Quantum-state extraction from high-Q cavities

0.71 MB·English

by  M. Khanbekyan

Quantum-state extraction from high-Q cavities
M. Khanbekyan,1,∗ L. Kno¨ll,1 A. A. Semenov,2,3 W. Vogel,2 and D.-G. Welsch1
1Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena, Max-Wien-Platz 1, D-07743 Jena, Germany
2 Fachbereich Physik, Universit¨at Rostock, Universit¨atsplatz 3, D-18051 Rostock, Germany
3Institute of Physics, National Academy of Sciences of Ukraine, 46 Prospect Nauky, UA-03028 Kiev, Ukraine
(Dated: February 9, 2008)
Theproblemofextractionofasingle-modequantumstatefromahigh-Qcavityisstudiedforthe
case in which the time of preparation of the quantum state of the cavity mode is short compared
with its decay time. The temporal evolution of the quantum state of the field escaping from the
cavity is calculated in terms of phase-space functions. A general condition is derived under which
the quantum state of the pulse built up outside the cavity is a nearly perfect copy of the quantum
state the cavity field was initially prepared in. The results show that unwanted losses prevent
4 the realization of a nearly perfect extraction of nonclassical quantum states from high-Q optical
0 microcavities with presently available technology.
0
2 PACSnumbers: 42.50.Dv,42.50.Gy

n
a
J I. INTRODUCTION the various decoherence effects associated with, e.g., the
1 motion and spontaneous emission of the atoms as well
2 High-QcavityQEDhasofferedanumberofnovelpos- as scattering, absorption, and transmission of the pho-
ton field. Several proposals have been made to reduce
sibilities of quantum-state engineering (see, e.g, Refs. [1,
3 the effect of decoherence due to the atomic motion [19].
2,3, 4] andreferencestherein). Inparticular,it provides
v In order to reduce unwanted spontaneous emission in
promising tools to generate nonclassical quantum states
1
schemesthatexploitmulti-levelatoms,adiabatictransfer
2 of atoms and light for further use. Accordingly various
techniques have been promising [16, 17, 18]. Moreover,
1 applications have been proposed, with special emphasis
8 on quantum communication and computation [5, 6, 7]. the adiabatic passageis the main idea of the proposalof
0 Incontrasttoatomicstates,theusageofquantumstates quantumnetworksoftrappedatoms,wherecavitymodes
3 provide communication channels, by leaking out of the
of light as carriers of quantum information is especially
0 cavitiesandpropagatingviaopticalinterconnectors[20].
appropriate, due to the reliability of light to propagate
/
h over long distances [8, 9]. Scattering and absorption losses, which are unavoidably
p connected with any material system, may be reduced by
Various schemes for generating nonclassical light in
- well designed cavities and the use of materials showing
t cavities have been considered. They are typically based
n extremely weak absorption,and almost perfectly reflect-
on effective two-level atoms. A generator of single pho-
a ingmirrorsreduce the transmissionlosses. Inparticular,
ton Fock states in an active microcavity with pump self-
u
in Ref. [21] a method is presented for the protectionof a
q regularization has been presented [10]. It has also been
genericquantumstateofacavitymodeagainstthedeco-
: shownexperimentallythatanentangledstateoftwonon-
v heringeffectsofphotonlossesbyfeedbackatomscrossing
degenerate cavity modes can be produced with means
i the cavity mode.
X of a sequence of differently tuned interactions of a pair
r of single atoms with the two cavity modes [11], and a On the other hand, transmission losses are necessar-
a scheme for entangling two modes of spatially separated ily required to be taken into account in order to imple-
cavities by consecutively passing through them atoms ment high-Q cavities that can serve as sources that emit
has been purposed [12]. A scheme has been proposed nonclassicalradiationfor further use outside the cavities
[13] and experimentally realized [14] for the generation [17, 22, 23, 24, 25]. The natural question arises whether
of photon number states on demand, by subjecting sin- or not nonclassical states of light, once generated inside
gle two-level atoms passing through a cavity to π pulse a high-Q cavity, can be extracted from the cavity and
interaction. Similarly, a proposal has been made for en- what the ultimate limits are. In the schemes considered,
tangling two cavity modes via interaction with a bunch it is often made the ad hoc assumption of nearly perfect
of two-level atoms assisted by a strong classical driving extraction(see,e.g.,Refs.[10,18,20,26]). Thefacthow-
field[15]. Schemesthatexploitmultilevelatomshavealso ever is that even very small material absorption may be
been studied both theoretically [16, 18] and experimen- expected to lead to drastic quantum state degradation
tally [17], with special emphasis on effective three-level [27].
atoms of Λ type. Recently,homodyne detectionofthe quantumstateof
The main obstacles to generate nonclassicalstates are the field leaving a high-Q cavity has been studied theo-
retically[28]. Fromthe resultsitmightbe expectedthat
the quantum state, in which an excited cavity mode is
preparedat some initial time, can be perfectly extracted
∗E-mailaddress: [email protected] from the cavity, so that after sufficiently long time, i.e.,
2
when the cavity is effectively empty, the pulse which has Let̺ˆ bethequantumstateanexcitedcavitywaveis
cav
leftthecavityisinthe samequantumstateasthe cavity prepared in at some initial time t . For times t t , the
0 0
≥
mode initially was. However, in the analysis the effects corresponding Langevin equation for the photon annihi-
of unwanted losses (such as absorption and scattering lation operator associated with the excited mode then
losses) on the extracted quantum state have not been reads
considered. Moreover, instead of calculating the quan-
tum state of the outgoingfield directly, the authors base aˆ˙ =−i ωcav− 2i (γrad+γabs) aˆ
thederivationonanoperationaldefinitionoftheWigner
(cid:2) c 1/2 c (cid:3)1/2
functionintermsofcollectivemodeoperatorsintroduced + Tˆb (t)+ Acˆ(t). (1)
in
withinthe frameofthe homodynedetectionschemecon- 2l 2l
(cid:16) (cid:17) (cid:16) (cid:17)
sidered. The reason is that they claim that due to the
In the first term,
mode continuum outside the cavity the Wigner function
would be ill defined. c
γ = T 2 (2)
Inthispaper,wedirectlycalculateasafunctionoftime rad 2l| |
thequantumstateofthepulsewhichleavesahigh-Qcav-
ity and may be used for further processing. The calcu- is the decay rate of the cavity mode which results from
lationsare performedfor arbitrarys-parametrizedphase the transmissionlossesdueto theradiativeinput-output
space functions, including the Wigner function. Taking coupling, and
into account both transmission and unwanted losses of
c
the cavity mode, we show that the crucial parameter for γ = A2 (3)
abs
2l| |
the efficiency of quantum state extraction is the ratio
of absorption losses to transmission losses of the cavity isthe decayrate whichresults fromthe unwantedlosses,
mode. As we will see, a quantum state can be almost briefly referred to as absorption losses in the rest of the
perfectlyextractedaftersufficientlylongtime,onlyifthe paper,suchas the unavoidablyexisting materialabsorp-
value of this ratio is sufficiently small, thereby the truly tion andscattering. For a high-Q cavity, both the trans-
required smallness sensitively depending on the nonclas- missioncoefficientT andtheabsorptioncoefficientAare
sical features of the state. very small comparedwith unity (T 1, A 1). Note
The outline of the paper is as follows. In Sec. II the that T and A are taken at the |cav|≪ity-m|od|e≪frequency
modelisexplainedandthebasicequations,includingthe ω . The second term in Eq. (1) is the Langevin noise
cav
operatorinput-outputrelations,aregiven. Thequantum force arising from the input radiation field, where
state of the outgoing field is calculated in Sec. III, and
Sec. IV presents two examples. Finally, a summary and 1
ˆb (t)= dωˆb (ω,t)
some concluding remarks are given in Sec. V. in √2π in
Z∆ω
1
= dωˆb(ω,t )e−iω(t−t0), (4)
0
II. BASIC EQUATIONS √2π Z∆ω
andthe thirdtermisthe Langevinnoiseforceassociated
A. Quantum Langevin equation
with absorption, where
Let us consider a one-dimensional high-Q cavity 1
cˆ(t)= dωcˆ(ω,t)
boundedwithaperfectlyreflectingmirroratx=0andan √2π
almost perfectly reflecting mirror at x=l. For a high-Q Z∆ω
cavity, the widths γk of the cavity modes at frequencies = 1 dωcˆ(ω,t )e−iω(t−t0). (5)
0
ω =kπc/lareverysmallcomparedwiththeirseparation √2π
k Z∆ω
∆ω=ω ω =πc/l, where c is the velocity of light.
k+1 k
Being inter−ested in resolving times that are large com- Hereandinthefollowing,thenotation dω...isused
∆ω
pared with the time of propagation of light through the to indicate that the integration runs over frequencies in
R
cavity, we may expand the intracavity field in terms of the interval [ωcav ∆ω/2, ωcav+∆ω/2]. The operators
−
standing waves at frequencies ω , where the associated aˆ(t),ˆb(ω,t),andcˆ(ω,t)satisfythefamiliarbosonicequal-
k
photon creation and annihilation operators aˆ† and aˆ , time commutation relations
k k
respectively, obey quantum Langevin equations [29, 30].
For sufficiently large Q values, we may further assume aˆ(t),aˆ†(t) =1, (6)
that the time of excitation and preparation of a cavity
(cid:2) (cid:3)
waveina(desired)quantumstateisshortcomparedwith
itsdecaytime(butstilllongcomparedwiththepropaga- ˆb (ω,t),ˆb† (ω′,t) =δ(ω ω′), (7)
in in −
tiontimethroughthecavity). Inthiscase,theprocessof
(cid:2) (cid:3)
preparationofthecavityquantumstateiswellseparated
fromthe processofits transmissiontothe outsidespace. cˆ(ω,t),cˆ†(ω′,t) =δ(ω ω′). (8)
−
(cid:2) (cid:3)
3
It is not difficult to see that the solution of Eq. (1) can ofEq.(16)(withrealT)isbasedonquantumnoisetheo-
be given in the form of rieswhereasinRef. [30]amore rigorousQEDderivation
is given (also see Ref. [31]). Equation (16) corresponds
aˆ(t) = aˆ(t0)e−i[ωcav−2i(γrad+γabs)](t−t0) tothefollowingequationforthecontinuous-modeoutput
operatorsˆb (ω,t):
+ 2cl 1/2T tdt′e−i[ωcav−2i(γrad+γabs)](t−t′)ˆbin(t′) ˆb (ω,to)ut
(cid:16) (cid:17) Zt0 out
+ c 1/2A tdt′e−i[ωcav−2i(γrad+γabs)](t−t′)cˆ(t′). (9) = c 1/2T lim 1 t1dt′e−iω(t−t′)aˆ(t′)
(cid:16)2l(cid:17) Zt0 (cid:16)2l(cid:17) t1→t+ √2π Zt0
+Rˆb(ω,t )e−iω(t−t0). (18)
0
B. Input-output relation
The proof of this equation is straightforward. Substi-
tuting Eq. (18) into Eq. (10), performing the frequency
In close analogy to Eq. (4), output operators
integralasbefore(i.e.,extendingtheintegrationlimitsto
1 ),andrecallingEq.(4),weexactlyarriveatEq.(16).
ˆbout(t)= √2π Z∆ωdωˆbout(ω,t) f∓Nuo∞lfitelltthhaetcˆboomutm(ωu,tta)tiaonndruˆb†oleut((1ω1,)t.), as given by Eq. (18),
= 1 dωˆb(ω,t )e−iω(t−t1) (t<t ) (10) We now substitute Eq. (9) together with Eqs. (4) and
1 1
√2π (5) into Eq. (18) to obtain
Z∆ω
canbe introduced, where,similar to Eq.(7), the bosonic ˆb (ω,t)=F∗(ω,t)aˆ(t )+Bˆ(ω,t), (19)
out 0
commutation relation
where the function F(ω,t) is defined by
ˆb (ω,t),ˆb† (ω′,t) =δ(ω ω′) (11)
out out − i c 1/2
F(ω,t)= T∗eiω(t−t0)
isvalid. Tak(cid:2)ingintoaccounttha(cid:3)t,onthetimescaleunder √2π 2l
consideration, the lower and upper integration limits of (cid:16) (cid:17)
thefrequencyintegralscanbeextended,withlittleerror, exp −i ω−ωcav−2i(γrad+γabs) (t−t0) −1,
to and + , respectively, from Eqs. (4) and (10) × ω ω i(γ +γ )
tog−et∞her with t∞he commutation relations (7) and (11) it (cid:8) (cid:2) − cav− 2 rad a(cid:3)bs (cid:9)
(20)
then follows that the commutation relations
and the operator Bˆ(ω,t) is a linear functional of the op-
ˆb (t),ˆb† (t′) =δ(t t′) (12)
in in − eratorsˆb(ω,t ) and cˆ(ω,t ) according to
0 0
and (cid:2) (cid:3)
Bˆ(ω,t)= dω′G∗(ω,ω′,t)ˆb(ω′,t )
ˆb (t),ˆb† (t′) =δ(t t′) (13) 0
out out − Z∆ω
may be regard(cid:2)ed as being val(cid:3)id. In a similar way, from + dω′H∗(ω,ω′,t)cˆ(ω′,t ). (21)
0
Eqs. (5) and (8) we derive Z∆ω
cˆ(t),cˆ†(t′) =δ(t t′). (14) Here, the functions G(ω,ω′,t) and H(ω,ω′,t), respec-
− tively, are defined by
Other important(cid:2)commutat(cid:3)ion rules are
G(ω,ω′,t)=T∗ξ(ω,ω′,t)+R∗eiω′(t−t0)δ(ω ω′) (22)
aˆ(t),ˆb (t′) = aˆ†(t),ˆb (t′) =0 if t<t′. (15) −
in in
and
T(cid:2)heoutputo(cid:3)pera(cid:2)torˆbout(t)ca(cid:3)nberelatedtothecavity H(ω,ω′,t)=A∗ξ(ω,ω′,t), (23)
operator aˆ(t) and the input operatorˆb (t) according to
in
the input-output relation where ξ(ω,ω′,t) reads
ˆb (t)= c 1/2Taˆ(t)+Rˆb (t), (16) ξ(ω,ω′,t)= 1 c T∗ 1
out 2l in 2π2l ω−ωcav− 2i(γrad+γabs)
(cid:16) (cid:17)
where eiω′(t−t0) ei[ωcav+2i(γrad+γabs)](t−t0)
−
R= T . (17) × ( ω′−ωcav− 2i(γrad+γabs)
−T∗
eiω(t−t0) eiω′(t−t0)
We renounce to repeat its derivation here, but refer the − . (24)
readertotheliterature[29,30]. InRef.[29]thederivation − ω ω′ )
−
4
It is not difficult to see that from Eq. (19) together with where F (t) and Bˆ (t) being defined according to
n n
the commutation rules (6) and (11) it follows that Eqs.(30)and(31),respectively,withF(ω,t)andBˆ(ω,t)
instead of β(ω) andˆb (ω,t), respectively, and Eq. (25)
out
Bˆ(ω,t),Bˆ†(ω′,t) =δ(ω ω′) F∗(ω,t)F(ω′,t). (25) changes to
− −
h i
Note that Bˆn(t),Bˆn†′(t) =δnn′ −Fn∗(t)Fn′(t). (34)
aˆ(t ),Bˆ(ω,t) = aˆ(t ),Bˆ†(ω,t) =0. (26) Let us a(cid:2)ssume that th(cid:3)e (initial) density operator ̺ˆ is
0 0
factorable as
(cid:2) (cid:3) (cid:2) (cid:3)
̺ˆ=̺ˆ ̺ˆ ̺ˆ (35)
III. QUANTUM STATE OF THE OUTPUT cav⊗ in⊗ abs
FIELD
(̺ˆ , density operator of the cavity mode; ̺ˆ , density
cav in
operator of the input field; ̺ˆ , density operator of the
abs
A. Characteristic functional
dissipative system responsible for absorption). Substi-
tuting Eq. (33) into Eq. (29), we may write, on recalling
To calculate the quantum state of the output field Eq. (26),
inthefrequencyinterval[ω ∆ω/2, ω +∆ω/2],we
cav cav
−
start from its characteristic functional N
C (β,t)=Tr ̺ˆ exp β F (t)aˆ†(t ) H.c.
out cav n n 0
Cout[β(ω),t] ( "nX=1 − #)
N
=Tr(cid:26)̺ˆexp(cid:20)Z∆ωdωβ(ω)ˆb†out(ω,t)−H.c.(cid:21)(cid:27), (27) ×Tr(̺ˆin⊗̺ˆabsexp"n=1βnBˆn†(t)−H.c.#). (36)
X
where ̺ˆ is the density operator of the initial quantum
In what follows we consider the case in which both the
state of the overall system, i.e., its quantum state at
input field and the dissipative system are initially in the
t=t . To further handle the functional, it is convenient
0 vacuum state. In this case, the second trace in Eq. (36)
to regard the integral as the limit of a sum, perform the
simply reduces
calculationsforthe sum, andtakethe limitatthe endof
the calculations. That is to say, we write N
Tr ̺ˆ ̺ˆ exp β Bˆ†(t) H.c.
C [β(ω),t]= lim C (β,t) (28) ( in⊗ abs " n n − #)
out out n=1
N→∞ X
N N
[β (β ,β ,...,β )], where
≡ 1 2 N =exp−21 |βn|2− βnFn(t)βn∗′Fn∗′(t),
N  nX=1 n,Xn′=1 
C (β,t)=Tr ̺ˆexp β ˆb†(t) H.c. . (29)  (37)
out ( " n n − #)  
n=1
X
which can be easily proved to be correct by recall-
Here, β andˆb (t), respectively, are defined by ing the commutation rule (34) and applying the Baker-
n n
Campbell-Hausdorffformulatowritetheexponentialop-
ωn+δω/2 erator in normal order. Combining Eqs. (36) and (37),
1
β = dωβ(ω)=√δωβ(ω ) (30) we may rewrite Cout(β,t) as
n n
√δω
Z
ωn−δω/2 Cout(β,t)
and N
ωn+δω/2 =exp"−12 |βn|2#e21|λ(β,t)|2Ccav[λ(β,t)]. (38)
n=1
1 X
ˆb (t)= dωˆb (ω,t) (31)
n √δω out Here,
Z
ωn−δω/2
C (β)=Tr ̺ˆ eβaˆ†(t0)−β∗aˆ(t0) (39)
(δω=∆ω/N). Note that cav cav
h i
is the characteristicfunction ofthe quantum state ofthe
ˆbn(t),ˆb†n′(t) =δnn′. (32) cavity mode, and the function λ(β,t) is defined by
The discrete versi(cid:2)on of Eq. (19(cid:3)) then reads
N
λ(β,t)= F (t)β . (40)
ˆb (t)=F∗(t)aˆ(t )+Bˆ (t), (33) n n
n n 0 n n=1
X
5
Equation (38) relates the multidimensional character- In fact, this transformationcorresponds to the introduc-
istic function of the quantum state of the multimode tion of non-monochromaticmodes, the phase-spacevari-
output field (at time t t ) to the characteristic func- ables of which are given by
0
≥
tion of the cavity-mode quantum state (at time t ). Let
0 N
Cout(β,t;s) and Ccav(β;s0) be the respective character- α′ = U α . (48)
istic functions in arbitrary s- and s -order, respectively. m mn n
0
n=1
TheextensionofEq.(38)validfors= s =0toarbitrary X
0
In order to diagonalize the quadratic form in the last
values of s and s is straightforward:
0
exponental in Eq. (44), we set
N
F (t)
C (β,t;s)=exp 1 β 2(s 1) U = n , (49)
out "2 | n| − # 1n η(t)
n=1
X
e21|λ(β,t)|2(1−s0)Ccav[λ(β,t);s0]. (41) where p
× N
η(t)= F (t)2, (50)
n
| |
n=1
B. Phase-space functions X
so that, according to Eq. (45), β′ is expressed in terms
1
of the β as
Letus now turnfromthe relationbetweenthe charac- n
teristic functions C (β,t;s) and C (β;s ) to the re- N
out cav 0 1
lation between the corresponding phase-space functions β1′ = Fn(t)βn. (51)
η(t)
n=1
X
1
P (α,t;s)= d2NβC (β,t;s) In this case, the multpimode phase-space function in the
out π2N out new variables, simply reduces to the product of single-
Z
modephase-spacefunctions[P (α,t;s) P (α′,t;s)],
N out out
→
× exp" (αnβn∗ −α∗nβn)#, (42) Pout(α′,t;s)=Pout(α′1,t;s)Pout(α′2,t;s)···
n=1
X P (α′ ,t;s)P (α′ ,t;s), (52)
and ··· out N−1 out N
as it is easily seen from Eq. (44). Obviously, only the
P (α;s )= 1 d2βC (β;s )eαβ∗−α∗β, (43) firstoftheseoutputmodesisrelatedtothe cavitymode,
cav 0 π2 cav 0 whereas all other modes are in the vacuum state. From
Z
Eq. (44) it then follows that the phase-space function of
respectively. TakingCout(β,t;s)fromEq.(41),wederive the relevant output mode is given by (α′≡α′1, β′≡β1′)
1
Pout(α,t;s)= π21N d2Nβ d2αPcav(α;s0) Pout(α′,t;s)= π2 Z d2β′Z d2αPcav(α;s0)
Z Z
N N exp η(t)α∗ α′∗ β′ η(t)α α′ β′∗
exp (α β∗ α∗β ) exp 1 β 2(s 1) × − − −
× " n n− n n # "2 | n| − # nhp i hp i
nX=1 nX=1 −12[1−s−η(t)(1−s0)]|β′|2 , (53)
× exp[λ(β,t)α∗−λ∗(β,t)α]exp 21|λ(β,t)|2(1−s0) . which after integration over β′ yields o
(cid:2) ((cid:3)44)
2 1
P (α′,t;s)=
out
π1 s η(t)(1 s )
To perform the 2N-fold integral over the βn, we change − − − 0
the variables by means of a unitary transformation
2 η(t)α α′ 2
d2αP (α;s )exp | − | , (54)
cav 0
N ×Z "−1−ps−η(t)(1−s0)#
β′ = U β , (45)
m mn n provided that
n=1
X
1 s η(t)(1 s ) 0. (55)
0
− − − ≥
U−1 =U∗ , (46) Note that the caseof equalitysignshouldbe understood
mn nm
as limiting process. We compare Eq. (54) with the well-
thus (cid:0) (cid:1) known relation
2 2β α2
N P(α;s)= d2βP(β;s′)exp | − | ,
β = U∗ β′ . (47) π(s′ s) − s′ s
n mn m − Z (cid:20) − (cid:21)
(56)
m=1
X
6
which is valid for
s′ s 0, (57)
− ≥
and see that the quantum state of the relevant output
mode can be expressed in terms of the quantum state of
the cavity mode in the compact form of
1 α
P (α,t;s)= P ;s′ , (58)
out cav
η(t) " η(t) #
where, for chosen value of s, the vaplue of s′ is given by
1 s
s′ =1 − . (59)
− η(t)
To calculate η(t), we recall that in the limit N
→∞
N
η(t)= lim F (t)2 = dω F(ω,t)2, (60)
n
N→∞n=1| | Z∆ω | |
X
with F(ω,t) from Eq. (20). Straightforward calculation
yields
γ
η(t)= rad 1 e−(γrad+γabs)(t−t0) . (61)
γ +γ −
rad abs
h i
Setting in Eq. (54) s=s =0, we see that the Wigner
0
functionoftherelevantoutputmodeisthefollowingcon-
volution of the Wigner function of the cavity mode with
a Gaussian:
2 1
W (α,t)=
out
π1 η(t)
−
2 η(t)β α2
d2βW (β) exp | − | . (62)
cav
× Z "− p1−η(t) #
This equation reveals that for perfectly extracting a
quantum state from a high-Q cavity, the condition
η(t)
1 (63)
1 η(t) ≫
−
should be satisfied, i.e., the value of the extraction effi- FIG. 1: Wigner function of the quantum state of the pulse
ciency η(t) must be sufficiently close to unity. How close thatleavesthecavity,themodeofwhichis(initially)prepared
to unity – it really depends on the characteristic quan- in a single-photon state. (a) η(t)=0.99; (b) η(t)=0.71; (c)
tum features of the state to be extracted. On the other η(t)=0.5.
hand, from Eq. (61) it follows that
γ
η(t) rad . (64) A. Fock states
≤ γ +γ
rad abs
Note that η(t) γ /(γ +γ ) for sufficiently long A typical nonclassicalstate is ann-photon Fock state,
rad rad abs
times t t '(γ≃ +γ )−1. whose Wigner function reads
0 rad abs
−
W(n)(α)= 2( 1)ne−2|α|2L 4α2 , (65)
IV. EXAMPLES cav π − n | |
(cid:0) (cid:1)
whereL (x) isthe Laguerrepolynomialofordern. Sub-
n
The really required efficiency for nearly perfect quan- stituting Eq. (65) into Eq. (62) and employing the inte-
tumstateextractionsensitivelydependsonthequantum gral representation of the Laguerre polynomials [34],
state that is desired to be extracted. To illustrate this,
letusconsidertwoexamplesofhighlynonclassicalstates, 1 e−xz/(1−z)
L (x)= dz , (66)
namely Fock states and Schr¨odinger catlike states. n 2πi zn+1(1 z)
Iγ −
7
wherethecontourγ enclosestheoriginbutnotthepoint
z = 1, after straightforward calculations we obtain the
Wigner function of the output pulse as
W(n)(α,t)= 2( 1)ne−2|α|2[2η(t) 1]n
out π − −
4η(t)
L α2 . (67)
n
× 2η(t) 1| |
(cid:20) − (cid:21)
From Eq.(67) it is not difficult to see that the condition
1
η(t)>1 (68)
− 2n
must be satisfied to guarantee that the n-photon Fock
state prevails in the mixed output quantum state. In
the simplest case of a one-photon Fock state, n=1, the
conditionreducestoη(t)>0.5. Thatistosay,theweight
of the one-photon Fock state exceeds the weight of the
vacuum state in the mixed state of the outgoing field,
W(1)(α,t)=[1 η(t)]W(0)(α)+η(t)W(1)(α), (69)
out −
only if the extractionefficiency exceeds 50%. The condi-
tion(68)clearlyshowsthatwithincreasingvalueofnthe
required extraction efficiency rapidly approaches 100%.
The dependence on the extraction efficiency of the
quantumstateoftheoutgoingfieldisillustratedinFig.1
forthecaseinwhichasingle-photonFockstateisdesired
to be extracted. Figure 1(a) reveals that nearly perfect
extraction requires an extraction efficiency that should
be not smaller than η(t)=0.99, which for t corre-
→∞
sponds to the requirementthat γ /γ .0.01. As long
abs rad
as η(t)>0.5, the single-photon Fock state is the domi-
nantstateinthemixedoutputstate,ascanbeseenfrom
Fig.1(b)[η(t)=0.71,i.e.,γ /γ =0.429(t )]. For
abs rad
→∞
η(t) 0.5,i.e.,γ /γ 1(t )],thefeaturestypical
abs rad
≤ ≥ →∞
of a single-photon Fock state are lost, Fig. 1(c).
B. Schr¨odinger catlike states
Another example of typically nonclassical states are
Schr¨odinger catlike states, e.g.,
ψ = (α + α ), (70)
| icav N | 0i |− 0i
with α real, and FIG. 2: Wigner function of the quantum state of the pulse
0
thatleavesthecavity,themodeofwhichis(initially)prepared
−1/2 in a Schr¨odinger catlike state given by Eq. (70) with α0=3.
= 2 1+ e−4α20 . (71) (a) η(t)=0.998; (b) η(t)=0.952; (c) η(t)=0.84.
N
h (cid:16) (cid:17)i
The Wigner function of such a state is given by
W (α)= 2N2 e−2|α−α0|2
cav
π
h
+e−2|α+α0|2 +2e−2|α|2cos(4α Imα) . (72)
0
i
8
SubstitutionofEq.(72)intoEq.(62)yieldsthefollowing state in which the cavity mode was prepared. In par-
expression for the Wigner function of the output fields: ticular, the output Wigner function can be given as a
convolution of the cavity Wigner function with a Gaus-
2 2 sian reflecting the unwanted losses.
W (α,t)= N
out
π The crucial parameter for nearly perfect extraction of
e−2|α−√η(t)α0|2 +e−2|α+√η(t)α0|2 a quantum state from a high-Q cavity is the extraction
× efficiency, which in the long-time limit is determined by
+2ne−2|α|2cos 4 η(t)α0Imα e−2α20[1−η(t)] . (73) twhaenrtaedtiolobsestewseaenndthtehecavcaitvyi-tmy-omdoeddeecdaeycaryateradtueedtuoeutno-
h p i o wanted(i.e.,transmission)losses. Thisratiomustbesuf-
FromEq.(73)itfollowsthatnearlyperfectextractionof
ficientlysmallinordertorealizeanearly100%extraction
the state requires the condition
efficiency, where the really requiredsmallness sensitively
depends on the quantum state to be exctracted. In par-
1
1 η(t) (74) ticular, extracting highly nonclassical states can require
− ≪ 2α 2
| 0| extremely small values of this ratio.
to be satisfied. It should be pointed out that evenfor the best optical
Figure 2 illustrates the dependence on the extraction high-Qmicrocavitiesavailabletherequiredefficienciesfor
efficiency of the quantum state of the outgoing field for nearly perfect extraction of nonclassical quantum states
a Schr¨odinger catlike cavity state with α =3. Compar- have not been reached, because the unwanted losses are
0
ing Fig. 2 with Fig. 1, we see that, as expected, the ef- ofthesameorderofmagnitudeasthetransmissionlosses
ficiency for extracting such a Schr¨odinger catlike state [23, 24, 25]. So, in the simplest case of extracting from
is required to be substantially higher than that for ex- a cavity a one-photon Fock state, the weight of the one-
tracting a single-photonFock state. For a nearly perfect photonFockstateexceedstheweightofthevacuumstate
extraction of the chosen Schr¨odinger catlike state, the inthemixedoutputquantumstateonlyiftheextraction
efficiency should be not smaller than η(t)=0.998, i.e., efficiencyisbiggerthan50%. However,the biggestvalue
γ /γ .0.002 for t [Fig 2(a)]. The nonclassi- that has been realized so far in the production of trig-
abs rad
cal interference fringes→typ∞ical of a Schr¨odinger catlike gered single photons by coupling a single semiconductor
state can be observed, at least rudimentarily, as long quantum dot to an optical mode in a micropost micro-
as η(t)>0.84, i.e., γ /γ <0.19 (t ) [Fig 2(b); cavity is about 38% [25]. On the contrary, in case of
abs rad
η(t)=0.952, i.e.,γ /γ =0.05 (t →)].∞For smaller high-Q microwave cavities the absorption losses may be
abs rad
values of the extraction efficiency, t→he∞quantum interfer- small compared with the transmission losses [32].
ences are effectively destroyed [Fig. 2(c)]. We have concentrated on the calculation of the quan-
tum state of the field that leaves a single cavity that is
initiallyexcitedinsomesingle-modequantumstate. The
V. SUMMARY AND CONCLUSIONS theory can also be extended to multimode excitation in
a single cavity as well as multi-cavity systems. We have
We have derived an input-output relation that relates further assumed that the input field is in the vacuum
thequantumstateofthepulseleavingahigh-Qcavityto quantum state. Clearly, the underlying formalism can
the quantum state in which an excited cavity mode was also be applied to the case, in which the input field is
prepared at some initial time. Performing the calcula- prepared in another than the vacuum state. Needless to
tionsinthe phasespace,wehaverepresentedthe respec- say that when the input field is in a thermal state, then
tive quantum states in terms of s-parametrized phase- additional noise is fed into the cavity, and the quantum
space functions and derived a formula that relates the state of the output field also carries additional noise. As
phase-space functions of the outgoing field and the cav- canbe seenfrom Eq.(16), the operatorinput-output re-
ity mode to each other. Taking into account unwanted lationused in this paper does not take into accountthat
lossesofthecavitymode,wehavestudiedthe conditions the input field could be absorbedin the entrance port of
under which a nearly perfect extraction of non-classical the cavity,whichwouldalsogiverisetoadditionalnoise.
quantum states from high-Q cavities should be possible. To include this effect in the theory, the input-output re-
To calculate the quantum state of the outgoing field, lation (16) should be generalized, e.g., by following the
we started from its time-dependent continuous multi- line in Ref. [33].
mode characteristic functional. By appropriate diago- Finally, we have assumed that the process of prepa-
nalization, it can be rewritten in terms of non-mono- ration of the quantum state of the cavity mode is suffi-
chromatic modes, one of which is related to the cav- ciently shortcomparedwith the decaytime ofthe cavity
ity mode, while all other modes remain unaffected by mode, so that the time scales of quantum state prepa-
the cavity mode. In this way, the s-parametrized phase- rationand extractionare well separatedfromeach other
spacefunctionsofthequantumstateoftherelevantnon- and the preparationprocess can be ignoredin the calcu-
monochromatic output mode can be expressed in terms lations. At this point it should be mentioned that one
of s-parametrized phase-space functions of the quantum possible way to reduce the effect of unwantedlosses may
9
be the use of cavities of deliberately enlarged transmis- the question of which quantum state is really obtained
sion,sothattheunwantedlossesbecomesmallcompared outsidethecavityinsuchacase,thepreparationprocess
with transmission losses. When, for example, the radius must necessarily be included in the calculations.
ofamicrospherecavityisdiminished, thenthetransmis-
sionlossesincrease,therebytheabsorptionlossesremain-
ing nearly constant [35, 36]. Since, on the other hand,
the quality factor is reduced, the preparation time may
Acknowledgments
be comparable with the cavity decay time, which is now
determined by the transmission time. So, in the single-
photonemitterexperimentsinRef.[22],inwhichacavity M.K. and D.-G.W. would like to thank Chris-
of a Q value of 6 104 is used, the measured transmis- tian Raabe and Stefan Scheel for valuable discussions.
×
sion time of several microseconds is of the same order of A.A.S. and W.V. gratefully acknowledge support by the
magnitude as the cavity decay time. In order to answer Deutsche Forschungsgemeinschaft.
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