Quantum dot single photon sources: Prospects for applications in linear optics quantum information processing

0.3 MB·English

by  A. Kiraz

Quantum dot single photon sources: Prospects for applications in linear optics
quantum information processing
A. Kiraz1, M. Atatu¨re2,3, and A. Imamog¯lu2,3,4
1 Department Chemie, Ludwig-Maximilians Universit¨at Mu¨nchen, Butenandtstr. 11, D-81377 Munich, Germany
2 Quantenelektronik, ETH-H¨onggerberg, HPT G12, CH-8093 Zurich, Switzerland
3 IV. Physikalisches Institut, Universitat Stuttgart,
Pfaffenwaldring 57, D-70550 Stuttgart, Germany
4 Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey
(Dated: February 1, 2008)
An optical source that produces single photon pulses on demand has potential applications in
linear optics quantum computation, provided that stringent requirements on indistinguishability
and collection efficiency of the generated photons are met. We show that these are conflicting
requirementsfor anharmonic emitters that are incoherently pumped via reservoirs. As a model for
4
0 a coherently pumped single photon source, we consider cavity-assisted spin-flip Raman transitions
0 in a single charged quantum dot embedded in a microcavity. We demonstrate that using such a
2 source, arbitrarily high collection efficiency and indistinguishability of the generated photons can
beobtained simultaneously with increased cavitycoupling. Weanalyzetherole of errors thatarise
n
fromdistinguishabilityofthesinglephotonpulsesinlinearopticsquantumgatesbyrelatingthegate
a
fidelitytothestrengthofthetwo-photoninterferencedipinphotoncross-correlationmeasurements.
J
We find that performing controlled phase operations with error < 1% requires nano-cavities with
6 PurcellfactorsFP ≥40intheabsenceofdephasing,withoutnecessitatingthestrongcouplinglimit.

2
PACSnumbers: 03.67.Lx,42.50.Dv,42.50.Ar
v
7
1 I. INTRODUCTION (a)
1 Beam Splitter
two-level emitter
8
0
Asignificantfractionofkeyexperimentsinthe emerg-
3 exciation laser
ing field of quantum information science [1], such as
0
Bell’s inequality violations [2], quantum key distribution
/ two-level emitter
h [3, 4] and quantum teleportation [5] have been carried
p
- out using single photon pulses and linear optical ele- (b) |E (+)(k ,r)> |E (+)(k ,r)>
t ments such as polarizers and beam splitters. However, 1 1 3 3
n
it was generally assumed that in the absence of photon-
a |E (+)(k ,r)> |E (+)(k ,r)>
u photon interactions, the role of optics could not be ex- 2 2 4 4
q tended beyond these rather limited applications. Re-
: cently, Knill, Laflamme, and Milburn have shown the- FIG. 1: (a) Configuration assumed in the analysis of two-
v
oretically that efficient linear optics quantum computa- photoninterference: Twoindependentidenticalsinglephoton
i
X tion (LOQC) can be implemented using on-demand in- sources excitedbythesame laser field. (b)Inputandoutput
r distinguishable single-photon pulses and high-efficiency fieldsof thebeam splitter.
a
photon-counters [6]. This unexpected result has initi-
ated a number of experimental efforts aimed at realizing
suitablesingle-photonsources. Impressiveresultsdemon-
withanarbitrarilyhighcollectionefficiencyandindistin-
strating a relatively high degree of indistinguishability
guishability. While our resultsapply to allsingle-photon
and collection efficiency have been obtained using a sin-
sources based on two-levelemitters, our focus will be on
gle quantum dot embedded in a microcavity [7]. Two-
quantum dot based devices. First, we show that single
photoninterferencehasalsobeenobservedusingasingle
photon sources that rely on incoherent excitation of a
coldatomtrapped ina high-Q Fabry-Perotcavity[8]. A
single quantum dot (througha reservoir)cannotprovide
necessarybutnotsufficientconditionforobtainingindis-
high collection efficiency and indistinguishability, simul-
tinguishablesinglephotonsondemandisthatthecavity-
taneously. To achieve this goal, the only reservoir that
emittercoherentcouplingstrength(g)exceedsthesquare
the emitter couplesto has to be the radiationfield reser-
root of the product of the cavity (κ ) and emitter (γ)
cav voirthatinducesthecavitydecay. Weshowthatasource
coherence decay rates. When the emitter is spontaneous
based on cavity-assisted spin-flip Raman transition sat-
emissionbroadenedandthe cavitydecaydominatesover
isfies this requirement and can be used to generate the
other rates, this requirement corresponds to the Purcell
requisitesingle-photonpulsesinthePurcellregime. This
regime (g2/κ γ >1).
cav analysis is done in section II where we calculate the de-
In this paper, we identify the necessary and suffi- gree of interference (indistinguishability) of two photons
cient conditions for generation of single photon pulses and the theoretical maximum collection efficiency, as a
2
functionofthecavitycouplingstrength,laserpulsewidth, flipRamanscattering,i.e. coherentlypumpedsinglepho-
and emitter dephasing rate for different single photon ton source.
sources. Previous analysis of two-photon interference among
Interference of two single-photon pulses on a beam- photons emitted from single emitters were carried out
splitterplaysacentralroleinallprotocolsforimplement- for two-level systems driven by a cw laser field [10, 11].
ingindeterministictwo-qubitgates,whichareinturnkey In contrast, we treat the pulsed excitation, and analyze
elements of linear optics quantum computation schemes currently available single photon sources based on two
[6]. Observability of two-photon interference effects nat- andthree-levelemitters. Wenotethatextensiveanalysis
urallyrequiresthatthetwosingle-photonsarrivingatthe of two-photon interference phenomenon was carried out
twoinputportsofthebeam-splitter be indistinguishable for twin photons generated by parametric down conver-
in terms of their pulsewidth, bandwidth, polarization, sion [9, 12, 13, 14], and single photon wave-packets [15],
carrier frequency, and arrival time at the beam-splitter. without considering the microscopic properties of the
Thefirsttwoconditionsaremetforanensembleofsingle- emitter.
photon pulses that are Fourier-transform limited: this
is the case if the source (single atom or quantum dot)
transition is broadened solely by spontaneous emission A. Calculation of the degree of two-photon
process that generates the photons. While the radiative interference
lifetime(i.e. thesingle-photonpulsewidth)oftheemitter
doesnotaffecttheobservabilityofinterference,anyother
Weconsidertheexperimentalconfigurationdepictedin
mechanism that allows one to distinguish the two pho-
Figure1(a). Twogeneralindependentidenticaltwo-level
tonswill. Asimpleexamplethatisrelevantforquantum
emitters are assumed to be excited by the same laser.
dot single photon sources is the uncertainty in photon
We assert no further assumptions on two-level emitters;
arrival(i.e. emission)time arisingfromtherandomexci-
they are considered to be light sources that exhibit per-
tationoftheexcitedstateoftheemittertransition: iffor
fect photon antibunching. Single photons emitted from
example this excited state is populated by spontaneous
the two-level emitters are coupled to different inputs of
phonon emission occuring with a waiting time of τ ,
relax a beam splitter which is equidistant from both sources.
then the starting time of the photon generation process
Inthe idealscenariowhere the input channelsaremode-
will have a corresponding time uncertainty of ∼ τ .
relax matched and the incoming photons have identical spec-
We refer to this uncertainty as time-jitter. Since the in-
tralandspatialdistributions,two-photoninterferencere-
formationaboutthephotonarrivaltimeisnowcarriedby
veals itself in lack of coincidence counts among the two
the phonon reservoir,the interference will be degraded.
output channels. This bunching behavior is a signature
Eventhoughtheroleofsingle-photonlossonlinearop- of the bosonic nature of photons.
tics quantum computation has been analyzed [6], there
Recent demonstration of two-photon interference us-
has been to date no analysis of gate errors arising from
ingasinglequantumdotsinglephotonsourcereliedona
distinguishability of single photons. To this end, we first
similar scheme based on a Michelson interferometer [7].
note that while various sources of distinguishability can
In this experiment, the interferometer had a large path
beeliminated,theinherentjitterinphotonemissiontime
length difference between its two branches. Such a dif-
remains as an unavoidable source of distinguishability.
ference, in excessof single photon coherencelength, pro-
Hence, in section III, we analyze the performance of
videdtheinterferenceamongphotonssubsequentlyemit-
a linear-optics-controlled phase gate in the presence of
ted from the same source. Two-photon interference in
time-jitterandrelatethegatefidelitytothedegreeofin-
this experiment is quantitatively similar to interference
distinguishability of the generatedphotons, as measured
obtainedamongphotonsemitted by twodifferentidenti-
by a Hong-Ou-Mandel [9] type two-photon interference
cal sources.
experiment.
Input-output relationships for single mode photon an-
nihilation operators in the beam splitter (Fig. 1(b)) are
defined by the unitary operation
II. MAXIMUM COLLECTION EFFICIENCY
AND INDISTINGUISHABILITY OF PHOTONS aˆ (ω) cosξ −e−iφsinξ aˆ (ω)
3 = 1 . (1)
GENERATED BY SINGLE PHOTON SOURCES (cid:20) aˆ4(ω)(cid:21) (cid:20) eiφsinξ cosξ (cid:21)(cid:20) aˆ2(ω)(cid:21)
In this section we first develop the general formalism aˆ (ω), aˆ (ω), aˆ (ω), and aˆ (ω) represent single mode
1 2 3 4
forcalculatinganormalizedmeasureoftwo-photoninter- photonannihilationoperatorsinchannelsk1,k2,k3,and
ference based on the projection operators of a two-level k4 respectively. k1, k2,k3, andk4 haveidenticalampli-
emitter. We then compare and contrast the case where tudes and polarizations while satisfying the momentum
theemitterispumpedviaspontaneousemissionofapho- conservation. We will abbreviate the unitary operation
ton or a phonon from an excited state, i.e. an incoher- in the beam splitter as u(B ).
ξ,φ
entlypumpedsinglephotonsource,tothecasewheresin- Assuming that u(B ) is constant over the frequency
ξ,φ
gle photon pulses are generated by cavity-assisted spin- range of consideration, Eq. (1) can be Fourier trans-
3

formed to reveal 80
-2 -1 0 1 2
aˆ (t) aˆ (t) 70
3 =u(B ) 1 . (2)
(cid:20)aˆ4(t) (cid:21) ξ,φ (cid:20) aˆ2(t)(cid:21) 60
aˆ1(t), aˆ2(t), aˆ3(t), and aˆ4(t) now represent time depen- 50
dent photon annihilation operators. )
areCqoiunacnidtiefinecdebeyvetnhtes acrtotshs-ecoorurteplauttioonf tfhuencbteioanmbseptlwiteteenr ~(2)tG(34 3400
channels 3 and 4 which is given by
20
G(2)(t,τ) = haˆ†(t)aˆ†(t+τ)aˆ (t+τ)aˆ (t)i, (3)
34 3 4 4 3 10
g(2)(t,τ) = G(324)(t,τ) , (4) 0 -30 -20 -10 0 10 20 30
34 haˆ†(t)aˆ (t)ihaˆ†(t+τ)aˆ (t+τ)i t (1/G )
3 3 4 4 spon
in its unnormalized (G(324)(t,τ)) and normalized FIG. 2: Unnormalized coincidence detection rate, G(324)exp,
(g(2)(t,τ)) form. By substitution of Eq. (2) in (3), of an incoherently pumped quantum dot. Parameter values
34 e
G(2)(t,τ) is expressed as are: Γrelax = 100Γspon, γdeph = Γspon, each laser pulse is
34 a Gaussian with pulsewidth 0.05/Γspon and peak Rabi fre-
G(2)(t,τ) = sin4ξhaˆ†(t)aˆ†(t+τ)aˆ (t+τ)aˆ (t)i quency35Γspon.
34 2 1 1 2
+ cos4ξhaˆ†(t)aˆ†(t+τ)aˆ (t+τ)aˆ (t)i
1 2 2 1
− cos2ξsin2ξ haˆ†(t)aˆ†(t+τ)aˆ (t+τ)aˆ (t)i This is the expression of the unnormalized second order
1 2 1 2
(cid:16) coherence function in terms of the dipole projection op-
+ haˆ†(t)aˆ†(t+τ)aˆ (t+τ)aˆ (t)i . (5) erators that we will use in the remaining of this section.
2 1 2 1
(cid:17) Underpulsedexcitationfurtherconsiderationsneedto
Inwhatfollowsweassumeidealmode-matchedbeamsin be takeninto accountto normalizethis equation. Before
inputs 1 and 2. Hence the bracket notation corresponds this discussion however, we note that under continuous
to time expectations only. wave excitation, Eq. (4) reveals the normalized second
In Eq. (5), photon annihilation operators of channels order coherence function
1 and 2 are due to the radiation field of a general single
2
two-level emitter. In the far field, this field annihilation G(1)(t,τ)
1
operator is given by the source-field relationship as g(2)(t,τ) = 1− (cid:12) (cid:12) 
34 2 (cid:12)(cid:12)heσˆee(t)i2ss(cid:12)(cid:12)
|r|  
aˆ(t)=A(r)σˆ t− , (6)  
ge(cid:18) c (cid:19) = 1 1− g(1)(τ) 2 , (10)
where A(r) is a time-independent proportionality fac- 2(cid:18) (cid:12)(cid:12) (cid:12)(cid:12) (cid:19)
(cid:12) (cid:12)
tor [16]. This linear relationship allows for substitution where hσˆ (t)i represents the steady state population
ee ss
of photon annihilation and creation operators by dipole density of the excited state.
projection operators σˆge and σˆeg respectively in Eq. (5). Experimental determination of the cross-correlation
Using the assumption that both of the emitters are in- function relies on ensemble averaging coincidence detec-
dependent and have identical expectation values and co- tion events. Hanbury Brown and Twiss setup is fre-
herence functions, we arrive at quently used in these experiments where the experimen-
tally relevant cross-correlationfunction
G(2)(t,τ) = cos4ξ+sin4ξ hσˆ (t)ihσˆ (t+τ)i
34 ee ee
−(cid:2)(cid:0)2cos2ξsin2ξ|G(cid:1)(1)(t,τ)|2] |A(r)|4 . (7) G(2) (τ)= lim T G(2)(t,τ)dt, (11)
34exp T→∞Z0 34
In this equation G(1)(t,τ) repreesents the unnormalized e e
is measured. The total detection time T is long com-
first-order coherence function
e paredto the single photon pulsewidth (T →∞) in these
G(1)(t,τ)=hσˆ (t+τ)σˆ (t)i. (8) experiments.
eg ge
(2)
InFig.2weplotanexemplarycalculationofG (τ)
For a balanceed beam-splitter, θ =π/4, Eq. (7) simplifies 34exp
foranincoherentlypumped, dephasedquantumdotcon-
to e
sidering a series of 6 pulses. This calculation is done by
G(2)(t,τ) ≡ G(324)(t,τ) the integration of G(324)(t,τ) (Eq. (11)), while G(324)(t,τ)
34 |A(r)|4 is calculated using the optical Bloch equations and the
e e
e 1 quantum regressiontheorem. We will detail these calcu-
= hσˆee(t)ihσˆee(t+τ)i−|G(1)(t,τ)|2 (.9) lationsinthefollowingsubsections. Insuchcalculations,
2
(cid:16) (cid:17)
e
4
the area of the peak around τ ∼ 0 (0th peak) gives the | p >
unnormalizedcoincidencedetectionprobabilitywhentwo Γ
photonsareincidentindifferentinputsofthebeamsplit- relax
| e >
ter. This area should be normalized by the area of the
Ω (t)
other peaks: Absence of two-photon interference implies L
0th peak and other peaks to be identical, whereas in to- Γspon Fp. Γspon
taltwo-photoninterference,0th peak hasvanishing area.
This normalized measure of two-photon interference is | g >
∞ G(2)(t,τ)dtdτ
t=0 τ,0 34 FIG. 3: Model of an incoherently pumped single quantum
p = . (12)
34 R∞ R Ge(2)(t,τ)dtdτ dot. Dashed line demonstrates the generated single photons
t=0 τ,n 34 via cavity leakage.
R R
Inthenumerator,integralinτeistakenoverthe0th peak,
whereasinthedenominatorthisintegralistakenoverthe
nth peak where n=±1,±2,.... B. Single photon source based on an incoherently
We nowsimplify Eq.(12)further usingthe periodicity pumped quantum dot
with respect to τ and t. First simplification is due to
periodicity in τ which is apparent in the periodicity of Variousdemonstrationsofsinglephotonsourcesbased
the peaks other than 0th peak in Fig. 2. The area of on solid-state emitters have been reported in recent
these peaks is given by years. Single quantum dots [17, 18, 19, 20, 21], single
∞ molecules[22,23,24],andsingleNvacancies[25,26]were
hσˆee(t)ihσˆee(t+τ −nTpulse)idt, (13) used in these demonstrations where pulsed excitation of
Z
0 a high energy state followed by a fast relaxation and
for n=±1,±2,... This is due to the vanishing G(1)(t,τ) excited state recombination proved to be a very conve-
for absolute delay times largerthan single photon coher- nient method to generate triggered single photons. This
e
ence time. Hence the normalizedcoincidenceprobability method of incoherent pumping ensured the detection of
can also be represented as at most one photon per pulse, provided that the laser
hadsufficientlyshortpulses,andlargepulseseparations.
∞ G(2)(t,τ)dtdτ
t=0 τ,0 34 In the following, we extensively consider the case
p = . (14)
34 t∞=0 Rτ,0hσˆRee(te)ihσˆee(t+τ)idtdτ of quantum dots and analyze two-photon interference
among photons emitted from an incoherently pumped
R R
Periodicity of G(2)(t,τ) and hσˆ (t)ihσˆ (t +τ)i in t quantum dot. In such a three-level scheme (Fig. 3),
34 ee ee
further simplifies Eq. (14) to time-jitterinducedbythefastrelaxation(Γrelax)andde-
e
phasing in |ei-|gi transition are the sources of non-ideal
p = N tT=p0ulse τ,0G(324)(t,τ)dtdτ two-photoninterference. Weinvestigatetheseeffectsfirst
34 N TpulRse hσˆR (te)ihσˆ (t+τ)idtdτ under continuous wave, then under pulsed excitation.
t=0 τ,0 ee ee
R R
Tpulse G(2)(t,τ)dtdτ
= t=0 τ,0 34 , (15) 1. Continuous wave excitation
TpulRse hσˆR (te)ihσˆ (t+τ)idtdτ
t=0 τ,0 ee ee
where N reprResents Rthe number of pulses considered in Under continuous wave excitation, G(1)(t,τ) is calcu-
the calculation. latedbyapplyingquantumregressiontheorem[16]tothe
e
Eq. (15) is the final result of the simplifications and is optical Bloch equation for hσˆeg(t)i, revealing
used in the rest of this section. It is important to note
that this equation enables us to obtain the normalized dG(1)(t,τ) =−γG(1)(t,τ), (16)
coincidence probability, p , by considering only a single dτ
34 e
laser pulse. This greatly improves the efficiency of the e
whereγ = Γspon +γ isthe totalcoherencedecayrate
simulations. 2 deph
of|ei-|gitransition. Hereγ denotesdephasingcaused
Therearetwolimitationsofourmethodofcalculation. deph
by all reservoirs other than that of the radiation field.
Firstly, the optical Bloch equation description does not
FollowingthesolutionofEq.(16),usingtheinitialcon-
takeintoaccountlaserbroadeninginducedbyamplitude
orphasefluctuations. Secondly,inthecaseofaquantum dition G(1)(t,0)=hσˆee(t)iss, the normalized coincidence
dot, an upper limit to laser broadening may arise due to detection probability is obtained by Eq. (10) as
e
the biexciton splitting (∼3.5 meV at cryogenic temper-
1
atures) and Zeeman splitting (∼ 1 meV for an applied g(2)(τ) = 1−e−2γτ . (17)
34 2
field of 10 Tesla). Overall, these restrictions should put (cid:0) (cid:1)
a lower limit of ∼ 1×10−12 s to the laser pulsewidth. Hence, for the continuous wave excitation case, indistin-
This lower limit is always exceeded in our calculations. guishability is solely determined by the total coherence
5

1.0 1.0
0.9
0.8
0.5
0.7

0.6
1-p 1-p
34
0.5 b b 34
0.0
1 10 100 0.1 1 10 100
F (2g2/(k G )) g (G )
P cav spon deph spon
FIG. 4: Dependence of indistinguishability and collection ef- FIG. 5: Dependenceof indistinguishability and collection ef-
ficiency on the cavity-induced decay rate ((FP +1)Γspon) of ficiency on dephasing (γdeph). Γspon = 109 s−1, FP = 9,
a quantum dot. Parameter values are: Γspon = 109 s−1, Γrelax = 1011 s−1, excitation laser is a Gaussian beam
Γrelax = 1011 s−1, γdeph = 0, excitation laser is a Gaussian with a pulsewidth of 10−11 s. Peak laser Rabi frequency is
beamwithapulsewidthof10−11s. PeaklaserRabifrequency 1.03×1011 s−1 achieving π-pulseexcitation.
is changed between 1.1×1011 and 0.93×1011 s−1.
decay rate in |ei-|gi transition. Decay time of the nor-
malized coincidence detection probability is 1/2γ. Thisassumptionclearlyconstitutesanupperlimitforthe
actual collection efficiency for typical microcavities [27].
2. Pulsed excitation Fig.4depictsoneofthemainresultswepresentinthis
paper. Due to the time-jitter induced by the relaxation
fromthethirdlevel,thereisatrade-offbetweencollection
A more detailed study of Bloch equations is necessary
efficiencyandindistuingishability. ForaPurcellfactorof
for the case of pulsed excitation. The interaction Hamil-
100wecalculateamaximumindistuingishabilityof44%
tonian of the system depicted in Fig. 3 is
with a collection efficiency of 99 %.
Hˆ =i~Ω (σˆ −σˆ ). (18)
int L pg gp
Thedependenceofindistinguishabilityondephasingis
The master equation depicted in Fig. 5. As expected, dephasing has no effect
on the collection efficiency. On the other hand, indistin-
d 1 Γ guishability vanishes for γ >Γ . Since dephasing
ρˆ = Hˆ ,ρˆ + relax (2σˆ ρˆσˆ −σˆ ρˆ−ρˆσˆ ) deph spon
dt i~ int 2 gp pg pp pp iseffectivelyanon-referredquantumstatemeasurement,
h i
Γ it results in additional jitter in photon emission time.
spon
+ (2σˆ ρˆσˆ −σˆ ρˆ−ρˆσˆ ) , (19)
ge eg ee ee
2
Tounderstandthiseffect,weshouldrecallthatdephas-
is used to derive the optical Bloch equations. As de- ingofanopticaltransitionisequivalenttoanon-referred
scribed previously,calculationof p follows the solution quantum state measurement that projects the emitter
34
of the optical Bloch equations and Eq. (16) considering into either its excited or ground state. Reciprocal de-
a single laser pulse. phasingrateγ−1 thengivestheaveragetimeintervalbe-
deph
We now study the dependence of indistinguishability, tween these state projections. In this case, photon emis-
(1−p ),onthecavity-induceddecayrate((F +1)Γ ) sionisrestrictedtotakeplaceinbetweentwosubsequent
34 P spon
anddephasing. InFig.4,weplotthecollectionefficiency measurement events, first (second) of which projects the
and indistinguishability as a function of the Purcell fac- emitter into the excited(ground)state. While the band-
tor, F , for a quantum dot with γ = 0. We as- width of the emitted photon is then necessarily givenby
P deph
sume Γ = 109 s−1 and Γ = 1011 s−1. Peak γ due to energy-time uncertainty, its emission (i.e.
spon relax deph
laser Rabi frequency is changed between 1.1×1011 and arrival) time will be randomly distributed within Γ−1 .
spon
0.93×1011 s−1 in order to achieve π-pulse excitation for Since the information about the random emission times
different Purcell factors. Collection efficiency is calcu- ofanytwophotonsiscarriedbythereservoirthatcauses
lated by β =F /(F +1), assuming that photons emit- thedephasingprocess,thephotonswillnolongerbecom-
P P
tedtothecavitymodearecollectedwith100%efficiency. pletely indistinguishable.
6
| 3 > (or |m =−3/2i) are strongly coupled via a resonant y-
z
polarized cavity mode. Considering the number of cav-
g
ity photons to be limited to 0 and 1, the electronic en-
Ω (t)
L Γspon cos2 θ | 2,1 > | 2 > e|mrgxy=lev−el1|/m2,x1=i a−n1d/|2mi,xca=n−be1/r2e,p0riesceonrtreedspboyntdhinegletvoel1s
and0 cavityphoton respectively. We will abbreviatethe
| 1 > κ energy levels |m = 1/2i, |m = 3/2i, |m = −1/2,1i,
x z x
cav
Γ sin2 θ and |mx =−1/2,0ias |1i, |2i, |3i, and |4i respectively.
spon
| 2,0 > | 4 >
Insuchathree-levelsystem,Ramantransitioninduced
bythelaserandcavityfieldstogetherwiththefinitecav-
FIG. 6: Single photon source based on cavity-assisted spin-
flip Raman transition in a single quantum dot. Dashed line ity leakage rate, κcav, enable the generation of a single
demonstratesthegeneratedsinglephotonsviacavityleakage. cavity photon per pulse. For large field couplings, level
|3i can be totally bypassed resulting in ideal coherent
population transfer between levels |1i and |2i. This sin-
gle photon source has therefore the potential to achieve
C. Quantum dot single photon source based on a 100%collectionefficiencytogetherwithidealtwo-photon
cavity-assisted spin-flip Raman transition
interference. This scheme is to a large extent insensitive
toquantumdotsizefluctuations andmayenablethe use
Raman transition in a single three-level system of different quantum dots in simultaneous generation of
strongly coupled to a high-Q cavity provides an alter- indistinguishable photons, provided that the cavity res-
native single photon generation scheme [28, 29, 30]. In onances and the electron g-factors are identical. Varia-
contrast to the incoherently pumped source discussed in tions in the electron g-factor between different quantum
subsectionIIB,thisschemerealizesacoherentlypumped dots would limit the photon indistinguishability due to
single photon source that does not involve coupling to spectral mismatch between the generated photons: We
reservoirs other than the one into which single photons do not consider this potential limitation in this paper.
are emitted. It allows for pulse-shaping, and is suitable In general, spontaneous emission and dephasing in |3i-
for quantum state transfer [31]. In this part we discuss |2i transition are the principal sources of non-ideal two-
the application of this scheme to quantum dots, and photon interference and decreased collection efficiency
demonstrate that arbitrarily high collection efficiency in this scheme. The ultimate limit for photon indistin-
and indistinguishability can simultaneously be achieved guishabilityduetojitterinemissiontimeisgivenbyspin
in this scheme. decoherence of the ground state.
A quantum dot doped with a single conduction-
band electron constitutes a three-level system in the Λ- Such a single photon source has been recently demon-
configuration under constant magnetic fields along x- strated using single cold atoms trapped in a high-Q
direction(Fig.6)[32]. Lowestenergyconductionandva- Fabry-Perot cavity [34]. Due to the limited trapping
lence band states of such a quantum dot are represented times, at most only 7 photons were emitted by a sin-
by |m = ±1/2i and |m = ±3/2i respectively due to gle atom in this demonstration. Practical realizations
x z
the strong z-axis confinement, typical of quantum dots. of this scheme also require a means to bring the system
The magnetic field results in the Zeeman splitting of the from level |4i to |1i at the end of each single photon
spin up (|m =1/2i) and down (|m = −1/2i) levels in generation event. In Ref. [34] this was achieved by a
x x
the conduction band. Considering an electron g-factor recycling laser pulse. The applied recycling laser pulse
of 2 and an applied field of 10 Tesla which is available determines the endofthe single-photonpulse andcanin
from typical magneto-optical cryostats, the splitting is principle limit the collection efficiency for systems with
expected to be ∼1 meV.At cryogenictemperatures,this longspontaneousemissionlifetimes. Inthecaseofquan-
splittingismuchlargerthanotherbroadeningsinconsid- tum dots, recycling can be achieved by a similar laser
eration, thus a three-level system in the Λ-configuration pulse applied between levels |4i and |3i. An alternative
is obtained. We emphasize that none of the experimen- recycling mechanism can be the application of a Raman
talmeasurementscarriedoutonself-assembledquantum π-pulse,generatedbytwodetunedlaserpulsessatisfying
dots yield any signatures of Auger recombination pro- the Raman resonance condition between levels |4i and
cesses for trion(2 electronand one hole system) or biex- |1i.
citon transitions. In particular, lifetime measurements
carried out on biexcitons gave τ ∼ τ /1.5, indi- We now discuss the numerical analysis of this system
biexc exc
cating the absence of Auger enhancement of biexciton which is described by the interaction Hamiltonian
decay [33].
We assume that an x-polarized laser pulse is applied
resonantlybetweenlevels|m =1/2iand|m =3/2i(or
x z
|m = −3/2i) while levels |m =−1/2i and |m = 3/2i Hˆ =i~g(σˆ −σˆ )+i~Ω (σˆ −σˆ ). (20)
z x z int 32 23 L 31 13
7

We use the master equation 1.00 1-p
34
d 1 0.95 b
ρˆ = Hˆ ,ρˆ +κ (2σˆ ρˆσˆ −σˆ ρˆ−ρˆσˆ ) 2F /(1+2F )
dt i~h int i cav 42 24 22 22 0.90 P P
Γ cos2θ
spon
+ 2 (2σˆ13ρˆσˆ31−σˆ33ρˆ−ρˆσˆ33) 0.85
Γsponsin2θ 0.80 0.980 1b -p34
+ (2σˆ ρˆσˆ −σˆ ρˆ−ρˆσˆ ) , (21)
43 34 33 33
2 0.975
0.75
to derive the optical Bloch equations. In the presence of 0.970
dephasing caused by reservoirs other than the radiation 0.70 k 20 (G 40)
field (γ ), we define the total coherence decay rate in cav spon
deph 0.65
transitionsfromlevel|3iasγ = Γspon+γ . Branching
2 deph 1 10 100
ofspontaneousemissionfromlevel|3itolevels|1iand|4i
g (F )
is indicated by cos2θ and sin2θ, respectively, as shown P
in Fig. 6.
FIG. 7: Dependenceof indistinguishability and collection ef-
G(1)(t,τ) = hσˆ24(t+τ)σˆ42(t)i is calculated by apply- ficiencyoncavitycoupling. Parametervaluesare: Γspon=1,
ingethe quantumregressiontheoremto the opticalBloch κcav = 10, γdeph = 0, θ = π/4, a Gaussian pulse with
equations for σˆ14, σˆ24, and σˆ34. The following set of dif- pulsewidth=10,peaklaserRabifrequencyischangedbetween
ferential equations are then obtained 0.75 and 2.8 . Inset: Dependence of indistinguishability and
collection efficiencyonκcav foraconstantFP of20. Parame-
d F(t,τ) = −Ω (t)H(t,τ), tervaluesare: Γspon=1,γdeph=0,θ=π/4,laserpulsewidth
dτ L of 10/Γspon,peak laser Rabifrequency of 1.9 - 2.1 .
d
G(1)(t,τ) = −gH(t,τ)−κ G(1)(t,τ), (22)
cav
dτ
de e
H(t,τ) = Ω (t)F(t,τ)+gG(1)(t,τ)−γH(t,τ).
L
dτ In contrast to the incoherently pumped single photon
e source,Fig. 7 shows that arbitrarilyhigh indistinguisha-
The variables: G(1)(t,τ) = hσˆ (t+τ)σˆ (t)i, F(t,τ) =
24 42 bility and collection efficiency can simultaneously be
hσˆ (t+τ)σˆ (t)i, and H(t,τ)=hσˆ (t+τ)σˆ (t)i have
14 42 e 34 42 achieved with better cavity coupling using this scheme.
initialconditionsG(1)(t,0)=hσˆ (t)i, F(t,0)=hσˆ (t)i,
22 12 For a cavity coupling that corresponds to a Purcell fac-
and H(t,0)=hσˆ (t)i.
32e tor of 40 (FP = 2g2/(κcavΓspon) = 40), our calcula-
Following the solutions of the optical Bloch equations
tionsreveal99%indistinguishabilitytogetherwith99%
andthesetofEqs.(22),normalizedcoincidencedetection
collection efficiency. This regime of operation is read-
probability,p ,is calculatedusingEq.(15)asdescribed
34 ily available in current state-of-the-art experiments with
in section IIA. Assuming ideal detection of the photons
atoms [36]. While sucha Purcellfactor has not been ob-
emitted to the cavity mode, we calculate the collection
served for solid-state emitters in microcavity structures
efficiency by the number of photons emitted from the
to date, recent theoretical [37] and experimental [35, 38]
cavity
progress indicate that the aforementioned values could
∞ be well within reach.
n=2κcav hσˆ22(t)idt. (23) As expected, the dependence of β on cavity coupling
Z
0 is exactly given by 2F /(1 + 2F ). This is due to
P P
Our principal numerical results are depicted in Fig. 7 the spontaneous emission from level |3i to |4i, namely
where we consider a dephasing-free system, and ana- Γ sin2θ =Γ /2,whichdefinestherelevantPurcell
spon spon
lyze the dependence of the collection efficiency and in- factor. As shown in the inset in Fig. 7, our calculations
distinguishability on the cavity coupling. In these cal- considering different κ values for a constant Purcell
cav
culations we assume a potential quantum dot cavity- factor revealed similar collection efficiency and indistin-
QED system with relatively small cavity decay rate of guishablity values. Hence Purcell factor is the most im-
κ = 10Γ [35]. Laser pulse is chosen to be Gaus- portant parameter in determining the characteristics of
cav spon
sian with a constant pulsewidth. The peak laser Rabi this single photon source.
frequency is increased with increased cavity coupling in Achieving the regime of large indistinguishability and
ordertoreachtheonsetofsaturationintheemittednum- collectionefficiencytogetherwithsmalllaserpulsewidths
ber of photons. The large pulsewidth of 10 ensures the is alsohighly desirable for efficient quantum information
operation in the regime where collection efficiency and processingapplications. Inthissinglephotonsourcethat
indistinguishability are independent of the pulsewidth. relies on cavity-assisted Raman transition, lower limits
All other parameters are kept constant at their values forthelaserpulsewidthareingeneralgivenbytheinverse
notedinthefigurecaption. Wechoosebothspontaneous cavity coupling constant (g−1) and cavity decay rate
emission channels to be equally present (θ =π/4). (κ−1) [30]. We analyzethe effectof the laser pulsewidth
cav
8

1.00 1.00
0.99 0.99
0.98 0.98

0.97 0.97
0.96 1-p34 0.96
b 1-p
34
b
0.95 0.95
1 10 0 10 20 30 40 50 60 70 80 90
Pulsewidth (1/G ) q (deg)
spon
FIG. 8: Dependence of indistinguishability and collection ef- FIG. 9: Dependenceof indistinguishability and collection ef-
ficiency on the Gaussian laser pulsewidth. Parameter values ficiency on θ. Parameter values are: Γspon = 1, g = 10,
are: Γspon = 1, g = 10, κcav = 10 (FP = 20), γdeph = 0, κcav = 10 (FP =20), γdeph = 0, θ = π/4. A Gaussian pulse
θ = π/4. Peak laser Rabi frequency is changed between 2.1 is assumed with pulsewidth=1 and peak Rabi frequency of
and 10.5 . 6.2 .

1.00
0.95
to indistinguishability and collection efficiency in Fig. 8.
In this figure we consider the potential quantum dot 0.90
cavity-QED system analyzed in Fig. 7 (κ = 10Γ )
cav spon
while assuming a Purcell factor of 20 (g = 10Γ ). As
spon 0.85
inthepreviouscases,wechangethemaximumlaserRabi
frequencyfordifferentpulsewidthvaluesinordertoreach
theonsetofsaturation. Forthissystem,weconcludethat 0.80 1-p
34
a minimum pulsewidth of 1/Γspon is sufficient to achieve b
maximum indistinguishability and collection efficiency. 0.75
0.0 0.5 1.0 1.5
g (G )
The two spontaneous emission channels from level |3i deph spon
have complementary effects on collection efficiency and
FIG.10: Dependenceofindistinguishabilityandcollectionef-
indistinguishability. Spontaneous emission from level |3i
to |1i reduces indistinguishability while having no effect ficiencyonthedephasingrate. Parametervaluesare: Γspon=
1, g = 10, κcav = 10 (FP = 20), θ = π/4, a Gaussian laser
oncollectionefficiency. Thisspontaneousemissionchan-
pulseisassumedwithpulsewidth=1andpeakRabifrequency
nel,Γ cos2θ,effectivelyrepresentsatime-jittermech-
spon of 6.2 .
anism for single photon generation. In contrast, sponta-
neous emissionfrom level|3i to level |4i has no effect on
indistinguishability while reducing collection efficiency.
These effects are clearly demonstrated in Fig. 9 where
we plotthe dependence ofcollectionefficiency andindis- III. INDISTINGUISHABILITY AND
tinguishability on θ. NONDETERMINISTIC LINEAR-OPTICS GATES
Finally in Fig. 10 we analyze the dependence of in- Having determined the limits and dependence of pho-
distiguishabilityandcollectionefficiencyondephasingof ton collection efficiency and indistinguishability on sys-
transitions from level |3i. In contrast to the case of an tem configuration and cavity parameters, we turn to
incoherently pumped quantum dot (Fig. 5), there is a the issue of photon distinguishability effects on the per-
small but non-zero dependence of collection effciency on formance of LOQC gates. Related question of depen-
dephasing. For the parameters we chose, collection effi- dence on photon loss [6, 39] and detection inefficiency
ciencies of0.975and0.970were calculatedfor dephasing [40] have previously been analyzed. For semiconductor
rates of 0 and 1.5Γ respectively. single photon sources, photon loss can be minimized by
spon
9
1 p ppp q qqq q qqq 1 number resolving single photon detectors. We now pro-
1 3
ceed to investigate the effects of photon distinguishabil-
2 p ppp 2 ity arising from physicalconstraints of the single photon
sources in consideration. In the presence of a temporal
jitter, ǫ, in the photon emission time, a single photon
state can be represented as
1 n 1
1 q qqq q qqq n 1
2 4
|1i= dωf(ω)eiωǫa†(ω)|0i, (26)
Z j
FIG. 11: Optical network realizing CS180◦.
where f(ω) is the spectrum of the photon wave-packet.
Forphotonsfromaquantumdotinacavity,thefunction
f(ω) is a Lorentzian yielding a double-sided exponential
dip in the Hong-Ou-Mandelinterference [9]. In the pres-
increasingcollectionefficiency,inprinciple,tonearunity
ence of relative time jitter, the visibility of interference
value. Therefore, close to ideal photon emission can be
is obtained after ensemble averaging over the time-jitter
achieved with better cavity designs and coupling. How-
ǫ in the range [0,ǫ ] yielding the relation
ever, as we have shown in previous sections, an incoher- 0
entlypumpedsemiconductorphotonsourcesuffersheav-
ily from emission time-jitter, especially for large values V(ǫ )= 1 (1−e−ǫ0/τ), (27)
0
ofPurcellfactor,while asemiconductorsystembasedon ǫ /τ
0
cavity-assistedspin-flipRamantransitionshowspromise
for a uniform distribution. In order to analyze time-
for near unity collection efficiency and indistinguishabil-
jittereffectsonthefidelityofthequantumgateshownin
ity. To assess the cavity requirements for the latter sys-
Fig. 11, we introduce a time-jitter for the helper photon
tem,weanalyzethereductioningatefidelityarisingfrom
inmode4. Forclarity,wekeeptheremainingphotonsin
photon emission time-jitter in a linear optics controlled
other modes ideal and indistinguishable. The symmetry
phase gate, a key element for most quantum gate con-
of the gate ensures that eachintroducedtime-jitter adds
structions.
to the power dependence of the overall error.
Thisnon-deterministicgateoperatesasfollows: Given
Rewriting Eq. (26) as
a two-mode input state of the form
|1i= dωf(ω)[1−(1−eiωǫ)]c†(ω)|0i, (28)
|Ψini= α|00i+β|01i+δ|10i+γ|11i , (24) Z j
(cid:2) (cid:3)
where |α|2+|β|2+|γ|2+|δ|2 = 1, the state at the two allows us to represent the output state in terms of the
output modes transforms into ideal output state and the time-jitter dependent part
|Φ(ǫ)i:
|Ψ i= α|00i+β|01i+δ|10i+eiΦγ|11i , (25)
out |Ψ i=|Ψ i−|Φ(ǫ)i. (29)
out out
(cid:2) (cid:3)
with a certain probability of success, |p|2. A realization Using the definition of the gate fidelity for a particular
ofsuchagateusingalllinearopticalelements,twohelper |Ψ i
out
single photons on demand, and two photon-number re-
solving single-photon detectors is depicted in Fig. 11 for
the special case of Φ = π [41, 42]. This realization con- |hΨout|Ψouti|2
F = , (30)
sists oftwo input modes forthe incoming quantum state |Ψouti hΨ |Ψ ihΨ |Ψ i
out out out out
to be transformed and two ancilla modes with a single
with Eq. 29 we obtain
helper photon in each mode. After four beam splitters
with settings θ = θ = −θ = 54.74◦ and θ = 17.63◦,
1 2 3 4
postselection is performed via photon-number measure- |p|2−2Re[hΨ |Φ(ǫ)i]+ |hΨout|Φ(ǫ)i|2
ments on output modes 3 and 4. Conditional to single- F = out |p|2 , (31)
photon detection in each of these modes, the quantum |Ψouti |p|2−2Re[hΨout|Φ(ǫ)i]+hΦ(ǫ)|Φ(ǫ)i
stateinEq.(24)istransformedintoEq. (25). Theprob-
where |p|2 = hΨ |Ψ i. Given the particular realiza-
ability of success for this construction is 2/27, which is out out
tion of this gate as depicted in Fig. 11, the overall gate
slightlybetterthen1/16,theprobabilityofsuccessofthe
fidelity takes the form
original proposal using only one helper photon with two
ancilla modes [6].
Thisistheprobabilityofsuccessforidealsystemscom- F =min F ǫ0 = c0+c1V(ǫ0)+c2V2(ǫ0) ,(32)
prising indistinguishable photons, and unity efficiency |Ψouti ǫ d +d V(ǫ )+d V2(ǫ )
h(cid:10) (cid:11) i 0 1 0 2 0
10
and collection efficiency are both shown to increase in
1.00
Fig.7asthePurcellfactorincreases. This,inturn,casts
0.98
a single constraintonthe cavityquality factor,requiring
0.96
F ≥ 40, in order to achieve both indistinguishability
P
y 0.94 and collection efficiency required for gate operations for
delit 0.92 LOQC. This threshold for cavity quality factor is within
Fi
e 0.90 the realistic values to date. We emphasize that so far
at
G there has been no calculation on the maximum allowed
0.88
time-jitter error for LOQC scheme [6].
0.86
IV. CONCLUSIONS
0.84
0.82
We analyzed the effects of cavity coupling, sponta-
0.80
neous emission rate, dephasing, and laser pulsewidth on
1E-3 0.01 0.1
Time Jitter e(eee /t ttt) indistinguishability and collection efficiency for two dis-
0
tinct types of single photon sources based on two and
FIG. 12: Dependence of nonlinear sign gate fidelity F, on three-level emitters. We showed that, in contrast to in-
normalized time-jitter ǫ. The horizontal line indicates the coherentlypumpedsystems,asinglephotonsourcebased
99% fidelity threshold. The vertical line indicates tolerable oncavity-assistedspin-flipRamantransitionhas the po-
time-jitter threshold. tential to simultaneously achieve high levels of indistin-
guishability and collection efficiency. For this system,
in the absence of dephasing, 99 % indistinguishability
and collection efficiency are achieved for a Purcell fac-
tor of 40. Our analysis revealed that strong coupling
where h·iǫ0 denotes ensemble averagingovertime-jitter ǫ
ǫ regime of cavity-QED (g > {γ,κ }) is not a require-
in the range [0, ǫ ] using an appropriate weight function cav
0
mentforoptimumoperationwhile,inthepresenceofde-
andV(ǫ )isthedegreeofindistinguishability,orthecor-
0
phasing, the characteristics of the system is determined
responding visibility in a Hong-Ou-Mandel interference.
by g2/κ γ rather than the Purcell factor. The de-
The coefficients c and d in Eq. (32) depend not only cav deph
i i
sired regime of operation, i.e. Purcell factor of 40 in the
on the gate properties suchas the probabilityof success,
absence of dephasing, is readily available for atoms in
butalsoontheinitialinputstatethroughthecoefficients
high-Q Fabry-Perot cavities. It is also within the reach
α, β, and γ. Consequently, the gate fidelity becomes a
for solid-state based single photon sources embedded in
function of the properties of the initial input state.
microcavitystructuresgivencurrenttechnology. Wealso
A plot for minimum gate fidelity (corresponding to a
analyzed the reduction in gate fidelity arising from pho-
|11i input state) found after extensive search over a set
ton emission-time-jitter in a linear optics controlled sign
of initial input states is shown in Fig. 12 as a function
gate. We found that the aforementioned Purcell regime
of time-jitter normalized to photon pulsewidth (ǫ /τ).
0
providesgateperformancewitherror< 1%usingthesin-
As is evident from the graph, time-jitter on the order of
gle photon source based on cavity-assisted Raman tran-
0.3% is the limiting case in order to achieve fidelity of
sition.
99%. For an incoherently pumped quantum dot single
photon source as analyzed in section IIB, the emission
time-jitter isonthe orderof1×10−11 s. Thus,forsingle
photonpulsewidthontheorderof1×10−9s,thisfidelity Acknowledgments
thresholdcannotbesatisfied. AsisalsoclearfromFig.4,
this corresponds to a Purcell factor of order unity and We acknowledge support from the Alexander von
collection efficiency of about 50%. In a cavity-assisted HumboldtFoundation,andthankG.GiedkeandE.Knill
spin-flip Raman transition, however, indistinguishability for useful discussions.
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