Distributed quantum computation via optical fibres

0.17 MB·English

by  Alessio Serafini

Distributed quantum computation via optical fibres
Alessio Serafini,1 Stefano Mancini,2 and Sougato Bose1
1 Department of Physics & Astronomy, University College London,
Gower Street, London WC1E 6BT, United Kingdom
2 INFM and Dipartimento di Fisica, Universit`a di Camerino, I-62032 Camerino, Italy
We investigate the possibility of realising effective quantum gates between two atoms in distant
cavities coupled by an optical fibre. We show that highly reliable swap and entangling gates are
achievable. We exactly study the stability of these gates in presence of imperfections in coupling
strengths and interaction times and prove them to be robust. Moreover, we analyse the effect of
spontaneous emission and losses and show that such gates are very promising in view of the high
level of coherent control currently achievable in optical cavities.
PACSnumbers: 03.67.Lx,03.67.Mn,42.81.Qb
6
0
0
2 The study of the possibilities allowedby coherentevo- surate and the mediating field does not exactly decouple
lutions of quantum systems is central to quantum infor- from the qubits at short enough times. However, as we
n
mation science. Most notably, exploiting suitable coher- will show, times do exist for which the qubits are de-
a
J ent dynamics to implement deterministic quantum gates coupled from the field at a high degree of accuracy. The

9 between separate subsystems is a basic aim for quantum resultingeffectivedynamicsofthetwoqubitscanthenbe
computation. Several proposals have been suggested to describedinterms ofquantumoperationswhichapprox-

3 engineer entanglement or quantum communication be- imateunitarygateswithahighfidelity. Thediscrepancy
v
tween atoms trapped in distant optical cavities, either between such approximategates and the desired unitary
4
through direct linking of the cavities [1, 2, 3, 4], or ones would be negligeable with respect to the errors in-
0
2 through detection of leaking photons [5, 6]. The realisa- volvedbyanexperimentalimplementationofthescheme.
7 tionofquantumgatesbetweendistantqubitsinquantum
We consider two two-level atoms in distant optical
0 optical settings has also been recently envisaged [7, 8].
5 cavities, interacting with the local cavity fields through
Such proposals are very promising and highly inventive.
0 dipole interactions in rotating wave approximation. The
However, they are either probabilistic or relying on ac-
/ two cavities will be henceforth labeled by the indexes
h curately tailored sequences of pulses (thereby requiring
p 1 and 2. We will allow for a detuning ∆ of the tran-
a considerable degree of control). In this paper, an al-
- sition of atom 2 from the resonance frequency ω of
t ternative to such schemes is proposed, with a particu-
n the cavities (whereas atom 1 will be assumed to be
lar focus on the implementation of distributed quantum
a at resonance). The cavities are connected by an op-
u computation. To this aim, we investigate the possibility
tical fibre, whose coupling to the modes of the cav-
q of realising deterministic gates between two-level atoms
ities may be modeled by the interaction Hamiltonian
:
v in separate optical cavities, through a coherent resonant H = ∞ ν b (a† +( 1)jeiϕa†)+ h.c. [2], where
Xi coupling mediated by an optical fibre. The only con- If Pj=1 jh j 1 − 2 i
trolrequiredwouldbethesynchronisedswitchingonand bi are the modes of the fibre, a1 and a2 are the cavities’
ar off of the atom-field interactions in the distant cavities, modes, νi is the coupling strength with the fibre mode
achievable through simple control pulses. The study of i and the phase ϕ is due to the propagation of the field
such a system (which would constitute the basic cell of through the fibre of length l: ϕ=2πωl/c [12].
scalable optical networks) is crucial in view of the out-
Now, let ν¯ be the decay rate of the cavities’fields into
standing improvements currently achievedin the control a continuum of fibre modes. Taking into accounta finite
of single atoms trapped in optical cavities [9] and of the lengthlofthefibreimpliesaquantizationofthemodesof
recentrealisationofmicrofabricatedcavity-fibresystems
thefibrewithfrequencyspacinggivenby2πc/l. Onehas
[10]. thenthatthenumberofmodeswhichwouldsignificantly
In the considered system the interaction between the interact with the cavities’ modes is of the order of n =
qubits is mediated by the bosonic light field. It has been (lν¯)/(2πc) [2]. We will focus here on the case n .1, for
showed that, in principle, an exact deterministic gate which essentially only one (resonant) mode of the fibre
may be realized if the interaction between two qubits will interact with the cavity modes (“short fibre limit”)
is mediated by another two-level system through XY [13]. Notice that such a regime applies in most realistic
nearest neighbour interactions [11]. If the central sys- experimental situations: for instance, l . 1m and ν¯
tem is a bosonic field though, interacting with the two 1GHz (natural units are adopted with ~ = 1) are i≃n
qubits through a rotating wave Hamiltonian, a perfect the proper range. We recall that the coupling ν to the
gate is not possible, as the Rabi frequencies in the two- modes of a fibre of finite length can be estimated as ν
≃
and single- excitation subspaces are no longer commen- 4πν¯c/l. Let us also notice that the coupling strength
p
2
ν can be increased by decreasing the reflectivity of the
cavity mirror connected to the fibre. In the specified 0.99
limit, the Hamiltonian HIf reduces to Hf 0.98
H =ν b(a† +eiϕa†)+ h.c. , (1) 0.97
f h 1 2 i
F 0.96
where b is the resonant mode of the fibre. The total
0.95
Hamiltonian of the composite system can be written, in
a frame rotating at frequency ω, as 0.94
2 0.93
H =∆|12ih12|+X(gj|0jih1j|a†j + h.c.)+Hf, (2) 3.1 3.2 3g.t3 3.4 3.5
j=1
where 1 and 0 are the excited and ground states of
j j
| i | i FIG. 1: Fidelities of an emulated swap gate as a function of
atom j, g is the dipole coupling between atom and field
j
in cavity j (generally complex, as local coupling phases, time. The gate is obtained for |g1|=|g2|=g and ∆=0; the
diamonds refer to ν/g = 1.1, the squares refer to ν/g = 1.2,
depending on the positions of the atoms in the cavities,
whilethetrianglesrefertoν/g=1. Allthequantitiesplotted
mightbepresent)and∆isthedetuningofthetransition are dimensionless.
ofatom2. Theaddressedsystemisthusequivalenttotwo
qubitsconnectedbyachainofthreeharmonicoscillators.
F|go2Br|−eefao|srgee1|poarfnoncdeoeσtdaj−itni=ogn,|,lgejlteihtuesju|rsefmoarlasjrok=dae1nfi,in2ne.tegres≡tin|gg1f|e,aδtur≡e ib,yj,̺k(t=) =0,P1,2i2,j,akn=d0Ethijek(stt)a̺t(e0)oEfi†jtkh(et)a.toImnspa̺r(tti)cuislagr,ivwene
areinterestedinsinglingout“decouplingtimes”atwhich
of the Hamiltonian H, which unveils some significantin-
the state of the atoms will be highly decoupled from the
sight about the dynamics we intend to study. Let us
light field so that their evolution will be approximately
consider the normal modes c and c of the three inter-
∓ unitary. Atsuchtimesthefieldhasaveryhighprobabil-
acting bosonic modes. One has c = (a1 e−iϕa2)/√2,
− ity of being in the vacuum state in both the single- and
withfrequencyω,andc∓ =(a1+ e−iϕa2∓√2b)/2,with two-excitation subspaces (the global vacuum is a trivial
frequencies ω √2ν. The three normal modes are not
∓ eigenvector of H). This condition is fulfilled when the
coupled with each other but interact with the atoms be-
Kraus operators E 0 for i,j,k = 0, so that the
cause of the contributions of the cavity fields. However, ijk ≃ 6
for ν g , the interaction of the atoms with the non Kraus operator E000 approximates a unitary evolution.
j
resona≫nt|mo|desishighlysuppressed(itisessentiallylim- More precisely, the fidelity of a Kraus operation {Eijk}
emulatingaunitarygateU canbe properlyestimatedas
itedtothesecondorderintheDysonseries)andthesys-
follows. Suppose a pure two-qubit state ψ enters the
tem reduces to two qubits resonantly coupled through a | i
operation as input: a measure of the reliability of the
single harmonic oscillator. Remarkably,as the dominant
gate is given by the overlap
interacting mode c has no contribution from the fibre
mode b, the system gets in this instance insensitive to
2
fibre losses. On the other hand, note that fulfilling the f(ψ )= ψ U† E ψ ψ E† U ψ .
condition ν g might require weak couplings, thus | i h | X ijk| ih | ijk | i
≫ | j| i,j,k=0 
implying larger operating times.
Let us now discuss the computational possibilities al- The fidelity F of the gate may then be obtained by av-
lowedby the coherentevolutiondescribed by the Hamil- eraging over all pure input states: F f(ψ ) .
|ψi
≡h | i i
tonian (2). To this aim, we will be interested in the Setting ∆ = 0, δ = 0 and g ν yields a highly reli-
≃
reduced dynamics of the two distant atoms. We will as- able swap gate at the decoupling time t π/g. The fi-
≃
sume that the system can be ‘initialized’ bringing all delity ofthe proposedswapoperationis showninFig. 1.
the field modes in the vacuum state and allowing for As apparent, such a fidelity can exceed the value 0.99
any initial state of the qubits. The Hamiltonian H and is remarkably stable with respect to possible imper-
clearly conserves the number of global excitations and, fections in the coupling strengths and in the temporal
for our aims, one can restrict to the zero-, single- and resolution needed to switch off the interaction once the
two-excitation subspaces. The quantum operation de- desiredevolutionisachieved. Letusremarkthattheval-
scribing the effective dynamics of the atoms can thus be ues g ν 1GHz (at hand with present technology in
≃ ≃
exactly worked out determining its Kraus operators for opticalcavities)wouldgrantanoperatingtimeτ 1Ns.
≃
any values of ν, g and ∆. Denoting by ijk the state We also report that, after a time t 3.4/g, a swap gate
j
| i ≃
of the field given by the number state i in the mode withfidelityF 0.98canbe obtainedforν 100g (and
≃ ≃
of cavity 1, k in the mode of cavity 2 and j in the fi- ∆ = δ = 0), i.e. in the range of parameters for which
bre mode, one has E (t) = ijk exp( iHt)000 for thesystemgetsinsensitivetofibrelosses. Thisagreeable
ijk
h | − | i
3
0.99 1
0.98
0.8
0.97
0.6
0.96
F F
E
0.95 0.4
0.94 0.2
0.93
43.6 43.8 44 44.2 44.4 0 5 10 15 20 25 30
Dt gt
FIG. 2: Fidelities of an emulated cphase gate (U0.15π) as FIG. 3: Entanglement of formation in ebits as a function of
a function of time. The diamonds refer to ν/∆ = 10, time (in units g−1) for ν = 100g = 10∆ and δ = g (dashed
|g1|/∆=0.1and|g2|/∆=0.15;thesquaresandthetriangles line), δ = 0.5g (continuous line) and δ =0 (dotted line). At
refer,respectively,toarelativevariationof−5%and+5%in thepeaks, cphase gates are emulated.
|g1|, |g2| and ν. The fidelities of the successive (more entan-
gling)cphasegatesaresimilar. Allthequantitiesplottedare
dimensionless.
strated in Fig. 2. The operating time of the gates would
range, for ν 10GHz, from 3µs to 0.3µs, according to
≃
the desired entangling power. Figure 3 shows the entan-
advantage is thus achieved by allowing a longer operat- glement of formation between the two atoms generated
ing time (due to the condition on g) and a slightly lower for an initial state (0 + 1 ) (0 + 1 )/2 (which gets
| i | i ⊗ | i | i
(but still almost perfect) fidelity. maximallyentangledifprocessedbyacontrolled-Z)with
several choices of parameters. As apparent, a speed-up
Moreover, this model allows for a reliable emula-
in the creationof entanglementis achievedby increasing
tion of an entangling gate. To fix ideas, we focus
on a ‘controlled-phase’ (cphase) gate between the two therelativedifferenceδ/g. However,toolargedifferences
(δ/g & 0.5) affect the fidelity and stability of the emu-
qubits, described by the unitary matrix U in the
ϑ
computational basis: U = Diag(1,1,1, eiϑ). This lated gate and thus, while advantageous for building up
ϑ
entanglement,arenotconvenienttoperformactualcom-
gate is equivalent, up to local unitaries, to the gates
Diag(1, eiϑ1, eiϑ2, eiϑ+ϑ1+ϑ2) for any ϑ1, ϑ2 [0,2π], putation.
∈ We now take into account dissipation due to sponta-
since the phases ϑ1 and ϑ2 can be cancelled out by local
neusemissionoftheatomsandtocavityandfibrelosses.
phase gates. We will thus henceforth refer to all such
gates as “cphase” gates. The entangling power of such The global system is then governed, in Schr¨odinger pic-
ture, by the following master equation
gates increasesas the phase ϑ increasesbetween 0 and π
(for which a controlled-Z gate is achieved). Let us also
2 2
γ κ β
recall that any of these entangling gates, together with ̺˙ = i[H,̺]+ L[a ]̺+ L[σ−]̺+ L[b]̺, (3)
localunitaryoperations,makeupauniversalsetofgates − 2 X j 2 X j 2
j=1 j=1
(asanytwo-modegatecanberecoveredasapropercom-
bination of the entangling gate and of local gates [14]). wherethesuperoperatorL[oˆ]isdefinedasL[oˆ]=2oˆ̺oˆ†
−
The symmetry of the Hamiltonian (crucial in realising a oˆ†oˆ̺ ̺oˆ†oˆ for operator oˆ and κ, γ and β stand, re-
−
swap gate), must be broken here because it prevents a spectively, for the spontaneous emission rate and for the
phase ϑ to appear at decoupling times. In point of fact, cavity and fibre decay rates (assumed for simplicity to
if the transition of atom 2 is detuned (e.g., by Stark or be equal in the two cavities). The thermal contributions
Zeemaneffect),aphasedoesarise,thusallowingforanef- of the bath have been neglected, as is possible at optical
fective entangling gate. Reliable decouplings allowing to frequencies. Consideringdecoherenceanalyticallyforone
emulate such a gate are achieved for ν g , for which excitation and numerically for two excitations (by inte-
j
≫| |
the fibre is “bypassed” and fibre losses do not affect the gratingEq.(3)), the operatortomographyofthe process
performance of the gate. For ν 100g 200δ 10∆ encompassingdecoherencehas beenreconstructedin the
a sequence of cphase gates – se≃parated≃by a pe≃riod of cases interesting for emulating gates.
about4.4g−1–withincreasingϑ(rangingfromϑ 0.15π In the regime ν g the fidelities of the gates have
j
to ϑ 0.93π)is emulated. The most entangling≃cphase been consistently f≫oun|d|to be essentially unaffected by
≃
gate(U0.93π)isachievedaftersix“Rabi-like”oscillations fibre losses. In general, moreover, the ‘direct’ effect of
inthetwoexcitationsubspace. The fidelityF ofthe em- spontaneousemissionprovestobemorerelevantthanthe
ulatedgateexceedsthevalue0.99. Itsstabilityisdemon- ‘indirect’ effect of cavity losses. For the swap gate with
4
ν 1.2g (with maximum fidelity F 0.997 without tem,notonlyentanglingandswapgates,butalsoperfect
≃ ≃
dissipation), the maximum fidelity drops to F 0.956 quantumstatetransferispossible. Besides,theproposed
for κ = 10−2g, thus allowing for a still relativ≃ely reli- setupwouldalsoallowfortheunitarygenerationofclus-
able gate, while a fidelity F 0.989 is maintained for terstatesbetweendistributedatomsorions[8],andcould
κ=γ =β =10−3g. Lowerdec≃ayrateleavesthegatevir- thus find application not only in gate-based but also in
tually unaffected, while higherratescompletely spoils it. “one-way” quantum computation. More generally, our
Noticethatvaluespermitting aneffectiveswapwouldbe results strongly emphasize the potentialities of quantum
already at hand for rubidium atoms in integrated fibre- opticalsystems towardsthe realisationof effective quan-
cavitysystems(seedatafromRef.[10],withlengthofthe tum networking schemes.
cavity L 100µm). The case ν = 100g = 200δ = 10∆,
We thank K. Jacobs, M. Trupke, C. Foot, J. Metz,
selectedt≃odemonstratethepossibilityofacphasegate,
Y.L. Lim, W. Lange and S. Bergamini for useful discus-
provedtobeslightlymoresensitivetospontaneousemis-
sions. This research is part of QIP IRC www.qipirc.org
sionandcavitylosses. Letusfocusonthefirstgate(after
(GR/S82176/01). SM thanks UCL Department of
one Rabi-like oscillation): for κ = 10−2g, the fidelity of
Physics&Astronomyforhospitality(fundedthroughthe
the gate falls to F 0.93 (in which case the fidelity of
grant GR/S62796/01).
≃
theoptimalmostentanglinggate,achievedaftersixoscil-
lations, is completely spoiled), while for κ = γ = 10−3g
(recall that this regime is insensitive to fibre losses), the
fidelityofthefirstgateisstillF 0.97. Generally,decay
ratesaslowas10−4ghaveanegl≃igeableeffectontheper- [1] J. I. Cirac et al.,Phys. Rev. Lett. 78, 3221 (1997); S. J.
formances of the gates, while decay rates of the order of van Enk et al.,ibid. 79, 5178 (1997).
10−2g would allow for remarkable experimental demon- [2] T. Pellizzari, Phys. Rev.Lett. 79, 5242 (1997).
[3] S. J. van Enk et al.,Phys.Rev. A 59, 2659 (1999).
strations of swap and entangling gates. In view of the
[4] S. Clark et al.,Phys.Rev. Lett.91, 177901 (2003).
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cavities,themaintechnicalissueleftseemstobelimiting Mancini and S. Bose, Phys. Rev. A 64, 032308 (2001);
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(withnegligible‘intrinsic’spontaneousemissionrates)of [6] L.-M. Duan and H. J. Kimble, Phys. Rev. Lett. 90,
effective two-level lambda systems could thus be good 253601 (2003).
[7] Y. L. Lim et al., Phys. Rev. Lett. 95, 030505 (2005);
candidatesfortheimplementationofsuchcomputational
S. D. Barrett and P. Kok, Phys. Rev. A 71, 060310(R)
schemes. In fact, let us consider a lambda system (refer
(2005); X.-F. Zhou et al., ibid. 71, 064302 (2005); Y.-F.
to Ref. [2] for details), where one transition is driven by Xiao et al., ibid. 70, 042314 (2004); Y.-F. Xiao et al.,
a laser of strength h with detuning d and the other is Phys. Lett. A 330, 137 (2004).
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oftheexcitedlevel,whichwillbeadiabaticallyeliminated Maunz et al., ibid. 94, 033002 (2005); S. Nußmann et
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under the condition d h. Let us suppose to exploit
≫ [10] M. Trupkeet al.,quant-ph/0506234.
such a two-level system for the proposed scheme. In our [11] M.-H. Yunget al.,Quant.Inf. Comp. 4, 174 (2004).
previous notation, one would have [2] g dh2/(d2+ξ2) [12] Actually, the description of the system in terms of dis-
and κ ξh2/(d2+ξ2), with g/κ d/ξ:≃a large enough tinct cavity and fibre modes – interacting through the
≃ ≃
detuning would thus allow to coherently implement the Hamiltonian H – holds for high finesse cavities and
If
scheme with these effective two-level systems. near resonant operations over time scales long with re-
We have investigated the implementation of quantum specttothefibre’sround-triptime(seeRef.[3]).Allthese
assumptions are matched by the system at hand in the
computation and entangling schemes for atoms trapped
operatingconditionswewillconsider(inparticularinthe
in distant cavities coupled by an optical fibre. Imperfec-
“short fibrelimit”, see next paragraph).
tionsanddissipationhavebeenconsideredshowingthat,
[13] Let us remark that the fibre could be also filtered in
in the short fibre regime, reliable gates with promising frequency to ensure the selection of a single interacting
operating times could be at hand with present technol- mode (suggestion by M. Trupke).
ogy. Let us also mention that, in the considered sys- [14] M.J.Bremneretal.,Phys.Rev.Lett.89,247902(2002).