Improved storage of coherent and squeezed states in imperfect ring cavity

0.13 MB·English

by  Petr Marek

Improved storage of coherent and squeezed states in imperfect ring cavity
Petr Marek and Radim Filip
Department of Optics, Palack´y University,
17. listopadu 50, 772 07 Olomouc,
Czech Republic
(Dated: February 9, 2008)
We propose a method of an improving quality of a ring cavity which is imperfect due to non-
unit mirror reflectivity. The method is based on using squeezed states of light pulses illuminating
the mirror and gradual homodyne detection of a radiation escaping from the cavity followed by
single displacement and single squeezing operation performed on the released state. We discuss
contribution of this method in process of storing unknown coherent and known squeezed state and
generation of squeezing in the optical ring cavities.
4
PACSnumbers: 03.65.Ud
0
0
2
imperfect ring cavity
I. INTRODUCTION

Jan Recently, quantum information processing and quan- OUT ST DT (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)C(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) IN
M2 M1
4 tum communicationexperiments utilizing photons asin-
1 formationcarrier(forareview[1])areperformedwithout
an utilizing advantageof quantum memories. The quan- M(R)
1
tum memories can store a quantum state carrying infor-
v LO
mation for a further processing as well as they can store
0 S BS1
an entanglement resource for an actual time quantum M
8
HD
0 communication. However,the photons are relatively dif- D D
1 ficulttostoreandanimplementationofapracticalquan-
0 tummemoryforphotonsremainsachallengingproblem.
4 Earlier proposals for a quantum memory were mainly i
0 Pk
dedicated to a storing quantum states of individual pho-
/ LOGIC
h tons in a high-Q cavity [2], in collective atomic excita-
p tions [3] or in a fiber loop [4]. To enhance storing indi-
- vidualphotoninthe fiber loop,alinear-opticalquantum
t FIG. 1: Setupfor protection of thecoherent, squeezed states
n computing circuit that runs an error-correction code [6] andsqueezedstategeneration: C–nonlinearcrystal,M1,M2
a wasproposed[5]. Toprotectaqubitagainstdecoherence, –high-qualitymirrors, M(R)–imperfect mirrorwith there-
u
the schemes based on the decoherence-free subspace [7] flectivity R, S – source of the squeezed states, LO – local
q
werealsoproposedandimplementedfortrappedions[8]. oscillator, BS1 – 50:50 beam splitter, HD – homodynemea-
:
v Quantuminformationprocessingwithcontinuousvari- surement,D – detectors, DT – displacement operation, ST –
i squeezing operation.
X ables (CVs) based on a manipulation with Gaussian
r statesofmany-photonsystems[9]representsaninterest-
a ing alternative to the quantum information and commu-
nication protocols exploiting individual photons. In the losses at mirrors deteriorate information encoded in the
CV quantum information protocols, the coherent states coherent state and typically restricts the storage time.
are mainly utilized as the information carrier and the Further, to produce a pronouncedly squeezed state for
squeezedstatesarebasicresourceforaproductionofthe efficientCVquantuminformationprocessing,aringcav-
CV entangled states. To perform more complex and col- ity containing a nonlinear crystal is frequently used in
lective CV quantum informationprotocolswe wouldlike acontinuous-waveopticalparametricoscillator/amplifier
to implement a quantum memory which is able to store (OPO/OPA) [10]. Here an unavoidable losses of the im-
unknown coherent state or known squeezed state for a perfectmirrorsrestrictsamaximalvalueoftheproduced
longtime. Tostoreacontinuous-variableinformationen- squeezing. A method of the squeezing generation in the
coded as simultaneous amplitude and phase modulation ring cavity was used, for example, to produce entangled
of coherent state for a long time we can utilize a sim- state for a teleportation of coherent states [11].
ple quantum memory device based on optical-fiber loop Inthispaperweproposeamethodhowtoincreasethe
or a ring cavity. We lock the continuous-wave field into quality of the ring cavity if we are able to inject an aux-
the high-fidelity fiber loop or ring cavity and release it iliary squeezed light to the cavity through the imperfect
when we need. Recently, a similar method has been pro- mirror and detect a field leaving outside from the mir-
posed to store the polarization qubit in a long length ror by a homodynne detection in every cycle of the field
fiber [4]. However, in the ring cavity an unavoidable in the ring cavity. According to the measured data, a
2
simply joint feed-forward correction by a modulation of a pair of the conjugate quadrature operators X and P
the finalstate canbe performedafter many cycles inthe satisfyingthe commutationrelations[X,P]=i, the k-th
ring cavity and we can preservean unknown coherentor passthroughthe cavitymirrorM(R)canbe represented
knownsqueezedstate for a longtime as willbe shownin by transformation relations
Sec.II.Forthestoringofthecoherentstateitisevennec-
essaryonlytoperformanadditionalsqueezingoperation Xk =RXk−1+TXMk, Pk =RPk−1+TPMk
to obtain a fidelity of the storing which is independent XM′ k =TXk−1−RXMk, PM′ k =TPk−1−RPMk, (1)
on unknown input amplitude. The presented method is
based on CV quantum erasing procedure [12] which al- whereX ,P denote the quadraturevariablesofthe sig-
k k
lows us to restore at least partially the input state after nal mode after k-th round and X , P stand for the
Mk Mk
aninteraction. Thusa reflectivityofthe mirrorisinfact quadrature variable of the meter mode used to inject a
enhanced using the erasing procedure. It can also used squeezedstateintothemirror. Detectingafieldfromthe
′
to enhance maximal amount of the squeezing generated mirrorbythehomodynedetection,theoperatorP col-
Mk
in OPO/OPAin which the ring cavity is filled by a non- lapses on a real number i . Using relation (1) we can
Pk
linear crystal as will be demonstrated in Sec. III. Thus straightforwardly derive an evolution of the quadrature
we could consequently stimulate an increase of fidelity operators
of CV quantum information protocols. This method is
differentto the errorcorrectingcodes since itworksonly N
with information which leaves from the memory unit. XN =RNXin+T XRk−1XMk,
k=1
N
II. PROTECTION OF CV STATE PN =R−NPin−T XR−kiPk (2)
k=1
InthisSection,wedemonstrateusefulnessofquantum after N round trips in the cavity. Here X and P de-
in in
erasing to achieve an enhancement of a time of the stor- scribetheinitialquadratureoperatorsofthesignalmode
age of unknown coherent state or known squeezed state. which we are trying to protect. Now to suppress the de-
Weconsideranemptyimperfectringcavity(withoutthe coherenceeffectinthe signalmode after the seriesofthe
crystal C) depicted in Fig. 1 which consists of two mir- N passages through the mirror,we can use all measured
rorsM1,M2with almostunit reflectivity at a frequency values iP1,...,iPN and implement on signal mode the
of stored field. The third imperfect mirror M(R) has a total displacement operation
lessreflectivityRthanM1,M2howeverstillR>0.99as
is typical in this kind of experiments experiments. The N
rest of the losses and imperfections are assumed to be XN′ =XN, PN′ =PN +T XR−kiPk, (3)
negligible in our analysis. k=1
To demonstrate a protection by the erasing effect, the
followed by additional squeezing operation
standard setup of imperfect cavity (in the box) is com-
pletedbythegeneratorSofsqueezedstates,balancedho- X′
modynnedetectionHD,displacementcorrectionDT and Xout = RNN, Pout =RNPN′ (4)
squeezing correction S . The erasing procedure can be
T
performedasfollows. Ineveryround-tripofthequantum to achieve an universal character of the protection. Uni-
fieldincavity,astatesqueezedinthequadraturevariable versalitymeansthatthemeanvaluesofbothcomplemen-
X ismixedwithastateofaninternalcavitymode and taryvariablesarepreserved. Thustheresultinguniversal
M
the output state leaving from the mirror BS is detected transformation
byhomodynemeasurementproducingthecurrenti pro-
P
N
pmoarntyioncyalclteosainmtheaescuarvedityq,utahderacuturrreenotfstihPe1,fiiePl2d,...D.u,riPinNg Xout =Xin+ RTN Xk=1Rk−1XMk, Pout =Pin (5)
correspondingtoeverycycleareregisteredinacomputer
memory. After N cycles corresponding to storage time, of the quadrature operators is obtained. Through our
a quantum state leaving the cavity by a reducing the re- activities any unknown input state was fully restored in
flectivity of the mirror M2 can be corrected by a total the momentum. Further, the mean value of both the
displacement DT calculated from the measured values coordinateandmomentumareunchangedif the injected
iP1,iP2,...,iPN, knownreflectivity R and total number states have vanishing mean values of the quadratures.
N of the round trips. In addition, we can convert the However the variance of the quadrature X will differ
out
corrected state to another having the same mean values from the variance X . If we consider the injected states
in
as the input state by total squeezing operation ST, de- to be independent and having the same variance σXM ,
pendingonthereflectivityRandnumberN ofthecycles. we can write the variance of X in the following form
out
Now, let us look at this procedure in detail. Using
Heisenberg picture representing each mode of light by σ2 =σ2 +(R−2N −1)σ2 . (6)
Xout Xin XM
3
1
A
0.9 A 1
B
0.8
0.8
0.7
n n
o o
of protecti00..56 B of protecti0.6
elity 0.4 elity D
d d0.4
Fi Fi
0.3
0.2 C 0.2 C
0.1
0 0
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
Number of cycles Number of cycles
FIG. 2: Fidelity of a coherent state in the protected ring FIG. 3: Fidelity of a squeezed vacuumstate in theprotected
cavity: R = 0.99, variance of squeezed quadrature of the ringcavity: R=0.99, thevarianceofsqueezedquadratureof
meter modes σ2 =0.5exp(−2). themeter modes σ2 =0.5exp(−2), signal mode squeezed in
M M
thequadratureP with the variance σ2 =0.5exp(−5).
Pin
From this follows that we can reduce the noise in the
quadratureX aswell if we use a state squeezedin the proposed increases efficiency of storage of these states.
out
coordinatesX andthusachievealmostperfectprotec- We consider and compare four different strategies: (A)
Mk
tion of an unknown coherent or known squeezed state. protectionwithvacuumsqueezedinXk injectedintothe
The reason is that the mixing the signal with appropri- modes Mk and homodynne detections, (B) protection
atelysqueezedstatesonthemirrorM(R)approachesthe with unsqueezed vacuum injected in the modes Mk and
ideal quantum non-demolition measurement of the cer- homodynne detections, (C) unprotected storing and (D)
tain fieldquadrature,whichcanbe reversedperfectly by only injecting vacuum squeezed in Pk in into the modes
the erasing procedure [12]. Note that an amount of fluc- Mk without the homodynne detections.
tuations in the complementary variables P does not To compare the different strategies we use the fidelity
Mk
influencetheproposedprotectionprocedureandonlythe of protection
squeezing of single quadrature is relevant.
For a comparison, without any measurements per- F = (σ2 +σ2 )(σ2 +σ2 ) −1/2, (9)
(cid:2) Xin Xout Pin Pout (cid:3)
formed, the resulting variances would look as
which is the overlap between initial and resulting Gaus-
σX2out =R2NσX2in+(1−R2N)σX2M, sian state. Due to universality of storage only the vari-
σ2 =R2Nσ2 +(1−R2N)σ2 , (7) ances of the initial and resulting state must to be cal-
Pout Pin PM
culate to infer the fidelity. For unknown initial coher-
if the protected state has mean values equal to zero. ent state, the fidelity of protection strongly depends on
However, if this was not true, we would have to per- squeezing of meter mode. In Fig. II we use for illustra-
formphase-insensitiveamplificationtoachieveuniversal- tion a feasible squeezed state generated with the vari-
ity and the variances would be ance σ2 = 0.5exp(−2) corresponding to 3dB squeezing
M
injected to the imperfect mirrorwith R=0.99. It is evi-
σ2 =σ2 +(R−2N −1)σ2 +(R−2N −1)σ2 , dentthatusingerasingwiththesqueezedstateisthebest
Xout Xin XM Xz
σ2 =σ2 +(R−2N −1)σ2 +(R−2N −1)σ2 ,(8) strategy for a long time preservation of unknown coher-
Pout Pin PM Pz
entstate. We canseeaqualitativechangefromanexpo-
with X ,P being the operatorsofancillarymode ofthe nentialdecoherenceinthe caseC tothe non-exponential
z z
amplifier. Qualitatively, the protection method reduces one in the case A. The strategy of squeezed vacuum
noisecompletelyinthequadratureP andalsopartially injection(D) is counterproductivesince anyunbalancing
out
in the quadrature X as can be seen from Eq. (7,8) noiseinthecomplementaryquadraturesatoutputresults
out
Todemonstrateanusefulnessoftheprotectionquanti- in formidable decrease of fidelity.
tatively we fix the reflexivity R of the mirror M(R) and On the other hand, squeezed states are asymmetri-
the squeezing in the auxiliary modes Mk since we will cal in the variances so that we can greatly benefit from
compare different strategies with the same resources. In asymmetry of our method, if we know the orientation
CV quantum computing there are important two basic of the squeezed state which could be protected. In our
types of quantum states, coherent and squeezed vacuum method,anoiseinaquadratureissuppressedatthecost
state, and our aim was to show how the protection we ofblurringtheotherone. Sinceourcorrectioncompletely
4
preserves the squeezed quadrature and the noise added signalfieldduringeachtripinthecavitycanbedescribed
into the conjugate one is diminutive compared to noise by the following transformation relations
already present, the fidelity is almost unity. Also the
squeezingofthe mode Mk is notofgreatimportancefor Xk =RGXk−1+TXMk, Pk = GRPk−1+TPMk
aAlmlaorsgtepredrifsetcitncrteifloenctbivetitwyeeRn, tahsecastnrabteegsieeesn(iAn)Faing.d((IBI)). XM′ k =GTXk−1−RXMk, PM′ k = GTPk−1−RPMk(,10)
arises for a large number N of the rounds in the cavity. forthequadratureoperatorsafterthek-thcycle. HereG
Both other strategies (C,D) are worse than (A,B). Here isa gainofsqueezingpersingle cycleinthe cavitywhich
the fidelity strongly depends on the proportion of vari- is typically small. In an analogy with the previous case,
ances of squeezed quadratures of the signal and meter after the displacement operation
modes. If we consider the input state is squeezed sub-
stantially more then the meter modes, what is reason- T N
able demand, we never find the fidelity for strategy (D) XN′ =XN, PN′ =PN + RX(RG)1−kiPk, (11)
comparable with the protected one. It turned out that k=1
for both unknown coherent and known squeezed state
we find the resulting state have the variances
protection is always the best strategy.
1 1−(RG)2N
σ2 = (RG)2N +T2 σ2 ,
Xout 2 1−(RG)2 XM
III. PROTECTION OF SQUEEZING
1
GENERATION σ2 = (RG)−2N. (12)
Pout 2
Because of a strong squeezing producing a large en- if we considerunsqueezed vacuumto be the initialstate.
tanglement is required for the CV quantum informa- Note, in contrast to protection of the state, we have
tion protocols, the optical cavity often used to form not performed any additional squeezing ST of the signal
OPO/OPA so that the down-converted fields pass the mode after it leaved the cavity.
nonlinear medium numerous times. For the single pass Again we will compare few different strategies, so for
case, the interaction in continuous-wave regime is weak evaluation is needed to find the variances of signal field
and requested gain is obtained after sufficient number of quadrature operators
rounds in the cavity, thus equivalently lengthening the
1 1−(RG)2N
interactiondistance. As anexample weassumeringcav- σ2 = (RG)2N +T2 σ2 , (13)
ity filled by a nonlinear crystal exhibiting (degenerate Xout 2 1−(RG)2 XM
or non-degenerate) down-conversion process. It consists 1 R 1−(R)2N
of two mirrors M1,M2 with almost unit reflectivity at σP2out = 2(G)2+T2 1−(GR)2 σP2M (14)
a frequency of down-converted beams and almost unit G
transitivity for the pump beam. This setup operating
if no active protection was performed. Recall now, we
as a continuous-wave frequency degenerate but polar-
are trying to protect the squeezing operation. That is, if
ization non-degenerate subtreshold OPO with collinear
ourinitialstatewasvacuumthenourtargetstateshould
phase-matchedtype-II down-conversionin KTP was fre-
be a minimal uncertainty squeezed state. There are two
quently used in the many experiments [10]. It produces
criteriaofa qualitythat wehavetakeninto ouraccount.
a squeezed state in two linear polarization modes which
First one is overall fidelity between the target and ac-
canbesimplyconvertedtosinglemodesqueezingbyλ/2-
tually produced state, the second one is an amount of
wave plate along a direction π/4 relative to these polar-
achievable single quadrature squeezing. Again, four dif-
izations. The nonlinear crystal is pumped by a pulse
ferentstrategies(A,B,C,D)wereconsideredasinthepre-
from the ring laser with intra-cavityfrequency doubling.
vious Section.
Forsimplicity,apumping partofthe experimentalsetup
Consider a pure squeezed state as a target state and
was omitted in Fig. 1.
ask how is it close to our prepared state. To quantify it
NextweconsiderthatanonlinearcrystalC producing we can count the fidelity
squeezed light pumped by an intensive laser pulse is in-
serted into the cavity to obtain a source of a sufficiently F =[(σ +σ )(σ +σ )]−1/2. (15)
Xout Xtarget Pout Ptarget
squeezed state. Here the imperfection of the cavity mir-
ror decreases the efficiency of the squeezing generation, between target and obtained state and compare it us-
possibly hindering the cumulation of effect at all. We ing the strategies (A,B,C,D). The result is depicted in
can use the previous method of protection to effectively Fig. III. For a small number of cycles the strategy of
reduce a losses in mirror M(R) and thus enhance an ef- squeezed vacuum injection (D) looks better but only be-
ficiency of the squeezing generation. cause meter mode squeezing is still comparable or even
Toprotecttheoperationweapplyaslightmodification better than the squeezing of signal field, so interaction
ofabovedescribedprocedure. Theringcavitycontaining on the mirror actually improves the state in the cavity.
nonlinear medium performing a weak squeezing of the However, at some point the relevant quadrature cannot
5
1 gain RG. From this interpretation is apparent we need
A
reflexivityandgainchoosesothatRG>1toobtainhigh
0.9
fidelity results.
0.8 Next we can study how the evolution of noise in the
0.7 B quadrature which we are trying to squeeze depend on
g
n the number N of cycles. The results can be understand
ezi0.6
e fromFig.III. Thelogarithmicscalewaschosentoclearly
u
q
of s0.5 showwhatishappening. First, forprotectedcase(A,B),
y thevarianceofmomentumisnotdependentonsqueezing
elit0.4
d in modes Mk and it has obviously no theoretical lower
Fi
0.3 bound. Here also is not necessary to use a squeezed me-
D
ter modes for a generation of the squeezing in a single
0.2
quadrature with no respect to the noise in the comple-
0.1 C mentaryone. Ontheotherhand,forthestrategies(C,D)
0 when the protection is not performed the squeezed vari-
0 50 100 150 200 250 300
Number of cycles ances saturate at a point
1−R2
FIG. 4: Fidelity of squeezed state produced in the protected lim σ =G2 σ (16)
ring cavity: R=0.99, the gain of squeezing in a single cycle N→∞ Pout G2−R2 PM
G = exp(0.02), the variance of the squeezed quadrature of
meter modes σ2 = 0.5exp(−2). A target mode is squeezed determined by the proprieties of the cavity and the used
M
in quadratureP with variance σ2 =0.5exp(−5). squeezinginthemodesPk. InFig.III weuseinthecase
Ptarget
(D)thesameamountofsqueezingσ P =0.5exp(−2)as
M
inthecases(A,B)howeverinthecomplementaryquadra-
0
ture.
From this follows, if we are interested only in achiev-
−1
ing squeezing in one quadrature, what is often sufficient
C
in the CV quantum information protocols, such as opti-
−2
mal CV teleportation, or protection of state depicted in
previous section, it is not necessary use a squeezing in
P−3
ar the modes Mk to protect the squeezing. We need only
V D
log −4 performaneffectivehomodynemeasurementfollowedby
a single displacement operation at the end.
−5 A=B
target
−6 IV. CONCLUSION
−70 50 100 150 200 250 300 We have shown how collective quantum erasure per-
Number of cycles
formed on single mode in different times can be used to
increase the quality of the cavity and the efficiency of
FIG. 5: Logarithm of squeezed variance of state produced in
the protected ring cavity: R = 0.99, the gain of squeezing storing and squeezing generation processes within. Our
in a single cycle G = exp(0.02), the variance of squeezed methodformidablyincreasesthefidelityofstoringofun-
quadrature of meter modes σ2 = 0.5exp(−2), A target knowncoherentstate,butduetoitsasymmetricalnature
M
mode is squeezed in quadrature P with variance σ2 = allowsalmostperfectstorageofknownsqueezedstate. If
Ptarget
0.5exp(−5). ourprotocolisappliedtoprocessofsqueezinggeneration
itremovesthe lowerboundforonequadraturesqueezing
attainable and also allows production of pure squeezed
be squeezedfurther due to the cavity lossesandeachcy- states with high fidelity.
cle in cavity only adds additional noise to the conjugate
quadraturewhatresultsinafidelitydecrease. Ifweapply Acknowledgments
the correctionprocedurethe squeezingbuildupisslower, The work was supported by project 202/03/D239 of
but inevitable. We can representsqueezing protected by TheGrantAgencyoftheCzechRepublic,bytheprojects
our method as nearly ideal (squeezed quadrature ideal, LN00A015andCEZ:J14/98oftheMinistryofEducation
antisqueezedquadratureslightlydisturbed),butwiththe of Czech Republic.
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