Fock-state stabilization and emission in superconducting circuits using dc-biased Josephson junctions

0.51 MB·

by  J. -R. Souquet

Fock-statestabilizationandemissioninsuperconductingcircuitsusingdc-biasedJosephson
junctions
J.-R. Souquet, A.A. Clerk
DepartmentofPhysics,McGillUniversity,3600rueUniversity,Montreal,Quebec,H3A2T8,Canada
(Dated:January21,2016)
We present and analyze a reservoir-engineering approach to stabilizing Fock states in a superconducting
microwavecavitywhichdoesnotrequireanymicrowave-frequencycontroldrives. Instead,oursystemmakes
useofaJosephsonjunctionbiasedbyadcvoltagewhichiscoupledbothtoaprinciplestoragecavityanda
secondauxiliarycavity. OuranalysisshowsthatFockstatescanbestabilizedwithanextremelyhighfidelity.
6 Wealsoshowhowthesamesystemcanbeusedtoprepareon-demandpropagatingFockstates,againwithout
1 theuseofmicrowavepulses.
0
2
Introduction– The ability to prepare, stabilize and ulti-
n
a matelytransmitnon-trivialquantumstatesiscrucialtoquan-
J tum information processing [1]. In the presence of dis-

0 sipation, state stabilization can be accomplished either via
2 measurement-plus-feedback schemes (see e.g. [2–9]), or

autonomously, using quantum reservoir engineering (QRE)
] FIG.1. Schematicoftheproposedsetup. AdcbiasedJosephson
l techniques[10,11]. Therehasbeenconsiderableprogressin
l junctionofenergyEJ isinserieswithtwocavities.Eachcavityhas
a implementingQREideasinsuperconductingcircuits,includ- animpedanceclosetoR =h/e2,andisdampedviaacouplingto
h ingexperimentswhichhavestabilizedqubitstates[11–13]as atransmissionline. K
-
s wellasphotonicFockstates[14]insidemicrowavefrequency
e cavities. Theseschemesaretypicallycomplex, requiringthe
m
useofseveralhigh-frequencymicrowavecontroltones. statesondemand. IncircuitQEDsetups,propagatingsingle-
.
t Inthiswork,weanalyzeanalternativeapproachtothesta- photonstatesareusuallygeneratedbydrivingmicrowavecav-
a
m bilization of Fock states in a superconducting resonant cav- ities coupled to a qubit with high-frequency control pulses,
ity which requires no microwave control tones. The starting seee.g.[26]. Amongthemanychallengesinthestandardap-
-
d pointforourschemeisasetupdiscussedinanumberofrecent proachistherequirementthatthegeneratedphotonshouldbe
n theoreticalstudies[15–17]andexperiments[18–21],wherea far detuned from the frequency of the control pulse [27–29].
o microwave cavity is coupled to a dc-biased Josephson junc- Our system is capable of on-demand Fock state generation
c
tion. Thedrivenjunctiondoesnotactlikeaqubit,butrather without any microwave control pulses: one simply needs to
[
asahighlynon-lineardrivingelement. Thatsuchasetupcan pulseadccontrolvoltage.

2
produce nonclassical photonic states was first pointed out in Biased junction as a nonlinear drive– To set the stage
v
7 Ref. [16], which showed that states could be produced hav- for our two-cavity system, we start by quickly reviewing the
7 ingsuppressednumberfluctuationsandavanishingg(2)inten- physicsoftheone-cavityversion,asstudiedin[15,16]. The
0 sitycorrelationfunction. Whilethesestatesviolateaclassical system consists of a Josephson junction coupled to a cavity
5 Cauchy-Schwarzinequality,theyaremixturesofthevacuum (frequency ω , modeled as an LC resonator), such that the
c
0
andtheonephotonFockstate,andhavealimitedfidelitywith voltage across the junction is the sum of an applied external
.
1 apureFockstate. Asaresult,theydonotexhibitanynegativ- dc voltage V and fluctuating voltage Vˆ associated with
dc cav
0 ityintheirWignerfunctions[22]. thecavitymode. Thecavityisalsocoupledtoatransmission
6
1 Here, we show how optimally coupling the junction to a line, which is treated (as standard) as a Markovian reservoir,
: secondcavity(asdepictedinFig.1)letsonetranscendthelim- andwhichgivesrisetoanenergydampingrateκ. Wefocus
v
itationsofthesinglecavitysystem,andpreparesingle-photon exclusivelyontemperatureandvoltagessmallenoughthatsu-
i
X Fockstateswithanextremelyhighfidelity. Suchtwo-cavity- perconductingquasiparticlesareneverexcited.
r plus-junctionsystemshaverecentlybeenrealizedexperimen- Working in an interaction picture at the cavity frequency,
a
tally[18],andhavealsobeendiscussedtheoretically[23–25], theHamiltonianofthecavityplusbiasedjunctionis[15,16,
largelyinthecontextofgeneratingcavity-cavitycorrelations. 25,30]
Armouretal.[24]foundnumericallythatsuchasystemcould
generatecavitystateswithweaklynegativeWignerfunctions; E (cid:16) (cid:17)
they did not however discuss Fock state stabilization, or the HˆJ =− 2J e2ieVdctDˆ[α(t)]+e−2ieVdctDˆ†[α(t)] , (1)
particularmechanismweelucidateandoptimize.
Inadditiontoefficientandhigh-fidelityFockstatestabiliza- where E is the Josephson energy, and Dˆ[α(t)] is the cav-
J
tion,weshowthatoursetupcanalsoplayanothercrucialrole: itydisplacementoperator(correspondingtoatime-dependent
itcanactasanefficientmeansforproducingpropagatingFock displacementα(t)). Itisdefinedintermsofthephotonanni-
2
state is reached [14]. For concreteness, suppose we chose
V =kΩ/2e(kaninteger),sothattoleadingorder,Cooper-
dc
pairs can only tunnel by emitting or absorbing k cavity pho-
tons. Ifwetakeω (cid:29)E ,κwecanrestrictattentiontothese
c J
processesandmakearotating-waveapproximation(RWA)to
ourfullHamiltonian. WithintheRWAwehave:
∞
Hˆ =(−1)k+1EJ (cid:88)w [λ]|n+k(cid:105)(cid:104)n|+h.c.. (6)
RWA 2 n+k,n
n=0
Considerthesimplestcasewherek =1,andeachresonantly-
tunnellingCooperpairemitsorabsorbsasingecavityphoton.
√
FIG.2. Matrixelementswn+l,n[λ]forjunction-drivencavityFock Ifwethensetλ = 1/ 2 ≡ λ˜ , thereisnomatrixelement
state transitions, as a function of the zero-point voltage fluctuation 2,1
inEq.(6)foratransitionfrom1to2cavityphotons. Asfirst
amplitude λ for different values of n and l. These functions are
discussed in Ref. [16], the junction induced drive now can
highlynon-linearwithrespecttoλ,andcancancelforspecificval-
ues.Rootsofw [λ],denotedλ˜ ,areindicatedinthefigure. addatmostonephotontothecavity,implyingthatthesystem
n+l,n n+l,n
effectivelyactslikeadriventwo-levelsystem.
The cavity steady-state is found by solving the Linblad
hilationoperatoraˆas masterequationforthedensitymatrixofthecavityρˆ ,
c
Dˆ[α(t)]=eα(t)aˆ†−α∗(t)aˆ, α(t)=2λeiωct. (2) ρˆ˙c =−i[HˆRWA,ρˆc]+κL[aˆ]ρˆc, (7)
which includes the dissipation from the (zero-temperature)
λdeterminestheamplitudeofthezero-pointvoltagefluctua-
(cid:112) transmissionline;here,L[aˆ]ρˆ=aˆρˆaˆ†− 1{ρˆnˆ+nˆρˆ}. When
tions in the cavity, and is given by λ = πe2Z/h where Z 2
λissettoλ˜ ,thestationaryintracavitystatecanbetermed
theimpedanceoftheLCresonator. 2,1
Fromthepointofviewofthecavity,Hˆ describesahighly- non-classical, in that it results in a vanishing g(2) intensity-
J
intensity correlation function [16]. This simply reflects the
nonlinear(butcoherent)cavitydrive[16]:eachtermtunnelsa
factthatthereiszeroprobabilityforhavingtwoormorepho-
Cooper-pair,andalsodisplacesthecavitystateby±α(t). To
tons in the cavity. We are still far however from our goal of
see clearly how these displacements can result in Fock state
producingasingle-photonFockstate. Asthecavityiseffec-
generation,wefollowadifferentroutefrom[16],andexpress
Dˆ[α(t)]directlyintheFockbasis(see,e.g.[31]): tively a driven two-level system, population inversion is im-
possible,andatbestthesteadystateisanincoherentmixture
∞ ∞ havinganequalprobabilityofvacuumandsinglephoton. We
Dˆ[α(t)]= (cid:88)(cid:88)w [λ]|n(cid:105)(cid:104)n+l|e−ilωct stress that such a state exhibits no negativity in its Wigner
n,n+l
n=0l=0 function.
∞ Fock state stabilization– Heuristically, the poor perfor-
(cid:88)
+ (−1)l|n+l(cid:105)(cid:104)n|wn+l,n[λ]eilωct. (3) manceofthesingle-cavitysetupiseasytounderstand:evenif
l=1 weeliminatethematrixelementfor|1(cid:105)→|2(cid:105)transitions,the
junction-driven cavity continues to oscillates back and forth
Here,thetransitionamplitudew [λ]isnothingmorethan
n,n+l
betweenthevacuumandthe|1(cid:105)Fockstate(eventuallyrelax-
ageneralizedFrank-Condonfactor. Forl≥0wehave:
ingintoamixedstate). ToachievetrueFockstategeneration,
w [λ]=e−2λ2(2λ)l(cid:113) n! L(l)(4λ2), (4) one needs to shut off the oscillation dynamics when the sys-
n+l,n (n+l)! n temisinthe|1(cid:105)state.Aswenowdiscuss,thiscanbeachieved
rathersimplybycouplingthejunctiontoasecond“auxiliary”
where L(k) is a Laguerre polynomial [32], and w [λ] =
n n+l,n cavity (see Fig. 1) whose damping rate κ is taken to be
aux
w [−λ].
n,n+l sufficientlylarge.
Asiswellknown,Frank-Condonfactorsarehighlynonlin-
In what follows, we denote quantities for the main “stor-
ear functions of the magnitude of the displacement (here set
age”cavitywithasubscript“s”,whileauxiliarycavityquan-
by λ), and can even exhibit zeros [33]; the behavior of rele-
titieshaveasubscript“aux”;Fockstatesofthetwo-modesys-
vantfactorsisshownFig.2.Weletλ˜ denotethesmallest
n+l,n temaredenoted|n,m(cid:105)=|n(cid:105) ⊗|m(cid:105) . Workinginaninter-
s aux
valueofλwhichmakesw [λ]vanish:
n+l,n actionpicturewithrespecttothefreecavityHamitonians,the
systemHamiltonianis(hereandthroughoutthetext,(cid:126)=1),
w [λ˜ ]=w [λ˜ ]=0. (5)
n+l,n n+l,n n,n+l n+l,n
HˆJ =−E2Je2ieVdctDˆ[αs(t),αaux(t)]+h.c.. (8)
The route to preparing single Fock states now seems
clear: by tuning the value of both V and λ (via the cavity Here Dˆ[α (t),α (t)] = Dˆ [α (t)]⊗Dˆ [α (t)] is the
dc s aux s s aux aux
impedance Z), one can arrange for the effective driving of tensorproductofdisplacementoperatorsforeachcavity,with
the cavity by the the junction to shut off when a given Fock respectivedisplacementamplitudesαj(t)=2λjeiωjt.
3
FIG. 3. Schematic depictions of biased-junction cavity pumping
processes. The boxed digit indicates the number of Cooper pairs
thathavetunnelled,whereasdotsintheparabolasindicateintracavity
photon number. a) Single cavity setup for an impedance yielding
λ = λ˜ . Cooper-pairtunnellingcantakethecavitybetweenthe0
2,1
and1photonstates,buttransitionstothe2-photonstateareblocked.
b)Two-cavitysetupforFockstatestabilization, whereλ = λ˜ .
s 2,1
Startingfromvacuum,Cooper-pairtunnelingcancauseoscillations
betweenthe|0,0(cid:105)and|1,1(cid:105)photonFockstates.Photondecayfrom
theauxcavityhoweverfreezesthesystemintothedesired|1,0(cid:105)state.
Whenthestoragecavityphotondecays(duetoκ ),thecyclerepeats.
s
FIG.4. Steady-stateprobabilityp [n]forthestoragecavitytobe
s
in a Fock state |n(cid:105), as obtained from the RWA master equation in
Eq.(10).WetakeE =κ (cid:28)ω ,ω ,andλ =1/2;λ and
Fortheauxiliarycavitytoplaythedesiredrole,wetunethe J aux s aux aux s
κ areindicatedintheplots.(a)Probabilitieswhenλ istunedclose
s s
voltagesothatVdc =(ωs+ωaux)/2e. Cooper-pairtunnelling toλ˜ (forstabilizing|1(cid:105)).(b)Same,forλ tunedclosetoλ˜ (for
2,1 s 3,2
thusrequiressimultaneouslyemittingasinglephotontoeach stabilizing |2(cid:105)). Even with imperfect tuning, the target Fock state
cavity(orabsorbingaphotonfromeachcavity). Wealsotune isstabilizedwithahighfidelity. (c)and(d)CorrespondingWigner
thestoragecavityimpedancesothatλ =λ˜ ;westressthat functionW[α]forthestoragecavitystate,showingthatnegativeval-
s 2,1
nospecialtuningofλ isneeded.Theresultingdynamicsis uesareobtained;notethatW[α]isrotationallyinvariantinallcases.
aux
sketchedinFig.3b.Similartothesingle-cavitysystem,theef-
fectivedrivingfromthebiasedjunctioncouples|0,0(cid:105)to|1,1(cid:105),
Linbladmasterequation:
butnottostateswithhigherphotonnumber. Wewouldseem
yetagaintohaveaneffectivetwo-levelsystem,andmightex- ρˆ˙ =−i[Hˆ ,ρˆ]+κ L[aˆ ]ρˆ+κ L[aˆ ]ρˆ. (10)
RWA aux aux s s
pectcoherentoscillationsbetweenthesetwostates. However,
the large damping rate κaux of the auxiliary cavity prevents Considerfirsttheidealcase,whereλs istunedperfectlyto
this: ifthesystemisinthe|1,1(cid:105)state,κauxwillcausearapid equal λ˜2,1. In the relevant limit κs (cid:28) κaux, the steady state
decay to |1,0(cid:105). In the absence of storage cavity damping, probabilitythatthesystemisinthedesiredstate|1,0(cid:105)is
the system is then effectively stuck: Cooper-pair tunnelling
against the voltage is impossible (as there are no photons in Γ (E w [λ ]w [λ ])2
(cid:104)1,0|ρˆ|1,0(cid:105)(cid:39) , Γ= J 1,0 s 1,0 aux ,
theauxcavity),whiletunnellingwiththevoltageisimpossi- Γ+κ κ
s aux
bleasthereisnomatrixelementconnecting|1,0(cid:105)and|2,1(cid:105). (11)
Includingstoragecavity-dampingdoesnotruinthephysics:if whereΓplaystheroleofaneffectivepumpingratefrom|0,0(cid:105)
thestoragecavityphotonleaksout,oneisbackinthevacuum to|1,0(cid:105),andwehavedroppedtermsassmallasκs/κaux.The
state,andtheprocessstartsagain. Onethusseesthepossibil- probability to be in the desired state tends to 1 in the limit
ity for having a steady state that has a high fidelity with the Γ (cid:29) κaux (cid:29) κs. The large |1,0(cid:105) population here is anal-
state|1,0(cid:105),i.e. astabilizedsingle-photonstateinthestorage ogous to the population inversion possible in a driven three-
cavity. levelsystem[34].
Tomaketheabovepicturequantitative,wefocusonavolt- Theaboveprocesscanbeefficientevenifthestoragecavity
ageVdc = (ωaux+ωs)/2e,andmakeaRWAonourHamil- impedance is not perfectly tuned. Suppose λs = λ˜2,1 + ε
tonian,yielding: with ε (cid:28) 1. Assuming again κs (cid:28) κaux, one finds that the
probabilitytobein|1,0(cid:105)ismodifiedtobe:
(cid:32) ∞ (cid:33)
Hˆ = EJ (cid:88) w [λ]|n +1(cid:105)(cid:104)n | Γ
RWA 2 ns+1,ns s s (cid:104)1,0|ρˆ|1,0(cid:105)(cid:39) . (12)
Γ+κ +4ε2Γ2/κ
ns=0 s s
(cid:32) ∞ (cid:33)
(cid:88) Becauseoftheimperfecttuningofλ ,itisnolongeradvanta-
⊗ w [λ]|n +1(cid:105)(cid:104)n | +h.c.. (9) s
naux+1,naux aux aux geoustohaveΓ (cid:29) κ (i.e.largeE ),astransitionstohigher
s J
naux=0 states will corrupt the dynamics. Eq. (12) suggests that an
Eachcavityisalsocoupledtoitsownzerotemperaturebath, optimalchoicewouldbetohaveκ = 2εΓ(whilestillmain-
s
andthereduceddensitymatrixρˆofthetwocavitiesobeysthe tainingκ (cid:29)κ ).
aux s
4
To complement the above analytical results, we have per-
formedafullnumericalsimulationoftheRWAmasterequa-
tioninEq.(10)usingtheQuTiPpackage[35]. Fig.4ashows
thestationarystoragecavityFockstatedistributionasafunc-
tionofλ ,anddemonstratesthathigh-fidelityFockstategen-
s
eration is possible despite an imperfect tuning of the storage
cavity impedance. The fidelity is sufficient to give rise to
storage-cavity Wigner function that exhibit large amounts of
negativity(asshowninFig.4c).
The above protocol can also be used to stabilize higher
Fock states with a good fidelity. One keeps the voltage set
to V = (ω +ω )/2e, but now tunes the storage cavity
dc s aux
impedance such that λ = λ˜ for some chosen n > 1.
s n+1,n
The system dynamics will now effectively get stuck in the
state |n,0(cid:105). Numerical results for the case n = 2 are shown
FIG.5. Timedependenceofstoragecavityphotonnumberoccu-
inFig.4bandd.
panciesp [n,t],wherethebiasV =(ω +ω )/2eisturnedon
Westressthatthegeneralschemeherecanbeviewedasan s dc aux s
att=0,andthecavitiesstartfromvacuum.Dashedline:κ chosen
s
example of reservoir engineering [10], with the biased junc- to optimize the steady-state value of p [1,t] (the Fock-state stabi-
s
tionandauxiliarycavityactingasaneffectivedissipativeen- lizationprotocol). Solidline: alternatechoiceofκ whichoptimize
s
vironment which stabilizes the storage cavity in the desired p [1,t]atintermediatetimes(pulsedprotocol).Inthislatterprotocol,
s
Fockstate. Inthatrespect,ourprotocolhassimilaritiestothe thevoltagecanbeturnedofneartoff,resultingwithhigh-probability
Fock-statestabilizationschemedescribedandimplementedin intheproductionofapropagatingFockstateinthetransmissionline
coupledtothestoragecavity. Inbothcases,weassumetherealistic
Ref.[14]. Inthatwork,theengineeredreservoiralsoinvolved
situation where the cavity impedance has not been tuned perfectly
anauxiliarycavity,butusedaqubitandtwomicrowavecon- (here,λ=0.95λ˜ ). Theinsetshowsp [n,t→∞]forthepulsed
21 s
trol tones (or more, if the target Fock state has n > 1). In
protocol.
ourwork,theauxiliarycavityisstillthere,butthemicrowave
control tones and qubit have been replaced with a Joseph-
son junction biased by a dc-voltage. Note that conversely to
having a single storage-cavity photon is a rather broad func-
Ref. [14], reaching higher Fock states does not require addi-
tionoftime,meaningthatonedoesnotneedprecisecontrolof
tionalresources.
theshut-offtimeofthedc-voltage. Thisisinstarkcontrastto
Itinerant Fock states on demand– The above protocol for
standardprotocolsforpreparingaFockstateusingtwo-level
stabilizingFockstatescannaturallybeusedtoproduceprop-
system dynamics (e.g. in the one-cavity version of our sys-
agatingFockstates: afterthedesiredFockstatehasbeensta-
tem),whereoneneedsaprecisecontrolofthedurationofthe
bilized,onesimplyturnsoffthedcbiasvoltagewhentheitin-
pulsescontrol.
erantphotonisdesired,andthestoragecavityFockstatewill
Conclusion– We have shown how a system where a
beemittedintothetransmissionlinecoupledtoit(inatempo-
voltage-biased Josephson junction is coupled to two cavities
ralmodehavinganexponentialprofile[36,37]). Analternate
can be used to stabilize Fock states with a high efficiency.
strategy is to exploit the transient dynamics of our scheme,
While our approach does require one to carefully tune the
andproduceanitinerantFockstatewithapulseddcvoltage.
impedance of the main storage cavity, it does not require
Inthiscase,oneoptimizesparameterstohaveahigh-fidelity
any microwave-frequency control tones. We also discussed
intra-cavityFockstateatanintermediatetime,asopposedto
howthesamesetupcouldbeusedtoproduceitinerantsingle-
in the long-time steady state. The voltage is then turned off
photonsondemand.
atthisintermediatetime. Thismakesitpossibletoachievea
ThisworkwassupportedbyNSERC.
highfidelityFockstateevenifthetuningofthestoragecavity
impedanceisnotperfect,i.e.λ (cid:54)=λ˜ .Inaddition,bychoos-
s 2,1
ing parameters such that the oscillation dynamics associated
with Cooper-pair tunnelling is slightly overdamped, one can
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