2ϑ)/|x1− x2|3,
where C depends on the specific Rydberg states used and
ϑ is the angle between x1− x2and the external field (along
which the electric dipoles of the Rydberg states are aligned).
This is an attractive system because Rydberg states have large
dipolemoments,leadingtopotentiallyverystronginteractions
between the polaritons [26–28].
We consider the situation where the external field
is perpendiculartothe
assumetheinitial pulse
ψ0(x1)ψ0(y1)ψ0(z1) and f2(x2) = ψ0(x2)ψ0(y2)ψ0(z2− l),
(11)
direction
profiles
of
to
motion.
be
We
f1(x1) =
d
e
g
0.050.100.15 0.20
0.75
0.85
0.90
1.00
F
0.95
0.80
FIG. 2. (Color online) Schematic setup for a photon-photon gate
working with counterpropagating single photon pulses in a medium
under EIT conditions. The pulses interact with each other through the
dipole-dipole force between the Rydberg states |di?. The inset shows
the relevant energy levels of the atoms. The pulses collide head-on,
but they have a transverse extent σ. This leads to a dependence of
the interaction-induced phase for the two-particle wave function on
the relative transverse position, resulting in a trade-off between the
effective phase φ of the two-photon operation and its fidelity F. Only
very small phases are compatible with high fidelities.
whereψn(x) = [
Hermitepolynomials.TheevolutionaccordingtoEq.(8)gives
the output two-particle wave function
1
σ√π2nn!]
1
2Hn(x
σ)e−1
2(x
σ)2withHn(x
σ)beingthe
f1(z1− l,xT,1)f2(z2+ l,xT,2)e−iϕ(x1,x2,l/v).
The interaction-induced phase in the above is given by
(12)
ϕ(z,xT,l/v) =C
2v
1
x2
T
?
z3+ 2zx2
?z2+ x2
?(z − 2l)2+ x2
T
?3
T
2
−(z − 2l)3+ 2(z − 2l)x2
T
T
?3
2
?
.
(13)
By Eq. (6), the conditional phase φ and the fidelity F in this
case are determined as follows:
√Feiφ=
?
2v
T
?
d3x?
1d3x?
2f2
1(x?
?
1)f2
2(x?
2)exp
×− iC
1
x?2
(z?+ l)3+ 2(z?+ l)x?2
?(z?+ l)2+ x?2
T
?3
T
T
?3
2
−(z?− l)3+ 2(z?− l)x?2
?(z?− l)2+ x?2
T
2
??
,
(14)
where x?
lation we have chosen l = 4πσ and σ = 10λ. The results are
shown in Fig. 2 . Increasing the parameter C/(2vσ2), which
indicatestheinteractionstrength,increasestheeffectiveoutput
phaseφ butdiminishestheoutputfidelityF.Asaconsequence,
significantphaseshiftsarecompletelyoutofreachifonewants
to achieve high fidelities. In the numerical calculations, we
haveincludedthefirst-ordercorrectionduetothethirdfactorof
Eq.(5).Suchacorrectioncomesfromthecommutators[l∇2
and [[l∇2
1= x1− lˆ ez, x?
2= x2, and x?= x?
1− x?
2. In this calcu-
T
2k,ϕ]
T
2k,ϕ],ϕ]; see Eq. (9). The commutators involving
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HE, MACRAE, HAN, LVOVSKY, AND SIMONPHYSICAL REVIEW A 83, 022312 (2011)
higher powers of ϕ are shown to vanish. The exponentials
of these commutators effect a modification of the intensity
profilef2
respectively.
The main cause for the trade-off between φ and F is the
dependence of the interaction-induced phase ϕ(z,xT) on the
transverse relative position xT. In fact, most of the behavior
shown in Fig. 2 can be understood by setting the phase in the
integrand of Eq. (14) as −C
l ? σ. It gives rise to transverse mode mixing in the form
e−iϕ(z,xT)ψ0(x1)ψ0(y1)ψ0(x2)ψ0(y2)
?
??
leading to the deviation from the ideal output two-particle
function eiφψ0(x1)ψ0(y1)ψ0(x2)ψ0(y2).
The above analysis shows that the transverse mode mixing
(or transverse mode entanglement) will develop even if the
pulses are initially in a single transverse mode. Our analysis
also clarifies that the pulse diffraction in the transverse
direction has no direct impact on the performance of a
photon-photongate.Itinfluencesgateperformancethroughits
interplay with the interaction between pulses, i.e., by the third
factor in Eq. (5). Compared with the effect of the transverse
mode mixing shown in Eq. (15), such a diffraction-interaction
interplay is insignificant in the regime considered here, where
the medium length l is much smaller than the Rayleigh length
r.Forexample,forl/r = 0.2(asinourcalculations),itinduces
corrections only at the few-percent level.
i(xi)andamodificationofthephaseprofilee−iϕ(z,xT),
v
1
x?2
T, which is quite accurate for
=
m,n,l,k
Cmnlkψm(x1)ψn(y1)ψl(x2)ψk(y2)
?=
m,n
cmnψm(x1)ψn(y1)
???
l,k
dlkψl(x2)ψk(y2)
?
,
(15)
V. POTENTIAL SOLUTIONS
Wehaveseenthatthetransversemultimodecharacterofthe
quantum fields, which leads to a transverse relative position
dependence of the interaction-induced phase shifts, has very
significant consequences for the fidelity and phase achievable
in photon-photon gates. We will now discuss two potential
solutions for this problem. The first is applicable only to the
case of long-range interactions. It consists of separating the
paths of the two photons by a transverse distance D which
is much greater than the transverse size σ of the pulses; i.e.,
the initial profiles of the pulses will be, for example, f1(x1) =
ψ0(x1)ψ0(y1)ψ0(z1) and f2(x2) = ψ0(x2− D)ψ0(y2)ψ0(z2−
l). With increasing D one will approach a situation where the
transversedegreesoffreedomofthephotonscaneffectivelybe
treated as pointlike. As a consequence, the effect studied here
will diminish. This is shown in Fig. 3. We have again chosen
a medium length l = 4πσ. Interaction-diffraction interplay
effects are at or below the 10−3level in this case because they
dependonthegradientsintheinteraction-inducedphaseacross
the wave packets, which decrease with increasing transverse
separation.
When adopting this solution, one has to take into account
that the interaction-induced phase decreases as the transverse
separation is increased. For example, achieving a conditional
phase shift φ = π with a fidelity F = 0.9 requires R =D
σ=
F
E
D
0.0 0.51.01.52.02.5 3.0
0.8
R = 10
R = 20
R = 30
0.7
1.0
0.9
FIG. 3. (Color online) Introducing a transverse separation be-
tween the two pulses greatly relaxes the trade-off between F and
φ, such that a phase of order π becomes compatible with high F.
The dimensionless separation R is defined as
strength has to be increased significantly in order to compensate for
the transverse separation.
D
σ. The interaction
26. By comparing to Ref. [16], this means that one could
work with a principal number of the Rydberg state n ? 75, a
transverse wave packet size σ = 7 µm, and a group velocity
v = 4m/s.Achievingφ = π withafidelityF = 0.99requires
R = 79, which is possible provided that, for example, n can
be increased to 100, v reduced to 1 m/s, and σ reduced to 5
µm. These requirements are realistic with current technology;
in particular Rydberg states with n = 79 were already used in
the experiment of Ref. [26].
The second potential solution, which is applicable both to
short-range and long-range interactions, consists of imposing
strong transverse confinement. If the confinement energy is
much greater than the interaction energy, then excitations
to higher-order transverse modes are largely suppressed.
All that the interaction can do in this case is multiply the
lowest-order transverse mode by an almost uniform phase
factor (since a nonuniform phase would imply non-negligible
amplitudes in higher-order transverse modes), thus allowing
high-fidelity quantum gates. Sufficiently strong confinement
could be achieved for example using hollow core photonic
crystal fibers [29] or optical nanofibers [30].
VI. SUMMARY
The importance of the multimode character of quantum
fields for the implementation of photon-photon gates has been
recognized in the past, but only for the longitudinal degree
of freedom. Here we have shown that transverse degrees
of freedom also play a significant role, imposing important
constraintsontheperformanceofpotentialquantumgates.For
contact interactions the effective phase is essentially always
zero, no matter how strong the interaction. For long-range
interactions the situation is more favorable, but there are still
significanttrade-offsbetweentheachievablephaseandfidelity.
We discussed two potential solutions. One is to have a sig-
nificant transverse separation between the two wave packets,
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PHYSICAL REVIEW A 83, 022312 (2011)
which is possible only for long-range interactions, and which
requires an even stronger interaction. The second potential
solution is to impose very strong transverse confinement,
whichmaybepossibleusinghollowfibersornanofibers.Inany
case it will be essential for future implementations of photon-
photongatestotaketransversemultimodeeffectsintoaccount.
ACKNOWLEDGMENTS
We thank M. Afzelius, H. de Riedmatten, A. Rispe,
and B. Sanders for useful discussions. This work was sup-
ported by AI-TF, NSERC DG, CFI, AIF, Quantum Works,
and CIFAR.
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[20] This undesirable effect was previously discussed for the longi-
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phase shift” and “spectral broadening.”
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[23] Thisactual outputstate
definitionofthetwo-particle
?0ˆU†(t)ˆ?1(x1)ˆ?2(x2)ˆU(t)?? = ?0ˆ?1(x1)ˆ?2(x2)?out?.?out?
isthusdetermined to
relation?0ˆ?i(x)ˆ?†
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M. Hafezi, A. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev.
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A. Rauschenbeutel, Phys. Rev. Lett. 104, 203603 (2010).
?out? =ˆU(t)??
isfromthe
functionψ(x1,x2,t) =
the uniqueformbythe
i(x?)0? = δ(3)(x − x?).
84, 1419
M.Saba, G.B. Jo,
022312-5