We describe a scheme for producing conditional nonlinear phase shifts on
two-photon optical fields using an interaction with one or more ancilla
two-level atomic systems. The conditional field state transformations are
induced by using high efficiency fluorescence shelving measurements on the
atomic ancilla. The scheme can be nearly deterministic and is of obvious
benefit for quantum information applications

Gilchrist, A.

Milburn, G. J.

English

arXiv

We describe a scheme for producing conditional nonlinear phase shifts on two-photon optical fields using an interaction with one or more ancilla two-level atomic systems. The conditional field state transformations are induced by using high efficiency fluorescence shelving measurements on the atomic ancilla. The scheme can be nearly deterministic and is of obvious benefit for quantum information applications

Gilchrist, Alexei

Milburn, Gerard J.

University of Queensland eSpace

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Conditional phase shifts using trapped atoms
A. Gilchrist and G. J. Milburn∗1
1Centre for Quantum Computer Technology and Department of Physics,
The University of Queensland, St Lucia, QLD, 4072 Australia.
∗ Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge,UK
(Dated: August 26, 2002)
Abstract
We describe a scheme for producing conditional nonlinear phase shifts on two-photon optical
fields using an interaction with one or more ancilla two-level atomic systems. The conditional field
state transformations are induced by using high efficiency fluorescence shelving measurements on
the atomic ancilla. The scheme can be nearly deterministic and is of obvious benefit for quantum
information applications.
PACS numbers:
1
It has recently been shown [1] that nonlinear phase shifts on two photon states can be
produced by coupling the mode of interest to ancilla modes via a beam splitter and mak-
ing photon counting measurements on the ancilla modes. Such conditional nonlinear phase
shifts can be used to perform two qubit operations for logical states encoded in photon num-
ber states. If such conditional gates are used to prepare entangled states for teleportation,
efficient quantum computation can be performed which with suitable error correcting codes
can be made fault tolerant [1]. In this paper we show that if the ancilla modes are replaced
with a two level atom similar conditional nonlinear phase shifts can be achieved by near
deterministic postslection on atomic measurements. The atomic measurements can made
with fluorescence shelving techniques which are very much more efficient than single pho-
ton counting measurements, thus reducing the need for new photon counting technologies
inherent in the KLM scheme. Recently, a different scheme for conditional quantum gates
based on atomic systems was presneted by Protsenko et al.[2]
Consider a single optical mode prepared in an arbitrary two photon state
|ψ〉 = c0|0〉+ c1|1〉 + c2|2〉 (1)
Our objective is to find a way to produce the nonlinear phase shift transformation defined
by
|ψ〉 → c0|0〉 + c1|1〉 − c2|2〉 (2)
To achieve this result we assume that at some fixed time an interaction between the field
mode and a single two level atom is switched on. After some interaction time t the interaction
is turned off and the atomic state is measured by fluorescence shelving. The resulting
conditional state of the field will then depend on the initial state of the two-level atom and
the interaction time. We will show that these can be so arranged as to effect the nonlinear
phase shift required.
We have in mind a quantum computing communication protocol in which the optical field
mode is derived from a transform limited pulsed field which israpidly switched into the cavity
mode containing the atomic systems at fixed times determined by the pulse repetition rate.
Similar systems have been proposed as a quantum memory for optical information processing
[7]. When the atomic measurement yields the required result the field may be switched out
again for further analysis or subsequent processing through linear and conditional elements.
2
|1〉
|2〉
|3〉
|4〉
∆ {
si
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a
l
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FIG. 1: Level scheme for an effective two-level transition controlled by a stimulated Raman process.
Once the cavity field is prepared, we need to switch on the interaction with the atomic
system. In order that we can switch this interaction at predetermined times we propose that
an effective two level transition connected by a Raman process with one classical field and
the quantised signal field, be used. A similar scheme has recently been proposed as the basis
of a high efficiency photon counting measurement [3, 4]. The process is also used in the EIT
schemes for storing photonic information [5] and for quantum state transfer between distant
cavities [6]. The level diagram is shown in figure 1. The nearly degenerate levels |1〉 and |2〉
are connected by a stimulated Raman transition to level |3〉. The detuning of the Raman
pulse from the excited state |3〉 is ∆, which is approximately the same as the detuning of the
signal mode form the same transition. An advantage of using a stimulated Raman process
of this kind is that the excited state |2〉 can be a metastable, long lived level. We thus do
not need to consider spontaneous emission from this level back to the ground state. The
readout of the atomic system may be achieved by using a cycling transition between the
excited state |2〉 and another probe level |4〉. Such measurements are routinely performed
in ion trap studies [8] and can have efficiencies greater than 99%.
The interaction between the single mode field and a two-level atom is described by the
effective Hamiltonian
Hˆ = κ(aˆ†σˆ− + aˆσˆ+) (3)
The interaction strength is given by κ = Ωg/2∆ where Ω is the Rabi frequency for the
Raman pulse and g is the one photon Rabi frequency for the signal field. The unitary
transformation that acts when this interaction is applied for a time t is
Uˆ(τ ) = exp[−iτ(aˆ†σˆ− + aˆσˆ+)] (4)
where τ = κt.
3
If the atom is prepared in the ground state and found in the ground state after the
interaction, the conditional state of the field is given by
Υˆgg(τ )|ψ〉 = cos(τ
√
aˆ
†
aˆ)|ψ〉 (5)
On the other hand if the atom is prepared in the exited state and found in the excited state
after the interaction, the conditional state is given by
Υˆee(τ )|ψ〉 = cos(τ
√
aˆaˆ
†)|ψ〉 (6)
There is considerable practical advantage to using the excited state preparation rather than
the ground state as there is always a signal for correct operation, however the analysis is
the same.
Now suppose we send in a generic two photon state
|ψ〉 = c0|0〉+ c1|1〉 + c2|2〉 (7)
which interacts with a single two level atom, prepared in the ground state, and after the
interaction the atom is found still to be in the ground state. In this case we need to apply
the measurement operator Υˆgg, and the resulting state of the field is
|φg〉 = A0c0|0〉 +A1c1|1〉 +A2c2|2〉 (8)
where A0 = 1, A1 = cos(τ ) and A2 = cos(
√
2τ ). Note that the frequencies of the A1 and A2
terms are irrational multiples of each other. As we sweep τ , A1 and A2 should explore their
entire phase space. It should be possible to find values of τ for which (A1, A2) approaches
arbitrarily close to (1,−1). These solutions can be found trivially for small τ by plotting A1
and −A2 and visually inspecting for intersections near 1. Some high-probability solutions
are summarised in table I.
With two atoms, one initially prepared in the ground state and the second prepared in
the excited state, the conditional state given that both atoms are found in their initial state
after the interaction is
|φge〉 = Υˆgg(τ1)Υˆee(τ2)|ψ〉 (9)
= A0c0|0〉+A1c1|1〉 +A2c2|2〉 (10)
where A0 = cos(τ2), A1 = cos(τ1) cos(
√
2τ2) , and A2 = cos(
√
2τ1) cos(
√
3τ2).
4
τ A0 A1 A2
6.5064 1 0.97519 -0.97516
37.73742 1 0.9992663 -0.9992665
219.918 1 0.999979 -0.999978
TABLE I: High-probability results for a single atom initially in a ground state and measured in a
ground state after an interaction time τ .
By using two atoms, each with a different interaction time, we have more freedom in
locating solutions for which the magnitude of the Ai’s are closer together. This occurs at the
expense of a more complex experimental scheme. To find solutions we employed a simulated
annealing algorithm on an initial ensemble of randomly chosen points. After the points
where suitably ‘cooled’ relative to a penalty function, we applied a simplex minimisation
method on the top contenders to find the local minima. Some interaction times that result
in implementing nearly ideal nonlinear-sign gates with high probability are given in table II.
τ1, τ2 A0 A1 A2
477.60911391, 197.78326606 -0.9906204535 -0.9906204532 +0.9906204537
37.79300921, 197.78109842 -0.9903219354 -0.9903219357 +0.9903219350
TABLE II: High-probability results for two atoms, one initially in a ground state the other in an
exited state and detected in their initial states after interaction times τ1 and τ2.
In an experiment it would be necessary to find a way to calibrate the interaction time until
the desired phase shift had been reached. One way to do this is depicted in figure 2. Two
single modes, each prepared in a one photon state are incident on a 50/50 beam splitter. The
two photon interference then results in the state |2〉|0〉 − |0〉|2〉. The conditional phase shift
can then be inserted on one arm so that the state is transformed to |2〉|0〉 − eiθ|0〉|2〉. This
state is then run through an identical 50/50 beam splitter to the first and the probability for
coincidence counts is sampled. This probability is given by Pc = cos
2 θ/2. The coincidence
detection rate then drops to zero at the required conditional phase shift of θ = pi.
We now estimate some typical values for the parameters. In a recent experiment a
similar stimulated Raman process was observed using single rubidium atoms falling through
a high finesse optical cavity [9]. The following parameters are typical of that experiment:
5
NS
|1〉
|1〉
coincidence
counts
FIG. 2: An interferometric scheme for calibrating the conditional nonlinear sign shift gate (NS) by
searching for coincidence counts from the photon detectors (PD).
g = 2pi× 4.5 MHz, Ω = 2pi× 30 MHz and ∆ = 2pi× 6 MHz. This gives a coupling constant
of the order of 70 Mhz. To achieve effective interaction constants of the order of those in the
table I requires interaction times of the order of 0.1− 5 µs. In this paper we have neglected
cavity decay which obviously needs to be kept small over similar time scale, which while
difficult is not impossible.
We would like to thank the Computer Science department of theUniversity of Waikato for
making available their computing resources. AG was supported by the New Zealand Foun-
dation for Research, Science and Technology under grant UQSL0001. GJM was supported
by the Cambridge-MIT Institute while a visitor at University of Cambridge.
[1] E. Knill, R. Laflamme and G.J. Milburn, Nature,409, 46 (2001).
[2] I.E.Protsenko, G.Reymond, N. Schlosser and P.Grangier, quant-ph/0206007 (2002) .
[3] A. Imamoglu,arXiv:quant-ph/0205196 (2002).
[4] D.F. James and P.G. Kwiat arXiv:quant-ph/0206049 (2002).
[5] M. Fleischhauer and M.D. Lukin, Phys. Rev. Letts. 84, 5094, (2000).
[6] J.I. Cirac, P. Zoller, H.J. Kimble and M. Mabuchi, Phys. Rev. Letts. 78, 3221 91997).
[7] T.B. Pittman and J.D. Franson, arXiv:quant-ph/0207041(2002).
[8] M.A. Rowe et al., Nature, 409, 791 (2000).
[9] M. Henrich, T. Legero, A. Kuhn and G. Rempe, Phys. Rev. Letts. 85, 4872 (2000).
6


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