Quantum processing photonic states in optical lattices

0.82 MB·English

by  Christine A. Muschik

Quantum processing photonic states in optical lattices
Christine A. Muschik, In´es de Vega, Diego Porras, and J. Ignacio Cirac1
1Max-Planck–Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse, D-85748 Garching, Germany
The mapping of photonic states to collective excitations of atomic ensembles is a powerful tool
whichfindsausefulapplication intherealization ofquantummemories andquantumrepeaters. In
thiswork weshow thatcold atoms in optical lattices can beusedto perform an entangling unitary
operation onthetransferred atomicexcitations. Aftertherelease ofthequantumatomicstate,our
protocolresultsinadeterministictwoqubitgateforphotons. Theproposedschemeisfeasiblewith
current experimental techniquesand robust against the dominant sources of noise.
PACSnumbers: 42.50.-p,03.67.Lx,03.67.Mn,32.80.Qk
7
0 Photonic channels are ideally suited for the transmis- rotations this operation is sufficient for universal quan-
0
sion of quantum states, since current technology is able tum computation [5]. Each incoming photon creates a
2
todistributephotonsbetweenremotelocationsbymeans collective atomic state, within the subspace of excita-

n of optical fibers. For this reason, they play a key role tions coupling to the light state. This state has to be
a
in practicalapplications ofquantum informationsuchas manipulated in such a way that the resulting state be-
J
quantumcryptography. Thestorageandmanipulationof longs to the same subspace. Using controlled collisions
1
photons is, however, more problematic. Storage of pho- betweenatoms,thistaskwouldrequirealargenumberof
3
tonic quantum states can be efficiently implemented by O(N2) operations, since each atom has to interact with

4 interfacing a photonic channel with an atomic system. all the others. We face here the problem of implement-
v This idea can be used to realize quantum repeaters [1], ingefficientlyanonlinearoperationwithcollectivestates,
3
and thus, to overcome the problem of losses in the pho- having only local interactions at our disposal. With our
9
tonic channel by applying entanglement purification at scheme we manage to reduce the number of operations
0
1 intermediate locations. Manipulation of photonic states to O(N1/3), by reducing the effective dimensionality of
1 requires the ability to perform entangling operations. the problem. The main idea is to map collective exci-
6 One possibility is to make use of materialswhich exhibit tations from the three dimensional Mott insulator to a
0
optical nonlinearites, but so far, available nonlinearities plane of particles, then to a line and finally to a single
/
h are too weak to provide us with short gate times. A atom, which can be directly manipulated. The plane,
p completely different approach which only requires linear the line and the single atom are created by means of an
-
t opticaloperationsandmeasurementswasproposedin[2]. initialization protocol, which has to be performed once,
n
However, this scheme is not very efficient in practice. before quantum gates can be run on the lattice. Re-
a
u markably,ourproposaldoesnotrequireaddressabilityof
In this work we show how to perform a deterministic
q individual atoms, and involves only two internal atomic
entangling gate between photons by interphasing a sys-
: levels. It comprises four kinds ofbasic operations,which
v tem of cold neutral atoms in an optical lattice with a
i are all within the experimental state of the art. Finally,
X photonic channel; that is, we show that this atomic sys-
the scheme is robust against the main sources of errors
temcanperformatthesametimethetasksofstoringand
r in a realistic setup.
a processing quantum information. The atomic ensemble
is assumed to be in a Mott insulating phase such that Atoms are assumed to possess two internal states a
| i
the lattice is filled with approximatelyone atom per site and b and to be initially prepared in a . As in quan-
| i | i
[3]. The photonic input state is mapped to a collective tum memory protocols, for example in [4], the photonic
atomicstatefollowinglight–matterinterfaceschemes[4]. input state is transferred to the atomic ensemble such
The ability to control atomic interactions in the optical that photonic Fock states n L are mapped to collective
| i
lattice allows us to perform a gate on collective atomic atomicstateswithn excitations n A. Theinitialatomic
| i
stateswhicharethenreleasedbacktothephotonicchan- stateisthereforegivenby Ψin A =α0 A+β 1 A+γ 2 A,
| i | i | i | i
nel. Inthisway,ourproposalprofitsfromtheadvantages where 1 A is a superposition of all permutations of
| i
oftwodifferentexperimentaltechniqueswhichhavebeen N particle product states containing one atom in b ,
| i
recently demonstrated. 1 = PN f a ...b ...a with PN f 2 = 1. An
| iA j=1 j| i1 | ij | iN j | j|
analogous definition holds for 2 .
We considerqubits definedbytheabsenceorthepres- | iA
enceofaphoton,butwithafewmodificationsourscheme Atomic states are processedby means of the following
is also well suited to process polarization qubits. Our four operations. (1) State–dependent transport. Atoms
gate transforms the light input state Ψin = α0 + are displaced depending on their internalstate using op-
L L
| i | i
β 1 +γ 2 into Ψout =α0 +iβ 1 +γ 2 ,where ticallatticeswithdifferentpolarizations[6,7]. (2)Popu-
L L L L L L
| i | i | i | i | i | i
n isthenphotonFockstate. Togetherwithone-qubit lation transfer between atomic states. Coherentcoupling
L
| i
2
ofthetwoatomiclevelsisachievedbydrivingRabioscil-
lations. A π/2 pulse creates the coherent superposition
a (b a )/√2, b (b + a )/√2, while a π
| i 7→ | i− | i | i 7→ | i | i
pulseinvertstheatomicpopulation. (3)Collisionalphase
shift. Controlled collisions between particles in different
states are induced by spin dependent transport. If two
particles occupy the same lattice site a collisional phase
FIG.1: Mappingofexcitationsinthebulktotheplane. (a)A
φ = ∆E t is accumulated [7], where ∆E is the on–
col int π/2pulseisappliedtotheplane. (b)The b latticeisshifted
site interaction. By controlling the interaction time tint, along xˆ such that atoms in b in the bulk| iinteract with the
φcol can be tuned. (4) State-dependent phase shift. A a part of the plane. The t|imi e spent after each single site
| i
state-dependentsingle particle rotationcanbe obtained, displacement is chosen suchthat a phase π/2 is accumulated
for example, by applying a magnetic field. if a collision occurs. Then this lattice movement is reversed.
Thus the initial positions of the atoms are restored and each
By combining these elements, the two qubit CNOT
target atom which is located on the path of a control atom
operation can be implemented (see [7] for details). It
in b experienced two collisions and picked up a total phase
transfers a target atom from its initial state a to b | i
t t of π. (c) Finally theplaneis subjected to another π/2 pulse,
| i | i
if the control atom is in b c. More specifically, consider which transfers most of the atoms back to a . Only atoms,
| i | i
control and target atoms placed along the x–axis at x which suffered a collision are transferred to b .
c
| i
and x (>x ), respectively. First, a π/2 pulse is applied
t c
to the target atom a (b a )/√2. Then the b
t t t
| i → | i −| i | i
lattice is displaced along xˆ, such that the control atom
collides with the target atom and induces a π phase on
a . Finally, the initial positions of the atoms are re-
t
| i
stored and a second π/2 pulse is applied to the target
atom (b + a )/√2 b .
t t t
| i | i →| i
The key idea in our scheme is to move control atoms
in b through a set of target atoms in a , thus trans-
| i | i FIG. 2: Creation of a line of atoms in b . (a) A π/2 pulse
ferring the atoms along its path to state b . This tool | i
is applied to thetarget atoms. (b) a and b componentsof
is employed in two related procedures, wh|icih lie at the thetarget qubits are separated spat|iailly by|ai b lattice shift
| i
heart of the proposed scheme and are introduced now. along zˆ. (c) The b lattice is further displaced along zˆ,
− | i −
(I) Mapping of collective excitations from an atomic en- such that the control atom in b interacts successively with
| i
semble of dimensionality d to a sample of dimensionality the a partof target atomsalong itspath,eachtimeleading
| i
to a collisional phase π/2. Both lattice shifts are reversed
d 1. A set of control qubits acts upon a set of target
− leaving all atoms in their original positions. (d) A π/2 pulse
qubits. An example is illustrated in Fig. 1, where con-
is applied to the target qubits. Atoms which have interacted
trol atoms in a three dimensional Mott insulator act on
with thecontrol atom are transferred to b .
targetatomsinaplane. Ifanatominthebulkisinstate | i
b , a collision is induced and the target atom hit by the
| i
control atom along its path through the plane is trans-
formed to b . In this way atoms in b are projected qubit in b acts successively on several target atoms in
| i | i | i
from the bulk to the plane. More precisely, the proce- a row, which are accordingly transferred to state b as
| i
dure maps a state with n atoms in b to a state with n explained in Fig. 2. [9] For the purpose of producing a
| i
atoms in b , except if two atoms in b in the bulk are plane(d=2),weproceedanalogouslywithalineofatoms
| i | i
located in a line along xˆ, leaving the corresponding tar- in state b instead of a single control qubit.
| i
get atom in a (CNOT2 =1). In any case an even/odd As mentioned above,the whole scheme can be decom-
| i
numberofexcitationsismappedtoaneven/oddnumber posed into two phases. During the initialization phase,
of excitations in the target object. This method allows the atoms are divided into four sets, namely the bulk, a
us to reduce stepwise the dimensionality of the problem. plane, a line, and a dot, which are spatially separated.
In the last step excitations are mapped from a line to a Thissetuphastobepreparedonceandcanafterwardsbe
single site (d = 1), and an odd number of excitations in used many times to perform gates. In the second phase
the line transfers the target atom to state b , while in the quantum gate protocol itself is performed. We ex-
| i
case of aneven number of excitations this atom is left in plain first the processing part and then how the setup is
state a . Thus the parity information is encoded in the prepared. The quantum gate protocol is summarized in
| i
state of a single atom. (II) Creation of a d dimensional Fig. 3. Theparityofthenumberofexcitationscontained
structure from a d 1 dimensional one. Starting from a in the bulk is mapped to the dot by means of procedure
−
control atom in b and an ensemble of target atoms in (I), such that the isolated atom is in state b in case of
| i | i
a , a line of atoms in b can be produced by running one excitation, while it is in state a otherwise. Now a
| i | i | i
many CNOT operations in series, such that the control phase shift π/2 is applied to the dot if it is in state b .
| i
3
In this way, the atomic state is transformed according
to 0 0 , 1 i1 , 2 2 . Then, the
A A A A A A
| i 7→ | i | i 7→ | i | i 7→ | i
previous steps are reversed and the excitations are con-
verted to light, leaving the setup in the original state.
Note that none of these steps requires addressability of
singlesites. Nowweconsiderthetruthtablecorrespond-
ing to the protocol. Let us denote by Bn the initial
| kib FIG. 3: Quantumgate protocol transforming theinput state
state of the bulk containing n = 0,1,2 atoms in |bi, lo- Ψin A =α0 A+β1 A+γ 2 Ainto Ψout A =α0 A+iβ1 A+
catedat certainlattice sites accordingto a configuration |γ 2 Ai. (a)-|(ci) Exc|itiations| iin the M| ottiinsulat|oriare su|cices-
| i
k and by P p, L l the state of the plane and the line sively mappedtostructuresoflower dimensionality resulting
| i | i
with all atoms in state a . Procedure (I) produces the in asingle atom being in state a /b in case of an even/odd
map | i number of excitations in the M|otit|inisulator. (d) A state de-
pendent phase is applied to the isolated particle such that
|Bknib|Pip|Lil|aid 7→|Bknib|Pkn′ip|Lnk′′il|anid, |1iA 7→i|1iA. Subsequentlysteps(a)-(c)havetobereversed.
where Pn′ and Ln′′ refer to the states of the plane
| k ip | k il
and line after the excitations have been mapped and
a describes the state of the dot with a = a = a
n d 0 2
| i
and a =b. Thus, the whole protocol results in
1
Bn P L a inmod2 Bn P L a . (1)
| kib| ip| il| id 7→ | kib| ip| il| id
The initialization protocol is summed up in Fig. 4.
Firstacollectiveexcitation 1 iscreated[10]. Thisstate
A
| i
contains one atom in b , which is separated from the
FIG.4: Initialization of thelattice. (a) Acontrolatom in b
ensembleandsubsequen|tilyusedtocreatealineofatoms isplacedoutsidetheensemble. (b)Thecontrolqubitinterac|tis
in b followingprocedure(II).Next,thislineisseparated successively witharowoftarget atomsintheensemble,thus
| i
from the bulk and utilized to produce a plane of atoms transferring them to state b , as explained in figure 2. We
| i
in b employing the same method. Finally the plane is obtainalineofatomsin b ,whichisalignedalongzˆ. (c)The
| i line is separated from th|eiensemble along ˆy . (d) The line
displaced such that the constellation shown in Fig. 4e is −
of control qubits is now used to create a plane of atoms in
obtained, and a π pulse is applied to the plane, the line
b . For this purpose a π/2 pulse is applied to the ensemble,
and the dot transferring these atoms to state a . | i
collisionsareinducedbya b latticeshiftalongyˆandanother
Note that collective excitations are not loca|liized, but π/2 pulse is applied to the| biulk. Since each control atom in
a superposition of states with atoms in b at different thelineleadstoalineofatomsin b ,whichisalignedalongyˆ
| i | i
sites. Moreover,we have a superposition of different po- weobtainaplaneinthexˆyˆplane. (e)Theplaneisseparated
sitionsoftheplane,thelineandthedot,astheexcitation from the ensembleby a b lattice shift along xˆ.
| i
createdatthe beginning ofthe initializationis alsodelo-
calized. Foranyterminthesuperposition,thefinalstate
differs only in a phase from the initial state. By adding
thetermsinequation(1)withrespecttothe positionsof in a three dimensional lattice is accomplished by a one
theexcitations,k,andthepositionsoftheplane,theline dimensional projection scheme.
and the dot we obtain the desired quantum gate trans- We address now transitions from a to b or to an-
| i | i
formation. other trapped state affected by b lattice shifts and give
| i
In the following we analyze the main sources of errors an example how a judicious choice of atomic levels al-
in our scheme [11]. It has been carefully designed in or- lows us to sidestep this source of errors, while still be-
der to minimize decoherence,first of all, by avoiding the ing able to perform the operations that are necessary
presenceofcatstatesintheinternalatomicstates,which for the quantum gate. By employing alkali atoms with
would give rise to errors if few particles are lost. Apart nuclearspin1/2forinstance,theatomicqubitcanbeen-
fromthat,runtimesareveryshortsuchthatdecoherence coded in hyperfine states of the S shell by identifying
1/2
has not much time to act. In particular, the time re- a F =1,m = 1 , and b F =1,m =1 . By
F F
| i≡| − i | i≡| i
quired to perform the scheme is essentially given by the choosing appropriately the detuning of two off–resonant
timeneededtorunthe collisionalsteps,sincepopulation standing waves with different polarizations [7], state–
transfersandseparationscanbe donemuchfaster. Each dependent transport can be implemented by trapping
collisonalstephasto be performedalongawholeensem- F = 1,mF = 1 and F = 1,mF = 1 by σ− and
| − i | i
ble length and requires therefore a time t N1/3, where σ light respectively. Transitions a b cannot be
int +
| i 7→ | i
t is the time spent in a single collision. Remarkably, induced by means of the off–resonant laser fields, since
int
the three dimensional problem scales like a one dimen- a correspondstothenuclearmagneticquantumnumber
| i
sionalone intime, since the task ofscanningN particles m = 1/2,while b correspondstom =1/2. π/2orπ
I I
− | i
4
pulses can be applied by means of resonant two-photon tentials forming standing waves are moved in opposite
Raman or microwave transitions. Finally, the standing directions V±(x)=cos2(kx φ) for some wave vector k,
±
waves do induce transitions to the other trapped states spatialvariablexandangleφ. Latticeshiftsaffectthere-
F = 1,m = 0 and F = 0,m = 0 . However, the fore both atomic species in the same way and lead only
F F
| i | i
optical potential experienced by these levels is given by to global phases of the resulting quantum states.
the equallyweightedsumof contributionsfromboth po-
larizations. Whileshiftingonelatticewithrespecttothe We thank Eugene Polzik for discussions and acknowl-
other, the optical potential vanishes at some point, and edge support from the Elite Network of Bavaria (ENB)
these two levels are emptied, which ensures that they do project QCCC, the EU projects SCALA and COV-
not affect the protocol. AQUIAL, the DFG-Forschungsgruppe 635 and Ministe-
Among the remaining noise mechanisms, the most rio de Educacion y Ciencia EX-2006-0295.
important ones are imperfect population transfer and
dephasing of quantum states due to spontaneous
emission[12] between two π/2 pulses. The correspond-
ing probability of error is proportional to the number of
target atoms in the mapping steps N2/3. This failure [1] W. Du¨r, H.-J. Briegel, J.I. Cirac, P. Zoller, Phys. Rev.
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probabilitycanbe reducedbyusing anelongatedatomic Zoller, Nature414, 413 (2001)
ensemblehavingaspatialextendLalongthedirectionof [2] E. Knill, R. Laflamme, G.J. Milburn, , Nature 409, 46
the first lattice shift in the quantum gate protocol and a (2001)
length l <L along the other directions. In this case the [3] D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, P.
probabilityofobtainingawrongresultisproportionalto Zoller, Phys. Rev. Lett. 81, 3108 (1998); M. Greiner,
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The probability of error due to the remaining noise
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mechanismscalesatworstlike N1/3,i.e. proportionalto Polzik,Nature432,482(2004); C.W.Chou,H.deRied-
the runtime of the protocol. First we consider imperfec-
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are occupation number defects. We only have to deal
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b in the course of the processing protocol. The failure Zoller, Phys. Rev.Lett. 82, 1975 (1999); O. Mandel, M.
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the Mott insulatorare givenby the probability for a sin-
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quence of atomic transitions into untrapped states. This [9] Theseparationstepshownin2bcanbeperformedeither
dynamicalparticlelossinduces anerrorwhichscaleslike by displacing the lattices first by half a lattice spacing
the duration of the gate, N1/3. Another limiting fac- alongxˆ(oryˆ)andthenbyadistanceexceedingthelength
of the ensemble along zˆ, or by moving the lattice fast
tor are imperfect collisions. The phase acquired in each −
along zˆ, which can be done such the atoms start and
lattice shift during the collisional steps may differ from −
end upin their motional ground state [8].
φ = π. However, as in the case of unoccupied lattice
col [10] This can forexamplebedoneusingheralded single pho-
sites, the probability of obtaining a wrong result due to tons from an EPR source or a weak coherent field to-
such an event is given by the probability on the single- gether with a postselecting photon detection.
site level. The fidelity of the scheme is also decreasedby [11] A quantitativeanalysis will be given elsewhere.
undesired collisional phases. The corresponding failure [12] Inhomogeneous background fields lead to uncontrolled
probabilityisproportionaltoN1/3,sincethesephasesare relative phases(|ai+eiβ|bi)/√2duringthestatedepen-
dent transport, thusinducing dephasing. This effect can
accumulatedinonedimensionaloperationseachcovering
be suppressed, since all transport shifts are reversed in
one ensemble length. Finally, kinetic phases acquired by
each mapping step. By swapping a and b by means
the atoms during lattice shifts do not play a role in the of a π pulsebetween thefirstand t|hei secon|di(reversing)
proposed scheme. Employing the common technique for shift both parts of the superposition a and b acquire
| i | i
state dependent transport, the nodes of two optical po- the same phase.