AIP Con e ence P oceedings 922, 507 (2007); h ps://doi.o g/10.1063/1.2759730 922, 507
© 2007 Ame ican Ins i u e o Physics.
Fo ced synch oniza ion o a
quan um dissipa i e dynamics
Ci e as: AIP Con e ence P oceedings 922, 507 (2007); h ps://
doi.o g/10.1063/1.2759730
Published Online: 20 July 2007
Igo Goychuk, Jesús Casado-Pascual, Manuel Mo illo, Jö g Lehmann, and Pe e
Hänggi
Fo ced synch oniza ion o a quan um dissipa i e
dynamics
Igo Goychuk*, Jesus Casado-Pascual^ Manuel Mo ilk^, Jo g Lehmann**
and Pe e Hanggi*
*Ins i u ii Physik, Uni e si a Augsbu g, Uni e si d ss . 1, D-86135 Augsbu g, Ge many
^Fisica Ted ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, Se illa 41080, Spain
**Depa emen u Physik und As onomie, Uni e si a Basel, Klingelbe gs asse 82,
CH-4056 Basel, Swi ze land
Abs ac . We gene alize he phenomenon o o ced s ochas ic synch oniza ion in o he quan um
domain wi hin he amewo k o a pa adigma ic spin-boson model ( unneling cha ge, o lipping
spin 1/2 coupled o an en i onmen ) which is d i en by an ex e nal pe iodic ec angula ield. The
o e damped egime o dissipa i e quan um unneling is s udied. The mal noise assis ed synch o-
niza ion o a e y high quali y is shown o occu in a b oad ange o empe a u es, d i ing s eng hs
and equencies, i he ex e nal d i ing equency exceeds he ze o- empe a u e limi o dissipa i e
unneling a e, he dissipa ion s eng h exceeds a c i ical alue, and he d i ing is su icien ly s ong.
A simple c i e ion o such s ochas ic synch oniza ion is es ablished. Bo h he simila i ies and he
p o ound di e ences wi h he akin phenomenon o quan um s ochas ic esonance a e ou lined.
Keywo ds: o ced s ochas ic synch oniza ion, wo-s a e quan um dynamics, dissipa ion
PACS:
05.60.Gg, 05.40-a, 05.45.X , 82.20.Gk
INTRODUCTION
Synch oniza ion is a e y uni e sal phenomenon in classical physics [1]. In pa icula ,
o ced synch oniza ion e e s o he locking o he phase o a nonlinea d i en oscilla o
o ha o a pe iodic d i ing ield. In he p esence o noise, i is also possible o de ine a
s ochas ic clock wi h some mean phase equency which depends on he noise s eng h.
The o de ing and locking o his s ochas ic clock o he pacemake d i ing clock wi hin
some noise ange is called s ochas ic synch oniza ion [2]. The less p obable phase slips
be ween he d i e and he d i en sys em
a e,
he be e is he quali y o synch oniza ion.
Can a quan um s ochas ic clock p o ided by a unneling cha ge jumping a andom
imes be ween wo di e en si es o localiza ion in a dissipa i e en i onmen , o a
quan um spin 1/2 lipping a andom be ween wo o ien a ions be synch onized wi h
a pe iodic ield and unde wha condi ions? We answe his in iguing ques ion wi hin
he amewo k o he pa adigma ic spin-boson model in he p esence o an ex e nal
ec angula d i ing.
CP922,
Noise and
Fluc ua ions,
I1?11
In e na ional
Con e ence,
edi ed by M. Tacano, Y. Yamamo o, and M. Nakao
© 2007 Ame ican Ins i u e o Physics 978-0-7354-0432-8/07/S23.00
507
MODEL, THEORY, AND RESULTS
The model is desc ibed by he ollowing Hamil onian
A( ) =
-e( )dz
+ -Mdx + -dz^Kj(h) + hj) + ^n<oj(h)hj +
-), (l)
whe ein, he ope a o s az and ax deno e he s anda d Pauli ope a o s, e( ) is a ime-
dependen ene gy bias, and
HA
is he unnel ma ix elemen . The ba h Hamil onian [las
e m in Eq. (1)] is exp essed in e ms o he ope a o s
c+
and bj associa ed o he j' h
ba h no mal mode wi h equency
COj.
The s ochas ic in luence o he quan um he mal
ba h is cap u ed by an ope a o andom o ce |( )
=
£/ Kj(P-e"°^ + bje~l<°' ). I can
be cha ac e ized by he spec al densi y /( o)
=
(n/Ti) Ly K?5( o
—
o,). We assume ha
J((o) acqui es he Ohmic o m, /( o) =
27zhoc(De~m>mc,
wi h ic ion s eng h
a
and an
exponen ial
cu o .
We conside he o e damped limi a > 1/2 and weak unneling limi
A
<C
oc. Then, he dynamics o he diagonal elemen s o he educed densi y ma ix,
Pp
(0 =
P±( )>
is gi en by he Pauli mas e equa ion [3]
Pp( ) =W_p( )p_p( )-Wp( )pp( ) ,
(2)
whe e he ime-dependen elaxa ion a es
Wp
( ) wi hin he Golden Rule app oxima ion
and he app oxima ion o adiaba ically slow a ying bias e( ) ead:
1
°°
W±( ) = -A2 dTexp[-g'(T)]cos
2
Jo
The unc ions Q'( ) and Q"( ) deno e he eal and imagina y pa s o he dissipa ion
ke nel
2(0 = %
+
i
/V
*2
(l( e)l(0))
,
(4)
n Jo
Jo
whe ein X =
J0°°
dcoJ( o) /
(%co)
is he ba h eo ganiza ion ene gy
[3].
Fo he conside ed
model one inds ha X = 2ahcoc and
m
=
2«iB{^T^| (1^^0|2} (5)
g"( )
=
2aa c an( oc ) .
(6)
In Eq. (5), T{£) deno es he Gamma unc ion,
COT
= ksT/h, and K = (OT/(OC. The
quan um a es sa is y W_( )
=
exp(—e( )/( cB ))W+( ) o each ins an o ime
.
A
ze o empe a u e,
.2
/ i
/. 2a—l
W±,M )
=
0^(0]^^
(^J exp[T£(0/(. oc)],
whe e 0( )
is
he Hea iside s ep unc ion [4], i.e. one
o
he a es
is
non-
ze o,
while ano he one
is
exac ly ze o.
In
he limi
o
high- empe a u es (clas-
sical en i onmen )
and
quasi-s a ic en i onmen al noise limi ,
he
a es
a e
e"(T)T^(o* (3)
508
well app oxima ed by he celeb a ed Ma cus-Le ich-Dogonadze a e exp ession
W±{ ) = {n/2)hA2/V4nnBTexpl-(±e( ) - I /{AXkBT) . In a ully quan um
egime o kgT <
hcoc,
he a es ha e o be e alua ed nume ically.
Ou hough -expe imen al se up mimics he quan um analogue o a classical (phase)-
synch oniza ion beha io elabo a ed in Re s. [5, 6]. A quan um pa icle wi h he cha ge
q (elec on, o p o on) makes unneling (ins an ) jumps o h and back be ween wo si es
o localiza ion sepa a ed by he dis ance 0. The ene gy bias is modula ed in ime by
he applied elec ic ield <?( ) yielding e( ) = q 0S'( ) (al e na i ely, i could be a spin
1/2 and a magne ic ield). The ex e nal ield al e na es also dicho omously, changing i s
di ec ion, howe e , pe iodically in ime. The unneling jump p ocess can be desc ibed
ma hema ically as a classical eleg aph noise de ined by he mas e equa ion (2) wi h
a es which a e ully quan um-mechanical and ime-dependen .
We a e in e es ed in he synch oniza ion o unneling e en s sepa a ed by andom
ime in e als wi h he ex e nal d i e al e na ions. One hen coun s he numbe o
jumps n( ) wi hin a ime window
[ o, ).
Following Re . [5], we in oduce he an-
dom phase
(j>( , o)
= nn{ ), which inc eases by
%
a each swi ching e en ( wo sub-
sequen swi ches co espond o a 27 -cycle o andom du a ion), and de ine he a e -
age equency and di usion coe icien s associa ed o he s ochas ic phase-p ocess as
Qph := lim^oo(<KMo))/(?-?o) and2Dph := lim,^,
[(<j>2( , 0))
-
(<p{ , 0))2]
/( - 0),
espec i ely. Using a s ochas ic pa h-in eg al desc ip ion o he d i en eleg aph p o-
cess [7], he ollowing exac esul s we e ob ained o a pe iodic ec angula d i ing
e( ) = ±8o wi h ampli ude
EQ
and equency
Q.
=
2%j
2?
[6, 8]
— nW i ,
Qph = —
^ 1
- 8piq
and
2n2
4 anh(W^/4) (7)
2Dph = 7 Qph-^5^q anh3(W^/4)
-^8p2eq(l-8p2eq){mmHW^/4)
-W^[l+2sech2(W^/4)] . (8)
He e, W deno es he sum o he o wa d and backwa d a es in Eq. (3) o a ixed alue
o he ield <?o, i.e., o e( ) = eo = q oSb, and 8peq = anh(eo/(2 c£ )) is he absolu e
alue o he di e ence o he equilib ium popula ions. The in e se Fano ac o o he
coun ing p ocess R := 7 Qph/(2Dph) p o ides a eliable quali y measu e o o ced syn-
ch oniza ion [2, 5]. I co esponds o he numbe o unneling jumps synch onized o he
d i ing al e na ions. Desynch oniza ion occu s, when he a iance o he synch onized
coun ing p ocess becomes abou uni y, (8n2(Synch o)} = 1 (a phase slip occu s). F om
his condi ion, he (de)synch oniza ion ime ollows as
'synch o ^ K--J /£ •
509
C i e ion o s ochas ic synch oniza ion
Fo s ochas ic synch oniza ion o occu he pa icle mus ha e su icien ime o make
a ansi ion o he lowe ene gy s a e du ing he il ing hal -pe iod. This equi es ha he
mean esidence ime (T+(e)) = l/W+(6o) in he highe ene gy s a e should be much less
han ST jl. On he o he hand, he backwa d ansi ions du ing such a o wa d il mus
be p ohibi ed. The synch oniza ion c i e ion hus eads,
l/W+{eo) < 572 < l/W-{eo) = exp{e0/kBT)/W+{e0) . (9)
I should be con as ed wi h he c i e ion o S ochas ic Resonance (SR) eading [9]
l/W+(eo) = STjl o 7 W+(e0) = Q. (10)
Clea ly, o a weak d i ing, wi hin he linea esponse (LR) app oxima ion, eo
<C
ksT,
he c i e ion (9) can ne e be jus i ied, while SR can occu [9]. Mo eo e , SR is e-
quen ly no ela ed o synch oniza ion a all. In pa icula , he ea lie wo ks on quan-
um SR (QSR) [10] wi hin he same spin-boson model unco e ed, wi hin he pa ame e
egime a < 1, ha QSR in he conside ed symme ic (in he absence o d i ing) sys em
is impossible, and an addi ional s a ic bias is equi ed [9]. This e en was hough o be a
main ea u e o QSR, as compa ed wi h he classical SR [9]. Such QSR in a biased spin-
boson sys em has, howe e , no ela ion o synch oniza ion, as i does no co espond o
any ma ching o he ime scale o d i ing and ha o he dissipa i e unneling dynamics.
In [11] i was shown, howe e , ha QSR is also possible wi hin he LR heo y o he
s udied symme ic model p o ided ha quan um ic ion is su icien ly s ong, a > 1.
This pa ame e egime is ele an , e.g. o nonadiaba ic elec on unneling in condensed
molecula sys ems [3], whe e a can be as la ge as a ~ 5
—
10 and e en la ge .
We we e guided by his ea lie QSR esea ch o ind he pa ame e egime, whe e
he he mal noise-assis ed Quan um S ochas ic Synch oniza ion (QSS) can occu . A
undamen al di e ence o QSS, as compa ed wi h i s classical coun e pa , is ha QSS
is expec ed o occu always a T = 0, i he d i ing is su icien ly slow,
Q.
<C
W =o(£o).
This is so because one o he a es is exac ly ze o, as i ollows om he de ailed balance
condi ion. Thus, a desynch oniza ion ansi ion occu s wi h enhancing he empe a u e
abo e some h eshold alue [8]. The ques ion is, howe e , whe he he he mal noise
can help o synch onize when
Q.
> W =o(so) and he pa icle canno ollow he bias
al e na ions a T = 0. As in he case o QSR, his ques ion is highly non i ial because
o he non-A henius, powe law empe a u e dependence o he unneling a es. Ou
nume ical s udy based on he ou lined analy ical heo y e ealed ha such a quan um
he mal noise assis ed QSS is only possible when a > 1, in ag eemen wi h he QSR
esea ch. Mo eo e , la ge a ~ 5
—
10 a e p e e able.
Nume ical esul s
Le us i s illus a e a undamen al di e ence be ween QSR and QSS o a la ge
ic ion a = 10, some ixed unnel ma ix elemen
A
= 4
•
10~4, equency Q = 10~n and
510

o a mode a e d i ing s eng h eo = 0.5 which is al eady beyond he LR condi ion. All
he dimensional quan i ies a e scaled (h = I,kg = 1) in e ms o he quan um c osso e
empe a u e Tc =
Hcoc/kB
o he mal ba h oscilla o s. The pa ame e s a e chosen o
mimic nonadiaba ic elec on unneling in he molecula dime s o azu in [8].
i-H
o
o
«J
Pi
1 3
1.2
1.1
1 II
u.y
0.8
0.7
1 1
_(b) /
~ , ,
-
-
V —
X.
'0.3 0.4 0.5 0.6
K 0.7 0.8
FIGURE 1. (a) Mean s ochas ic equency 2ph, mean phase di usion coe icien Dph and (b) he in e se
Fano ac o R e sus he scaled empe a u e
K
o he d i ing s eng h
£o
= 0.5. O he pa ame e s a e gi en
in he ex .
Fig. 1(a) showsjha synch oniza ion o he chosen pa ame e s is absen . The mean
phase equency Qph as a unc ion o he scaled empe a u e
K
c osses he line
Q.
= 10~n
a a ce ain poin . Howe e , any equency locking supplemen ed by a minimum o
he phase di usion coe icien is absen . The e o e, his is no a synch oniza ion, pe
de ini ion, as any synch oniza ion equi es a equency locking. The c ossing poin
a ound K « 0.5 co esponds, howe e , o QSR as i can be deduced om Fig. 1(b).
Indeed, he quali y, o cohe ence ac o R displays a smoo h maximum when quan um
s ochas ic esonance occu s. The ampli ude o his maximum is, howe e , so small ha
desynch onizing phase slips occu pe manen ly, e en whe e a d i ing-induced phase
cohe ence, R > 1, exis s. Such QSR can be conside ed also as a kind o cohe ence
esonance [12] induced by he ex e nal d i ing. I cons i u es a p ecu so o quan um
s ochas ic synch oniza ion ha occu s wi h a u he inc ease o he d i ing s eng h
abo e some h eshold eo « 2.5, o o he pa ame e s ixed. Fo eo = 5 he quali y o
synch oniza ion is al eady e y high as e idenced by Fig. 2.
10
* io-10
IQ
^ io"12
OH
IG
.14
10
(a)"1
iiph
-- 2Dph
x"
y
V,;
10 10 10
K
FIGURE 2. The same as Fig. 1 o he d i ing s eng h
£Q
= 5.
511
In his igu e, he mean phase equency Qph is clea ly locked o he ex e nal equency
Q in a b oad ange o empe a u es, while he mean phase di usion coe icien Dpb
displays a p onounced minimum a some empe a u e. A his minimum, he quali y o
he synch oniza ion is as onishingly high [see Fig. 2(b)] eaching he maximum abou
105
phase-locked unneling jumps be o e a phase slip occu s.
Ou es ima ions o elec on unneling in molecula dime s like azu in show [8] ha
o unneling dis ances
TQ
~ 15 A, c osso e empe a u es Tc ~ 150 K, unnel couplings
TiA
~ 5
•
10~3 meV, he equi ed d i ing equencies a e in he ange om se e al Hz o
se e al hund eds Hz, while he elec ical ield s eng h should be abou 5
•
104 V/cm o
a i e a a high quali y synch oniza ion. Such expe imen al s udies should be easible in
labo a o ies.
ACKNOWLEDGMENTS
J. C.-P. and M. M. acknowledge he suppo o he Minis e io de Educacion y Ciencia
o Spain (FIS2005-02884) and he Jun a de Andalucia. I. G. and P. H. acknowledge
suppo by he DFG h ough SFB 486 and by he Ge man Excellence Ini ia i e ia he
Nanosys ems Ini ia i e Munich (NIM).
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