Table Of ContentSyn
hronous Quantum Memories with Time-symmetri
Pulses
∗
Q. Y. He, M. D. Reid, and P. D. Drummond
Centre for Atom Opti
s and Ultrafast Spe
tros
opy,
Swinburne Un∗iversity, Melbourne VIC 3122, Australia and
Email: pdrummondswin.edu.au
We propose a dynami
al approa
h to quantum memories using a syn
hronous os
illator-
avity
model, in whi
h the
oupling is shaped in time to provide the optimum interfa
e to a symmetri
inputpulse. This over
omes theknown di(cid:30)
ulties of a
hieving high quantuminput-output(cid:28)delity
with storage times long
ompared to the input signal duration. Our generi
model is appli
able
to any linear storage medium ranging from a super
ondu
ting devi
e to an atomi
medium. We
9 show that with temporal modulation of
oupling and/or detuning, it is possible to mode-mat
hto
0 time-symmetri
pulses that have identi
al pulse shapes oninputand output.
0
2
Quantum memories (QM) are key devi
es both for ferent shapes on input and output
n
a quantum information and fundamental tests of quantum An essential feature of our treatment is that we show
J me
hani
s. A QM will write, store then retrieve a quan- how a smooth, time-symmetri
se
h-pulse
an be stored
TI
9 tum state after an arbitrary length of time. QM devi
es for times longer than , and re
alled with high quan-
1 are
onsidered vital for the implementation of quantum tum (cid:28)delity. This means that the output pulse-shape
networks, quantum
ryptographyand quantum
omput- repli
ates the input pulse. Hen
e, this type of quantum
]
h ing. At a morefundamental level, they
ould enable one logi
an be
as
aded, with inter
hangeable inputs and
p to generate an entangled quantum state in one devi
e, outputs. Thisisavitalfeatureofalllogi
devi
es. Impor-
-
then test its de
oheren
e properties in a di(cid:27)erent lo
a- tantly, we do not use a slowly-varying pulse approxima-
t
n tion. This would allow one to test the equivalen
e of tion[5,6,7,8,9,10℄,aswasrequiredinearlierproposals.
a
the quantum state des
ription for more than one physi- Thisisessential,toallowtheuseoffastpulseswhi
h
an
u
q
al environment. For example, there are proposals that bestoredfortimes mu
hlongerthanthepulse-duration.
[ gravitational de
oheren
e may o
ur beyond the stan- Our theoreti
al
al
ulations are
arried out with sim-
dard model of quantum measurement theory [1℄. This ple non-saturating linear os
illator models that are an-
1
v would be testable with
ontrolled ways to input, store, alyti
ally soluble. Cru
ially, this allows us to
al
ulate
6 then readout a quantum state in di(cid:27)ering environments. pulse-shapes that are dynami
ally mat
hed in time to
5 Theben
hmarksforaQMarestoragetimeandinput- the
avity-os
illator system. This strategy
an be
om-
T
9
output(cid:28)delity. Thememorytime mustbelongerthan bined withavarietyofotherte
hnologies. Thesein
lude
2 TI T >TI
the duration of the input signal: . Otherwise, quantum nondemolition (QND) [11, 12, 13, 14℄ inter-
.
1 the memory is more like a phase-shifter than a memory. a
tions, Raman and ele
tromagneti
ally indu
ed trans-
0
The(cid:28)nalquantumstatemustalsobea
loserepli
aofthe paren
y(EIT)[15,16,17,18,19℄,inhomogeneousbroad-
9
original. In quantitative terms, the mean state overlap ening (CRIB) [20, 21, 22, 23, 24, 25℄, super
ondu
ting
0
v: [2℄ between the intF¯ended and a
hieF¯ve>d Fq¯uCantum sF¯tCates transmission lines and squids [26, 27, 28℄, magneti
on-
(the mean (cid:28)delity ) must satisfy . Here is trolwithatwo-levelatom[29,30℄,nanome
hani
alos
il-
i
X thebestmean(cid:28)delityobtainablewitha
lassi
almeasure latorstorage[31, 32℄, and even intra-
avityBEC devi
es
r andregeneratestrategy[3℄. Furthertothis,anidealQM [33℄. Thisopenssomeex
itingexperimentalpossibilities,
a
proto
olmustenablenumeroussequentialquantumlogi
in
luding
omparisonsof(cid:28)delity in QMdevi
eswith dif-
operationstobeperformed, meaningmanyinput-output ferente(cid:27)e
tivemasses,asafundamentaltest ofde
oher-
(cid:16)quantum states(cid:17),
arried on ingoing and outgoing pulse en
e in quantum me
hani
s.
waveforms. This means that the output pulse envelope Previous QM experiments were frequently limited by
should be identi
al to that of the input. relativelyshortstoragetimes[34℄. Otherdemonstrations
In this paper we proposea new QM proto
ol(Fig. 1), fo
us on retrieval e(cid:30)
ien
y at very high photon number
satisfying all of these
onstraints, in whi
h the (cid:16)state(cid:17) [35℄. However,theseusuallyhaveaverylow(cid:28)delity,sin
e
is stored in a dynami
ally swit
hed
avity-os
illatorsys- the (cid:28)delity at a (cid:28)xed e(cid:30)
ien
y de
reases exponentially
tem. The
avity a
ts as an input-output bu(cid:27)er whi
h withphotonnumber. Asarule,previousproposalseither
syn
hronouslymode-mat
hesthe external input pulse to ignore (cid:28)delity, or use
riteria only appli
able to spe
ial
a long-lived internal quantum linear os
illator. We de- knownstates,like
oherentorsqueezedstates[34,36,37℄.
rivea
onditiononthetime-dependen
eoftheos
illator- It is more useful in both appli
ations and fundamental
avity
oupling required to mat
h to any external pulse- tests to allow for arbitrary input states. Our analysis is
shape, in
luding time-symmetri
pulses. This
ontrasts not restri
ted to any
lass of states, ex
ept for an upper
with previous work, in whi
h the
oupling was a step bound on the input photon number.
fun
tion[4℄,resultinginnon-symmetri
pulseshavingdif- Model. Thequantuminformationinatemporalmode
2
voirsareassumedheretobeintheva
uumstate,without
ex
ess phase or thermal noise. √ηM =aoout/aion
Hen
e, we
an solve Eq (1) to obtain Aout
byintegratingaRoverthepositive-Poutput(cid:28)eld . This
isvalidsin
e
orrespondstoabosoni
operatorwhi
h
onlya
tsonazero-temperaturereservoir,andistherefore
equal to zero for the va
uum state in the positive P-
representation.
Figure 1: (Color online) Proposed dynami
al atom-
avity We will analysethe mode-mat
hing
onditionsfortwo
κ
QM. Tuhi0en
avituyo0u
touples to only one ingoing and outgoing di(cid:27)erent dynami
almodels with (cid:28)xed
avity damping .
mode, and ,anditisthequantumstateofthismode
In order to obtain dynami
al mode-mat
hing we require
that is stored. The pulse shape is optimized for e(cid:30)
ient t < T
an outgoing va
uum state for . In the positive P-
writing and reading of the state onto and from the os
illa-
tor medium inside the
avity. A symmetri
pulse shape is represAeonuttat=ion√th2κisatranAsilnate=s t0o the simplAeirneq=uir√em2κenat
used, so thetime-reversedoutputis identi
al to the input. that − , so that .
Thetwomodels usestrategiesofeithervariable
oupling
or variable detuning to swit
h on and o(cid:27) the
ouping
oAˆfinth(te)propagating single transverse-mode operator (cid:28)eld between the os
illator and the intra-
avity (cid:28)eld. For
is(cid:28)rsttransferredtoaninternal
avitymodewith simpli
ity, we treat the
ase of zero internal damping
aˆ(t) γT <<1
operator , thenwˆbr(itt)teninto theots=
il0latorormemory ( ) in the equations, while still in
luding os
illa-
with mode operator up to time . Subsequently, tordampinginthegraphstodemonstratethatthise(cid:27)e
t
theintera
tionisturnedo(cid:27) ordetuned fora
ontrollable
an be made small if ne
essary. With no loss of general-
T κ=1
storage time . The intera
tion is swit
hed on again af- ity, we
onsider units for whi
h .
T
ter tiAmˆoeut(t,)allotw>ingTreadout into an outgoing quantum
(cid:28)eld at (Fig. 1). We treat quantum infor- 2
b
mation en
oded into single propagating modes that are
1.5 Ain
temporally and spatially mode-mat
hed to the memory
1
oudpeouvetri(
aiento)[(3rts8),a3re9℄aˆ.o0Hute(rine)t=heRreule0ouvta(nint)⋆in(tp)uAˆtoaunt(dino)(utt)pdut,twmhoedree outA, b 0.5
0 is understood to represent the output (input) inA, 0
temporal mode shape.
−0.5
We use theAˆpouots(iitni)v,eaˆoPut-(rienp)raˆe,seˆbntation [40℄, in whi
h −1
all operators Aout(in),a0out(in)a, bare formally repla
ed Aout
by
-numbers 0 . Using input-output −1−.510 −5 0 κ t 5 10 15
theory[41℄, the resulting dynami
al equations are:
a˙(t) = (κ+iδ(t))a(t)+g(t)b(t)+√2κAin(t)
− Figure 2: (Color online) Case 1: Cavity input (dashed) and
b˙(t) = (γ+i∆(t))b(t) g(t)a(t)+ 2γBin(t). output amplitudes (solid). The dotted line gives the os
illa-
− − p (1)
κ t
oorupalminpglisthuadpee. inThtiemdeags(hte)d.-Hdoetrteedt0
=ya−n5l,inTe=re5p,reas0e=nts1.the
Here is the
avity damping (assumed (cid:28)xed), with
δ(t).
dg(ett)uning The internal
avity-os
illator
oupling is ∆,δ = 0
(assumed variable),γw, ∆hi(let)the damping and detun- 1. Variable
oupling ( ). In this approga(
th),,
ing of the os
illatorinare respe
tively, with an os- we propose that the
avity de
ay is (cid:28)xed, and that
B
illator reservoir . These equations
an be applied theintera
tionofthe
avity(cid:28)eldwiththelinearmeAdiinum=,
to a range of experiments ranging from solid-state
rys- √is2sκwait
hed. Durinag(tth)e input stage, the relation
tals or
old atoms to super
ondu
ting
avities or nano- meansAthina(tt) is predetermined for any degs(itr)ed
os
illators. The
ompleteness of the representation al- mode-shape . Thisγgives0an expressgio(tn)f=or b˙/a,
lows us to treat any quantum state or memory proto
ol. sin
e from Eq (1), with → , one has − .
Sin
e the equations are linear, the overall time-delayed Hen
e:
input-output relationship must be given by: [a˙ gb]/a=a˙/a+(b˙2)/(2a2)=1.
− (3)
out in R
ao =√ηMao +p1−ηMa . (2) a =Inaorsdeecrht(otreatli)ze a time-symmetri
input mode with
0 0
ηM − , from Eq (b3)=wea0seete−tt0hsaetchth(te intte0r)-
Here an amplitude retrieval e(cid:30)
ieanR
y is introdu
ed nal (cid:28)eld amplitude must be − .
forthetime-delayedread-out,and representstheover- The optimal shapge(to)f=the b
˙/aavit=y-os
siellcaht(otr
otup)ling in
0
all e(cid:27)e
ts of the loss reservoirs. For simpli
ity, all reser- time is therefore − − − . This
3
a0 F¯n¯ = 1/[1+n(1 √ηM)2]
is independent of the amplitude whi
h en
odes the mean (cid:28)delity is − . These (cid:28)-
quantuminformation. Here the
ouplingissyn
hronized delities may
orrespond to quite high e(cid:30)
ien
ies, sin
e
to t0, whi
h is the pulse arrival time. The quantum ηM >[1−p1/(n+1)]2isneededforaQM.Withn=20,
ηM >0.61
memory readout is obtained simply by time-reversal af- QM should be a
hieved for , provided there is
ter half the memory storage time has elapsed, so that no other de
oheren
e.
g(t)=g(T t)
− . The resulting output mode is alsotime- For many quantum information appli
ations, a larger
rAeovuetrs=ed a√n2daissecuhn(
Tha+ngted ta)part from being inverted:
lass of possible quantum inputs is needed. In the most
0 0
− − . Atypi
alresult isshown general
ase, we
an de(cid:28)ne the input state as any state
nm
in Fig. 2, from integrating Eq (1). withaphotonnumberboundedby . This
orresponds
to an arbitrary state (cid:12)(cid:12)Ψ~E of 1+nm levels. Fnm is then
(cid:12)
2
1.5 Ain b tthhee
aovnersatrgaein(cid:28)tdethliatyt(cid:12)(cid:12)oΨ~ve(cid:12)(cid:12)r=al1l.pToshseib(cid:28)lede
loitey(cid:30)l
imienittsfosra(tiimsfypienrg-
(cid:12) (cid:12) 1+nm
outA, b 0.15
fisea
nFt¯)nb
mlmeau≤rllty2ip/gl(eenn
melro+antien2)gan[o4yf5,annu4m6a℄.brbeSirtirnoa
fre
yoap
ielassosif
aalleqvmuealenmsttouartmye
inA, 0 state, it must be
onstrained by this (cid:28)delity bound also.
−0−.15 Detunings−1100000 ∆ δ δ’ ∆’ a1nWdnem2
a=nn2o,wwh
ai
lh
ualalltoewtshfeo(cid:28)rdaerlbitiytrianrythseta
taessewoiftnhmup=to1
−5 0 5 10 15 Aout and photonsrespe
tively. Aftertra
ingoverthereser-
−1.5
−10 −5 0 κ t 5 10 15 voir modes, we obtain the predi
ted memory (cid:28)delities in
our beam-splitter model of [4℄:
ηM +2√ηM +3
Figure 3: (Color online) Case 2: Cavity input (dashed) and F1 =
6
output amplitudes (solid). Other lines and parameters as in
Fig (δ2()t.) The inset gives the detuning shapes in time: ∆(t) F2 = ηM2 +2ηM√ηM +3ηM +2√ηM +4.
and . 12 (5)
2. Variable detuning. In this approa
h, the
ou- An arbitrary quantum state (cid:28)delity measure gives a
∆(t) δ(t)
pling is
hanged by varyingthe detunings and . better indi
ation of the power of a QM than a measure
g = κ = 1
We
onsider the simplest
ase with and a
onstrained to the
oherent states. For example, any
a = a0sech(t t0) ηM >0.23
symmetri
pulse − . To give a va
uum storagedevi
ewith
anpotentiallybeaquan-
2
output during the writing phase we must have: tum memory for arbitrary states with up to photons.
∆ = i(b˙+a)/b
δ = i(a˙ a b)/a 2 b1
− − (4) 1.5 Ain b3 b2
b = b + ib δ = i(a˙ a
bW)e/asu+ppbos/eat,h∆at = [(b˙1 + a)2b, so tb˙habt]/b2 + i[−(b˙ −+ b 1
a1)b1 + b˙2b22]/b2. 1Im(∆) 2=−Im2(δ1) |=| 0 1 outA, 0.5
tthhaatt bb12 ==(aa˙0|−e|t−at)0Ssi=enc
h−e(at0−et−t0t0)tsaenchh2((tt−−tt00)).,Fa,inndwalehlye,(cid:28)nn
tdoe inA, −0.05 γ / κ = 0.05 A3out
−1 γ / κ = 0.0125 A2out
eore−bat(ltai−zinte0)sftryaomnmhm(tEet−qrit
(04)i)n+ptsuhetechar(netqd−uiotr0ue)dt,pδud=tetpueutn−lisnte0gtsasnhoahfp:(te−s∆,tw0=)e. −1−.510 −5 0 κ γt / κ = 05 Ao1ut 10 15
After a
ontrollable storage time, the intera
tion is
sδw′ it
hedδba
k b∆y′time r∆eversal of the detunings, so that Figure 4: (Color online) Case 1 with losses: Cavity input
→ − and → − . The readout is obtained as (dashed) and output amplitudes (solid) with various loss ra-
γ/κ = 0, 0.0125, 0.05
before, as shown in Fig. 3. tios during the storage time: . Other
Memory Fidelity. Coherentstates haveproved use- lines and parameters as in Fig (2).
fulinquantumappli
ationssu
hasteleportation[42℄and
quantum state transF¯fecr=fro(m1 +lignh)t/(o2nnto+a1t)oms [11℄. It Os
illator Loss: Figure 4 shows the typi
al input-
is well known that n¯ serves as a output relation for various loss ratios during a storage
T = 5 γ/κ = 0, √ηM = 1
ben
hmark for a QM with a gaussian ensemble of
o- time of duration . For , for
γ/κ=0.0125, √ηM =0.84
herent states [43, 44℄ having a mean photon number we(cid:28)ndane(cid:30)
ien
yof ,while
n¯ γ/κ = 0.05, √ηM = 0.50
. For our beam-splitter solution Eq. (2), the output for we obtain . Here we have
ρˆout(α) = √ηMα √ηMα
for this proto
ol is | ih |, and the numeri
ally integrated Eq (1) and used the integral of
4
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