Single photon absorption by a single quantum emitter

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by  D. Pinotsi

Single photon absorption by a single quantum emitter
D. Pinotsi and A. Imamog˜lu
Institute of Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
(Dated: February 1, 2008)
We show that a three-level lambda quantum emitter with equal spontaneous emission rates on
both optically active transitions can absorb an incident single photon pulse with a probability
approachingunity,providedthatthefocusedlightprofilematchesthatoftheemitterdipoleemission
pattern. Evenwithrealisticfocusinggeometries, ourresultscouldfindapplicationsinlong-distance
entanglement of spin qubits.
8 Optically thick systems such as an ensemble of atoms ing van Enk [4], we introduce a parameter η to describe
0 or a single atom in a cavity play a key role in quantum theoverlapbetweentheincomingsingle-photonfieldand
0 optics. Various applications of those schemes have been the dipole emission profile. In this formalism, the pos-
2 proposed, such as nonlinear optics using slow light [1] itive frequency component of the electric field operator

n and flying qubit to stationary qubit conversion [2]. It is can be written as the sum of two parts, one that corre-
a typically assumed that a single emitter in free space will sponds to the relevant dipole mode and a second for the
J only have a small effect on incoming radiation. Recent remaining modes,

1 theoreticaleffortsaimedatunderstandingthefundamen-
2 tallimits onthestrengthofthis coupling[3,4]aswellas E(+) =E(+)+E(+) (1)
tot dip nondip

atdevelopingnovelmethodsforstronglightfocusing[5]. b b b
] If we consider an incoming coherent field such that
h Experimental results already show that laser extinction
p of up to 12% is possible [6, 7, 8]. DEd(+ip)E = β and DEn(+on)dipE = α then η is a dimension-
- It is already established [9] that matching the dipole lesbs number that reblates the intensity of the field at the
t
n emissionprofileofanemitterwithafocusedlaserisanec- target’s position to the contribution of the dipole field:
a
essary condition for achieving perfect absorption. How-
u ever, it was shown by van Enk [4] that this is not suffi- |β|2
q η = (2)
[ cient,sinceforatwo-levelemitterabsorptionisnecessar- |α|2+|β|2

ily followed by emission of a photon of identical first or-
3 η = 1 implies perfect overlap between the focused field
der coherence properties; as a consequence, the detected
v and the emission pattern of the emitter.
transmitted or reflected power is never zero.
7
The cascaded quantum systems approach [11, 12, 13,
3 We showhere thatif the emitter is a 3-levelΛ system,
14] provides a description of the dissipative dynamics of
0 then having equal spontaneous emission rates on both
the two emitters that is equivalent to the master equa-
2 optical transitions, along with matched dipole emission
. tion: here,a stochasticwave-functiondescribingthe sys-
7 and focused laser profiles is sufficient for perfect extinc-
tem evolves according to the Schr¨odinger equation:
0 tion of light. The scheme we describe does not require
7 anyadditionalcoherentcouplingandinthiswayitisdis- d 1
0 tinguishedfromEITtype systems[10]. In particular,we dt|ψc(t)i= ı~Heff |ψc(t)i (3)
:
v considertheinterestinglimit ofsingle photon absorption
i aswellaspossibilitiesofintra-conversionofaflyingqubit
X
toastationaryqubitandentanglementofdistantqubits.
r
We assume that the lower states of the 3 level Λ system
a
have a long coherence time; this is the case for a sin-
gle trapped atom/ion, or a solid-state emitter provided
that the two lower states are spin states. To analyze
single-photon absorption, we use the cascaded quantum
systems approach [11], since we have a quantum system
(targetemitter) drivenbyanotherquantumsystem(sin-
glephoton). Usingthesource-fieldexpression,weexpress
the incident photon annihilation/creation operators us-
ing the time-retarded dipole operators of its source. We
furtherassumethatthissourceSconsistsofathreelevel Figure 1: Schematicof thecascaded system analysed in the
text. The source atom on theleft, generates a single-photon
quantumsystemwhose|0i→|3itransitionisdrivenres-
pulse upon excitation by a coherent laser field. The
onantly via a classical laser pulse Ω (t).
L generated single photon is focused on thetarget atom to
The single-photon pulse emitted from the |3i → |1i ensuremaximum overlap with the target’s atom dipole
transition(ω(S))ofthesourcecouplestothetargetatom emission pattern. The Faraday rotator (F) ensures
31
anddrivesits |1i→|3itransition(ω(T) =ω(S)). Follow- unidirectional coupling.
31 31
2
with the non-Hermitian Hamiltonian: Initially,bothemitters(sourceandtarget)areassumed
to be in their ground states; |ψ i = |0 ,1 i. The con-
ı~ in S T
H =H +H +H − Cˆ†Cˆ (4) sequence of the desired deterministic photon absorption
eff S T ST 2 X k k processistheprojectionofthetwo-emitterwave-function
k
to a state where the source atom is in state |1i and the
This coherent evolution of the stochastic wavefunc- target atom is in state |2i (|ai = |1 ,2 i). By simu-
S T
tion is interrupted by gedanken measurement events de- lating the dynamical evolution of the system wavefunc-
scribed by the collapse operators Cˆ . In H , H ,H tion according to the effective non-Hermitian Hamilto-
k eff S T
denote the free source and target Hamiltonian and H nian, interrupted at random times by wavefunction col-
ST
describestheinteractionbetweenthem. Thecollapseop- lapses [11, 15, 16, 17], we determine the probability of
erators are in turn given by: single-photonabsorptionasafunctionofrelevantsystem
parameters, such as η and the ratio of the spontaneous
Cˆ1 = qΓ(3S1)σ1(S3)+qΓ(3T1)η σ1(T3) (5) ewmavisesfiuonnctriaotnes(oMfCthWeFta)rgneutmemeriitctaelr.siImnuoluatrioMnosnwteeChaarvloe
Cˆ2 = qΓ(3T1)(1−η) σ1(T3) (6) cΩhoswehnicahgiasuossfiathnelaosredrepruolfseΓw(Ti)thanpdeabkaRndawbiidftrheqwuehnicchy
o 31
Cˆ3 = qΓ(3T2)σ2(T3) (7) satisfies: τl−as1er <Γ3(T1). By keepingthe quantity Ωo/Γ(3S1)
small, we assure that the generated single photon pulse
By substituting the expressions for Cˆ ’s, we find [11]: is Gaussian and that it has a width τ−1 < Γ(T).
k ph−pulse 31
This assumption ensures that the target atom is never
Γ(S) saturated.
H = ~Ω (t)(σ(S)+σ(S))−ı~ 31 σ(S)−
eff L 03 30 2 33 First we consider the case η = 1: the only possible
Γ(T)+Γ(T) collapse events in this case are due to the operators C1
−ı~ 31 32 σ(T)− and C . If Γ(T) 6= Γ(T), there is a finite probability fbor
2 33 3 31 32
thetwbo-emittersbeingprojectedintoeitherstate|1S,1Ti
−ı~qΓ(3S1)Γ(3T1)η σ1(S3)σ3(T1) (8) or |1S,2Ti. However, if these two spontaneous emission
rates are equal, then the collapse events due to C never
1
where we assumed that the laser is resonant with the occur because of a destructive quantum interferbence be-
|0iS → |3iS transition of the source atom, σij = |iihj| tween the two paths leading to the state |1S,1Ti.
are the projection (i=j) or the lowering/raising(i6=j) To understandthe originof this interference effect, we
operatorsfor the atomic states and Γij denote the spon- first consider the simpler case of weak coherent light in-
taneous emission rates of the source and target atoms. cident on the target atom. Following van Enk [4] and
The last term of the effective Hamiltonian (8) is ex- assuming η =1, we find that the mean output field is
plicitlyunidirectional,sinceitisproportionaltoσ(S)σ(T)
13 31
which corresponds to the excitation of the target atom Γ(T)
E(+) =β+ Γ hσ i=β−2β 31 . (9)
by the single photon generated by the source. However, D outE p 31 31 Γ(T)+Γ(T)
thereisnotermσ(S)σ(T) whichwouldaccountforthere- b 31 32
31 13
verse process: the target atom absorbs photons emitted Here, we have evaluated hσ i using the Optical Bloch
31
earlier in time by the source atom, while the backward Equations for the 3-level target atom in steady state.
scattered photons from the target atom do not interact (+) (T) (T)
Clearly, E =0 provided Γ =Γ .
with the source. Experimental implementation of this D outE 31 32
To analbyze the single-photon absorption we could
unidirectional feature of coupling is achievable using a
alternatively use Optical Bloch Equations to evaluate
Faraday rotator placed in between the source and the
target emitters, as depicted in Figure 1. C†(t)C (t) = E(−)E(+) : in the limit Γ(T) = Γ(T)
D 1 1 E D out outE 31 32
Quantum jumps described by the collapse operators anbd Γ−b1 ≪ t ≪ Ωb−1,bτ , we find analytically that
correspond to the detection of a photon with a definite 31 o laser
C†(t)C (t) = 0, implying perfect destructive interfer-
energy and emission profile. The collapse operator C D 1 1 E
1
denotes the gedanken detection of a scattered photonbin enbce. Ebquivalently, the probability amplitude for detect-
the incoming mode; such a detection event has contri- ing a photon with energy ω(T) that is emitted by the
31
butions from photons emitted in the |3i → |1i and targethas the same magnitude but opposite signas that
S S
|3i → |1i transitions. However, a photon emitted in of detecting a source photon transmitted without scat-
T T
the latter transitionhas a probabilityη for coupling into tering.
the incoming mode. The collapse operator C2 accounts Given that for Γ3(T1) = Γ3(T2) and η = 1 the only pos-
for a photon that is emitted by the targetbatom, that sible collapse event is the projection of the system into
does not couple to the incoming radiation mode. Fi- |ai = |1 ,2 i and that all the other state occupancies
S T
nally, C denotes the detection of a photon emitted in are decreased by a ”no-collapse” event due to H , the
3 eff
the |3ib→|2i transition. system necessarily absorbs the incident photon. Since
T T
3
the target emitter occupancy changes as a result of this emission time of the single photon pulse [19]. This jitter
process,we couldalsoconsiderthe absorptionprocessas however, has consequences for single-photon absorption:
”mappingoftheincidentphotontoanatomicexcitation” wefindinoursimulationsthatchoosingΓ(S) =Γ(S)leads
30 31
[18]. Figure 2 shows the numericalcalculations based on to a ∼10% decrease in the absorptionprobability of the
anensembleaverageofmanyquantumtrajectoriesinthe targetatom. Thisdecreasecanbetracedbacktothefact
limit η =1: we observe that deterministic single-photon that destructive interference prohibiting collapse events
absorption is robust against small variations around the described by C , is no longer perfect.
1
value 1 of the ratio Γ(3T2)/Γ(3T1). One immebdiate application of near-deterministic
single-photonabsorptionisinintra-conversionofasingle-
photonpolarizationqubittoalocalized(spin)qubit. We
consider a generalization of the target emitter energy
level diagram for the case of polarization-spin conver-
sion; we would in this case need two optically excited
targetatomstates|3+i and|3−i whicharecoupledto
T T
the state |1i by a right and left hand circularly polar-
T
ized photon, respectively (the relevant target atom en-
ergy level diagram is depicted in Figure 4). If these ex-
citedstatesdecayintometastablestates|2+i and|2−i
T T
by emission of an identical photon, respectively, then we
would achieve the state transformation:
Figure 2: Γ(3T2)/Γ3(T1) dependanceof theprobability of single |Ψini=|1iT ⊗(κ1aˆ†++κ2aˆ†−)|vaci−→
photon absorption, definedas the steady state expectation |Ψ i=(κ |2+i +κ |2−i )⊗ˆb†|vaci (10)
valueof thedesired state (|ai=|1S,2Ti) population final 1 T 2 T
(ρaa=limt→∞hσaa(t)i). Here, Γ(3T1) =1, Γ3(S1) =10Γ(3T1), upon deterministic absorption of the incident photon.
γ2(T1) =0, ΩL(t)=Ωoexp(−(t−t0)2/τ2), Ω0 =1, τ =10, Here, aˆ†+, aˆ†− correspond to the photon creation oper-
t =20.
0 ators, where the index + (-) labels right (left) circular
polarization and |vaci denotes the vacuum state of the
Next, we consider the dependence of the absorption photonfield. Similarly,ˆb†|vacidenotesaphotonemitted
probability on the dipole-emission overlap parameter η: onthe|3+i →|2+i or|3−i →|2−i transitionofthe
T T T T
simulationsdepictedinFigure3confirmthatthefidelity
target emitter.
of single-photon absorption is linearly proportionalto η.
Inthecaseofη <1,theintra-conversionwillfailwitha
finite probability; however,if the emitter has a recycling
transitiontypicalforalkaliatoms,thenwecandetermine
if the target is in ground state |1i or in a subspace or-
T
thogonal to it, after the absorption is completed. This
would allow us to determine if the single-photonabsorp-
tion succeeded without compromising the faithful intra-
conversion,should we determine that it was successful.
Another potential application of the scheme analyzed
inthisworkwouldbeintheentanglementofdistantspin
qubitswithahigh-successprobability[20,21]. Untilnow,
wehaveconsideredanidealsourcewhichgeneratesapho-
ton that can be collected with unity efficiency (η = 1).
S
Figure 3: Overlap parameter η dependanceof the This assumption has to be relaxed when we consider re-
probability of single photon absorption. The parameters alistic spin entanglement. To this end, we consider a
usedinthesimulationarethesameastheonesusedinFig.2. source,whichisinitiallyinstate|0i (Figure4)andupon
S
linearly-polarizedlaserexcitationgeneratesasingleright
To have a complete description of the dynamics, we (left) hand circularly polarized photon by Raman scat-
would also need a collapse operator C4 =qΓ(3S0)σ0(S3) de- toefritnhge ienmtoisssitoantep|r1o+ceisSs(i|s1−aniSe).miCttleera-rplhy,ottohneeenntda-nsgtaletde
scribing light scattering on the |3iSb→ |0iS transition. state |1+i ⊗ aˆ† |vaci+ |1−i ⊗ aˆ† |vaci. If the pho-
WehavesetΓ3(S0) =0andtherebyneglectedthesecollapse ton generaSted b+y such a sourSce em−itter is absorbed by
eventssince,uptothispoint,wehavebeenmainlyinter- a target emitter as described earlier, then the end-result
estedinabsorptionbythetargetatom,ofaphotonpulse of single-photon absorption is the generation of the en-
that is generated by an ideal source. If Γ(S) 6= 0, then tangled state, |1+i ⊗|2+i +|1−i ⊗|2−i . The suc-
30 S T S T
the source atom will still act as a unity-efficiency single- cess probability for generation of this state is p ∼ η2:
photon source, albeit with an uncertainty (jitter) in the for challenging but realistic values of η ∼ 0.3 [3, 4],
4
statetransferbasedonemittersembeddedincavities[2].
WeconsideravariantoftheschemeofRef.[2]andreplace
the coherent field coupling (Ω (t)) of the recipient (tar-
2
get)atomby spontaneousemissionΓ onthe |ri −|ei
re 2 2
transition[23]. Weassumeκ≫g andadiabaticallyelim-
inate the cavity mode: in the Purcell limit g2 ≫ κΓ ,
rg
we find that the cavity+atomsystem is equivalent to an
”effective single emitter” for which η ∼ 1. If the Pur-
cell enhancement is such that Γ = g2/κ, then the cav-
re
ity+atom system can absorb the incident single-photon
withunityefficiency,withouttheneedforacoherentfield
that should have been synchronized with the incoming
single-photon pulse.
Figure 4: Schematic of a free-space coupling scheme that Inconclusion,wehaveanalysedthe quantumstateex-
achieves entanglement of distant spin qubits. Following laser change between light and matter at the level of single
excitation, thesource atom emits a single photon whose quanta. Theformalismpresentedhereisgeneralandcan
polarization degree of freedom is entangled with the final be applied to various quantum systems such as atoms,
source atom state. Upon absorption of thesingle photon by QDs and molecules. The most important challenge at
thetarget, thetwo atoms are projected onto a this point is the enhancement of the overlap parameter
’spin-entangled’ state.
η;possiblemethodsincludechangeoftheemissionprofile
of the dipole/atom by using dielectrics. Another possi-
bility involves a scheme proposed by Leuchs et al. [5]
this would yield an entanglement generation probabil-
to achieve tight focus of an incident radially polarized
ity of ∼ 10%, which compares favorably with the previ-
field. For applications involving intra-conversion of po-
ously proposed probabilistic long-distance entanglement
larization and spin qubits, an additional challenge is to
schemes [20, 21]. The ability to carryoutrecyclingtran-
achieve high η for both polarizations at the same time.
sitions to determine whether or not the incoming single-
photon is absorbed, is crucial for this application [22]. We wish to acknowledge many useful discussions with
Lastbutnotleast,wepointoutthatourfindingscould H. Tureci, S. van Enk, G. Leuchs, A. Sorensen and V.
substantially simplify previous protocols for quantum Sandoghdar.
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