Quantum computing of molecular magnet Mn$_{12}$

3 Pages·0.12 MB·English

by  Bin Zhou

Quantum computing of molecular magnet Mn
12
Bin Zhou1,2,3, Ruibao Tao1,†, Shun-Qing Shen4, and Jiu-Qing Liang5
1. Department of Physics, Fudan University, Shanghai 200437, China
2. State Key Laboratory of Magnetism, Institute of Physics , Chinese Academy
of Sciences, P.O. Box 603-12, Beijing 100080, China
2 3. Department of Physics, Hubei University, Wuhan 430062, China
0 4. Department of Physics, The University of Hong Kong, Hong Kong, China
0 5. Institute of Theoretical Physics, Shanxi University, Taiyuan 030006,China
2 (February 6, 2008)

n
a
Quantum computation in molecular magnets is studied by solving thetime-dependent Schr¨odinger
J
equation numerically. Following Leuenberger and Loss (Nature (London) 410, 789(2001)), an ex-
3
ternal oscillating magnetic field is applied to populate and manipulate the spin coherent states in

molecular magnet Mn . The conditions to realize parallel recording and reading data bases of
] 12
l Groveralgorithsm in molecular magnetsarediscussed in details. Itisfoundthat anaccuratedura-
l
a tion time of magnetic pulse as well as theamplitudes are required to design thedevice of quantum
h computing.
-
s
e
m
Quantum phase was proposed to store information in s−1 gµ H (t)
B m
t. connectionwithanew classofcomputationalalgorithms Vhigh(t)= 2
ma based on the rules of quantum mechanics rather than mX=m0
classical physics.1 One example is the database search × ei(ωmt+Φm)S +e−i(ωmt+Φm)S . (4)
+ −
- problem proposed by Grover.2,3 Differing from other
d h i
quantum algorithms, the superposition of single-particle
n H is a single spin Hamiltonian for molecular magnet
spin
o quantum states is sufficient for Grover’s algorithm. The Mn .6 |miarethesimultaneouseigenstatesofH and
12 spin
c Grover’s algorithm was successfully implemented using thez-componentspinoperatorS withtheenergyε and
[ Rydberg atoms.4 Recently, Leuenberger and Loss theo- z m
1 retically proposed that molecular magnets Mn12 can be tthoedmesocrmibeentthme¯hc.ouVplolwinganbdetVwheigehnatrheetehxeteZrneeaml managtneremtics
v usedto realizethe Grover’salgorithmby utilizing multi- fields and the spin S. The low frequency is applied along
3 frequency coherent magnetic radiation.5 In the S-matrix
the easy axis, and V supplies the necessary energy for
low
2 andtime-dependenthigh-orderperturbationtheory,they
the resonance condition. The high frequency transverse
0 showed that it was possible to populate and manipulate
fields are introduced to induce the transition from the
1
spincoherentexcited statesby applying a singlepulse of
0 initial state |si to virtual states |mi, m=m0,···,s−1,
weakoscillatingtransversemagneticfieldwithanappro-
2 andthefrequenciesωm,m=m0,···,s−2,mismatchthe
0 priatenumberofmatchingfrequencies,and thecoherent levelseparationby ω ,thatis ,h¯ω =ε −ε +h¯ω ,
0 m m m+1 0
/ state can be applied for storing a multi-bites informa- and ω = ε −ε + (s −m − 1)h¯ω . The phases
t s−1 s s+1 0 0
a tion. In this paper the population and manipulation of Φ = m+1 Φ +ϕ (ϕ is the relative phase) and
m a spin coherent excited state in an external oscillating m k=s−1 k m m
- magnetic field is studied by solving the time-dependent P H , if t∈(−T,T) for m=0 and m≥m ,
nd Schr¨odinger equations numerically. The conditions for Hm(t)= 0,motherwise. 2 2 0
the field to implement the Grover’s algorithm are dis- (cid:26)
o
c cussed in details. We find that an accurate duration The time-dependent Schr¨odinger equation reads
: time of magnetic pulse as well as the oscillating frequen-
v cies and amplitudes are required to design the device of ∂
i i¯h |Ψ(t)i=H|Ψ(t)i, (5)
X quantum computing in the molecular magnets Mn12. ∂t
Following Leuenberger and Loss, we consider a molec-
r
where
a ular magnet with spin S (> 1/2) in presence of a weak
oscillating transverse magnetic field. The Hamiltonian s
for the system reads |Ψ(t)i= am(t)e−iεmt/h¯|mi (6)
H =H +V (t)+V (t), (1) mX=m0
spin low high
The eigenstates of m < m have been neglected. The
where 0
molecular magnets Mn behaves like single spin, and
12
H =−AS2−BS4+gµ δH S , (2) has a ground state with spin s=10.
spin z z B z z
The Grover scheme becomes adapted to describe the
V (t)=gµ H (t)cos(ω t)S , (3) quantum computational read-in and decoding of the
low B 0 0 z
1
quantum data register a=(a ,a ,...,a ). Firstly, as Grover’s algorithm is implemented successfully, accord-
s s−1 m0
a simplest example, let us discuss the case of m0 = 9. ing to Leuenberger and Loss, one should have a(2) ≈
10
At the first stage of the so called “read-in”, the initial
values of {a(m0) : m = 10,9} are set to be {a(100) = 1, 1, a(92) ≈ 2 a(91) ≈2η, a(82) ≈0, a(72) ≈2 a(71(cid:12)(cid:12)(cid:12)) ≈(cid:12)(cid:12)(cid:12)2η,
a(90) =0}. After irradiating the molecular magnets Mn12 a(6(cid:12)(cid:12)2) ≈(cid:12)(cid:12) 2 a(6(cid:12)(cid:12)1) ≈(cid:12)(cid:12) 2η, a(5(cid:12)(cid:12)2) ≈(cid:12)(cid:12) 0 and(cid:12)(cid:12) η ≪(cid:12)(cid:12) 1. I(cid:12)(cid:12)nstea(cid:12)(cid:12)d, our
with a coherent magnetic pulse with the relative phase (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)nume(cid:12)rical(cid:12)calc(cid:12)ulation(cid:12)gives(cid:12)
ϕ(1) in the duration time T, one has the quantum data (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
9 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
register a=(a(1),a(1)) with a(1) ≈0.95, a(1) ≈0.31, a(1) ≈0.04,
10 9 10 9 8
a(110) =cos gµ2Bh¯H9h9|S−|10iT , (7) (cid:12)(cid:12)(cid:12)a(71)(cid:12)(cid:12)(cid:12) ≈0.01, (cid:12)(cid:12)(cid:12)a(61)(cid:12)(cid:12)(cid:12)≈0.02, (cid:12)(cid:12)(cid:12)a(51)(cid:12)(cid:12)(cid:12)≈0.01
(cid:18) (cid:19) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
and (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
a(91) =−ie−iϕ(91)sin gµ2Bh¯H9h9|S−|10iT . (8) a(120) ≈0.82, a(92) ≈0.59, a(82) ≈0.08,
(cid:18) (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
At the second stage of the so called “decoding”, the so- (cid:12)a(2)(cid:12) ≈0.06, (cid:12)a(2)(cid:12)≈0.01, (cid:12)a(2)(cid:12)≈0.01.
(cid:12) 7 (cid:12) (cid:12) 6 (cid:12) (cid:12) 5 (cid:12)
lutions of (7) and (8) become the initial values of {a }
m (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
inEq.(5),andthe relativephaseissetto beϕ(92) =0.At Theva(cid:12)(cid:12)lues(cid:12)(cid:12)of{a(m1),a(m2(cid:12)(cid:12))}a(cid:12)(cid:12)recomplex(cid:12)(cid:12) num(cid:12)(cid:12) bers(notreal)
the end of decoding, one obtains
and we only present their absolute values here. There-
fore, the numerical solution shows that the encoding
−isin 2gµBH9h9|S |10iT , for ϕ(1) =0
a(2) = 2h¯ − 9 number likes 00012, but is not 11012 as expected by
9 (0, (cid:16) (cid:17) for ϕ(1) =π Leuenberger and Loss at the specific values of applied
9
field and duration time T . However, {a(1),a(2)} depend
(9) m m
onthedurationtimeT,andonemaybestillabletoreach
If the condition of gµBH9h9|S−|10iT/2h¯ ≪ 1 is sat- totheencodingnumber11012byaspecificdurationtime
isfied, the result in Eq. (9) is in a good consis- T of pulse.
tence with Ref.[5]. For a longer duration time T ≫ To clarify the feasibility of the method proposed by
[gµ H h9|S |10i/2h¯]−1,theconditionfailstoimplement Leuenberger and Loss, we study a simple, but nontrivial
B 9 −
case ofm =8 as a representativeexample. The key ap-
the Grover algorithm. 0
proximationsinLeuenbergerandLoss’smethodarethat
For m < 9, it is difficult to solve Eq.(5) analyti-
0
cally. However, as it is a single particle problem only Sm(j,)s = 0 for j < n(= s− m0). The transitions from
relating to a finite number of states, it is possible for us |ki to |mi, i.e. S(j) (m < k < s) and all higher order
m,k
to solve the differential equations numerically instead of amplitudes S(j) are neglected. In the case of m = 8,
evaluating the S-matrix perturbatively as Leuenberger m,s 0
the first order perturbation are required to be zero:
and Loss did. The results of explicit numerical calcu-
(1) (1) (1) (2) (2)
lation should be more reliable than other approximate S9,10 = S8,9 = 0, or at least S9,10 ≪ S9,10 (≃ S8,10 )
rTehsueltpsa.raFmoretielrlussutrsaetdiohne,rweearfeoctuhseosnamtheeacsatsheemo0ne=s i5n. and S8(1,9) ≪ S9(2,1)0 (≃ S8(2,1)0(cid:12)(cid:12)(cid:12)).Inf(cid:12)(cid:12)(cid:12)act,(cid:12)(cid:12)(cid:12)thequ(cid:12)(cid:12)(cid:12)antu(cid:12)(cid:12)(cid:12)mam(cid:12)(cid:12)(cid:12)-
Ref.[5]: ω = 5×107s−1, T = 10−7s, H = H = 2G, plitu(cid:12)des o(cid:12)fthe(cid:12) tran(cid:12)sitio(cid:12)nsby(cid:12)the perturbativeS−matrix
0 0 9 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
H /H = −0.04, H /H = −0.25, H /H = −0.61, form(cid:12)ula a(cid:12)re g(cid:12)iven b(cid:12)y (cid:12) (cid:12)
8 0 7 0 6 0
and H /H = −1.12. For the molecular magnets
rMesnp1e2c,tAi5vealyn0.d6ABccinorEdiqn.g(2t)oatrhee0R.5ef6.K[5],atnhder1e.1la1ti×ve1p0h−a3sKes, S9(2,1)0 = 2iπ g2µh¯B 2 H0Hω90eiΦ9h9|S−|10i2Tπ (10)
(cid:16) (cid:17)
(1) (1) (1) (1) (1)
are ϕ = ϕ = ϕ = 0 and ϕ = ϕ = π for
9 8 7 6 5
fienrscto-doirndgerthdeiffneruemntbiearl e1q3u10at=ion1s1f0o1r2.{aNmo(wt),}wiensEoqlv.e(s6i)x. S8(2,1)0 = 2iπ g2µh¯B 2 H8H9ωeiΦ8eiΦ9h8|S−|9ih9|S−|10i2Tπ
0
The initial values of the {a } are set to be a(0) = 1 (cid:16) (cid:17)
m 10 (11)
and a(0) = 0 for m 6= 10. After a duration time T, we
m
have a(1)=(a(1),a(1),a(1),a(1),a(1),a(1)). At the second
10 9 8 7 6 5
stage for decoding the number, a(1) becomes the initial S(1) = 1gµBH9eiΦ9h9|S |10isin(ω0T/2) (12)
values of Eq.(5) and the relative phases should be set by 9,10 i 2h¯ − ω /2
0
ϕ(2) =ϕ(2) =ϕ(2) =0andϕ(2) =ϕ(2) =π. Onceagain,
9 7 5 8 6
the sixfirst-orderdifferentialequationswiththe newini-
tial conditions are solved numerically. At the end of de- S(1) = 1gµBH8eiΦ8h8|S |9isin(ω0T/2) (13)
coding, we obtain a(2)=(a(2),a(2),a(2),a(2),a(2),a(2)). If 8,9 i 2h¯ − ω0/2
10 9 8 7 6 5
2
where Φ = ϕ and Φ = Φ +ϕ (ϕ and ϕ are the itative conclusion we drew from the case of m = 8.
9 9 8 9 8 9 8 0
relative phase). The first two terms {S(2) ,S(2) } result The more quantum states are used to store information,
9,10 8,10
from the second perturbation contributions with energy the higher precision are required to design the magnetic
conservation. From the condition S(2) ≃ S(2) , one pulse. Itisclearthatthelowerorderperturbationseries,
9,10 8,10 j < n , of the S-matrix do not vanish for general dura-
can deduce the field amplitude H8/(cid:12)(cid:12)H0 =(cid:12)(cid:12)0.1(cid:12)(cid:12)6. How(cid:12)(cid:12)ever, tion T eventhe relativetransitions do not satisfy energy
the lower order terms S(1) and S(cid:12)(1) ,(cid:12)in g(cid:12)enera(cid:12)l, are conservation. Therefore, to implement the encoding and
9,10 8,9
not exactly equal to zero even if they do not satisfy the decodinginthemolecularmagnetsbyapplyinganexter-
energyconservation. Theycanbe zeroonlyifω T =2πl nal oscillating magnetic field, one should be very careful
0
wherel isaninteger. Due tothese termsS(1) andS(1) , to choose the appropriate time duration T as well as the
9,10 8,9
the amplitudes of the a (t) and a (t) in the state |Ψ(t)i matching frequencies and amplitudes of the field.
8 9
areobviouslyoscillatingwith varyingT. We haveshown
them in Fig. (1).
Now, let us assume the parameters: T = 1.0×10−9s,
H = 200G, H = 20G and ω = 4π×109s−1. It gives
0 9 0
S8(2,1)0 ≃ 0.11. Following Leuenberger and Loss, we set 0.5
(cid:12)(cid:12)ϕ(92) =(cid:12)(cid:12) 0 and ϕ(82) = 0. Then, the recording number 002 0.4 |a9(2)|
(cid:12) (cid:12) (1) (1) (1)
corresponds to {ϕ = π, ϕ = π}, 01 to {ϕ =
9 8 2 9
0, ϕ(1) = π}, 10 to {ϕ(1) = π, ϕ(1) = 0}, and 11 de 0.3
ctoalc{uϕ8l(9a1t)io=n a0r,eϕs(8h1)2ow=n0i}n.9tThheeTraebsulelts18b(wyhoeurrenmum=eri9c,a82l amplitu 0.2 -------- |a8(2)|
0.1
represent the binary digits 20 and 21, respectively).
j (1)=0, j (1)=p
Table 1: The numericalresults for the case of m0 =8. 0.0 9 8
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
decoded number a(1) a(1) a(1) a(2) a(2) a(2) T (10-9s)
10 9 8 10 9 8
002 (cid:12)0.99(cid:12) (cid:12)0.13(cid:12) (cid:12)0.10(cid:12) (cid:12)1.00(cid:12) (cid:12)0.07(cid:12) (cid:12)0.01(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
012 (cid:12)0.99(cid:12) (cid:12)0.13(cid:12) (cid:12)0.10(cid:12) (cid:12)0.96(cid:12) (cid:12)0.26(cid:12) (cid:12)0.04(cid:12)
10 0.98 0.14 0.10 0.98 0.04 0.20
2
112 0.98 0.14 0.10 0.94 0.28 0.19 FIG.1. Amplitudes a(2) and a(2) as afunction of pulse
9 8
Using the above parameters, the proposal by Leuen- duration T (for ϕ(1) =(cid:12)0 and(cid:12) ϕ(1)(cid:12)=π)(cid:12)
9 (cid:12) (cid:12) 8 (cid:12) (cid:12)
berger and Loss can be implemented. However, the val- (cid:12) (cid:12) (cid:12) (cid:12)
ues {|a(2)|, |a(2)|, |a(2)|} are not constants and will be TheworkissupportedbytheNationalNaturalScience
10 9 8
changed with the duration time T of pulse. In a cer- Foundation of China, Shanghai Research Center of Ap-
tain range, S(1) or S(1) will have the close magni- plied Physics, Institute of Physics of Chinese Academy
9,10 8,9 ofSciences,andaCRCGgrantoftheUniversityofHong
tude to Sm(n(cid:12)(cid:12),)s , w(cid:12)(cid:12)hich(cid:12)(cid:12)can (cid:12)(cid:12)result in the deformation of Kong.
(cid:12) (cid:12) (cid:12) (cid:12) †E-mail: [email protected]
the recor(cid:12)ding,(cid:12)or even in the destruction. In Fig.1 we
(cid:12) (cid:12)
plot amp(cid:12)litud(cid:12)es of a(2) and a(2) with varying T (for
9 8
ϕ(1) = 0 and ϕ(1) =(cid:12) π)(cid:12)which(cid:12) cor(cid:12)responds to 01 . The
9 8 (cid:12) (cid:12) (cid:12) (cid:12) 2
amplitudes a(2) ∼(cid:12)0.26(cid:12)and a(cid:12)(2) (cid:12)∼0.04 for a duration
9 8
time T = 10(cid:12)−9s(cid:12). The value 0(cid:12).26 c(cid:12)an be considered as a
digitnumbe(cid:12)(cid:12)r1a(cid:12)(cid:12)nd 0.04as0.(cid:12)(cid:12)The(cid:12)(cid:12)values a(2) and a(2) 1D.Deutsch,Proc. R. Soc. (London) Ser. A 400, 97 (1985)
9 8 2L. K.Grover, Phys. Rev.Lett. 79, 325 (1997)
arevariedwiththepulsedurationT.The(cid:12)(cid:12)first(cid:12)(cid:12)maxi(cid:12)(cid:12)mum(cid:12)(cid:12) 3L. K.Grover, Phys. Rev.Lett. 79, 4709 (1997)
of a(82) ∼ 0.107 at T ∼ 1.16 ×10−9s, th(cid:12)e se(cid:12)cond(cid:12)max-(cid:12) 4J.Ahn,T.C.Weinacht,andP.H.Bucksbaum,Science287,
imu(cid:12)(cid:12)m (cid:12)(cid:12)a(82) ∼ 0.22 at T ∼ 1.66 ×10−9s which becomes 54M6.3N(2.0L0e0u)enberger and D. Loss, Nature 410, 789 (2001)
(cid:12) (cid:12)
larger (cid:12)(cid:12)than(cid:12)(cid:12)the first minium of a(92) ∼0.19 at T ∼1.37 6A.L.Barra, D.Gatteschi,andR.Sessoli, Phys.Rev.B56,
(cid:12) (cid:12) 8192 (1997)
×10−9s. Therefore, we conclud(cid:12)e th(cid:12)at a rather definite
(cid:12) (cid:12)
duration of pulse should be chos(cid:12)en fo(cid:12)r a stable and clear
quantum recording.
Onecanstraightforwardlyextendtheabovediscussion
to the case of m = 5, but it does not change the qual-
0
3