Cavity-assisted spontaneous emission of a single $\Lambda$-type emitter as a source of single-photon packets with controlled shape

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by  M. Khanbekyan

Cavity-assisted spontaneous emission of a single Λ-type emitter as a source of
single-photon packets with controlled shape
M. Khanbekyan∗
Institut fu¨r Theoretische Physik, Universita¨t Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany
D.-G. Welsch
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universita¨t Jena, Max-Wien-Platz 1, D-07743 Jena, Germany
(Dated: January 5, 2017)
The radiation emitted by a classically pumped three-level Λ-type emitter in a resonator cavity
featuring both radiative and unwanted losses is studied. In particular, the efficiency of one-photon
Fockstateexcitationoftheoutgoingwavepacketandthespatiotemporalshapeofthewavepacket
are investigated. It is shown that for given shape of the emitter–cavity interaction, adjusting the
shapeofthepumppulserendersitpossibletogeneratetheemissionofone-photonFockstatewave
7
packets of desired shape with high efficiency.
1
0
2
I. INTRODUCTION photonFockstatesinaschemebasedonvSTIRAPtech-

n nique is governed by an interplay of various system pa-
a
rameters, such as the transition dipole matrix element,
J The interaction of a single emitter with the quantized
intensityandamplitudeofthepumppulse,frequencyde-
4 radiation-field in a high-Q cavity serves as a basic in-
tuning etc. Further, it has also been reported that the
gredient in various schemes in quantum information sci-
spatiotemporal shapes of the outgoing wave packets of
] ence (for a review see, e.g., Ref. [1]). In this context, an
h the emitted one-photon Fock state radiation follow the
p essential prerequisite for quantum information process- driving laser pulse shapes [11, 12]. However, this behav-
- ing[2]isthedeterministicgenerationofindistinguishable
iorisonlyobservedforacertainrangeofparametersand
t
n light pulses of one-photon Fock states on demand with theefficiencyofone-photonFockstategenerationisvery
a well-controlledfeatures,suchasefficiency,frequency,po-
low,whichmakesitimpossibletousetheschemeforfur-
u larization, timing etc.. Among the various features of
ther applications in quantum communications. More re-
q
single-photon emission, control and manipulation of the
[ cently,thepossibilityofgenerationofsingle-photonwave
spatiotemporal shape of the single-photon light pulses is
packetswitharbitraryshapesinthecasewhentheatom–
1 essential for quantum network applications where sym-
cavity interaction remains constant during the genera-
v metric wavepackets are required for a time reversal of
tion process has been demonstrated both theoretically,
4 the emission process [3] or where time-bin entanglement
8 on using the master equation formalism for describing
is used [4].
9 the atom–photon system [13], and experimentally, with
0 The most prominent way to generate single-photon un-trapped atoms injected into a cavity by means of an
0 Fock states in high-finesse optical cavities interacting atomic fountain [14].
1. with single emitters is based on vacuum stimulated Ra- In an earlier work, we have studied the potential of
0 man adiabatic approach (vSTIRAP) technique [5, 6].
a single two-level atom in a high-Q cavity as a single-
7 A single transition of an emitter with a Λ-type energy
photon emitter, where the wave packet associated with
1 level scheme is excited by a pump laser stimulating the
: the emitted photon is shorter than the cavity decay
v emission into a strongly coupled cavity–emitter transi-
time[15,16]. Withintheframeofexactquantumelectro-
i tion. AlongsidetheprincipalrealizationsofvSTIRAPin
X dynamics, we have particularly shown the feasibility of
atomicsystems(seee.g.[7]),aΛ-typeschemecanbealso
the generation of single-photon wave packets with time-
r implementedinsemiconductorquantumdot–microcavity
a symmetric spatiotemporal shapes.
systems, where a Λ-type system is created by applying
a strong magnetic field to a singly charged quantum dot Inthepresentarticleweextendthetheorytotheinter-
resulting in Zeeman splitting of the spin states in the actionofapumpedthree-levelΛ-typeemitterwithareal-
conductionband[8]orbyapplyingalateralelectricfield isticcavity-assistedquantizedelectromagneticfield,with
which charges a single quantum dot with a single elec- the aim of well-defined and controlled one-photon Fock-
tron resulting in optical transitions between ground and stateemission. Inparticular, westudyindetailboththe
excited electron states and the charged exciton [9]. In properties of the excited outgoing wave packet and the
particular, in atomic systems control of the photon fre- efficiencytoprepareitinasingle-photonFockstate. The
quency [10], wavepacket length [10] and phase [7] have generalexplicitexpressionderivedforthespatiotemporal
been reported. The efficiency of the generation of single- shape of the excited outgoing wave packet allows us to
studydifferentcouplingregimesoftheemitter–pumpand
the emitter–cavity-field interactions. In this context, we
showthepossibilitytogeneratesingle-photonwavepack-
∗E-mailaddress: [e-mail] ets with requested spatiotemporal shapes, which can be
2
achieved by using appropriately designed driving laser is the electric dipole-moment operator (dAkk(cid:48) =
pulses. k dˆ k(cid:48) ),andtheoperatorofthemedium-assistedelec-
A
(cid:104) | | (cid:105)
The paper is organized as follows. The basic equa- tric field Eˆ(r) can be expressed in terms of the variables
tions for the resonant interaction of an emitter with a ˆf(r,ω) and ˆf†(r,ω) as follows:
cavity-assisted electromagnetic field are given in Sec. II.
In Sec. III the Wigner function of the quantum state the Eˆ(r)=Eˆ(+)(r)+Eˆ(−)(r), (7)
excitedoutgoingwavepacketispreparedinandthespa-
tiotemporal shape of the wave packet are studied for the
case of a classically pumped three-level Λ-type emitter. (cid:90) ∞
SectionIVisdevotedtotheproblemofgeneratingsingle- Eˆ(+)(r)= dωEˆ(r,ω), Eˆ(−)(r)=[Eˆ(+)(r)]†, (8)
0
photon wave packets of desired shapes. A summary and
some concluding remarks are given in Sec. V.
Eˆ(r,ω)=
II. BASIC EQUATIONS i(cid:114) (cid:126) ω2 (cid:90) d3r(cid:48)(cid:112)ε(cid:48)(cid:48)(r(cid:48),ω)G(r,r(cid:48),ω) ˆf(r(cid:48),ω). (9)
ε π c2 ·
0
We consider a single emitter (position r ) that inter-
A
In the above, the classical (retarded) Green tensor
acts with the electromagnetic field in the presence of a
G(r,r(cid:48),ω) is the solution to the equation
dispersing and absorbing dielectric medium with a spa-
tially varying and frequency-dependent complex permit-
ω2
tivity ∇ ∇ G(r,r(cid:48),ω) ε(r,ω)G(r,r(cid:48),ω)=δ(3)(r r(cid:48))
× × − c2 −
ε(r,ω)=ε(cid:48)(r,ω)+iε(cid:48)(cid:48)(r,ω), (1) (10)
together with the boundary condition at infinity,
with the real and imaginary parts ε(cid:48)(r,ω) and ε(cid:48)(cid:48)(r,ω), G(r,r(cid:48),ω) 0 if r r(cid:48) , and defines the struc-
→ | − | → ∞
ture of the electromagnetic field formed by the present
respectively. Applyingthemultipolar-couplingschemein
dielectric bodies.
electric dipole approximation, we may write the Hamil-
tonianthatgovernsthetemporalevolutionoftheoverall
system, which consists of the electromagnetic field, the
III. SINGLE-PHOTON GENERATION
dielectric medium (including the dissipative degrees of
EFFICIENCY AND WAVE PACKET SHAPE
freedom), and the emitter coupled to the field, in the
form of (for details, see Refs. [17, 18])
A. Pumped three-level Λ-type emitter in a cavity
(cid:90) (cid:90) ∞
Hˆ = d3r dω(cid:126)ωˆf†(r,ω) ˆf(r,ω)
· Let us consider a single atom-like emitter placed in a
0
(cid:88) resonatorcavityandassumethatonlyasingletransition
+ (cid:126)ω Sˆ dˆ Eˆ(r ). (2)
kk kk− A· A (2 3 , frequency ω23) is quasi resonantly coupled to
k a|n(cid:105)a↔rro|w(cid:105)-band cavity-assisted electromagnetic field (fre-
In this equation, the first term is the Hamiltonian of quencyωk),cf. Fig.1. Wefurtherassume,thatanexter-
thefield–mediumsystem,wherethefundamentalbosonic nal (classical) pump field with quasi resonant frequency
fields ˆf(r,ω) and ˆf†(r,ω), ωp and(time-dependent)RabifrequencyΩp(t)isapplied
to the 1 2 transition (frequency ω ), cf. Fig. 1.
21
| (cid:105)↔| (cid:105)
(cid:2)ˆf(r,ω),ˆf†(r(cid:48),ω(cid:48))(cid:3)=δ(ω ω(cid:48))δ(r r(cid:48)), (3) Forthesakeofsimplicityofpresentation, werestrictour
− − treatment to the one-dimensional case (z axis) and as-
(cid:2)ˆf(r,ω),ˆf(r(cid:48),ω(cid:48))(cid:3)=0, (4) sume that the resonator cavity is formed by an empty
body bounded by an outcoupling fractionally transpar-
play the role of the canonically conjugate system vari- ent mirror at z = 0 and a perfectly reflecting mirror at
ables. ThesecondtermistheHamiltonianoftheemitter, z = l, and (negative) z is the position of the emitter
where the Sˆkk(cid:48) are the flip operators, insid−e the cavity. A
Inthiscase, theone-dimensionalversionoftheHamil-
Sˆkk(cid:48) = k k(cid:48) , (5) tonian (2) in the rotating-wave approximation reads
| (cid:105)(cid:104) |
corresponding to the k k(cid:48) transition with the fre- (cid:90) (cid:90) ∞
quency ωkk(cid:48), where k| (cid:105)is↔th|e(cid:105)energy eigenstates of the Hˆ = dz dω(cid:126)ωfˆ†(z,ω)fˆ(z,ω)+(cid:126)ω21Sˆ22+(cid:126)ω31Sˆ33
emitter. Finally, the|l(cid:105)ast term is the emitter–field cou- 0
(cid:104) (cid:105)
pling energy, where g(t) d Sˆ† Eˆ(+)(z )+H.c.
− 23 32 A
dˆA =(cid:88)dAkk(cid:48)Sˆkk(cid:48) (6) − (cid:126)2Ωp(t)(cid:104)Sˆ1†2e−iωpt+H.c.(cid:105), (11)
kk(cid:48)
3
i
2 C˙ = Ω (t)e−i∆ptC (t)
2 p 1
| i 2
d23 (cid:90) ∞dω ω2 (cid:90) dz(cid:112)ε(cid:48)(cid:48)(z,ω)
− √π(cid:126)ε c2
0 0
A
G(z ,z,ω)C (z,ω,t)g(t)e−i(ω−ω23)t, (14)
ω × A 3
ω 23
k
3 C˙ (z,ω,t)= d∗23 ω2(cid:112)ε(cid:48)(cid:48)(z,ω)
ω21 ωp | i 3 √π(cid:126)ε0A c2
G∗(z ,z,ω)C (t)g(t)ei(ω−ω23)t, (15)
A 2
×
where is the area of the coupling mirror of the cavity,
A
and∆ =ω ω isthedetuningofthepumpfrequency
p p 21
−
from the 1 2 transition frequency. The Green func-
tion G(z,|z(cid:48)(cid:105),↔ω)|d(cid:105)etermines the spectral response of the
1 resonator cavity. For a sufficiently high-Q cavity, the ex-
| i citation spectrum effectively turns into a quasi-discrete
set of lines of mid-frequencies ω and widths Γ , accord-
k k
FIG.1: Schemeofenergylevelsandtransitionsforthethree- ing to the poles of the Green function at the complex
level Λ-type atomic emitter. frequencies
ω˜ =ω 1iΓ , (16)
k k− 2 k
with ω31= ω21 ω23. Here, the (real) time-dependent where the linewidths are much smaller than the line sep-
−
function g(t) defines the (time-dependent) shape of the arations,
interaction of the emitter with the cavity field, which
withoutlossofgeneralitycanbechosentobenormalized Γ 1(ω ω ). (17)
k (cid:28) 2 k+1− k−1
to unity. It may be used to simulate the (quasistatic)
motionoftheemitterthroughthecavityinthedirection In this case, assuming that the kth mode of the cavity is
perpendiculartothecavityaxisand/ormotionalongthe quasi resonantly coupled to the transition 2 3 with
| (cid:105)↔| (cid:105)
cavity axis. the transition frequency ω23 (cf. Fig. 1), we substitute
theformalsolutionstoEqs.(13)and(15)[withtheinitial
In what follows we assume that the emitter is initially
condition C (0)=1, C (0)=0 and C (z,ω,0)=0] into
(at time t=0) prepared in the state 1 and the rest of 1 2 3
| (cid:105) Eq. (14). Then, using the relations for the Green tensor
the system, i.e., the part of the system that consists of
G(z ,z ,ω)asgivenbyEqs.(A1)and(A2),wecanderive
the electromagnetic field and the dielectric media (i.e., 1 2
the integro-differential equation
the cavity), is prepared in the ground state 0 . Since,
|{ }(cid:105)
in the case under consideration, we may approximately i (cid:90) t
spantheHilbertspaceofthewholesystembythesingle- C˙ = Ω (t)e−i∆pt+ dt(cid:48)K(t,t(cid:48))C (t(cid:48)), (18)
2 p 2
2
excitation states, we may expand the state vector of the 0
overall system at later times t (t 0) as
≥ where the kernel function K(t,t(cid:48)) reads
|ψ(t)(cid:105)=C1(t)|{0}(cid:105)|1(cid:105)+C2(t)e−iω21t|{0}(cid:105)|2(cid:105) K(t,t(cid:48))=− 14Ωp(t)Ωp(t(cid:48))e−i∆p(t−t(cid:48))
(cid:90) (cid:90) ∞
+ dz dωC (z,ω,t)e−i(ω+ω31)tfˆ†(z,ω) 0 3 , 1
3 |{ }(cid:105)| (cid:105) α ω˜ g(t)g(t(cid:48))e−i(∆k−iΓk/2)(t−t(cid:48)), (19)
0 − 4 k k
(12)
with ∆ =ω ω and
k k 23
−
wherefˆ†(z,ω) 0 isanexcitedsingle-quantumstateof
the combined fi|{eld}–(cid:105)cavity system. α = 4|d23|2 sin2(ω z /c). (20)
k (cid:126)ε l k A
0
It is not difficult to prove that the Schr¨odinger equa- A
tionfor ψ(t) leadstothefollowingsystemofdifferential From Eq. (18) together with Eq. (19) we can conclude
| (cid:105)
equationsfortheprobabilityamplitudesC1(t),C2(t)and that Rk √αkωk can be regarded as vacuum Rabi fre-
≡
C3(z,ω,t): quency of emitter–cavity interaction.
Following Ref. [15], it can be shown that when the
i Hilbert space of the system is effectively spanned by a
C˙1 = Ωp(t)ei∆ptC2(t), (13) single excitation, on a time-scale that is short compared
2
4
to the inverse spontaneous emission rate of the emitter, B. Efficiency of single-photon Fock state generation
the multimode Wigner function of the quantum state of
the outgoing field can be derived to be (Appendix B) InsertingEq.(24)intoEq.(23)andusingEq.(A3),we
findtheefficiencyofone-photonFockstategenerationas
(cid:89)
W (α ,t)=W (α ,t) W(0)(α ,t), (21)
out i 1 1 i i
i(cid:54)=1 η(t)= Rk2 γkrad(cid:20)(cid:90) tdt(cid:48)g(t(cid:48))C (t(cid:48))
2
4 Γ
0
where
(cid:90) t(cid:48) (cid:21)
dt(cid:48)(cid:48)g(t(cid:48)(cid:48))C∗(t(cid:48)(cid:48))ei(∆k+iΓk/2)(t(cid:48)−t(cid:48)(cid:48))+C.c. , (28)
W (α,t)=[1 η(t)]W(0)(α)+η(t)W(1)(α), (22) × 2
1 − 1 1 0
where γ describes the wanted radiative losses due
with W(0)(α) and W(1)(α) being the Wigner functions krad
i i to transmission of the radiation through the fraction-
of the vacuum state and the one-photon Fock state, re-
ally transparent mirror. As we can see, the efficiency
spectively, for the ith nonmonochromatic mode. As we
is proportional to the ratio γ /Γ , where the damp-
see, the mode labeled by the subscript i=1, is basically krad k
ing parameter Γ can be regarded as being the sum
prepared in a mixed state of a one-photon Fock state k
Γ =γ +γ(cid:48), where γ(cid:48) describes the unwanted losses
and the vacuum state, due to unavoidable existence of k krad k k
due to absorption. For the numerical calculations, we
unwantedlosses. Theothernonmonochromaticmodesof
will assume γ /Γ =0.9, which corresponds to 10%
theoutgoingfieldwithi=1areinthevacuumstateand, krad k
(cid:54) unwanted losses. Apart from the unavoidably existing
therefore, remain unexcited.
unwantedlosses,theone-photonFockstateefficiencyde-
The Wigner function W (α,t) of the mode associated
1 pendsontheshapeg(t)andtheRabifrequencyR ofthe
k
with the excited outgoing wave packet reveals that η(t)
emitter–cavity interaction and the shape and the inten-
can be regarded as being the efficiency to prepare the
sityoftheexternalpumpfield. Finally,insertingEq.(25)
excited outgoing wave packet in a one-photon Fock state
together with Eq. (24) into Eq. (27) and using Eq. (A3),
(Appendix B):
wefindthespatiotemporalshapeoftheexcitedoutgoing
wave packet outside the cavity as (in the following, for
(cid:90) ∞ (cid:90) ∞
η(t)= dω F(ω,t)2 dω F(ω,t)2, (23) the sake of simplicity, we assume ω31=0)
| | (cid:39) | |
0 −∞
(cid:115)
R (cid:126)ω γ
with φ (z,t)= k k krad
1
2 2ε c η(t)
0
A
d23 (cid:114)c ω2 (cid:90) t−z/c
F(ω,t)= √π(cid:126)ε0A ω c2 × 0 dt(cid:48)C2∗(t(cid:48))g(t(cid:48))e−i(∆k+iΓk/2)t(cid:48)eiω˜k∗(t−z/c). (29)
(cid:90) t
dt(cid:48)G∗(0+,z ,ω)C∗(t(cid:48))g(t(cid:48))eiω(t−t(cid:48))eiω23t(cid:48)eiω31t, (24) To present explicit results, we restrict ourselves to
× A 2
0 timesmuchlargerthanthetimetheexcitedwavepacket
needs to almost completely exit the cavity. To find the
where 0+ indicates the position z=0 outside the cavity.
spatiotemporalshapeoftheexcitedoutgoingwavepacket
The excited outgoing wave packet (mid-frequency ωk) is φ (z,T) at time T Γ−1 for given pump intensity and
characterized by the mode function 1 (cid:29) k
emitter–cavity interaction shapes Ω (t) and g(t), respec-
p
tively, we first solve Eq. (18) together with Eqs. (19),
F(ω,t)
F1(ω,t)= (cid:112) . (25) (20) and the initial condition C2(0)=0. Equation (18)
η(t) isaVolterra-typeintegro-differentialequationofthesec-
ond kind, which can be solved using a standard fourth-
The intensity of the outgoing field at position z is given order Runge-Kutta algorithm. Then, inserting the solu-
by tion C (t) into Eqs. (28) and (29) we can find, respec-
2
tively, the efficiency of one-photon Fock state emission
I(z,t)=η(t)φ (z,t)2, (26) and the spatiotemporal shape of the wave packet leaving
1
| |
the cavity.
with In accordance with the vSTIRAP state-mapping pro-
posal [19, 20], in an ideal system the maximum one-
1(cid:90) ∞ (cid:114) (cid:126)ω photon Fock state efficiency can be achieved when the
φ (z,t)= dω e−iωz/cF (ω,t). (27)
1 2 ε cπ 1 dynamics of the system follows the ”dark state”, which
0 0 A is formed as a superposition of the states 0 1 and
|{ }(cid:105)| (cid:105)
Equation (26) reveals that φ (z,t) represents the spa- fˆ†(z,ω) 0 3 . This is the case if the process is
1
|{ }(cid:105)| (cid:105)
tiotemporal shape of the outgoing field associated with adiabatic and the emitter–cavity interaction is well-
the excited nonmonochromatic mode F (ω,t). established before the driving pulse is applied. The case
1
5
(a) (a)
k k
Γ 4 Γ
/ /
y y 1
c c
n n
e e
u u
q q
e 2 e
r r
f f
bi bi
a a
R R
0 0
Γ t Γ t
k k
(b) Efficiency (b) Efficiency
) 0.9 ) 0.081
s s
t t
ni 0.9 ni 0.69
u u
0.9 0.88
rb. rb. 0.9
a a
( ( 0.9
)| )|
T T
z, z,
( (
1 1
φ φ
| |
Γ z/c Γ z/c
k k
0 50 100 150 200 250 300 40 60 80 100 120 140 160
FIG. 2: (a) Various pulse shapes of the pump Ω (t) (dot- FIG. 3: (a) Various pulse shapes of the pump Ω (t)
p p
ted, dashed and dash-dotted lines) with the maximum value with the maximum value Ω = 0.1Γ (dotted red line),
pmax k
Ω = 5Γ and the shape of the emitter–cavity interac- Ω = 0.4Γ (dashed-dotted blue line), Ω = 0.7Γ
pmax k pmax k pmax k
tion g(t) (double line, shown as normalized to Ω ). (b) (dashedpurpleline),Ω =1Γ (dashed-dottedgreenline),
pmax pmax k
Correspondingspatiotemporalshapesoftheone-photonFock Ω =1.5Γ (dotted orange line). (b) Corresponding spa-
pmax k
state outgoing wave packet (R = 5Γ , ∆ = ∆ = Γ , tiotemporal shapes of the one-photon Fock state outgoing
k k k p k
T =350Γ ). wave packet (R =2Γ , ∆ =∆ =0, T =200Γ ).
k k k k p k
ofthevSTIRAPisillustratedinFig.2,whereaGaussian- interction in more detail. For the sake of transparency
type emitter–cavity interaction shape and three driving wefirstconcentrateonthecase,whentheemitter–cavity
pulse shapes are considered. Comparing the values of interaction is constant, i.e., it does not vary with time.
η(T), we see that η(T) 0.9 for all three cases of the
≈ InFig.3thespatiotemporalshapeofthesingle-photon
driving pulse shapes, which shows that the efficiencies of
outgoingwavepacket(solidlines)isshownforvariousin-
one-photon Fock state generation is only limited by the
tensities of a symmetric single-peak driving pulse. It is
unwanted losses in the system. Moreover, Fig. 2 reveals
seen that when the value of the maximum of the driving
that the excited outgoing wave packet carrying a single
pulse is low, Ω = 0.1Γ , the shape of the outgoing
photon is generated in the time interval immediately as pmax k
wave packet (red, solid line in Fig. 3(b)) perfectly co-
the driving pulse is applied. Thus, since the adiabatic
incides with the shape of the driving pulse (red, dotted
evolution required by the vSTIRAP is much longer than
line in Fig. 3(a)). However, the efficiency of one-photon
the cavity mode decay time, the generated single-photon
Fock state generation is quite low, η(T) = 0.081. It is
wavepacketescapesfromthecavityonceitisgenerated,
furtherseenthattheefficiencyincreaseswithΩ . On
and, hence, the shape of the wave packet φ (z,T) is pmax
| 1 | the other hand, the figure also reveals that with increas-
effectively independent of the driving pulse shape.
ing Ω the driving pulse induces the photon emission
pmax
earlier, and as a result, the shape of the outgoing wave
packetdoesnotfollowtheprofileofthedrivingpulsebut
C. Effects of emitter–cavity and emitter–pump features an asymmetric shape.
interaction strengths and shapes
Next, letusconsiderthedependenceofthespatiotem-
poralshapeofthesingle-photonoutgoingwavepacketon
In general, it is desirable to have the possibility to the Rabi frequency of the emitter–cavity interaction R .
k
control and manipulate the shape of the outgoing wave In Fig. 4 the shape is plotted for various values of R in
k
packetand,atthesametime,toexciteitinaone-photon the case of a symmetric single-peak driving pulse. For
Fockstatewithhighefficiency. Letusthereforestudythe R =Γ thevSTIRAPcaseisseentoberealized,result-
k k
dependenciesoftheefficiencyandthewavepacketshape inginhigh-efficiencysingle-photonFockstategeneration
on the emitter–cavity interaction and the emitter–pump and asymmetric spatiotemporal shape of the associated
6
IV. SHAPE-CONTROLLED SINGLE-PHOTON
Efficiency
WAVE PACKET EMISSION
0.3
units) 00..69 0.6 y/Γk Aswehaveseen,theshapeofthesingle-photonoutgo-
b. 0.4 nc ingwavepacketcanbechangedbychangingtheemitter–
ar ue cavity interaction and/or the driving pulse. In this con-
( q
)| re text, aquestionofparticularinterestisthegenerationof
T f
z, 0.2 bi single-photon wave packets of controlled shapes. To give
( a
1 R ananswer,letusassumethatthedesiredspatiotemporal
φ
| shape of the single-photon outgoing wave packet is given
0 byafunctionφ(z,T)atsometimeT,wherethecondition
T Γ−1 again ensures that the wave packet has almost
40 60 80 100 120 140 160 (cid:29) k
completely left the cavity.
Γ z/c(Γ t)
k k
From Eq. (29) [φ (z,t) φ(z,T)] we find that
1
→
FIG. 4: spatiotemporal shape of the one-photon Fock state (cid:115)
2 2ε c η(T)
(osuotlgidoinbgluwealvineep)a,cRkektf=or7RΓkk=(sΓolkid(sroeldidlginreee)ninlinteh)e,Rcaks=e o4fΓak C2(t)= Rkg(t) (cid:126)0ωkAγkrad e−iω23t
single-peak driving pulse (dotted red line), Ω = 0.7Γ ,
pmax k (cid:26) (cid:27)
∆k =∆p =0, T =200Γk. dφ[c(T −t),T] iω˜∗φ[c(T t),T] . (30)
× dt − k −
Further, inserting Eq. (19) into Eq. (18) we see that
Efficiency
units) 000...159 0.6 y/Γk D(t)=f(t)+(cid:90)0tdt(cid:48)f(t)f(t(cid:48))C2(t(cid:48)), (31)
rb. 0.4 enc where D(t) is defined by
a u
( q
T)| fre D(t) C˙ (t)
z, 0.2 bi ≡ 2
φ(1| Ra + Rk2 (cid:90) tdt(cid:48)C (t(cid:48))g(t)g(t(cid:48))e−i(∆k−iΓk/2)(t−t(cid:48)), (32)
2
4
0 0
and f(t) is related to the shape of the pump pulse Ω (t)
40 60 80 100 120 140 160 p
according to
Γ z/c(Γ t)
k k
i
f(t)= Ω (t)e−i∆pt. (33)
FIG. 5: spatiotemporal shape of the one-photon Fock state 2 p
outgoingwavepacketforR =Γ (solidgreenline),R =4Γ
k k k k
(solid blue line), R = 12Γ (solid red line) in the case of a Differentiation of Eq. (31) with respect to t then yields
k k
double-peak driving pulse (dotted red line), Ω = 0.7Γ , the following differential equation for f(t):
pmax k
∆ =∆ =0, T =200Γ .
k p k
D(t)f˙(t)+C (t)f3(t) D˙(t)f(t)=0. (34)
2
−
Hence, the following scheme of shape controlling may
outgoing wave packet. For higher values of R , the driv-
k be established. For a chosen (i.e., desired) spatiotem-
ing pulse is not strong enough to realize the vSTIRAP,
poral shape φ(z,T) of the single-photon outgoing wave
and the efficiency of one-photon Fock state generation is
packet, the probability amplitude C (t) is calculated
2
reduced, but the spatiotemporal shape of the associated
fromEq.(30). Theresultisthen,togetherwithEq.(32),
wave packet follows the shape of the driving pulse.
inserted into the differential equation (34). Its solution
An analogous situation can be observed for a double- eventually yields the sought shape of the driving pulse.
peak driving pulse as illustrated in Fig. 5. For emitter– To illustrate the scheme, we apply it first to the gen-
cavity Rabi frequencies of the same order of magnitude erationofsingle-photondouble-peakwavepacketsinthe
as the driving pulse maximum, the vSTIRAP is again case, when the emitter–cavity interaction is constant, as
seen to be realized and the efficiency of one-photon Fock showninFig.6fortwoone-photonFockstategeneration
stategenerationisclosetothemaximumvalueof0.9,but efficiencies. In the case of low efficiency the shape of the
the shape of the associated wave packet becomes asym- single-photonoutgoingwavepacketisseentocloselyfol-
metric. As expected, for higher rates of emitter–cavity low the shape of the driving pulse. It is also seen that
Rabifrequencyitfollowsthedrivingpulseshapebutthe whenanefficiencyclosetounityisrequired,thenthesec-
one-photonFockstategenerationefficiencyisratherlow. ond peak of the driving pulse must be higher than that
7
1.2
Efficiency Efficiency
0.8
0.9 0.9
1
nits) 0.2 0.8 /Γk nits) 0.2 0.6 /Γk
u y u y
φ(z,T)(arb.|| 000...246 Rabifrequenc φ(z,T)(arb.|| 00..24 Rabifrequenc
0 0
40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 200
Γkz/c(Γkt) Γkz/c(Γkt)
FIG. 6: Shapes of the driving pulse (solid lines) required FIG. 7: Shapes of the driving pulse (solid lines) re-
togenerateadouble-peakwavepacketcarryingaone-photon quired to generate a plateau-like wave packet carrying a
Fock state (dotted red line) for a high generation efficiency one-photon Fock state (dotted red line) for a high genera-
η(T) = 0.9 (solid red line) and a low generation efficiency tion efficiency η(T) = 0.9 (solid red line) and a low gen-
η(T) = 0.2 (solid blue line), Rk = 8Γk, ∆k =∆p =0, eration efficieny η(T) = 0.2 (solid blue line), Rk = 8Γk,
T =200Γk. ∆k =∆p =0, T =200Γk. The inset shows the required
shape of the driving pulse (solid green line) in the case of a
time-dependentshapeoftheemitter–cavityinteraction(solid
of the first one. This can be explained by the fact that black line).
during the generation of a single-photon wave packet, a
part of the wave packet (which corresponds to the first
peak) already escapes the cavity. Therefore, to generate
the second required peak the pump intensity should be
higher.
A similar behavior is observed in the case when a
plateau-like spatiotemporal shape of the single-photon
wave packet is desired. as it can be seen from Fig. 7.
Namely, in the case of constant emitter–cavity interac-
tion,forlowefficienciestherequiredprofileofthedriving
pulseissimilartothespatiotemporalshapeoftheoutgo-
ing radiation. For generation efficiencies close to unity,
the shape of the driving pulse needs to be asymmetric.
Forcomparison,acaseoftime-dependentemitter–cavity
interaction is shown in the inset of Fig. 7. In partic-
ular, the interaction shape is chosen to simulate time-
dependent variation of the location of the emitter in the
cavity, as small oscillations of the interaction due to the
FIG. 8: Shape of the driving pulse (solid red line) required
variationofemitterpositionaroundtheanti-nodesofthe
togeneratesingle-photonoutgoingwavepacketswhoseshape
cavitymode. Itisseenthattoobtainthedesiredshapeof (dotted red line) resembles the ”Tatik-Papik” monument.
thesingle-photonoutgoingwavepacket,theprofileofthe
driving pulse should follow the variation of the emitter–
cavity interaction. Let us finally have a look at Fig. 8. transition of the emitter resonantly interacts with the
Itshowsthegeneration ofanasymmetrictwo-peak wave cavity field and the other one is subjected to an external
packettheshapeofwhichresemblesthemonumentcalled classicalpumpfield. Wehaveappliedthetheorytodeter-
”Tatik-Papik” (”Grandma-Grandpa”). mine both the efficiency of one-photon Fock state gener-
ationandthespatiotemporalshapeoftheoutgoingwave
packet. In this context, we have shown that, even for
V. SUMMARY AND CONCLUDING REMARKS high one-photon Fock state generation efficiencies, wave
packets of arbitrary desired shape can be generated by
Based on macroscopic QED in dispersing and absorb- appropriately adjusting the pump.
ing media, we have presented an exact description of the Under the assumption that the Hilbert space of the
resonant interaction of a three-level Λ-type emitter in total system is spanned by a single-quantum excitation,
a high-Q cavity with the cavity-assisted electromagnetic thequantumstateoftheexcitedoutgoingnonmonochro-
field. In particular, we have studied the case when one matic mode is always a mixture of a one-photon Fock
8
state and the vacuum state. Due to the unavoidable ex- dependence of the emitter–cavity interaction, such as in-
istenceofunwantedlosses,suchasabsorptionorscatter- jecting the emitter into the cavity or fluctuating emitter
ing, the efficiency of one-photon Fock state generation is motion around the center of the interaction region.
always limited by the ratio of the cavity radiative decay
rate to the total decay rate, which includes both wanted
and unwanted losses. Acknowledgments
Thespatiotemporalshapeoftheexcitedoutgoingwave
packet and the efficiency of one-photon Fock state gen- M.K.acknowledgesvaluablediscussionswithJ.Wier-
eration depend on the interplay of various parameters of sig.
the emitter–cavity and emitter–pump interactions, such
as their strengths and shapes. In particular, in the case
Appendix A: Green tensor
of the vSTIRAP the efficiency of one-photon Fock state
generation may be close to the upper limit determined
by the unavoidable unwanted losses in the system. It The Green tensor has the following property:
requires that the process is adiabatic and the emitter– ω2(cid:90)
cavity interaction is well-established before the driving dxε(cid:48)(cid:48)(x,ω)G(z ,x,ω)G∗(z ,x,ω)=ImG(z ,z ,ω).
c2 1 2 1 2
pulse is applied—so that the system dynamics follows
(A1)
the dark state. However, in this regime, control of the
The Green tensors G(z ,z ,ω) and G(0+,z ,ω) read
spatiotemporalshapeoftheexcitedoutgoingwavepacket A A A
(see, Ref. [15])
is not possible, as the shape is nearly independent of the
driving pulse shape. (cid:88) c2 1
G(z ,z ,ω)= sin2(ω z /c) (A2)
In the case when the emitter-cavity interaction can be A A − ω˜ lω ω˜ k A
k k
k −
regarded as being time independent on the timescales of
the cavity decay time and the driving pulse length, two and
interactionregimesmaybeidentified. Intheregimewhen G(0+,z ,ω)= (cid:88)t13c2 i eiωkl/csin(ω z /c),
the emitter–cavity interaction Rabi frequency is suffi- A − 2ω˜ l ω ω˜ k A
k k
ciently high compared to the (maximum) emitter–pump k −
(A3)
interaction Rabi frequency, the spatiotemporal shape of
respectively, where t denotes the transmission coeffi-
the excited outgoing wave packet follows the shape of 13
cient from inside the cavity to outside.
the driving pulse. However, the one-photon Fock state
generation efficiency isverylow. This interaction regime
has been realized by means of the interaction of a high-
Appendix B: Quantum State of the Outgoing field
finesse cavity with both neutral atoms [11] and trapped
ions [12], where single-photon wave packet shapes that
To calculate the quantum state of the outgoing field,
follow the driving field shapes have been reported. In
following Ref. [15], we start from the one-dimensional
the regime when the emitter–pump interaction Rabi fre-
version of Eq. (9) and decompose the electric field into
quencyisofthesameorderofmagnitudeastheemitter–
incoming and outgoing fields. Then, we may represent
cavity Rabi frequency, the one-photon Fock state gen-
theoutgoingfieldatthepointz=0outsidethecavityas
eration efficiency is close to the upper limit. However,
tphaeckseptahtiaosteamsipnogrlael-psheaakpeporoffithleeaansdsoicsiiantdedepoeuntdgeonintgonwathvee Eˆ (z,ω)(cid:12)(cid:12) =i(cid:114) (cid:126) ω2
out z=0+ ε π c2
driving pulse profile. 0 A
(cid:90)
Desired shapes of outgoing wave packet with high effi- dz(cid:48)(cid:112)ε(cid:48)(cid:48)(z(cid:48),ω)G (0+,z(cid:48),ω)fˆ(z(cid:48),ω). (B1)
out
ciency of one-photon Fock state can be generated by ap- ×
propriately adjusting the emitter–pump interaction for It is useful to introduce the bosonic operators
given emitter–cavity interaction. Since the spatiotempo-
(cid:114)
ralshapeofthewavepacketisrobustagainstfluctuations ˆb (ω)=2 ε0cπA Eˆ (z,ω)(cid:12)(cid:12) , (B2)
ofthepumppulse,theresultsfoundimplythefeasibility out (cid:126)ω out (cid:12)z=0+
to produce identical wave packets of desired shape car-
where
rying a one-photon Fock state each. As an illustration
of the scheme, we have considered the generation of a (cid:2)ˆbout(ω),ˆb†out(ω(cid:48))(cid:3)=δ(ω−ω(cid:48)). (B3)
single-photondouble-peakwavepacket. Infact, theone-
To calculate the quantum state of the outgoing field, we
photonFockstateisequallydistributedamongtwotime-
start from the multimode characteristic functional
(space-)separated single-peak wave packets, which is a
realization of a time-bin entanglement that can be used C [β(ω),t]
out
to perform quantum key distribution [21]. Importantly, (cid:20)(cid:90) ∞ (cid:21)
theschemealsoallowsanoptimizationofthepumppulse = ψ(t) exp dωβ(ω)ˆb† (ω) H.c. ψ(t) . (B4)
profile to match with specific characteristics of the time (cid:104) | 0 out − | (cid:105)
9
To evaluate C [β(ω),t] for the state ψ(t) as given by Accordingly, applying the Baker-Campberll-Hausdorf
out
| (cid:105)
Eq. (12), we first note that from Eq. (12) together with formula, using the commutation relation Eq. (B3) to-
the relation fˆ(z,ω) 0 =0 it follows that gether with Eq. (B6) we may rewrite Cout[β(ω),t],
|{ }(cid:105) Eq. (B4) as
fˆ(z,ω)ψ(t) =C (z,ω,t)e−i(ω+ω31)t 0 3 . (B5)
3
| (cid:105) |{ }(cid:105)| (cid:105) (cid:34) (cid:35)
(cid:88)
Hence, on recalling Eqs. (B1) and (B2), it can be seen C [β (t),t]=exp 1 β (t)2
out i −2 | i |
that i
ˆb (ω)ψ(t) =F∗(ω,t) 0 3 , (B6)  (cid:12)(cid:12)(cid:88) (cid:90) ∞ (cid:12)(cid:12)2
out | (cid:105) |{ }(cid:105)| (cid:105) 1 (cid:12)(cid:12) βi(t) dωFi(ω,t)F(ω,t)(cid:12)(cid:12) . (B12)
where × −(cid:12) i 0 (cid:12)
(cid:114)c ω2 We now choose
F(ω,t)= 2i ei(ω+ω31)t
− ω c2 F(ω,t)
F (ω,t)= , (B13)
(cid:90) 1 (cid:112)
dz(cid:112)ε(cid:48)(cid:48)(z,ω)G∗ (0+,z,ω)C∗(z,ω,t). (B7) η(t)
× out 3
with η(t) being given by Eq. (23). In this way, from
To represent Cout[β(ω),t] in a more transparent form, Eq. (B12) we obtain Cout[βi(t),t] in a ’diagonal’ form
we introduce a time-dependent unitary transformation with respect to the nonmonochromatic modes:
according to
(cid:89)
C [β (t),t]=C [β (t),t] C [β (t),t], (B14)
(cid:88) out i 1 1 i i
β(ω)= F∗(ω,t)β (t), (B8)
i i i(cid:54)=1
i where
(cid:90) ∞
βi(t)= dωFi(ω,t)β(ω). (B9) C1(β,t)=e−|β|2/2(cid:2)1 η(t)β 2(cid:3) (B15)
0 − | |
and
Here, the normalized complex functions F (ω,t) repre-
i
sent nonmonochromatic modes of appropriately chosen
C (β,t)=e−|β|2/2 (i=1). (B16)
frequency distributions where associated nonmonochro- i (cid:54)
matic mode operators read
The Fourier transform of C [β (t),t] with respect to
out i
(cid:90) ∞ the β (t) then yields the (multi-mode) Wigner function
ˆbouti(t)= dωFi(ω,t)ˆbout(ω). (B10) Wout(iαi,t), Eq. (21).
0 Substitution of the formal solution of Eq. (15) for
Note that C3(z,ω,t) [with the initial condition C3(z,ω,0)=0] into
Eq. (B7) and the use of Eq. (A1) yields Eq. (24) (for
(cid:88)
ˆb (ω)= F∗(ω,t)ˆb (t). (B11) details, see Ref. [15]).
out i outi
i
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