Quantum Dynamics and Random Matrix Theory

6 Pages·0.09 MB·

by  Herve Kunz

Quantum dynamics and Random Matrix theory
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0 Herv´e KUNZ∗
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2 Institute of theoretical Physics, Swiss Federal Institute of Technology Lausanne

n Lausanne , CH-1015 EPFL, Switzerland
a
J
February 8, 2008
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2

]
D
C Abstract
.
n We compute the survival probability of an initial state, with an energy
li in a certain window, by means of random matrix theory. We determine its
n
probability distributionand show that is is universal, i.e. caracterised only
[
by the symmetry class of the hamiltonian and independent of the initial
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state.
v
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5
0
1
0 In classical mechanics, temporal chaos is caracterised by the extreme sensi-
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bility of a trajectory to variation of initial conditions. No direct analog of this
0
/ phenomenon has been found in quantum mechanics so far. On the other hand,
n
numerical evidence has been accumulated [1], showing that energy levels of a
i
l
n quantum system, whose classical counterpart is chaotic, have a statistical be-
:
v havior described by Wigner’s random matrix theory (RMT), on the mean level
i
X spacing scale. The question we want to adress is the following: are there specific
r predictions of RMT for quantum dynamics, which would caracterise the temporal
a
behavior of ”chaotic” quantum systems.
We consider the following situation: The system is prepared in an initial state ϕ
at time 0, with an energy in a certain window, centered at e and of width 2sl(e),
where l(e) is the mean level spacing, and we want to compute the probability to
find our system again in the state ϕ, at a later time t. This quantity that we call
the survival probability R is given by
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ϕ expitH P(∆)ϕ
R = | ¯h (1)
(cid:12)(cid:16) (ϕ, P(∆)ϕ) (cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
H is the hamiltonian of our sys(cid:12)tem and P(∆) is the(cid:12)spectral projector on ∆.
∗[email protected]fl.ch
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We have chosen to take an energy in a range of the order of the mean level
spacing in order to look at properties of the system which are independent of
specific details. If (λ , ψ ) denote respectively the jth eigenvalue and eigenvector
j j
of the Hamiltonian H, then the survival probability can be written as
y χ exp2πiτx 2
R = j=1 j j j Θ χ 1 (2)
j
(cid:12)(cid:12)P j=1 yjχj (cid:12)(cid:12) Xj − 
(cid:12) (cid:12)
(cid:12) P (cid:12)  
where (cid:12) (cid:12)
y = (ϕ, ψ ) 2 (3)
j j
| |
and if we define x by the relation
j
λ = e+x l(e) (4)
j j
1 if x s
χ χ (x ) = | j| ≤ (5)
j ≡ (−s,s) j ( 0 otherwise
The Heaviside function Θ ensures that there is at least one eigenvalue in ∆.
What appears naturally in this expression is the time measured in units of the
Heisenberg time
h
t = (6)
H
l(e)
so that
t
τ = . (7)
t
H
If we look at this problem from the point of view of RMT, we will replace the
Hamiltonian by a large N N self-adjoint matrix, whose probability distribution
×
is basis independent and therefore of the form
e−W(λ1,...,λN)dH (8)
Wigner’s gaussian model corresponds to the choice
N N
W = λ2 (9)
2 j
j=1
X
The first conclusion to be drawn is that the survival probability is statistically
independent of the initial state ϕ. This follows from the fact that the variables
y N have a probability distribution, independent of ϕ and given by:
{ j}j=1
1 N N β−1
µ (y)dy = δ y 1 y2 dy (10)
N C  j −  j
N j=1 j=1
X Y
 
The parameter β = 1, 2, 4 caracterise the symmetry class of the Hamilto-
nian, respectively orthogonal, unitary and symplectic. Equation (10) follows
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easily from the Haar measure on the corresponding groups. C is a normalising
N
constant.
The variables x N are statistically independent of the variables y N
{ j}j=1 { j}j=1
and have a distribution given by
1
exp W (e+xl(e))∆β(x)dx (11)
D −
N
where the Van der Monde determinant
∆(x) = x x (12)
i j
| − |
1≤i<j≤N
Y
comes from the change of variables H (λ , ψ )N [2]. D is a constant of
ij → j j j=1 N
normalisation.
We can take l(e) = 1 , where ρ(e) is the density of states when N = .
Nρ(e) ∞
The problem that we need to solve now is to find the probability distribution of
the survival probability p(R)dR in the N = limit. We find that R is not self-
∞
averaging i.e. p(R) is not a delta distribution concentrated on the mean value
of R. On the other hand its probability distribution p(R) is universal, i.e. it
depends only on the symmetry parameter β, at least for a large class of W.
There aretwo formulas forp(R), one more appropriate to small windows, another
one to large windows.
In the first case, we decompose p(R) into
∞ E
n
p(R τ) = p (R τ) (13)
n
| 1 E |
nX=1 − 0
where E is the probability to find exactly n eigenvalues in ∆ and p (R τ) is the
n n
|
conditional probability density of R knowing that there are exactly n eigenvalues
in ∆.
It can be expressed as
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s ∞ n
p (R τ) = Eˆ(x ,...,x )dnx µ (z ,...,z )dnzδ R z exp2πiτx
n | Z−s 1 n Z0 n 1 n  −(cid:12)(cid:12)j=1 j j(cid:12)(cid:12) 
 (cid:12)(cid:12)(cid:12)X (14)(cid:12)(cid:12)(cid:12) 
(cid:12) (cid:12)
E(x ,...,x )
Eˆ(x ,...,x ) = 1 n (15)
1 n E
n
s
E = E(x ,...,x )dnx (16)
n 1 n
Z−s
E(x ,...,x ) being the probability density of finding the n eigenvalues in ∆ at
1 n
(x ,...,x ).
1 n
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Useful expressions for E(x ,...,x ) and E can be found in [2] and [3]. It is
1 n n
expressible in terms of a determinant
E(x ,...,x ) = detL (x x ) ; (i,j) (1...n) (17)
1 n β i j
| ∈
where
K
β
L = (18)
β
1 K
β
−
K is an operator whose kernel in the simplest case β = 2 is given by
β
sinπ(x y)
K (x y) = − (19)
β
| π(x y)
−
defined on L2( s, s).
−
Universality comes from the fact that E(x ,...,x ) is expressible in terms of
1 n
the correlation functions and the latter ones depends only on β, for a large class
of W. W modifies only the density of states and therefore the mean level spacing
l(e).
This expression for p(R τ) is mostly useful in the small window limit, because
|
when s 0
→ E sβ2n2+n(1−β2) (20)
n
∼
Moreover in this case we have
limsnEˆ(sx ,...,sx ) = A x x β (21)
1 n n i j
s→0 | − |
1≤i<j≤n
Y
so that the probability distribution of R shows a scaling behavior
1 R 1
lim s−β−1pr − x = g (λ)dλ (22)
s 0 ((πτs)2 ≥ ) Zx β
→
R 1
→
the function g (λ) being given by
β
g (λ) = A λβ−21 1√1 λ+ 1 lnλ ln1+√1 λ (23)
β β
2 − 2 − −
(cid:20) (cid:21)
On the other hand, one can see from eq (13) and (14) that the probability
distribution of R is well defined at infinite times. Namely
1
p (R τ) = p (R )+O (24)
n n
| |∞ τ
(cid:18) (cid:19)
as can be seen by an integration by parts where
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∞ 2π n dφ n
p (R ) = µ (z ,...,z ) jδ R z eiφj (25)
n |∞ n 1 n 2π  −(cid:12) j (cid:12) 
Z0 Z0 j=1 (cid:12)j=1 (cid:12)
Y  (cid:12)(cid:12)X (cid:12)(cid:12) 
 (cid:12) (cid:12) 
(cid:12) (cid:12)
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Using an integral representation for the delta appearing in the definition (10) of
the µ , we can reexpress (25) as
n
1 ǫ+i∞ +∞ ∞ n
p (R ) = dueu drrJ √Rr dze−uzJ (rz)zβ2−1
n |∞ 4πcn Zǫ−i∞ Z0 0(cid:16) (cid:17)(cid:20)Z0 0 (cid:21)
(26)
ǫ being any positive number, and J (x) the Bessel function. This expression can
0
be simplified, considerably when β = 2, 4.
In the unitary case (β = 2) one finds
n 1 n−3
p (R ) = − (1 R) 2 (27)
n
|∞ 2 −
For a large window of energy, it is more appreciate to find another expression for
p (R τ). It is given as some integral over a Fredholm determinant .
n
| G
is a generating function for the variables y and x appearing in the defi-
j j
G { } { }
nition of R, eq (2).
N
( ;ϕ; ) = lim exp χ cos πτ +ϕ + (28)
| | |
G ∇ ‡ N→∞* −iN † ∇ ∈ § ‡ +
|X=∞ h (cid:16) (cid:17) i
It can be expressed in terms of the operator K appearing in eq (19), when
β
β = 1, 2 as
β
β ∈
= ′ det + β ∈ (29)
G E ∞ K }
h (cid:16) (cid:17)i
with
β
E = [det (1 K )]2 (30)
0 β
−
and g is the multiplication operator by the function
−1
2i
g = 1+ [z +rcos(2πτx+ϕ)] (31)
β
" #
When the window is large (s >> 1) we can expand the determinant in powers
of K , the first two terms of this expansion dominating the other ones [4].
β
One finds that the probability distribution is exponential.
1 R 1 R
lim p τ = exp (32)
s→∞ s s σ(τ) −σ(τ)
(cid:18) (cid:12) (cid:19)
(cid:12)
(cid:12)
In the orthogonal case (β = 1), for(cid:12)example
4 2 τ + τ ln1+2 τ if τ 1
σ(τ) = − | | | | | | | | ≤ (33)
2+ τ ln 2|τ|+1 if τ 1
( | | 2|τ|−1 | | ≥
One can notice the singularity at the Heisenberg time τ = 1 and the fact that
σ( ) exists.
∞
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However if we smooth out in time R(τ), taking for example
1 τ1
R = R(τ)dτ (34)
τ τ
1 − 0 Zτ0
then we get a selft-averaging quantity
1 R
lim p = δ R σ (35)
s→∞ s s! −
(cid:16) (cid:17)
with
1 τ1
σ = dτ σ(τ) (36)
τ τ
1 − 0 Zτ0
Some numerical work on chaotic billiards [5], in the large window limit, confirm
this exponential distribution. Integrable billiards show a very different behaviour
[5].
Finally, we would like to mention the fact that Wigner’s energy level statistics
canbeobtainedformodels,whereeigenvaluesandeigenvectorsarecorrelated. We
think therefore that the study of quantum dynamics could discriminate between
such models and those we have considered where they are uncorrelated.
Acknowledgements
This work is dedicated to F.Y. Wu for his 70th birthday.
References
[1] See for example the review by O. Bohigas, Les Houches 1989, Chaos et
physique quantique (M-J. Giannoni et al, North Holland, 1991).
[2] M.L. Mehta, Random Matrices (Ed. Academic Press Inc, 1991).
[3] C.A. Tracy and H. Widom, Journal of Statistical Physics 92, 809 (1998)
[4] H. Kunz, J. Phys. A : Math & Gen 32, 2171 (1999)
[5] R. Aurich and F. Steiner, Int. J. Mod. Phys. 13, 2361 (1999)
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