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Towards deterministic optical quantum computation with coherently driven atomic ensembles

a r X i v :q u a n t -p h /0501042v 1 10 J a n 2005

Towards deterministic optical quantum computation

with coherently driven atomic ensembles

David Petrosyan ?

Institute of Electronic Structure &Laser,FORTH,Heraklion 71110,Crete,Greece

(Dated:February 1,2008)

Scalable and e?cient quantum computation with photonic qubits requires (i)deterministic sources of single-photons,(ii)giant nonlinearities capable of entangling pairs of photons,and (iii)reliable single-photon detectors.In addition,an optical quantum computer would need a robust reversible photon storage devise.Here we discuss several related techniques,based on the coherent manipula-tion of atomic ensembles in the regime of electromagnetically induced transparency,that are capable of implementing all of the above prerequisites for deterministic optical quantum computation with single photons.

PACS numbers:03.67.Lx,42.50.Gy

I.INTRODUCTION

The ?eld of quantum information,which extends and generalizes the classical information theory,is currently attracting enormous interest in view of its fundamental nature and its potentially revolutionary applications to computation and secure communication [1,2].Among the various schemes for physical implementation of quan-tum computation [3,4,5,6,7,8],those based on photons [7,8]have the advantage of using very robust and ver-satile carriers of quantum information.However,the ab-sence of practical single-photon sources and the weakness of optical nonlinearities in conventional media [9]are the major obstacles for the realization of e?cient all-optical quantum computation.To circumvent these di?culties,it has been proposed to use linear optical elements,such as beam-splitters and phase-shifters,in conjunction with parametric dawn-converters and single-photon detectors,for achieving probabilistic quantum logic with photons conditioned on the successful outcome of a measurement performed on auxiliary photons [8].Yet,an e?cient and scalable device for quantum information processing with photons would ideally require deterministic sources of single photons,strong nonlinear photon-photon interac-tion and reliable single-photon detectors.In addition,a versatile optical quantum computer would need a robust reversible memory devise.

In this paper we discuss several related techniques which can be used to implement all of the above prereq-uisites for deterministic optical quantum computation.The schemes discussed below are based on the coherent manipulation of atomic ensembles in the regime of elec-tromagnetically induced transparency (EIT)[10,11,12,13],which is a quantum interference e?ect that results in a dramatic reduction of the group velocity of weak prop-agating ?eld accompanied by vanishing absorption [14].As the quantum interference is usually very sensitive

FIG.1:Schematic representation of the quantum computer with single-photon qubits.The operation of the computer consists of the following principal steps:Qubit initialization is realized by deterministic single-photon sources(SPhS).Information processing is implemented by the quantum processor with single-qubit U and two-qubit W logic gates.Read-out of the result of computation is accomplished by e?cient single-photon detectors(SPhD).

cally be composed of(a)quantum register containing a number of qubits,whose computational basis states are labeled as|0 and|1 ;(b)one-and two-qubit(and possi-bly multi-qubit)logic gates–unitary operations applied to the register according to the particular algorithm;and (c)measuring apparatus applied to the desired qubits at the end of(and,possibly,during)the program execution, which project the qubit state onto the computational ba-sis{|0 ,|1 }.Operation of the quantum computer may formally be divided into the following principal steps. Initialization:Preparation of all qubits of the register in a well-de?ned initial state,such as,e.g.,|0...000 .Input: Loading the input data using the logic https://www.doczj.com/doc/e5185296.html,puta-tion:The desired unitary transformation of the register. Any multiqubit unitary transformation can be decom-posed into a sequence of single-qubit rotations and two-(or more)qubit conditional operations,which thus con-stitute the universal set of quantum logic gates.Output: Projective measurement of the?nal state of the register in the computational basis.The reliable measurement scheme would need to have the?delity more than1/2, but ideally as close to1as possible.

A schematic representation of an optical quantum com-puter is shown in?gure1.In the initialization sec-tion of the computer,deterministic sources of single pho-tons generate single-photon pulses with precise timing and well-de?ned polarization and pulse-shapes(see sec-tion V).A collection of such photons constitutes the quantum register.The qubit basis states{|0 ,|1 }of the register are represented by the vertical|V ≡|0 and horizontal|H ≡|1 polarization states of the photons. The preparation of an initial state of the register and the execution of the program according to the desired quan-tum algorithm is implemented by the quantum proces-sor.This amounts to the application of certain sequence of single-qubit U and two-qubit W unitary operations, whose physical realization is described below.Finally, the result of computation is read-out by a collection of e?cient polarization-sensitive photon detectors(see sec-tion VII).

For the photon-polarization qubit|ψ =α|V +β|H ,the universal set of quantum gates can be con-structed from arbitrary single-qubit rotation operations U and a two-qubit conditional operation W,such as the controlled-not(cnot)operation|a |b →|a |a⊕b (a,b∈{0,1})or controlled-phase or Z(cphase or cz) operation|a |b →(?1)ab|a |b .In turn,any single-qubit unitary operation U can be decomposed into a product of rotation R(θ)and phase-shift T(φ)operations

R(θ)= cosθ?sinθ

sinθcosθ ,T(φ)=

100e iφ ,

and an overall phase shift e i?.As an example,the Pauli X,Y,Z and Hadamard H transformations can be repre-sented as X=R(π/2)T(π),Y=e iπ/2R(π/2),Z=T(π), H=R(π/4)T(π).

As shown in?gure2(a),for the photon-polarization qubit|ψ the R(θ)and T(φ)operations are implemented, respectively,by the rotation of photon polarization by an-gleθabout the propagation direction,and relative phase-shiftφof the|V and|H polarized components of the photon.Both operations are easy to implement with the standard linear optical elements,Faraday rotators,polar-izing beam-splitters or phase-retardation(birefringent) waveplates.A possible realization of the cphase two-qubit entangling operation is shown in?gure2(b).There, after passing through a polarizing beam-splitter,the ver-tically polarized component of each photon is transmit-ted,while the horizontally polarized component is di-

2 (b)

FIG.2:Proposed physical implementation of quantum logic gates.(a)Single-qubit logic gates U are implemented with a sequence of two linear-optics operations:R(θ)–Faraday rotation(FR)of photon polarization by angleθabout the propagation direction;T(φ)–relative phase-shiftφof the photon’s|V and|H polarized components due to their op-tical paths di?erence.(b)Two-qubit cz(or cphase)gate W cz is realized using polarizing

beam-splitters(PBS)andπcross-phase modulation studied in section VI.

rected into the active medium,wherein the two-photon state|Φin =|H1H2 acquires the conditional phase-shiftπ,as discussed in section VI.At the output,each photon is recombined with its vertically polarized com-ponent on another polarizing beam-splitter,where the complete temporal overlap of the vertically and horizon-tally polarized components of each photon is achieved by delaying the|V wavepacket in a?ber loop or sending it though a EIT vapor cell in which the pulse propagates with a reduced group velocity(see section III).

The remainder of this paper is devoted to the phys-ical realizations of the constituent parts of the optical quantum computer described above.

III.ELECTROMAGNETICALLY INDUCED

TRANSPARENCY

The propagation of a weak probe?eld Ee i(kz?ωt)with carrier frequencyωand wave vector k=ω/c in a near-resonant medium can be characterized by the linear sus-ceptibilityχ(ω),whose real and imaginary parts de-scribe,respectively,the dispersive and absorptive prop-erties of the medium:E(z)=E(0)e?κz e iφ(z),where κ=k/2Imχ(ω)is the linear absorption coe?cient,and φ(z)=k/2Reχ(ω)z is the phase-shift.In the case of light interaction with two-level atoms on the transi-tion|g →|e ,the familiar Lorentzian absorption spec-trum leads to the strong attenuation of the resonant?eld (?≡ω?ωeg=0)in the optically dense medium ac-cording to E(z)=E(0)e?κ0z,where the resonant ab-sorption coe?cientκ0=σ0ρis given by the product

?4?2024

Detuning δ

/γ

?0.6

?0.4

?0.2

0.0

0.2

0.4

0.6

0.0

0.2

0.4

0.6

0.8

1.0

(b)

FIG.3:Electromagnetically induced transparency in atomic medium.(a)Level scheme of three-levelΛ-atoms interact-ing with a cw driving?eld with Rabi frequency?d on the transition|s ?|e and a weak pulsed E?eld acting on the transition|g ?|e .The lower states|g and|s are long-lived(metastable),while the excited state|e decays fast with the rateΓ.(b)Absorption and dispersion spectra(δR=?)of the atomic medium for the E?eld in units of resonant absorp-tion coe?cientκ0,for?d/γge=1andγR/γge=10?3.The light-gray curves correspond to the case of?d=0(two-level atom).

of atomic densityρand absorption cross-sectionσ0=?2geω/(2 c?0γge),?ge= g|d|e being the dipole matrix element for the transition|g →|e andγge(≥Γ/2)the corresponding coherence relaxation rate.When,how-ever,the excited state|e having decay rateΓis coupled by a strong driving?eld with Rabi frequency?d and de-tuning?d=ωd?ωes to a third metastable state|s ,the situation changes dramatically(?gure3(a)).Assuming all the atoms initially reside in state|g ,the complex susceptibility now takes the form

χ(ω)=

2κ0

γge?i?+|?d|2(γR?iδR)?1,(1)

whereδR=???d=ω?ωd?ωsg is the two-photon Raman detuning andγR the Raman coherence(spin)re-laxation rate.Obviously,in the limit of?d→0,the susceptibility(1)reduces to that for the two-level atom. The absorption and dispersion spectra corresponding to the susceptibility of equation(1)are shown in?gure3(b) for the case of?d=γge and?d=0,i.e.δR=?.As seen,the interaction with the driving?eld results in the Autler-Towns splitting of the absorption spectrum into two peaks separated by2?d,while at the line center the medium becomes transparent to the resonant?eld,pro-videdγR?|?d|2/γge.This e?ect is called electromag-netically induced transparency(EIT)[10,11].At the exit from the optically dense medium of length L(op-tical depth2κ0L>1),the intensity transmission coe?-cient is given by T(ω)=exp[?k Imχ(ω)L].To determine the width of the transparency windowδωtw,one makes a power series expansion of Imχ(ω)in the vicinity of max-

imum transmission δR =0,obtaining [12,13]

T (ω)?

exp(

?δ

2R /δω2

tw ),

δωtw =

|?d |2

2κ0L

,(2)

where the usual EIT conditions,?d γR ,?2d γR /γge

|?d |2

,are assumed satis?ed.Equation (2)implies that for the absorption-free propagation,the bandwidth δωof near-resonant probe ?eld should be within the trans-parency window,δω<δωtw .Alternatively,the temporal width T of a Fourier-limited probe pulse should satisfy

T >～δω?1

tw .

Considering next the dispersive properties of EIT,in ?gure 3(b)one sees that the dispersion exhibits a steep and approximately linear slope in the vicinity of ab-sorption minimum δR =0.Therefore,a probe ?eld slightly detuned from the resonance by δR <δωtw ,dur-ing the propagation would acquire a large phase-shift φ(L )?κ0Lγge δR /|?d |2while su?ering only little ab-sorption,as per equation (2).At the same time,a near-resonant probe pulse E (z,t )propagates through the medium with greatly reduced group velocity v g =

?ω[

k 1+c κ0γge

κ0γge

?c,

(3)

while upon entering the medium,its spacial envelope is compressed by a factor of v g /c ?1.

So far,we have outlined the absorptive and disper-sive properties of the stationary EIT without elaborating much on the underlying physical mechanism.As eluci-dated below,EIT is based on the phenomenon of coherent population trapping [10,11],in which the application of two coherent ?elds to a three-level Λsystem of ?gure 3(a)creates the so-called “dark state”,which is stable against absorption of both ?elds.Since we are interested in quan-tum information processing with photons,the probe ?eld has to be treated quantum mechanically.It is expressed through the traveling-wave (multimode)electric ?eld op-erator ?E (z,t )= q

a q (t )e iqz ,where a q is the bosonic annihilation operator for the ?eld mode with the wave-vector k +q .The classical driving ?eld with Rabi fre-quency ?d is assumed spatially uniform.In the frame rotating with the probe and driving ?eld frequencies,the interaction Hamiltonian has the following form:

H =

N j =1

[??σj ee +δR ?σj ss ?g ?E (z j )e ikz j ?σj eg ??d (t )e ik d z j ?σ

es +H .c .],(4)

where N =ρV is the total number of atoms in the quan-tization volume V =AL (A being the cross-sectional area

of the probe ?eld),?σj

μν=|μj νj |is the transition oper-ator of the j th atom at position z j ,k d is the projection of the driving ?eld wavevector onto the e z direction,and

g =?ge

(?sin θ)m (cos θ)n ?m |(n ?m )q |s (m ) .

(5)

Here the mixing angle θ(t )is de?ned via

tan 2

θ(t )=

g 2N

√

?N z

N z

j =1

?σj μνis the collective atomic

operator averaged over small but macroscopic volume containing many atoms N z =(N/L )dz ?1around posi-tion z .The evolution of the atomic operators is governed by a set of Heisenberg-Langevin equations [16],which are

treated perturbatively in the small parameter g ?E

/?d and in the adiabatic approximation for both ?elds,

?σge =?

?t ?iδR +γR

?σgs +i ?d

1+δR (?+iγge )?d

ge ,(8b)

5 where?Fμνareδ-correlated noise operators associated

with the atomic relaxation.When the driving?eld is

constant in time andδR??|?d|2,equations(7-8)yield ?v

N?σgs,(12) whose photonic and atomic components are determined by the mixing angleθ,?E=cosθ?Ψand?σgs=sinθ?Ψ/√

√

+v g(t)

√√

(a)

FIG.4:Reversible memory for the photon-polarization qubit.

(a)After the basis transformation|V →|R ,|H →|L on a45?orientedλ/4plate,the photon enters the atomic medium serving as a memory device.(b)Level scheme of M-atoms interacting with the E L,R components of single-photon pulse and the driving?eld with Rabi frequency?d.

the pulse has fully accommodated in the medium,the driving?eld?d is adiabatically switched o?.As a result, the photon wavepacket is stopped in the medium,and its state|ψ is coherently mapped onto the collective atomic state according to[26,28]

α|L +β|R →α|s(1)1 +β|s(1)2 .(15) This collective state is stable against decoherence[28], allowing for a long storage time of the qubit state in the atomic ensemble.At a later time,the photon can be released from the medium on demand by switching the driving?eld on,which results in the reversal of map-ping(15).

V.DETERMINISTIC SOURCE OF

SINGLE-PHOTONS

Generation of single-photons in a well de?ned spa-tiotemporal mode is a challenging task that is currently attracting much e?ort[29].In a simplest setup,one typ-ically employs spontaneous parametric down conversion [30]to generate a pair of polarization-and momentum-correlated photons.Then,conditional upon the outcome of measurement on one of the photons,the other photon is projected onto a well-de?ned polarization and momen-tum state.In more elaborate experiments,single emit-ters,such as quantum dots[31]or molecules[32],emit single photons at a time when optically pumped into an excited state.Recently,truly deterministic sources of single photons have been realized with single atoms in high-Q optical cavities[33].In these experiments,single-photon wavepackets with precise propagation direction and well characterized timing and temporal shape were generated using the technique of intracavity stimulated Raman adiabatic passage(STIRAP).Notwithstanding these achievements,the cavity QED experiments in the strong coupling regime required by the intracavity STI-RAP involve sophisticated experimental setup which is very di?cult,if not impossible,to scale up to a large number of independent emitters operating in parallel. In this section,we describe a method for deterministic generation of single-photon pulses from coherently ma-nipulated atomic ensembles.As discussed in section III,

symmetric Raman(spin)excitations in optically thick atomic medium exhibit collectively enhanced coupling to

a single-photon anti-Stokes pulse E,whose propagation direction and pulse-shape are completely determined by the driving-?eld parameters.The question is then:How can one produce,in a deterministic fashion,precisely one collective Raman excitation?In a number of recent the-oretical and experimental studies,such excitations were produced by the process of spontaneous Raman https://www.doczj.com/doc/e5185296.html,ly,one applies a classical pump laser to the atomic transition|g →|e and detects the number of forward scattered Stokes photons[34].Since the emis-sion of each such photon results in one atomic excitation |s symmetrically distributed in the whole ensemble,the number of Stokes photons is uniquely correlated with the number of Raman excitations of the medium.However, due to the spontaneous nature of the scattering process, the production of collective single excitation state|s(1) requires the postselection conditioned upon the measured number of Stokes photons,which makes this scheme es-sentially probabilistic

Below we will describe a scheme that is capable of pro-ducing the single collective Raman excitation at a time. It is based on the dipole-blockade technique proposed in[35],which employs the exceptionally strong dipole-dipole interactions between pairs of Rydberg atoms.In a static electric?eld E st e z,the linear Stark e?ect re-sults in splitting of highly excited Rydberg states into a manifold of2n?1states with energy levels ?νnqm= 3

nqea0e z.A pair of atoms1and2prepared in such Stark eigenstates|r interact with each other via the dipole-dipole potential

V DD=

d1·d2?3(d1·e12)(d2·e12)

FIG.5:Level scheme of atoms for deterministic generation

of single-photons.Dipole-dipole interaction between pairs of

atoms in Rydberg states|r facilitates the generation of single

collective Raman excitation of the atomic ensemble at a time,

via the sequential application of the?(1)r and?(2)r pulses of

(e?ective)areaπ.This collective excitation is then adiabati-

cally converted into a single-photon wavepacket by switching

on the driving?eld?d.

disregard state|e ,and the Hamiltonian takes the form

H=V AF+V DD,(17)

with the atom-?eld and dipole-dipole interaction terms

given,respectively,by

V AF=?

j[?(1)r e ik(1)r z j?σj rg+?(2)r e ik(2)r z j?σj rs +H.c.],(18a)

V DD=

i>j?ij(R)|r i r j r i r j|,(18b)

where?ij(R)= r i r j|V DD|r i r j ≈?n4e2a20/(π ?0R3) is the dipole-dipole energy shift for a pair of atoms i and j separated by distance R.Suppose that initially all the atoms are in state|g ,while the second laser is switched o?,?(2)r=0.Then the?rst laser?eld,coupled symmetrically to all the atoms,will induce the transition from the ground state|g1,g2,...,g N ≡|s(0) to the col-lective state|r(1) ≡1N j e ik(1)r z j|g1,...,r j,...,g N representing a symmetric single Rydberg excitation of the atomic ensemble.The collective Rabi frequency on the transition|s(0) →|r(1) is√

N?(1)r T=π/2(an e?ectiveπpulse),one produces the single Rydberg excitation state|r(1) .At the end of the pulse,the probability of error due to popu-lating the doubly-excited states|r i r j is found by adding the probabilities of all possible double-excitations,

P double～1?2

≈N|?

(1)

r|2

N?(1)r is small compared to the average dipole-dipole energy shiftˉ?.Another source of errors is the dephasing given by P deph≤γr T～γr/(√

√

FIG.6:Cross-phase modulation of two single-photon pulses.

(a)Two horizontally polarized photons E1and E2,after pass-ing through the±45?orientedλ/4plates,are converted into the circularly left-and right-polarized photons,which are sent into the active medium.(b)Level scheme of tripod atoms in-teracting with quantum?elds E1,2,strong cw driving?eld with Rabi frequency?d and weak magnetic?eld B that re-moves the degeneracy of Zeeman sublevels|g1 and|g2 . however,a second weak(signal)?eld,dispersively cou-

pling state|s to another excited state|f ,is intro-duced in the medium,it causes a Stark shift of level

|s ,given by?St=|?s|2/?s,where?s is the Rabi frequency and?s>?s,Γf is the detuning of the sig-

nal?eld from the|s →|f resonance.Thus the EIT spectrum is e?ectively shifted by the amount of ?St,which results in the conditional phase-shift of the probe?eld,φ(z)?κ0zγge?St/|?d|2.Notwithstanding this promising sensitivity,large conditional phase shift of one weak(single-photon)pulse in the presence of an-other(also known as cross-phase modulation)faces se-rious challenges in spatially uniform media.In order to eliminate the two-photon absorption[21]associated with Doppler broadeningδωD of the atomic resonance |s →|f ,one either has to work with cold atoms,of choose large detuning?s>δωD,which limits the result-ing cross-phase shift.Another drawback of this scheme is the mismatch between the slow group velocity of the probe pulse subject to EIT,and that of the nearly-free propagating signal pulse,which severely limits their ef-fective interaction length and the maximal conditional phase-shift[22].This drawback may be remedied by us-ing an equal mixture of two isotopic species,interact-ing with two driving?elds and an appropriate magnetic ?eld,which would render the group velocities of the two pulses equal[23].Other schemes to achieve the group ve-locity matching and strong cross-phase modulation were proposed in[38,39,40].Here we discuss an alterna-tive,simple and robust approach[27],in which two weak (single-photon)?elds,propagating through a medium of hot alkali atoms under the conditions of EIT,impress very large nonlinear phase-shift upon each other. Consider a near-resonant interaction of two weak,or-thogonally(circularly)polarized optical?elds E1and E2 and a strong driving?eld with Rabi frequency?d with a medium of atoms having tripod con?guration of levels, as shown in?gure6.The medium is subject to a weak longitudinal magnetic?eld B that removes the degener-acy of the ground state sublevels|g1 and|g2 ,whose Zeeman shift is given by ?=μB m F g F B,whereμB is the Bohr magneton,g F is the gyromagnetic factor and m F=±1is the magnetic quantum number of the corre-sponding state.All the atoms are assumed to be optically pumped to the states|g1 and|g2 ,which thus have the same incoherent populations ?σg

1g1 = ?σg2g2 =1/2. The weak?elds E1and E2,having the same carrier fre-

quencyω=ω0eg equal to the|g1,2 →|e resonance frequency for zero magnetic?eld,and wavevector k par-allel to the magnetic?eld direction,act on the atomic transitions|g1 →|e and|g2 →|e ,with the detun-ingsδ1,2=???kv,where kv is the Doppler shift for the atoms having velocity v along the propagation di-rection.In the collinear Doppler-free geometry shown in ?gure6(a),the circularly polarized driving?eld couples level|e to a single magnetic sublevel|s ,whose Zeeman shift ?′=μB m F′g F′B is incorporated in the driving ?eld detuning,δd=ωd?ω0es+?′?k d v=?d?k d v. Assuming,as before,the weak-?eld limit and adiabati-cally eliminating the atomic degrees of freedom,the equa-tions of motion for the electric?eld operators?E1,2corre-sponding to the quantum?elds E1,2are obtained as[27] ?v

?t ?E2=?κ2?E2+i(???d)(s2?η2?I1)?E2

+?F2,(19b) where v g=c cos2θ?c is the group velocity,with the mixing angleθde?ned as tan2θ=g2N/(2|?d|2)(the factor1/2corresponds to the initial population of lev-els|g1,2 ),κ1,2=tan2θ/c[γR+γge(?±?d)2/|?d|2] and s1,2=tan2θ/c[1+?(?±?d)/|?d|2]are,respec-tively,the linear absorption and phase modulation co-e?cients,η1,2=g22?tan2θ/[c|?d|2(2??iγR)]are the cross-coupling coe?cients,and?I j≡?E?j?E j is the dimen-sionless intensity(photon-number)operator for the j th ?eld.In deriving equations(19),the EIT conditions |?d|2?(?+kˉv)(?±?d),γR(γge+kˉv),whereˉv is the mean thermal atomic velocity,were assumed satis-?ed.Note that if states|g1,2 and|s belong to dif-ferent hyper?ne components of a common ground state, the frequenciesωandωd of the optical?elds di?er from each other by at most a few GHz,and the di?erence (k?k d)v in the Doppler shifts of the atomic resonances |g1,2 →|e and|s →|e is negligible.

In what follows,we discuss the relatively simple case of small magnetic?eld,such thatγR??,?′??d and ?d=ωd?ω0es.When absorption is negligible,κ1,2z?1, z∈{0,L},which requires that v g/γR?L and?2d<γR|?d|2/γge,the solution of equations(19)is

?E1(z,t)=?E1(0,τ)exp[iη?d?E?2(0,τ)?E2(0,τ)z],(20a)?E2(z,t)=?E1(0,τ)exp[iη?d?E?1(0,τ)?E1(0,τ)z],(20b) where the cross-phase modulation coe?cient is given by

η?g 2/(v g |?d |2),while the linear phase-modulation is incorporated into the ?eld operators via the uni-tary transformations ?E

1,2(z,t )→?E 1,2(z,t )e i ?d z/v g .The multimode ?eld operators ?E j (z,t )= q a q

j (t )e

iqz (j =1,2),with quantization bandwidth δq ≤δωtw /c (q ∈{?δq/2,δq/

})

restricted by the width of the EIT win-dow δωtw [23],have the following equal-time commuta-tion relations

[?E i (z ),?E ?j

(z ′)]=δij Lδq L 2

dzdz ′Ψij (z,z ′,t )e ?iqz e ?iq ′z ′

(24)

Substituting the operator solutions (20)into (22),for the expectation values of the intensities one ?nds

j (z,t ) =|f j (?cτ)|2.(25)

For 0≤z

therefore ?I

j (z,t ) =|f j (zc/v g ?ct )|2,while outside the medium,at z ≥L and accordingly τ=t ?L/v g ?(z ?

L )/c ,we have ?I

j (z,t ) =|f j (z +L (c/v g ?1)?ct )|2.On the other hand,after the interaction at z,z ′≥L ,the

equal-time (t =t ′)two-photon wavefunction reads [27]Ψij (z,z ′,t )=f i [z +L (c/v g ?1)?ct ]

×f j [z ′+L (c/v g ?1)?ct ]×

f j [z +L (c/v

g ?1)?ct ]

2(z ′?z ) e iφ?1

,(26)where φ=η?d L 2δq/(2π).For large enough spatial sepa-ration between the two photons,such that |z ′?z |>δq ?1and therefore sinc[δq (z ′?z )/2]?0,equation (26)yields

Ψij (z,z ′,t )?f i [z +L (c/v g ?1)?ct ]f j [z ′+L (c/v g ?1)?ct ],

which indicates that no nonlinear interaction takes place between the photons,which emerge from the medium un-changed.This is due to the local character of the interac-tion described by the sinc function.In the opposite limit of |z ′?z |?δq ?1and therefore sinc[δq (z ′?z )/2]?1,for two narrow-band (Fourier limited)pulses with the duration T ?|z ′?z |/c ,one has f j (z )/f j (z ′)?1,and equation (26)results in

Ψij (z,z ′,t )?e iφf i [z +L (c/v g ?1)?ct ]

×f j [z ′+L (c/v g ?1)?ct ].Thus,after the interaction,a pair of single photons ac-quires conditional phase shift φ,which can exceed πwhen (δqL/2π)2>(v g /c )(|?d |2/g 2).To see this more clearly,we use equation (24)to calculate the amplitudes of the state vector |Φ(t ) :

ξqq ′

ij (t )=e iφξqq ′

ij (0)exp {i (q +q ′)[L (c/v g ?1)?ct ]}.(27)

At the exit from the medium,at time t ?L/v g ,the second exponent in equation (27)can be neglected for all q,q ′and the state of the system is given by

|Φ(L/v g ) =e iφ|Φin .

(28)

When φ=π,the output state of the two photons is

|Φout =?|Φin .

(29)Utilizing the scheme of ?gure 2(b),one can then realize

the transformation corresponding to the cphase logic gate between two photons representing qubits.

Before closing this section,we note that several impor-tant relevant issues,such as the spectral broadening of interacting pulses and the necessity for their tight focus-ing over considerable interaction lengths,were addressed in a number of recent studies [41,42,43].

VII.

SINGLE-PHOTON DETECTION

To complete the proposal for optical quantum com-puter,we need to discuss a measurement scheme capa-ble of reliably detecting the polarization states of single

FIG.7:Level scheme of atoms for e?cient photon detection.

Adiabatically switching o?the driving?eld,?d→0,results

in the conversion of photonic excitation E into the atomic

Raman excitation|g →|s .The latter is detected using the pump?eld acting on the cycling transition|s ?|f with

Rabi frequency?p and collecting the?uorescent photons. photons.When the photonic qubit|ψ =α|V +β|H

passes though a polarizing beam-splitter,its vertically and horizontally polarized components are sent into two

di?erent spatial modes–photonic channels.Placing e?cient single-photon detectors at each channel would

therefore accomplish the projective measurement of the qubits in the computational basis.The remaining ques-

tion then is the practical realization of sensitive photode-tectors.Avalanche photodetectors with high quantum

e?ciencies are possible candidates for the reliable mea-surement[44].Let us,however,outline an alternative

scheme[45],based on stopping of light in EIT media, whose potential e?ciency is unmatched by the state-of-

the-art photodetectors.

Consider an optically dense medium of four-level atoms with N-con?guration of levels,as shown in?gure7.Ini-

tially,all the atoms are in the ground state|g .Un-der the EIT conditions discussed in section III,a single-

photon pulse entering the medium can be fully stopped by adiabatically switching o?the driving?eld Rabi fre-

quency?d.As a result,the atomic ensemble is trans-ferred into the symmetric state|s(1) with single Raman excitation,i.e.,an atom in state|s .Next,to detect the atom in this state,one can use the electron shelving or

quantum jump technique[46].To that end,one applies a strong resonant pumping laser acting on the cycling tran-sition|s ?|f with Rabi frequency?p and collects the ?uorescent photons emitted by the atoms with the rate R f=Γf|?p|2/(2|?p|2+γ2sf),whereΓf is the sponta-neous decay rate of state|f .In the limit of strong pump?p?γsf,transition|s →|f saturates and R f～Γf/2.

In alkali atoms,the cycling transition with circularly

polarized pump laser can be established between the ground and excited state sublevels|s =|F=2,m F= 2 and|f =|F=3,m F=3 .To estimate the relia-bility of the measurement,assuming unit probability of photon trapping in EIT medium,let as suppose that a photodetector with e?ciencyη?1is collecting the?u-orescent signal S f=ηR f t during time t.This time is limited by the lifetime of state|s ,Γ?1s,which is related to the Raman coherence relaxation rate byγR≥Γs/2.

A reliable measurement requires S f=1

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