Strong Coupling between a Topological Qubit and a Nanomechanical Resonator

0.26 MB·English

by  Fang-Yu Hong

Strong Coupling between a Topological Qubit and a Nanomechanical Resonator
Fang-Yu Hong,1 Jing-Li Fu,1 and Zhi-Yan Zhu1
1Department of Physics, Center for Optoelectronics Materials and Devices,
Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China
(Dated: January 23, 2013)
Wedescribeaschemethatenablesastrongcoherentcouplingbetweenatopologicalqubitandthe
quantizedmotionofamagnetizednanomechanicalresonator. Thiscouplingisachievedbyattaching
an array of magnetic tips to a namomechanical resonator under a quantum phase controller which
coherentlycontrolstheenergygapofatopological qubit. Combinedwithsingle-qubitrotationsthe
strong coupling enables arbitrary unitary transformations on the hybrid system of topological and
mechanicalqubitsandmaypavethewayforthequantuminformation transferbetweentopological
3 and optical qubits. Numerical simulations show that quantum state transfer and entanglement
1 distributingbetweenthetopological andmechanicalqubitsmaybeaccomplished withhighfidelity.
0
2 PACSnumbers: 03.67.Lx,03.65.Vf,74.45.+c,85.25.-j

n
a
J Introduction.—A major challenge facing the field of state spin qubit [24].
9 quantum information processing (QIP) arises from the Thus the best solution is to make hybrid systems by
1 delicate nature of a quantum system, their tendency to combining the advantages of topological qubits, robust
decohereintoclassicalstatesthroughcouplingto the en- quantumstorageand protectedgates, with those of con-

] vironment. To address this obstacle there emerged some ventional qubits such as high fidelity readout, univer-
h
p interesting topological quantum computation schemes sal gates, and quantum network. Such hybrid schemes
- [1, 2], where quantum information is stored in nonlocal have recently been suggested for the anyons coupled to
nt (topological) degrees of freedom of topologically ordered superconducting flux qubits [25–27] and for the anyons
a systems. Beingdecoupledfromlocalperturbationsthese in atomic spin lattices [28], in optical lattices [29], and
u nonlocal degrees of freedom enable the topological QIP in Majorana nanowires [30] coupled to a semiconduc-
q approaches to obtain its extraordinary fault tolerance tor double-dot qubit [17]. Here we propose a scheme
[
and to have a huge advantage over conventional ones. for quantum information transfer between a magnetized

1 As the simplest non-Abelian excitation for topological nanomechanicalresonator[31–33]andatopologicalqubit
v qubits,thezeroenergyMajoranaboundstate(MBS)[3], encoded on Majorana fermions (MFs) on the surface
7 is conjectured to be exist in the spin lattice systems [1], of a topological insulator (TI) [6]. The motion of the
8
in the p+ip superconductors [4], in the filling fraction resonator under a quantum phase-controller (QPC) [34]
9
ν = 5/2 fractional quantum Hall system [2], in the su- modifies the energy gap between the two topological
4
perconductor Sr RuO [5], in the topological insulators qubit states, resulting in a strong coupling between the
. 2 4
1 coupled to s-wave superconductors [6, 7], and in some topological qubit and the quantized motion of the res-
0
semiconductors of strong spin-orbit interaction coupled onator with its strength conveniently controlled by the
3
to superconductors [8–11] where an experimental obser- QPC. Based on this strong coupling arbitrary quantum
1
: vation has recently verified its existence [12]. information transfer and quantum entanglement distri-
v
On the other hand, the nonlocal nature of topological bution between the topological qubit and the resonator
i
X qubits makes it tough to measure and manipulate them, can be performed with high fidelity. Considering the co-
r because they can only be controlled by globe braiding herentinteractionbetweenlightandananoscalemechan-
a operations, i.e., by physical exchange of the associated ical resonator [35–39], this scheme may lay the founda-
local non-Abelian anyons [13, 14]. Furthermore these tions for the coherent coupling between topological and
braiding operations for Ising anyons alone are not suffi- optical qubits.
cient to accomplish universalquantum computation and Hybrid system.—The prototype hybrid quantum sys-
have to be combined with topologically unprotected op- tem shown in Fig.1 consists of a topological qubit en-
erations [15, 16]. Implementing unprotected operations coded on four MFs, a QPC, and a nanomechanical res-
within a topological system proves to be very challeng- onator covered with an array of magnetic tips. The flux
ingduetotheexistenceofsignificantnonuniversaleffects QPC is made up of a Josephson junction (JJ) with two
[17]. At the same time, stead advancements have been superconducting islands a,b and a rf SQUID loop of in-
achievedinconventionalQIPsystems,suchastherecent ductanceLienclosinganexternallyappliedmagneticflux
progresses in a basic quantum network of single atoms Φx. The phase difference φ between superconducting is-
in optical cavities [18], in long lifetime of nuclear spins landsaandbisdeterminedbyφ= 2πΦx/Φ0[34],where
−
in a diamond crystal [19, 20], in high fidelity operations Φ0 = h/2e is the flux quantum. The MFs described
on trappedions [21] andon superconducting qubits [22], by Majorana fermion operators γi(i = 1,2,3,4) are self-
in distributing entanglement between single-atoms at a Hermitian, γi† = γi, and satisfy fermionic anticommuta-
distance [23] and between an optical photon anda solid- tion relation γ ,γ = δ . The Majorana fermion γ is
i j ij i
{ }
2
topological qubit 0.5 1.0
0.4 ρs22 ρs11 0.75 ρs22
0.3
0.50
0.2
resonator a JJ--p2 1 g TuI g 2 --p2 0.01 0.250 ρs11
b 1 2 0.5 1
m tiapgnetic --p2 3 g3 TduI g4 4 --p2 0000....4444999999999876 ρs00 0000....6428 x 10−3 ρs01 ρs0ρ0s02
0.4995 0
0.5 0.5
phase controller 00..43 ρs02 −iρs01 00..43 iρs12
0.2 iρ 0.2
FIG. 1. (color online). Schematics for a hybrid system com- s12
0.1 0.1
prising a topological qubit, a QPC, and a nanomechanical 0 0
resonator. The topological qubit is encoded on two pairs -0.5 -0.3 -0.10 0.1 0.3 0.5 -0.25 -0.1 0 0.1 0.25
tg/π tg/π
of Majorana fermions ((γ1,γ2) and (γ3,γ4)). Two Majorana
fermions (marked with circles) at two superconducting tri- (a) (b)
junctions are coupled though STIS quantum wire with cou-
pling strength dependent on the phase difference between FIG. 2. (color online). a) Numerical simulation of the
phase φu = −π of islands u and phase φd = θ of island d. state transfer: √12(|↓0i + |↑0i) → √12(|↓0i − i|↓1i).
The flux QPC consists of a JJ and a rf SQUID loop enclos- The state transfer fidelity is F1 = 0.990. b) Numerical
inganexternalfluxΦx whichdeterminesthephasedifference simulation of quantum entanglement generating, 0
φ between superconductor islands a and b. The resonator is ( 0 i 1 )/√2 with a fidelity F2 = 0.993. |T↑heip→a-
|↑ i − |↓ i
covered with an array of magnetic tips. The motion of the rameters used are g = 20(2π) MHz, g′ = 100(2π) MHz,
magnetized resonator modifies themagnetic fluxpenetrating T = 25 mK, Qr = 1 −103, γp = 1(2π) MH−z, ωp = 4.3(2π)
×
the plane enclosed by the QPC, resulting in changes in the GHz, and ωr = ωt = 1(2π) GHz. The corresponding matrix
phase differenceφ and in theenergy splitting of thetopolog- elements of the density matrix ρs of the hybrid system are
ical qubit. ρs00 = 0ρs 0 , ρs01 = 0ρs 1 , ρs02 = 0ρs 0 ,
h↓ | |↓ i h↓ | |↓ i h↓ | |↑ i
ρs11 = 1ρs 1 , ρs12 = 1ρs 0 ,ρs22 = 0ρs 0 .
h↓ | |↓ i h↓ | |↑ i h↑ | |↑ i
localized at trijunction i(i = 1,2,3,4), which comprises
three s-wave superconductors patterned on the surface
planeofareaS enclosedbytheQPCloop,andthecorre-
of a TI [6]. A pair of MFs operators γ ,γ can make
i j sponding annihilation and creation operations a and a .
up a Dirac fermion operator f = (γ iγ )/√2, which †
ij i− j Themotionoftheresonatorcauseamagneticfluxfluctu-
creates a fermion and fi†jfij = nij = 0,1 represents the ation∆Φr SGu0(a+a†), where Gis the averagemag-
occupationofthecorrespondingstate. Twologicalstates netic field g≃radient produced by the magnetic tips, and
of the topological qubit |0it and |1it are encoded on u0 is the amplitude of the resonator’s zero-point fluctu-
the four MFs with |0it = |012034i and |1it = |112134i. ations. The Hamiltonian for the QPC can be written as
The four MFs γi(i = 1,2,3,4) interacts through the Hp = ωpb†b with the plasma frequency ωp (CLi)−1/2
superconductor-TI-superconductor(STIS)wireofwidth [34],andthecorrespondingannihilationand≈creationop-
Wtiv,eleHnagmthiltLo,naianndfpohratsheseφtoup=olo−gπicaanldquφbdit=reθa.dTsh(e~e=ffe1c)- meraagtinoentsicbtaipnsdtbh†e. pThaaksiengθicnatno btheewcrointtternibuastion from the
H = E(θ)σz, where the coupling strength [27]
t − 2 t
E(θ)= vLFqΛ2θ+f02(Λθ), (1) θ =θ0+ξa√+2a† +ζb√+2b†, (3)
and Pauli operator σz = (0 0 1 1) . Here f (y)
0isththienvienrvteibrslee dfuonmcatiionnt, Λoθf =y| =i∆vh0FLx|/−stina|nθ2i(hxw)|itdthefithneedinidnu0cthede niwsihtiunerdieetsθo0efqisuqiutlihabenrticuuommrrepflsopusocittniudoanint,igoζnp≈shao2sf√etπwh(heEEeCLQn)Pt41hCeisr[t3eh4seo],nmaaatnogdr-
superconducting gap∆ and the effective Fermi velocity
0
ξ =2πSGu /Φ .
v [6] 0 0
F
The Hamiltonian for the whole hybrid system de-
v =v[cosµW + ∆0 sinµW] ∆20 , (2) scribed by a density matrix ρ has the form
F µ µ2+∆2
0
1
where µ is the chemical potential of the TI and v is the H =a aω +b bω E(θ)σz. (4)
† r † p− 2 t
velocity of an electron on the TI’s surface.
The nanomechanical resonator is described by the
Hamiltonian H = ω a a with the mechanical vibration ExpandingthecouplingstrengthE(θ)tofirstorderinthe
r r †
frequency ω along the direction zˆ perpendicular to the small parameters ξ dE(θ) and ζ dE(θ) gives
r ωr dθ |θ=θ0 ωr dθ |θ=θ0
3
the Hamiltonian To describe dissipative effects we introduce the quan-
1 tumLangenvinequationfortheQPCdegreesoffreedom
H =a†aωr+b†bωp− 2E(θ0)σtz in the limit g′ →0:
1 1 γ
− 2g(a†+a)σtz − 2g′(b†+b)σtz, (5) b˙ =−i[b,Hp]− 2pb−√γpς (13)
where
wherethenoiseoperatorςfulfills ς (t)ς(t) =N δ(t t)
† ′ p ′
ξ dE(θ) withN =[exp(ω /k T) 1] 1 ahndγ N iistherelev−ant
g = p p B − − p p
√2 dθ (cid:12) decoherence rate. From QLE (13) by Fourier transfor-
(cid:12)θ=θ0
(cid:12) mation the steady-state correlation functions J(ω ) and
ζ dE(θ)(cid:12) t
g′ = √2 dθ (cid:12) . (6) K(ωt) can be obtained as
(cid:12)θ=θ0
(cid:12) γ (N +1) γ γ
1B(y0 r+ew1rit)inagndHamil=ton1ia(n0(5(cid:12)) 1in) taenrmdsapopflyi|n↓gith=e J(ωt)= p 2ωpp [(iωt+iωp− 2p)−1−(iωt−iωp− 2p)−1],
√2 | i | i t |↑i √2 | i−| i t (14)
rotating-waveapproximationandthe interactionpicture
we obtain γ N γ γ
K(ω )= p p[( iω +iω p) 1 ( iω iω p) 1]
HI =−21g(a†σt−+aσt+)− 21g′(bσt+ei(ωt−ωp)t t 2ωp − t p− 2 − − − t− p− 2 (1−5)
+b†σt−e−i(ωt−ωp)t), (7) Rewritingequation(10)givesthefollowingeffectivemas-
ter equation
where the resonance condition ω = E(θ ) ω is as-
r 0 t
saunmdelodw,earnidngσot+p=era|t↑oirhs↓,r|easnpdecσtt−ive=ly.|↓Nioh↑w|waerec≡othnecernatirsaintge ∂∂tρs =−i∆2[σtz,ρs]+Γp(Np+1)D(σt−)ρs+ΓpNpD(σt+)ρs
on the experimentally relevant regime ω ω ,g,g , +γ (N +1)D(a)ρ +γ N D(a )ρ , (16)
p r ′ r r s r r † s
≫
where we can adiabatically remove the fast dynamics of
where we have included the dissipation of the res-
the phase controller degrees of freedom. Through pro-
onator modes for a mechanical quality factor Q =
jection operator techniques we have the following Born r
ω /γ , D[cˆ]ρ := (2cˆρ cˆ cˆ cˆρ ρ cˆ cˆ)/2, N =
adpenpsriotxyimmaattiroinx [o4f0]t:he master equation for the reduced [erxp(rω /k T)s 1] 1, ∆s=†γ−pg′2†, ansd−Γ s=†2γp2ωtg′2.r
r B − − 2ωp2 p ωp4
Example.—As an example we discuss a SiC beam of
∂ t
ρ (t)= dtTr [H (t),[H (t),ρ (t) ρ ]], (8) dimensions (l,w,t) = (1.1,0.12,0.075)µm with a basic
∂t s −Z ′ p I I ′ s ′ ⊗ p
t0 mode of frequency 1(2π) GHz, u0 15 fm, and Q=500
≈
at temperature T = 4.2 K [31, 48], or Q 2300 at T =
where ρp is the steady state of the QPC in the absence ≈
25 mK according to the temperature dependence of the
of the qubit-resonator system. We perform the Markov
quality factor Q 1 T0.3 [49]. A magnetic tip of size of
approximation on equation (8) by replacing ρs(t′) with 50 nm with hom−oge∝neous magnetization M 2.3 106
ρ (t) and by sending t , resulting in the Marko-
s 0 ≈ ×
→ −∞ [32, 43] attached on the resonator produces a magnetic
vian quantum master equation
gradient of G 1 108 T/m at a distance of 1 µm ,
∂ ∞ resulting in ξ ≈> 0.×002 for a surface S 1µm2. The
∂tρs(t)=−Z dτTrp[HI(t),[HI(t−τ),ρs(t)⊗ρp]], QPC comprises a large Josephson junctio≈n [44] and a rf
0
(9) SQUIDloopwithverysmallinductance [45],wemayset
This Markovapproximationholds if the QPC modes de- ζ 0.01 and ωp 4.3(2π) GHz [34]. For topological
≈ ≈
cthaye tmoupcohlofgaisctaelrqtuhbaintgb′y−1mourchifmthoeryeatrheafnarg′d−e1tu[3n6e]d. fSruobm- qanudbitvFwe≈m2a.2y×ch1o0o4sem∆/s0b≈y2a5d(j2uπst)inGgHtzhe[1T2]I,’sLc∼hem5µicma,l
stituting H (5) into equation (8) gives (neglecting tran- potential µ (2). From equations (1, 6) we obtain g
I ≈
sients by dispatching t0 →−∞) −20(2π) MHz and g′ ≈ −100(2π) MHz for θon = 0.09;
g 5 KHz and g 25 KHz for θ =3.1
∂ 1 ≈− ′ ≈− off
∂tρs =−4g′2(cid:2)J(ωt)(σt+σt−ρs−σt−ρsσt+) entAlyppcloicnattrioolnlesd.—byTmheodciofyuipnlgintghestprehnagsethθ:gtchaeninbteercaochtieorn-
+ K(ωt)(σt−σt+ρs−σt+ρsσt−)+H.c. , (10) betweenthequbitandtheresonatorisswitchedon(off)
(cid:3) by tuning θ to θ ( θ ). A unitary transformation
where on off
µ 0 +ν 0 µ 0 iν 1 , (17)
J(ω )= ∞ b(τ)b (0) eiωtτdτ (11) |↓ i |↑ i→ |↓ i− |↓ i
t Z h † i
0 can be performed by adiabatically turn on the coupling
for a durationcorrespondingto a π pulse g(t)dt= π.
−
K(ωt)=Z ∞hb†(τ)b(0)ie−iωtτdτ (12) Nishexat qausainntgulem-qustbaittertortaantsiofenrofrnomthethleatttoeprRoclaongictahlenqufibnit-
0
4
ergysplitting E(θ ) may be affectedby some processes,
on
such as dynamics modulations of the superconducting
0.98 gap and variation of the electromagnetic environment.
F Theerrorofthequantuminformationtransferbetween
0.96
the topological qubit and the resonator is estimated in
1
0.94 termsoffidelitybynumericalsolvingtheeffectivemaster
0.95 equation(16). Wemaychooseωr =E(θon)=ωt =1(2π)
0.92
GHz, ω = 4.3(2π) GHz, T = 20 mK [42], γ = 1 MHz
p p
0.9 0.9 [22], Qr = 2 103, g = 20(2π) MHz, and g′/2π =
× −
100(2π) MHz. The evolution of the state transfer
0.85 0.88
0.8 0.86 1 ( 0 + 0 ) Rtf1g(t)dt=−π ψ 1 ( 0 i 1 )
1
0 0 √2 |↓ i |↑ i −−−−−−−−−−→| i≡ √2 |↓ i− |↓ i
0.84 (19)
0.1 0.1
0.2 0.2 and the generating of a maximally entangled state
0.3 0.3
γr / g 0.4 0.4 γp / g 0 Rtf2g(t)dt=−π/2 ψ ( 0 i 1 )/√2 (20)
0.5 0.5 |↑ i−−−−−−−−−−−→| 2i≡ |↑ i− |↓ i
are shown in Fig. 2a) and b), respectively, with the cor-
FIG. 3. (color online). The effect of decoherence sources γr responding fidelity F1 = ψ1 ρs(tf1)ψ1 = 0.990 and
a√1n2d(|γ↓p0oin−thi|e↓fi1die)l.itOytohfesrtpataeratmraentsefresr:ar√e12a(s|↓in0Fii+g.2|.↑0i)→ Fco2h=erehnψc2e|ρsos(utrfc2e)s|ψγ2ria=nd0γ.9ph9o3n.|tThheestian|tfleuiternacnesfoefrtfihdeelditey-
F is shown in Fig.3. Finally we estimate the influence
1
of the fluctuations in the energy splitting E(θ ) on the
on
operationfidelitybyassumingunknownerrorsinE(θ ),
on
to the motion mode of the resonator,where µ and ν are
andg: thecorrespondingfidelityF decreasesfrom0.989
1
arbitrary complex numbers satisfying µ2+ ν 2 = 1. A
to 0.984 for 1% unknown errors in E(θ ) and g.
| | | | on
maximally entangled state 0 ( 0 i 1 )/√2 Conclusion.—Insummary,wehavepresentedascheme
|↑ i → |↑ i− |↓ i
can be generated if g(t)dt = π/2. The choose of for quantum information transfer between topological
−
g(t)dt = 3π/2 aRccomplishes a √SWAP gate, the qubit and the quantized motion of a nanomechanical
−
Rsquared root of SWAP gate, up to a single-qubit rota- resonator. Quantum state transfer, quantum entangle-
tion. Series of √SWAP gates and single-qubit 90 ro- mentgenerating,andarbitraryunitarytransformationin
◦
tations about zˆ on the subsystem i denoted by R (90) thetopological-qubit-resonatorsystemmaybeperformed
z,i
gives the controlled-phase (CP ) gate with high fidelity. Considering the advances in coherent
t,r
transfer of quantum information between the quantized
motionoftheresonatorandotherconventionalqubitsin-
CP =R (90)R ( 90)√SWAPR (180)√SWAP
t,r z,t z,r − z,t cluding optical qubits [32, 35, 36, 42, 47], this quantum
(18)
interface enables us to store conventional quantum in-
forthehybridsystem. Finallyanarbitraryunitarytrans-
formation on topologicalqubits for long time storage,to
formation on the hybrid system can be decomposed into
efficiently detect topological qubit states, to design par-
CP gates and single-qubit rotations [46].
t,r tially protected universaltopologicalquantum computa-
Numerical simulations.—The main sources of error of tion, where topological qubit can receive a single-qubit
thequantummanipulationsdiscussedabovearedecoher- state prepared by a conventional qubit with high accu-
ence from the resonatorand the QPC. Low temperature racy,compensatingthetopologicalqubit’sincapabilityof
is required to exponentially decrease the probability of generating some single-qubit states.
the occupation of the excitation modes of the STIS wire This workwassupportedbythe NationalNaturalSci-
by the factor γw ≡ exp(k−BvTFL) [27]: T = 20 mK gives ence Foundation of China ( 11072218and 11272287),by
γ < 10 3 for the aforesaid values of v and L. The fi- ZhejiangProvincialNaturalScienceFoundationofChina
w − F
nitelengthofSTISwirewillhaveverysmalleffectonthe (Grant No. Y6110314),and by Scientific Research Fund
coherencetime ofthe topologicalqubit itself: the energy ofZhejiangProvincialEducationDepartment(GrantNo.
splittingE(θ ) 1(2π) 103Hzforφ =0.13. Theen- Y200909693).
off off
∼ ×
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