arXiv:1006.3584v1 [quant-ph] 18 Jun 2010
Transverse multi-mode effects on the performance of photon-photon gates
Bing He1, Andrew MacRae1, Yang Han1,2, A.I. Lvovsky1, and 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
The multi-mode 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 lon-
gitudinal degree of freedom. Here we show that there are related significant constraints due to the
transverse degrees of freedoms of the photons, which will need to be taken into account in future
quantum gate implementations. We also discuss two potential solutions, namely separating the
two photons in the transverse direction, and transverse confinement that is strong compared to the
Photons are attractive as carriers of quantum infor-
mation because they propagate fast over long distances
and interact weakly with their environment. Their utility
for quantum information processing applications such as
quantum repeaters  or quantum computing  would
be further enhanced if it was possible to efficiently imple-
ment two-qubit gates between individual photons. Such
two-qubit gates can be implemented probabilistically us-
ing just linear optics and photon detection , but deter-
ministic gates require strong photon-photon interactions.
Several approaches to the implementation of interaction-
based photon-photon gates have been proposed, includ-
ing proposals based on Kerr non-linearities in fibers or
crystals , on electromagnetically induced transparency
(EIT) in atomic ensembles , and on the interaction of
both photons with an individual quantum system .
In the first theoretical papers the quantum fields were
idealized as single-mode, and the main focus was on how
to achieve sufficiently strong interactions. Later it was re-
alized that the (longitudinal) multi-mode character of the
fields imposes important constraints on the implementa-
tion of high-fidelity quantum gates [7–14]. The phase
shifts due to the interaction depend on the relative po-
sition 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. (This
undesirable effect was previously discussed for the longi-
tudinal degree of freedom under the names of “inhomo-
geneous phase shift” and “spectral broadening”.) 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’ exter-
nal degrees of freedom. For large phase shifts this leads
to low fidelities for the simplest quantum gate proposals
It has been suggested that this difficulty can be over-
come by more sophisticated quantum gate designs where
the two photons pass through each other.
be achieved by trapping one [8, 15] or both  of the
two photons, by having a counter-propagation geome-
try [10, 11], or by considering two photons with different
group velocities . Here we show that having the pho-
tons 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 are analogous to the lon-
gitudinal case, but that are not mitigated by counter-
propagation. In the following we describe these limita-
tions in detail. We also discuss potential solutions, in par-
ticular separating the two wave packets in the transverse
direction (which is possible for non-linearities based on
long-range interactions) or transverse confinement that
is very strong compared to the interaction (see below).
Let us consider two counter-propagating quantum
fields (e.g. photons in a Kerr medium or polaritons in an
EIT medium) in three spatial dimensions, represented by
field operatorsˆΨ1(x,t) andˆΨ2(x,t) (describing the an-
nihilation of the corresponding field quantum at location
x at time t), which satisfy the following equations,
2k)ˆΨ1(x,t) = −iˆ α(x,t)ˆΨ1(x,t)
2k)ˆΨ2(x,t) = −iˆ α(x,t)ˆΨ2(x,t)(1)
with the interaction potential
ˆ α(x,t) =
d3x′∆(x − x′)(ˆI1(x′,t) +ˆI2(x′,t)), (2)
tors satisfy the commutation relation [ˆΨi(x),ˆΨ†
δijδ(x − x′). Here we have made the slowly varying enve-
lope and paraxial approximations; v is the group velocity
in the positive or negative z direction, k is the associated
wave vector, ∇2
the real function ∆(x − x′) contains all the information
about the interaction between the two fields. Note that
ˆ α(x) includes both cross- and self-phase modulation. We
will consider concrete examples for the interaction term
below. The essential point for our subsequent discus-
sion is the dependence of the photon wave packets and
of the interaction on the transverse coordinates. We will
neglect the effect of the transverse Laplacian (and thus
n(x)ˆΨn(x), and the field opera-
Tis the transverse Laplace operator, and
diffraction), apart from the discussion of confinement at
the end of the paper.
Under these conditions, the solution of the above equa-
tions is (cf. Ref. )
ˆΨ1,2(x,t) = e−i?t
where ezis the unit vector in the z direction. The phase
term can be expressed in terms of the fields at t = 0 by
noting that Eq. (3) for real ∆(x−x′) impliesˆI1,2(x,t) =
ˆI1,2(x ∓ vtez,0), which implies that only the phase, but
not the shape, of the wave packets is affected by the
interaction. In particular this means that there is no
absorption in our model, which in the case of EIT corre-
sponds to the spectral support of the pulses lying well in-
side the transparency window. Suppose that the photons
are originally prepared in well-separated wave packets f1
and f2, such that the initial two-particle wave function is
ψ(x1,x2,0) = ?0|ˆΨ1(x1,0)ˆΨ2(x2,0)|Φ? = f1(x1)f2(x2),
where |Φ? is the quantum state of the two-photon sys-
tem (in the Heisenberg picture). Then the output two-
particle wave function can be shown to be
0dt′ˆ α(x∓v(t−t′)ez,t′)ˆΨ1,2(x ∓ vtez,0),(3)
ψ(x1,x2,t) = ?0|ˆΨ1(x1,t)ˆΨ2(x2,t)|Φ?
= f1(z1− vt,xT,1)f2(z2+ vt,xT,2)
where we have explicitly shown the dependence on the
transverse coordinates of the two photons, xT,1and xT,2.
For an ideal gate, the output wave function should be
ψid(x1,x2,t) = f1(z1−vt,xT,1)f2(z2+vt,xT,2)eiφ, (5)
i.e. there should just be a uniform phase φ (equal to π for
the standard C-PHASE gate), without any modification
of the photon wavepackets apart from the translation due
to their propagation. For large enough t such that the
wave packets are well separated again after the interac-
tion, Eq. (4) would correspond to an almost uniform
phase (thanks to the integral over t′) , if it weren’t
for the dependence of the phase on the transverse coor-
dinates. Unfortunately in general the transverse coordi-
nates cannot be neglected. We will now illustrate their
effect with two concrete examples.
Contact interaction. Our first example is a highly lo-
cal interaction described by a delta function potential,
∆(x1− x2) = V0δ(3)(x1− x2). This is a good model
for non-linearities that are due to the interaction of both
photons with the same atom in an atomic ensemble, or to
short-range atomic collisions [4, 5, 8–10]. Using Eq. (4),
this interaction gives an output two-particle function
ψ(x1,x2,t) = f1(z1− vt,xT,1)f2(z2+ vt,xT,2)
( ,x x)?
( )f x
() f x
FIG. 1: This qualitative plot shows the two-particle wave
function ψ as a function of the transverse coordinates x1 and
x2 of the two photons. The photons are distributed over a
multitude of transverse positions. For a contact interaction
the interaction leads to a non-zero phase only if the two pho-
tons happen to be at exactly the same position (the diagonal
line x1 = x2in the figure), see Eq. (6). The probability of this
occurring is infinitesimal. As a consequence, the effective out-
put phase φ defined by Eq. (7) is always zero, independently
of the strength of the interaction.
where again t has to be large enough such that the
two counter-propagating wave packets are well separated
again after having interacted. One can define the fidelity
F and effective phase φ of the two-photon operation by
projecting the actual output state onto the output state
that one would obtain in the absence of any interaction,
ψ0(x1,x2,t) = f1(z1− vt,xT,1)f2(z2+ vt,xT,2), i.e. one
For Eq. (6) one sees that the interaction-induced phase
is non-zero only if the transverse coordinates of the two
photons coincide, i.e. on a subset of configuration space
that has measure zero, see Fig. 1. As a consequence,
the output state is in fact indistinguishable from ψ0, i.e.
one has F = 1 and φ = 0. In the case of an ideal delta
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.
[7, 12] for the one-dimensional, but co-propagating case.
Note that this result is compatible with a non-zero phase
for classical fields (i.e. large-amplitude coherent states),
which corresponds approximately to replacing the field
operators by classical field functions in Eq. (3), cf. Refs.
[16, 17]. Let us also note that while the delta function
potential is of course an idealization, one finds essen-
tially equivalent results for any interaction whose range is
much shorter than the transversesize of the wave packets.
These results imply that there is little hope for the real-
ization of high-fidelity photon-photon gates due to Kerr
non-linearities in bulk crystals, for example. (See below
for potential solutions based on strong confinement.)
Dipole-dipole interaction. We now show that related
0.00 0.05 0.100.15
FIG. 2: Schematic setup for a photon-photon gate working
with counter-propagating 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 ex-
tent σ. This leads to a dependence of the interaction-induced
phase for the two-particle wave function on the relative trans-
verse 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.
(though not identical) difficulties arise also for more long-
range interactions. This can be seen from our second ex-
ample, which is motivated by Refs. [11, 18]. It concerns
the interaction between polaritons whose atomic compo-
nent 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 dipole moments, leading to potentially
very strong interactions between the polaritons .
We first consider the case where the external field is
perpendicular to the direction of motion. Using Eq. (4)
as before, the output two-particle wave function is now
ψ(x1,x2,t) = f1(z1− vt,xT,1)f2(z2+ vt,xT,2)
where in the limit of well separated pulses before and
after the interaction, and setting z = z1− z2− 2vt and
xT = xT,1− xT,2, the phase is given by
1 −z3+ 2zx2
By calculating the overlap between ψ and the two-
particle function without interaction ψ0 one can again
determine both the fidelity F and the effective phase φ
0.00.5 1.0 1.52.02.5 3.0
FIG. 3: Introducing a transverse separation between 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 di-
mensionless separation R is defined as
that the interaction strength has to be increased significantly
in order to compensate for the transverse separation.
σ. The price to pay is
for the two-photon operation. For simplicity, in our nu-
merical calculations we consider Gaussian wave packets
of size σ in all spatial dimensions, however our results
do not strongly depend on the size or shape of the wave
packets (note that the dipole interaction has no charac-
teristic length). Since the interaction-induced phase de-
pends on the transverse relative position, increasing the
interaction strength increases the effective output phase
φ, but diminishes the output fidelity F. This leads to
a trade-off between F and φ, see Fig.
phase shifts of order π are completely out of reach if
one wants to achieve high fidelities. Note that most of
the behavior shown in Fig. 2 can be understood by just
considering the first term in Eq. (10), i.e. by setting
tial point is the dependence of the phase on the relative
We briefly comment on the situation when the exter-
nal field is parallel to the z axis. In this case one finds,
under the same conditions and using the same notation
as above, ϕ(z,xT) =
tion of z. As a consequence, for Gaussian wave packets
the effective overall phase φ is always zero in this case,
independently of the interaction strength. This is in con-
trast with the results of Ref. , where the transverse
degrees of freedom were integrated out from the interac-
Potential solutions - Transverse separation. We have
seen that the transverse multi-mode character of the
quantum fields, which leads to a transverse relative posi-
tion 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.
first is applicable only to the case of long-range inter-
(xT,1−xT,2)2. The essen-
2, which is an odd func-
actions. It consists in separating the paths of the two
photons by a transverse distance D that is much greater
than the transverse size of their wave packets σ.
this case one approaches a situation where the trans-
verse degrees of freedom of the photons can effectively
be treated as point-like. As a consequence, the effect
studied here becomes negligible. This is shown in Fig.
3. The price to pay is that even for dipole-dipole inter-
actions the interaction-induced phase decreases quickly
as the transverse separation is increased. To create the
conditional phase shift of π with a fidelity F ≃ 0.9, for
example, one has to work with R =D
to the parameters proposed in Ref. , one would then
have to increase the principal number of the Rydberg
state from n ≃ 25 to n ≃ 75 (note that C ∝ n4), while
reducing the transverse dimension of the pulses from 30
µm to 9 µm, assuming that the group velocity is kept the
same (4 m/s).
Potential solutions - Strong transverse confinement.
The second potential solution, which is applicable both to
short-range and long-range interactions, consists in im-
posing strong transverse confinement. This corresponds
to adding a transverse potential in Eq.
to the virial theorem, the confinement energy (or more
precisely, frequency) can be estimated from the trans-
verse kinetic term in Eq. (1) and the transverse width σ
of the confined wave function to be of order ωc ≈
In particular this gives the (approximate) energy sepa-
ration between the lowest confined state and the first
excited state. If the confinement is much stronger than
the interaction, i.e. if ωc ≫ α, where α is the typical
magnitude of the interaction operator ˆ α in Eq. (1) for
the relevant quantum states, then the probability for the
photons to be excited to a transverse excited state is only
of order (α
tions the solutions Eq. (3) and Eq. (4) are no longer
valid. All that the interaction can do in this case is mul-
tiply the ground state wave functions by a uniform phase
factor (a non-uniform phase would imply non-negligible
amplitudes in transverse excited states), thus allowing
high-fidelity quantum gates. The phase induced by the
interaction is of order ατ, where τ is the duration of the
single photon pulses. If one wants this to be of order
π, one arrives at the condition
in the regime of strong confinement. This can also be
written σ2≪ λl, where λ is the wavelength and l = vτ
is the length of the pulses in the medium. Considering
the dipole-dipole interaction example of Ref. , for
realistic interaction strengths, and taking into account
that pulse durations are limited by Rydberg state co-
herence times, this means that σ should be at the level
of the wavelength. This could be achieved for example
using hollow core photonic crystal fibers  or optical
nanofibers . We intend to discuss the prospects for
photon-photon gates based on Rydberg states in more
detail in a future publication.
σ= 24. Compared
ωc)2(and thus negligible). Under these condi-
kσ2 ≫ 1 in order to be
Conclusion. The importance of the multi-mode char-
acter of quantum fields for the implementation of photon-
photon gates had been recognized in the past, but only
for the longitudinal degree of freedom. Here we showed
that the transverse degrees of freedom also play a sig-
nificant role, imposing important constraints on the per-
formance of potential quantum gates. For contact inter-
actions the effective phase is essentially always zero, no
matter how strong the interaction. For long-range inter-
actions the situation is more favorable, but there are still
significant trade-offs between the achievable phase and
fidelity. We discussed two potential solutions. One is to
have a significant transverse separation between the two
wave packets, which is possible only for long-range inter-
actions. The price to pay is the need for an even stronger
interaction. The second potential solution is to impose
very strong transverse confinement, which may be possi-
ble using hollow fibers or nanofibers. In any case it will
be essential for future implementations of photon-photon
gates to take transverse multi-mode effects into account.
We thank A. Rispe and B. Sanders for useful discus-
sions. This work was supported by NSERC, CFI, AIF,
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