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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

interaction.

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 [1] or quantum computing [2] 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 [3], 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 [4], on electromagnetically induced transparency

(EIT) in atomic ensembles [5], and on the interaction of

both photons with an individual quantum system [6].

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

[13].

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 [9] of the

two photons, by having a counter-propagation geome-

try [10, 11], or by considering two photons with different

group velocities [14]. Here we show that having the pho-

This can

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,

(∂

∂t+ v∂

(∂

∂z− iv∇2

∂z− iv∇2

T

2k)ˆΨ1(x,t) = −iˆ α(x,t)ˆΨ1(x,t)

∂t− v∂

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)

where ˆIn(x)

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

=

ˆΨ†

n(x)ˆΨn(x), and the field opera-

j(x′)] =

Tis the transverse Laplace operator, and

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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. [11])

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)

e−i?t

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

equal to

0dt′∆[z1−z2−2v(t−t′),xT,1−xT,2], (4)

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′) [11], 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)

e−iV0δ(2)(xT,1−xT,2)/v, (6)

1x

2x

12

( , x x)?

11

( )f x

22

() 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

can define

√Feiφ≡

?

d3x1d3x2ψ∗

0(x1,x2,t)ψ(x1,x2,t). (7)

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

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0.00 0.050.100.15

0.75

0.80

0.85

0.90

0.95

1.00F

?

?

?

d

e

g

??

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 [19].

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

given by

(8)

ψ(x1,x2,t) = f1(z1− vt,xT,1)f2(z2+ vt,xT,2)

e−iϕ(z1−z2−2vt,xT,1−xT,2), (9)

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

?

ϕ(z,xT) =C

2v

1

x2

T

1 −z3+ 2zx2

(z2+ x2

T

T)

3

2

?

. (10)

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

0.0 0.5 1.01.5 2.02.53.0

0.70

0.80

0.85

0.95

0.90

0.75

1.00F

?

R =?12

R =?24

R =?36

??

E

?

D

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.

D

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

ϕ(z1−z2−2vt,xT,1−xT,2) ≈

tial point is the dependence of the phase on the relative

transverse position.

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) =

2v

(z2+x2

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. [11], where the transverse

degrees of freedom were integrated out from the interac-

tion potential.

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-

2.Significant

C

2v

1

(xT,1−xT,2)2. The essen-

C

z

T)

3

2, which is an odd func-

The

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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. [11], 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. [11], 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 [20] or optical

nanofibers [21]. We intend to discuss the prospects for

photon-photon gates based on Rydberg states in more

detail in a future publication.

In

σ= 24. Compared

(1). Thanks

v

kσ2.

ωc)2(and thus negligible). Under these condi-

vτ

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,

Quantum Works and CIFAR.

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