Location- and observation time-dependent quantum-tunneling

0.33 MB·English

by  V. Fleurov

Location- and observation time-dependent quantum-tunneling
V. Fleurov1, R. Schilling2, and B. Bayani2
1Raymond and Beverly Sackler Faculty of Exact Sciences,
School of Physics and Astronomy,
Tel-Aviv University, Tel-Aviv 69978 Israel. and
2 Johannes Gutenberg University, Mainz, Germany
We investigate quantum tunneling in a translation invariant chain of particles. The particles
interact harmonically with their nearest neighbors, except for one bond, which is anharmonic. It
is described by a symmetric double well potential. In the first step, we show how the anharmonic
9
coordinate can be separated from the normal modes. This yields a Lagrangian which has been
0
used to study quantum dissipation. Elimination of the normal modes leads to a nonlocal action of
0
Caldeira-Leggetttype. Iftheanharmonicbonddefectisinthebulk,onearrivesatOhmicdamping,
2
i.e. thereisatransition ofadelocalized bondstatetoalocalized oneiftheelastic constantexceeds

n a critical value Ccrit. The latter depends on the masses of the bond defect. Superohmic damping
a occurs if the bond defect is in the site M at a finite distance from one of the chain ends. If the
J observation time T is smaller than a characteristic time τM ∼M, depending on the location M of
8 the defect, the behavior is similar to the bulk situation. However, for T ≫ τM tunneling is never
2 suppressed.

] PACSnumbers: 03.65.Xp,05.30.-d,61.72.-y
r
e
h
t I. INTRODUCTION
o
.
t
a The influence of environmental degrees of freedom (DOF) on quantum phenomena, like e.g. tunneling, has been
m
of great interest during the last decades [1, 2, 3]. A significant progress came from the investigation of a more or
- less phenomenological model where a particle in a one-dimensional potential V(q) or a free particle is coupled to a
d
bath ofharmonic oscillatorswith the spectraldensityJ(ω). The quantum dissipationgeneratedby the bath depends
n
qualitatively on the low frequency behavior of J(ω) [1, 2, 3]. A particular interesting case is Ohmic damping, when
o
c J(ω) ω for low enough frequencies. In that case and for a symmetric double well potential, the particle at zero
∼
[ temperature undergoes a transition from a delocalized state to a localized one if the coupling constant between the

particle and bath exceeds a critical value [4]. An interesting observation has been made by Caldeira and Leggett [5].
1
The exponentofthe exponentialfactorforthe tunneling probabilityis multiplied byη, the phenomenologicalfriction
v
8 coefficient of the corresponding classical dynamics. This relationship between classical and quantum dissipation
1 has been deepened and generalized by Leggett [6] for an arbitrary linear coupling between the particle and bath
5 coordinates.
4 Let q˜(ω) be the Fourier transform of the classical particle trajectory q(t) and
.
1
0 ∂V
9 K0(ω)q˜(ω)+ (ω)=0 (1)
∂q
0
f
: e
v the transformed classical equation of motion, where K (ω) contains the dissipative influence of the bath. Then
0
Xi the reduced Euclidean particle propagator GE(q′,T q,0) (where the harmonic DOF have been eliminated) can be
|
represented by a path integral in the imaginary time t=e iτ [7]
r −
a
GE(q′,T q,0)= [q(τ)]e−~1S[q(τ)] . (2)
| D
Z
q(0)=q
q(T)=q′
The action
S[q(τ)]=S [q(τ)]+S [q(τ)] (3a)
0 nonlocal
contains the local
T
M µ
S [q(τ)]= dτ pq˙(τ)2+V(q(τ))+ q(τ)2 (3b)
0
2 2
Z (cid:20) (cid:21)
0
2
and the nonlocal part
T τ
1 1
S [q(τ)]= dτ dτ′K(τ τ′)q(τ)q(τ′)= dωK(ω)q˜(ω)2. (3c)
nonlocal
− − −2π2 | |
Z Z Z
0 0
e
The second equality holds for T . K(ω) is the Fourier transform of the integral kernel K(τ) and it is related to
→∞
K (ω) by K(ω)= 1K ( iω ). If the Euclidean Lagrangianof the particle-bath system is
0 2 0 − | |
e
L=L +L , L =L +L (4a)
e e e 0 1 1 bath int
with
1
L (q,q˙)= M q˙2+V(q), (4b)
0 p
2
and
N 2
1 c
L (q,x;q˙,x˙)= m x˙2 +ω2 x α q (4c)
1 2α=1 α" α α(cid:18) α− mαωα2 (cid:19) #
X
then[1, 2, 3]
∞
2 J(ω)
µ= dω (5a)
π ω
Z
0
and
∞
1 cosh(ω(T τ ))
K(τ)= dωJ(ω) 2 −| | (5b)
π sinh(ωT)
Z 2
0
with the spectral density
π N c2
J(ω)= α δ(ω ω ) . (5c)
α
2 m ω −
α α
α=1
X
For a finite T the kernel K(τ) and the paths q(τ) can be periodically continued. Then the Fourier series for K(τ) is
given by the Fourier coefficients [3]
1 c2 1
K = α (6)
n T m ν2+ω2
α α n α
X
with ν =(2π/T)n, n = 0, 1, 2,.... Here M and m are the masses of the particle and harmonic oscillators,
n p α
± ± { }
respectively. ω are the oscillator frequencies and c are the coupling constants between q and the coordinates
α α
{ } { }
of the oscillators x . Note that x are not necessarily positions, but can represent normal mode coordinates of
α α
{ } { }
vibrations, etc.
In the following we will restrictourselvesto a system ofN particles whose potential energyV(~x ,...,~x ) includes
1 N
harmonic and anharmonic interactions. Without an external field, V must be translationally invariant. However,
since the coordinates of Langrangian (4) are not specified, it is not necessarily invariant under translations. This
has motivated Chudnovsky [8] to apply the Caldeira-Leggett approach to a system of two particles (i = 1,2) with
positions x and masses M , coupled to oscillatorswith coordinates x , frequencies ω and masses m ,α=1,...,N.
i i α α α
The corresponding Euclidean Lagrangianis of the form
N
M M 1
L= 1x˙2+ 2x˙2+V(x x )+ m x˙2 +ω2(x x )2 (7)
2 1 2 2 1− 2 2 α α α α− 2
α=1
X (cid:2) (cid:3)
withV(x x )beingtheinteractionenergybetweenbothparticles. Thecouplingofparticlei=1totheoscillatorsis
1 2
−
assumedtobezero. ItisobviousthatListranslationallyinvariantunderx x +a, x x +aprovidedthat
i i α α
−→ −→
3
x are realspace coordinates. Surprisingly the elimination ofharmonicDOF does notleadfor zerotemperature to
α
{ }
a nonlocal action of Caldeira-Leggett type (cf. Eqs. (3c), (5b),(6)). The Fourier coefficients of Chudnovsky’s kernel
have the form
2 M2ν2
Kc = 1 n , (8)
n T m ω2
M +M + α α
1 2 ν2+ω2
α n α
X
which differs qualitatively from the Caldeira-Leggetttype result (6). However, for M one gets
1
→∞
2 1 ν2
Kc (M M )ν2 2m ω2 n (9)
n → T 1− 2 n− T α αν2 +ω2
α n α
X
where the last term, up to the additional factor ν2, corresponds to the second order time derivative of the kernel,
− n
identical to K from Eq. (6), if one chooses cc =√2m ω [8].
n α α α
Then the question arises: Does a translationally invariant model in general lead to a nonlocal action, which is not
ofCaldeira-Leggetttype? This is one ofthe mainpoints we wantto investigateamongothers inourpaper. Itwill be
done for a microscopic(within a Born-Oppenheimer approximation), explicitly translationally invariant lattice model
withadefectwhichcannotdiffuse. WewillshowhowthenormalmodecoordinatesfortheharmonicDOFcanexactly
be separated from the anharmonic ones. This leads to a Lagrangian of the form of Eq. (4) with coupling constants
c , frequencies ω and a spectral density determined by the microscopic model parameters. Such a microscopic
α α
justification of Lagrangian (4) has been presented for quantum diffusion [9]. There a particle diffusing through an
elastic lattice is considered. If, however, that particle cannot diffuse and is an integral part of the lattice, e.g. an
impurity which can tunnel only between two positions, one has to separate the center of mass (COM) and relative
coordinates of all particles. For such a situation, Sethna [10] has estimated the coupling constants c by comparing
α
the strain field of an elastic monopole of an impurity with the displacement for a longitudinal mode. But, as far as
we know, there is no microscopic derivation available for the quantities c , ω and J(ω) for a non-diffusing impurity
α α
in a lattice. Besides such a microscopic derivation we will show that the quantum behavior of the defect is sensitive
to its location.
The outline of our paper is as follows: Section II presents our model and outlines the main steps leading to
Lagrangian (4). The implications for the quantum behavior will be discussed in the third section with a special
emphasisonthe roleofdefectlocation. AsummaryandconclusionsarecontainedinthefinalsectionIV. Appendices
A, B and C contain details on the separation of the harmonic and anharmonic DOF.
II. MODEL
WeconsideranopenchainofN particleswithmassesm , n=1,...,N andharmonicnearestneighborinteractions
n
andone anharmonicbond representinga defect. This model coulddescribe a linear macromoleculewith animpurity.
The classical Hamiltonian reads
N
1
H = p2 +V(x ,...,x ) (10a)
2m n 1 N
n
n=1
X
with the potential energy
N−1
C
V(x ,...,x )= (x x a)2+V (x x ) . (10b)
1 N n+1 n 0 M+1 M
2 − − −
n=1
(X6=M)
in which x is the position of n-th particle, C is the elastic constant of the harmonic nearest neighbor interaction,
n
a is the equilibrium length of the harmonic bonds and V (x x ) is the energy of the anharmonic bond defect
0 M+1 M
−
locatedbetween the sites M andM+1. The potentialenergy V is explicitly translationnallyinvariant. Severalways
exist to separate the harmonic and anharmonic DOF, which finally lead to the Langrangian (4) (see Appendices B
and C). One (see Appendix B), which is also applicable in higher dimensions, is to introduce the COM
1
X = (m x +m x ) (11a)
d M M M+1 M+1
m +m
M M+1
4
and relative coordinates of the bond defect
q =x x . (11b)
M M+1 M
−
Then, x ,...,x ,X ,x ,...,x are harmonic coordinates. Their kinetic and potential energy can be diago-
1 M−1 d M+1 N
nalized by introducing normal coordinates. q is linearly coupled to these coordinates. As a result, one obtains the
M
Lagrangian(4). Here we will choosea differentapproachapplicable to 1dsystems, which, however,leads to the same
Lagrangian. Let
N N
1
X = m x , M = m (12a)
c n n c n
M
c
n=1 n=1
X X
be the COM of all the particles and
q =x x a , i=1,...,N 1,
i i+1 i i
− − −
a =0, and a =a otherwise, (12b)
M i
be the relative coordinates, respectively. Using notation q =X , Eq. (12a),(12b) has the form
0 c
N
q +a = A x , i=0,1,...,N 1 . (13a)
i i in n
−
n=1
X
with
m
n
A = δ +(δ δ )(1 δ )
in 0,i i,n+1 i,n 0,i
M − −
c
Let π be the canonical conjugate momenta of q . It is easy to prove that Eq. (13a) implies
i i
N−1
p = A π , n=1,...,N . (13b)
n in i
i=0
X
Substituting p into Eq. (13a) yields
n
N−1 N−2
1 1 1 1 1
H = π2+ + π2 π π
2M 0 2 m m i − m i i+1
c i=1 (cid:18) i i+1(cid:19) i=1 i+1
X X
N−1
C
+ q2+V (q ) . (14)
2 i 0 M
i=1
(iX6=M)
Here the first term is the kinetic energy of COM, which will be dropped from now on. Note that the use of relative
coordinates introduces a coupling between the momenta. In the next step, we perform the canonical transformation
π =p˜ , i=1,...,N 1; i=M,
i i
− 6
1
π =p˜ + (m p˜ +m p˜ ), (15a)
M M M+1 M−1 M M+1
m +m
M M+1
q =q˜ , i=1,...,N 1; i=M 1
i i
− 6 ±
m
M+1
q =q˜ q˜ , (15b)
M−1 M−1 M
− m +m
M M+1
m
M
q =q˜ q˜ .
M+1 M+1 M
− m +m
M M+1
5
This leads to
H =H +H +H (16a)
d harm int
where
m +m C m2 +m2
H = M M+1p˜2 +V (q˜ )+ M M+1 q˜2 (16b)
d 2m m M 0 M 2 (m +m )2 M
M M+1 M M+1
is the defect Hamiltonian,
N−1 N−1
1 C
H = T p˜p˜ + q˜2 . (16c)
harm 2 ij i j 2 i
i,j=1 i=1
(X6=M) (X6=M)
is the harmonic part of the Hamiltonian and
m q˜ +m q˜
M+1 M−1 M M+1
H = C q˜ (16d)
int M
− m +m
M M+1
is the coupling between the two.
The matrix T = (T ) in (16c) depends on the masses m (see Appendix A). Let e(α) =
ij i
(e(α),...,e(α) ,e(α) ,...,e(α) )t and λ be the normalized eigenvectors and eigenvalues of T, respectively, and
1 M−1 M+1 N−1 α
S=(e(1),...,e(N−2))
be the orthogonalmatrix, which diagonalizes T, i.e.
S−1TS=Λ, Λ =λ δ . (17)
αβ α αβ
Then we can introduce the normal coordinates
N−1 N−1
x = S q˜ , p = S p˜ (18)
α αi i α αi i
i=1 i=1
(X6=M) (X6=M)
for α=1,2,...,N 2 such that H becomes diagonal
harm
−
N−2
1
H = λ p2 +Cx2 . (19a)
harm 2 α α α
α=1
X (cid:2) (cid:3)
The interaction term takes the form
N−2
H = c x q . (19b)
int α α M
−
α=1
X
The M dependent coupling constants are given by
1
c =C m e(α) +m e(α) . (20)
α m +m M+1 M−1 M M+1
M M+1
h i
The final step is the Legendre transformation of H, which leads to the Euclidean Lagrangian
L=L +L , L =L +L (21a)
d 1 1 harm int
where
1
L = mq˙2 +V (q ), (21b)
d 2 M 0 M
and
e
N−2 2
1 c
L = m x˙2 +ω2 x α q . (21c)
1 2 α=1 α" α α(cid:18) α− mαωα2 M(cid:19) #
X
6
Here we have used the equality q˜ =q (cf. Eq. (15b)) and
M M
N−2 c2 m2 +m2
α =C M M+1 , (22)
m ω2 (m +m )2
α=1 α α M M+1
X
which follows from the completeness of the set e(α) of eigenvectors and 1/(m ω2) = 1/C (see below). Eq. (22)
α α
allows us to include the counterterm C m2M+m2M+1 q2 in Eq. (16b) into L . This counterterm, the role of which
2 (mM+mM+1)2 M 1
has been discussed by Caldeira and Leggett [11], results from the canonical transformation, Eq. (15a), (15b). This
transformationeliminatesthecouplingbetweenthemomentaoftheharmonicDOFandthatofthedefectandgenerates
couplingbetweenthenormalmodecoordinates x andthecorrespondingdefectvariableq (cf.Eq.(16d)). Dueto
α M
{ }
the disappearance of the counterterm, there is no frequency renormalization for the bond defect[3]. The Lagrangian
(21) is identical to that of Eq. (4). The masses m and frequencies ω follow from
α α
1
mα = , ωα =(Cλα)12 (23a)
λ
α
and the reduced defect mass m˜ is given by
m m
M M+1
m= . (23b)
m +m
M M+1
Note that these results are exact for one dimeensional systems. It can be shown that for an arbitrary impurity in a
two- orthree-dimensionalsystemthe Langrangian(21) canbe derivedwithin a kind of harmonicapproximation[12].
III. QUANTUM TUNNELING
Inthissectionwewillinvestigatethezerotemperaturequantumbehavioroftheanharmonicbonddefectembedded
in the harmonic chain as shown in Fig. 1. We will assume that V (q ) is a symmetric double-well potential with
0 M
(cid:0)(cid:1)(cid:0)(cid:1)a (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)a(cid:0)(cid:1)s(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) ( Ι )
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
1 M M+1 N
(cid:0)(cid:1)(cid:0)(cid:1)a (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)al (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) ( ΙΙ )
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
1 M M+1 N
FIG.1: TwodegenerateclassicalgroundstatesoftheopenchainwithN particles. Themassesmn,n6=M,M+1arechosento
beequal. aistheequilibriumlengthoftheharmonicbondsandas,althetwodegenerateequilibriumlengthsoftheanharmonic
bond.
degenerate minima at q− =a >0 and q+ =a >a . Then the classicalgroundstate of V(x ,...,x ) (cf. Eq. 10b)
M s M ℓ s 1 N
is twofold degenerate (see Fig.1). Therefore the low-lying eigenstates form doublets. Neglecting the excited doublets
at zero temperature is justified if the bare tunneling splitting of the ground state doublet is much less than the
frequency of the upper phonon band edge ω (see Eq. (25a))[1]. If the total number N of particles in the chain
0
is macroscopically large and the particle number M is in the bulk of the chain, i.e. M = (N), one might have
expected a suppression of tunneling since the change from, e.g. q− = a to q+ = a , would rOequire a translation of
M s M ℓ
the macroscopicmass of the left and rightharmonic parts of the chain. We will see that this naive expectationis not
always correct.
On the other hand, if the defect is close to one of the free boundaries, i.e. either M = (1) or (N M) = (1),
O − O
only a finite mass M has to be translated. Consequently tunneling cannot be suppressed. This qualitative M-
∼
dependence should follow from that of the kernel K(τ), which itself results from the strong M-sensitivity of the
spectral density J(ω). Subsection IIIA discusses this phenomenon. The limit m (or m ), motivated
M M+1
→∞ →∞
by the conclusions drawn in Ref. [8], will be discussed in Subsection IIIB.
7
A. Location-dependent quantum tunneling
AswewillseeinSubsectionIIIBtheformofthenonlocalactionEq.(3c)doesnotdependqualitativelyonthemasses
m , evenif one of the defect masses m or m becomes infinitely large. Therefore we will choose for convenience
n M M+1
m m. Let us start with the situation where the bond defect is located within the bulk, i.e. M = (N), so that
n
≡ O
M
O< lim =ξ <1. (24)
N→∞ N
M→∞
One can prove that the tunneling phenomena do not depend on ξ if it is different from 0 and 1. Therefore we choose
M =N/2andwithoutlossofgeneralityN tobeeven. Thischoiceandtheassumptionm mallowsustodetermine
n
theeigenfrequenciesω andtheeigenvectorse(α),α=1,...,N 2exactlyforfiniteN. Ac≡alculation,whosetechnical
α
−
details are presented in Appendix A, results in
4 q± q C
λ± = sin2 α ω =ω sin α , ω =2 , (25a)
α m 2 → α 0 2 0 m
(cid:18) (cid:19) (cid:16) (cid:17) r
sin(q+j) , 1 j M 1= N 1,
2 α ≤ ≤ − 2 −
+(α)
e =
j N 1
r − sin(q+(j 1)) , M +1= N +1 j N 1,
α − 2 ≤ ≤ −

2 N
e−(α) = sin(q−j), 1 j N 1; j =M = , (25b)
j N α ≤ ≤ − 6 2
r
where
π
(2α 1) , σ =+,
N 1 −
q(σ) = − (25c)
α
π 2α , σ =
N −
forα=1,...,N/2 1. Itiseasytoseethate+(α) arethe symmetriceigenvalueswithrespectto j N j ande−(α)
− j → − j
are the antisymmetric ones. From this, m m and Eq. (20) it is obvious that the bond defect does not couple to
n
≡
the antisymmetric vibrational modes, as may be expected from the symmetry of the problem. With these results we
can calculate the spectral density. Making use of Eqs. (20), (23a), (25a), (25b) and (25c) we get from Eq. (5c) in the
thermodynamic limit N that
→∞
π
1 q q q
J(ω)= Cω dq cos2 sin δ ω ω sin
0 0
2 2 2 − 2
Z0 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
ω 2 ω
=C 1 . (26a)
s −(cid:18)ω0(cid:19) ω0
In the limit ω ω we obviously have
0
≪
ω
J(ω)=C , (26b)
∼ ω
0
whichcorrespondsto the situationofOhmic damping. The Ohmic damping results fromtwo facts: First, the density
of states g(ω) of the vibrational modes in a one-dimensional lattice is constant for ω ω and second the squared
0
coupling constant involves the factor sin2(q M) which, for N , M with M/N≪=ξ(=0,1), oscillates faster
α
→∞ →∞ 6
and faster so that it can be replaced by 1/2. It is emphasized that the condition M = (1), i.e. ξ = 0 or 1, will
O
change the shape of J(ω) qualitatively.
Now we can calculate the kernel K(τ). For ω T 1 and ω (T τ ) 1 the fraction in Eq. (5b) can be well
approximated by exp( ω τ sinq). The influen0ce o≫f the osci|lla0to2rs−o|n|tu|n≫neling of q is determined by the large
− 0| | 2 M
8
τ behavior, i.e. by the low frequency modes. Therefore, substituting J(ω) from Eq. (26b) into Eq. (5b) we find, of
−
course, the well-known result for Ohmic damping [1, 2, 3]
1 1
K(τ)= Cω , ω τ 1. (27)
∼ π 0(ω τ)2 0 ≫
0
Asaconsequence,thereexistsacriticalelasticconstantC sothattheanharmonicbondcantunnelforC <C ,
crit crit
despite macroscopic masses have to be moved (see Figure 1). For C > C symmetry is broken. If the anharmonic
crit
bond is prepared in its ground state, e.g. q− =a , it will remain there on average.
m s
If, however, the bond defect is located close to one of the boundaries, so that either M = (1) or N M = (1)
O − O
(ξ = 0 or 1 in (24)), the situation changes. Note that we perform first the thermodynamic limit N . Then
M = (1) means that M may equal 1,2,...,106 or even a larger, but still finite number. For m m o→ne ∞can easily
n
O ≡
show that
1
c = C e(α) +e(α) =C sin(q M) (28)
α 2 M−1 M+1 Nα α
(cid:16) (cid:17)
where is the normalization constant of e(α) (see Appendix A). Replacing by its low frequency behavior
α n α
(2/N)1N/2 and taking the limits ω T 1 and{ω (}T τ ) 1 yields N
0 ≫ | 0 2 −| | |≫
π
1 1
KM(τ)∼= 2Cω0π dqqsin2(qM)e−12ω0|τ|q (29)
Z
0
for the kernel. In order to be consistent we also replaced ω(q) = ω sin(q/2) by its low frequency dispersion ω(q) =
0 ∼
1ω q. The integrand of Eq. (29) involves two q-scales
2 0
1 1
q = and q = . (30)
M τ
M ω τ
0
| |
Equating q =q defines the timescale
M τ
τ =ω−1M. (31)
M o
The physical meaning of τ is as follows: The path integral formalism [7, 13] allows one to investigate quantum
M
tunneling by determining the instanton solutions, i.e. the solutions of the classical equation of motion for a double
well potential in imaginary time. The width τ of a single instanton is
kink
m 1/2
τ = .
kink V′′(q±)
(cid:18) 0 M (cid:19)
If we assume that V′′(q±) C, the elastic constant of the harmonic bonds, then τ ω−1 so that
0 M ≈ kink ≈ 0
τ Mτ . (32)
M kink
≈
Theτ-dependence ofK (τ)issensitivetowhether τ /τ M orviceversa. Letusstartwiththelongtimelimit
M kink
| | ≫
(i) τ /τ 1.
M
| | ≫
ThenitfollowsfromEqs.(30)and(31)thatq q . ThemajorcontributiontotheintegralinEq.(29)comes
τ M
from q < q q . Therefore, we are allowed≪to replace sin2(qM) = sin2(q/q ) by (q/q )2 = (qM)2, which
τ M M M
leads to the ≪spectral density J(ω) ω3 at low frequencies corresponding to superohmic damping. It implies
that K (τ) τ−4. The precise resu∼lt is
M
∼
48 1
K (τ)= Cω M2, τ τ . (33)
M ∼ π 0(ω τ)4 | |≫ M
0
9
1
0,0001
1e-08
τ)K(M
1e-12
1e-16
1e-20
1 10 100 1000 10000 1e+05
τ
FIG. 2: τ-dependenceof KM(τ) for M =5,10,40 and 160 (from bottom to top) on a log-log representation. The dotted and
dashed line, corresponds to τ−2 and τ−4, respectively. The crossover at τ ≈ τM from the τ−2 behavior to that of τ−4 can
clearly beseen. The parameters havebeen chosen as follows: T =105,C =m=1⇒ω0=2.
(ii) A different situation takes place for τ /τ 1.
M
| | ≪
qM π
Then q q andthe integralin Eq. (29) must be decomposed in two contributions dq + dq . The
τ M
≫ ··· ···
0 qM
first integral yields a constant in the leading order in ω τ/τ =ω τ/M. For the seconRd one we aRre allowed to
0 M 0
replacesin2(qM)=sin2(q/q )by1/2sincethemaincontributioncomesfromq q q sothatthefunction
M τ M
sin2(qM)=sin2(q/q ) is oscillating fast betweenzero and one, whereasexp 1≈ω τ≫q =exp 1q/q varies
M −2 0| | −2 τ
slowly. Accordingly τ τ corresponds to an ’effective’ spectral density J(ω) ω, i.e. to Ohmic damping. It
M
≪ (cid:0) ≈ (cid:1) (cid:0) (cid:1)
is straightforwardto estimate both integrals. As the final result we obtain
1 1 ω τ 1
0
K (τ)= Cω 1+ | | + (34)
M ∼ π 0"8M2(cid:18) O(cid:18) M (cid:19)(cid:19) (ω0τ)2#
for τ τ τ Mτ . Taking the limit M in Eq. (34) restores the result (27) for the kernel
kink M kink
≪ ≪ ≈ → ∞
K(τ).
For M = (1) or N M = (1), and N we obtain for K (τ) a crossover at τ from the power law τ−2
M M
for τ τ Oto τ−4 for τ− τ .OFigure 2 illu→str∞ates this behavior for K (τ), calculated numerically. On the log-log
M M M
≪ ≫
plot of Figure 2 the crossover between both power laws can easily be observed.
Thiscrossoverisrelatedtothe M-dependenceofthespectraldensitybecausethedefect-phononcouplingconstants
c are M-dependent. If the ’observation time’ T (which is the time T in G from Eq. (2)) is smaller than τ then
α E M
(τ τ′) will be smaller thanτ , as well. Consequentlythe kernelK (τ τ′)entering the nonlocalaction(Eq. (3c))
M M
− −
decays as 1/(τ τ′)2. If, however, T is larger than τ then it is possible that (τ τ′) becomes larger than τ , as
M M
well. For those−values of (τ τ′) the kernel K decays as 1/(τ τ′)4. This discus−sion reveals that the choice of the
M
− −
’observationtime’ T allowsto fix the ’large’-τ behaviorofK (τ), where,of course,τ T. If M is far awayfromthe
M
≤
chain end the crossover time τ is correspondingly large. Increasing M even more τ increases, too. Nevertheless,
M M
the ’observation time’ dependence still exists. It disappears for M , only. We remind the reader that the limit
→ ∞
N has to be taken first.
→∞
Now making use of the analogy [1, 3, 4] between the calculation of the action of a multi-instanton configuration
interacting via K (τ τ′) and a one-dimensional Ising model with the coupling constants J K (n m)
M nm M
− ∼ − ∼
n m−k we can conclude the following: If the ’observationtime’ T is smaller than τ then we have K (τ) τ−2,
M M
| − | ∼
i.e. Ohmic damping. In that case the bond defect may tunnel for C < C (T), whereas symmetry becomes broken
crit
forC >C (T). Note,thatthis isnotasharptransitionatC (T)sincefinite T correspondstoafinite Isingchain
crit crit
which does not exhibit a sharp phase transition. What reallyhappens when increasing the coupling constant C is an
increase of the correlation ’length’ ξ(C). As soon as ξ(C) equals the ’size’ T of the Ising-chain a ’long range’ order
occurs. However, if T is much larger than τ Mτ we have K (τ) τ−4 and tunneling is never suppressed
M kink M
≈ ∼
[1, 3].
One might be puzzled by these conclusions since the transition for Ohmic damping to decoherence for M =
(N),N occurs for large T, or to be more precise it becomes a sharp transition for T = , only. As we
O → ∞ ∞
already stressed above, the transition for T < τ is not sharp. The relevant phonons contributing to K (τ) have
M M
wavenumbers q q q = 1. This makes the ’effective’ spectral density Ohmic. Mapping the situation for
≈ τ ∼ T T
M = (1) again onto the Ising chain of length L results in the coupling constants J decaying like J n m−2
nm 0
O | − |
10
for n m < M and like J n m−4 for n m > M. It is clear that there is no sharp phase transition for finite
0
L. B|u−t if L| <M, the couplin|g−cons|tants d|eca−y as|J n m−2. For a fixed temperature T (not to be confused with
0
| − |
the ’observationtime’ T)there willbe nomagnetic order(=ˆ coherenttunneling)for J <J (T)(=ˆC <C (T)).
0 0,crit crit
For J > J (T)(=ˆ C > C (T)) a crossover to the ’long range order’ (=ˆ decoherent tunneling) takes place.
0 0,crit crit
Accordingly the quantum tunneling phenomenon is richer for M = (1) than for M = (N).
O O
Actuallywemaythinkalsointherealtimettermsthataslongasonlythephononswithrelativelyhighfrequencies
ω 1/t and hence short wave length such that 1/q < M participate in the interaction with the anharmonic defect,
≈
the latter does not ’feel’ that the chain is finite and behaves as in the Ohmic case. In the course of time the lower
frequency phonons with higher wave length ’reach’ the end of the chain and a crossover to a superohmic behavior
takes place.
B. Dependence on the masses of defect
In this subsection we will assume that
m , n=M,M +1,
6
m = M , n=M, (35)
n  1
M2 , n=M +1.
For M = (N) we may choose without loss of generality M =N/2 and N being even. It is easy to prove that e+(α)
O j
fromEq. (25b) remaineigenvectorsof T with q+, α=1,2,...,N/2 1 givenby Eq.(25c). The remaining (N/2 1)
α − −
eigenvectors are of the form
sin(q−j) , 1 j N 1,
α ≤ ≤ 2 −
e−(α) = − (36a)
j Nα 
sin(q−(N j)) , N +1 j N 1 .
α − 2 ≤ ≤ −
q−isasolutionoftranscendentalequation. Letusintroducethequantitiesβ =m/M ,i=1,2andδ =β β /(β +β ).
α i i 1 2 1 2
The limit M (or M ) implies δ 0. Since the results in [8] motivate us to study, e.g. M , we find
1 2 2
→∞ →∞ → →∞
1 2δ
q− =q++ + (δ2) (36b)
α α N 1tan(qσ+) O
− 2
in the limit δ 0. Substituting Eqs. (36a), (36b) into Eq. (20) yields
→
c− =c+ .
α ∼ α
in the leading order in δ. As a result J(ω) and therefore K(τ), too, are doubled as compared to the case of M =m,
i
i.e. we get
2 1
K(τ)= Cω . (37)
∼ π 0(ω τ)2
0
for M .
2
→∞
The only essentialresultofchangingM fromm to infinity is thatthe criticalelastic constantincreasesbya factor
2
of two.
IV. SUMMARY AND CONCLUSIONS
For a translationally invariant chain with one anharmonic bond and otherwise harmonic nearest neighbor interac-
tions,wehaveshownexactlyhowtheanharmonicdegreeoffreedomcanbeseparatedfromtheharmoniconesintheir
normal mode representation. As a result, we have obtained Lagrangian(21), which is of the form of Lagrangian(4).
Note, that this result can also be obtained for a three-dimensional system within the Born-Oppenheimer approxima-
tion starting with an arbitrary translationally invariant potential V(~x ,...,~x ) for a N-particle system [12]. Since
1 N
the Caldeira-Leggett type nonlocal action [1, 2, 3] is based on the form (4) (or (21)) of the Lagrangian it is not the
translationinvarianceandthereforenotthe conservationofmomentum,whichcanleadtoadifferenttypeofnonlocal