Classical limits for quantum maps on the torus

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 39, NUMBER 4

APRIL 1998

Classical limits for quantum maps on the torus Andrzej Lesniewski, Ron Rubin, and Nathan Salwen Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138-2901

~Received 11 March 1997; accepted for publication 26 March 1997! We provide a rigorous canonical quantization for the following toral automorphisms: cat maps, generalized kicked maps, and generalized Harper maps. For each of these systems we construct a unitary propagator and prove the existence of a well-defined classical limit. We also provide an alternative derivation of Hannay and Berry results for the cat map propagator on the plane. © 1998 American Institute of Physics. @S0022-2488~98!02404-9#

I. INTRODUCTION

The burgeoning field of ‘‘quantum chaos’’ or, more precisely, the quantum mechanics of classically chaotic dynamics has opened the door to several issues in quantization ~see e.g., Ref. 1!. One of these is the quantization of classical maps.2–5 Classical maps arise naturally in the study of chaos. In the Poincare´ surface of section maps ~see Ref. 6!, for instance, the flow on a d-dimensional phase space is ‘‘reduced’’ to a (d21)-dimensional map by considering the discrete evolution of points of intersection of the trajectory with a co-dimension of one hypersurface. Chaotic properties of the flow typically leave tell-tale signs on the surface of section. It is clear that the two problems are intimately connected, and numerical results indicate this. Another typical way in which classical flows lead to maps is via a coarse graining of Hamilton’s equation, p 8 5p1 f ~ x ! Dt,

x 8 5x1p 8 ,

where f (x)52¹ x u(x) is the classical force function for the Hamiltonian H5 p 2 /21u(x), and where in the second equation p 8 is used rather than p to ensure the map is area preserving. For maps corresponding to integrable flows, the parameter Dt is a type of ‘‘order parameter’’ for the transition to chaos in the corresponding family of maps. Typically for large Dt, the system is completely chaotic, while for small Dt, the system mimics its continuous time progenitor and remains on regular trajectories for all time. ~Examples abound. See, e.g., Ref. 6.! The quantization of classically chaotic maps has recently become an interesting issue in its own right. Hannay and Berry3 provide a quantization of the cat map ~ x 8 , p 8 ! 5 b ~ x,p ! mod 1,

b PSL ~ 2,Z! ,

~1!

using a semiclassical argument. First they impose the integrality condition of Planck’s constant h51/N. Then argue that on the torus the Hilbert space has dimension N. This is semiclassical in that it corresponds to the quantization condition

R

~2!

pdx5nh.

For the unit torus, the left-hand side of ~2! is unity, and the integrality condition for Planck’s constant follows. In fact, this is also the integrality condition of geometric quantization which requires the symplectic form divided by Planck’s constant to define a deRham cohomology class with integer coefficients.7 With the Hannay–Berry prescription for quantization, a further restriction was necessary: only a subset of SL(2,Z), the so-called ‘‘checkerboard’’ matrices

S 0022-2488/98/39(4)/1835/13/$15.00

odd even

even odd

D S ,

1835

even

odd

odd

even

D

~3! © 1998 American Institute of Physics

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Lesniewski, Rubin, and Salwen

could be quantized. In Refs. 8, 4, and 5, a different approach toward quantizing the cat maps was introduced, in which the dynamics is described in the Heisenberg picture as the evolution of a noncommutative algebra of observables. The Hilbert space remains infinite dimensional, and no restrictions on Planck’s constant or the allowed classical maps are made. The quantum torus is constructed by deforming the classical algebra of observables. We let U5Q \ ~ e 2 p ix ! ,

V5Q \ ~ e 2 p i p ! ,

where Q \ ( s ) is an operator on the Hilbert space H corresponding to the classical symbol s, and the algebra of observables A\ is then generated by $ U,V,U † ,V † % . Technically, the algebra is given with an appropriate closure. For example, taking A\ 5 $ U,V,U † ,V † % 9 , where R 9 is the bicommutant of R, the algebra generated is a von Neumann algebra. The quantum dynamics is implemented as a discrete automorphism group of A\ generated by a unitary operator ~called the propagator! F such that as \→0, F † Q \ ( s )F→Q \ ( s + b ) in a suitable topology. A quantization scheme of the kind just described was given in Refs. 4 and 5 for the cat map, along with the connection to the original quantization proposed by Hannay and Berry. In this paper, we extend the results to three classes of maps. We divide the remainder of the paper into three sections, corresponding to the different maps. In Sec. II, we review the relevant results of Refs. 4 and 5 for the cat maps, and also provide a canonical derivation of the propagator on the plane obtained by Ref. 3 using a ‘‘discrete time path integral.’’ In Sec. III, we provide a quantization scheme for ‘‘generalized kick maps’’ k (x,p) 5(x 8 , p 8 ), with p 8 5 p1 e f ~ x ! ,

x 8 5x1p 8 .

~4!

Here e is a real constant, and f is a continuous real function ~the force function! on the unit circle with Fourier coefficients f k obeying the bound

(

kPZ

k 2 u f k u ,`.

~5!

In Sec. IV, we quantize ‘‘generalized Harper maps’’: h (x,p)5(x 8 ,p 8 ), with p 8 5p1 e 1 f ~ x ! ,

x 8 5x1 e 2 g ~ p 8 ! ,

~6!

where e 1 and e 2 are real constants, and f and g are real functions on the unit circle with Fourier coefficients satisfying the bound:

(j

j 2 u f j u ,`,

(k k 2u g ku exp

S

2 p e 1k

S ( DD j

u f ju

,`.

Note for generic f in kick maps, and generic f and g in Harper maps, the coefficients e and e 1 , e 2 are the parameters which induce a transition to chaos as they are increased in magnitude. II. THE CAT MAP REVISITED

The cat map ~1! has become a paradigm for the study of quantum chaos ~see, e.g., Ref. 9!. The first quantization scheme for the cat map was given by Hannay and Berry3 in which a path integral argument from continuous time linear dynamics was used to write down the propagator for quantum evolution. In Ref. 4, a canonical quantization was proposed using Toeplitz operators in Bargmann space, and an explicit integral kernel for the unitary one-step evolution on the entire complex plane was given. Recall that Bargmann space H 2 (C,d m \ ),L 2 (C,d m \ ) is the Hilbert space of entire functions on C which are square integrable with respect to the measure d m \ (z) 5( p \) 21 exp(2uzu2/\)d2z. Formally, Toeplitz quantization T \ of a symbol over phase space is the operator of multiplication on L 2 (C,d m \ ) by this symbol followed by a projection onto H 2 (C,d m \ ). For calculations, we must simply note that this gives an anti-Wick ordering of the corresponding quantum operator: T \ 5(z n¯z m )5T \ (z¯m )T \ (z n ). The quantum torus is then conDownloaded 22 Apr 2008 to 128.103.60.225. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp

J. Math. Phys., Vol. 39, No. 4, April 1998

Lesniewski, Rubin, and Salwen

1837

structed by restricting the algebra of observables to be the Toeplitz quantization of Fourier series with classical generators exp(2pix) and exp(2pip). As in Refs. 4 and 5, we define the unitary operators U and V on H 2 (C,d m \ ) to be the quantization of classical generators for f PC(T2 ), i.e., U5exp~ 2 p ixˆ ! 5exp~ p 2 \ ! T \ ~ exp~ 2 p ix !! , V5exp~ 2 p ipˆ ! 5exp~ p 2 \ ! T \ ~ exp~ 2 p ip !! , where xˆ 5T \ (x), pˆ 5T \ (p), and z5(1/&)(x2ip). Observe also that the canonical position and momentum variables on the plane satisfy the usual Heisenberg relation @ xˆ ,pˆ # 5i\, and so UV5e 24 p

2 \i

~7!

VU.

Bargmann space is natural in that a phase space dynamics is given a phase space quantization. However, it is clear from the isomorphism between H 2 (C,d m \ ) and L 2 (R,dx) ~see Ref. 10! that we should be able to reformulate the problem in L 2 (R). Following Ref. 4, to the classical cat map b defined in Eq. ~1! we assign a corresponding quantum propagator C. For linear dynamics on the torus we have the following result. Theorem 1: There exists a unique (up to a phase) unitary quantum propagator C satisfying the following properties: (1) C † xˆ C5axˆ 1b pˆ 5xˆ 8 ,

C † pˆ C5cxˆ 1d pˆ 5 pˆ 8 ,

~8!

where ( ac bd )PSL(2,Z), such that C maps the harmonic oscillator ground state into the unique ground state of the transformed coordinates. (2) The mapping U→U 8 5C † UC, and V→V 8 5C † VC extends to an automorphism of the von Neumann algebra A\ . (3) For s PC(T2 ), i C † T \ ~ s ! C2T \ ~ s + b !i →0

as \→0.

(4) Acting on a state FPL 2 (R), C can be written explicitly as CF ~ x ! 5

S DE 1/2

1 hb

e i ~ ay

2 22yx1dx 2 ! /2b

R

2

F ~ y ! dy.

~9!

2

Remark 1: C(x,y)5(1/hb) 1/2e i(ay 22yx1dx )/2b is precisely the integral kernel proposed in Ref. 3 using a semiclassical path integral argument. Proof: The proofs of ~1!, ~2!, and ~3! are given in Ref. 4. We prove statement ~4!. Following Ref. 10, the isomorphism between H 2 (C,d m \ ) and L 2 (R) is implemented by the Bargmann transform. For xPR, and zPC, the integral kernel B:L 2 (R) →H 2 (C,d m \ ) is given by B ~ z,x ! 5

S D 1 p\

1/4

exp~ &xz/\2x 2 /2\2z 2 /2\ ! .

Note also that B 21 ~ z,x ! 5B ~ ¯z ,x ! . The integral kernel for the linear dynamics on the plane in Bargmann representation was found in Ref. 4 to be F ~ z,w ! 5

1

Au a u

H

exp 2

J

¯ z2 w ¯ z bw ¯2 b 1 1 , 2\ a \ a 2\ a

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Lesniewski, Rubin, and Salwen

where

a 5 12 ~ a1d1i ~ b2c !! ,

b 5 21 ~ a2d1i ~ b1c !! .

Then C ~ x,y ! 5 5

E

B ~ ¯z ,x ! F ~ z,w ! B ~ w,y ! d m \ ~ w ! d m \ ~ z ! 1

Ap \ u a u

E

¯ z 2 /2\ a 1w ¯ z/\ a 1 b w ¯ 2 /2\ a exp~ &xz¯/\2x 2 /2\2z¯2 /2\2 b

1&yw/\2y 2 /2\2w 2 /2\2 u z u 2 /\2 u w u 2 /\ ! d 2 zd 2 w. This integral is a straightforward but tedious Gaussian integral. It can be calculated directly with the following result. For complex numbers A, B, C, D, E such that Re A.0 and (Re A)2 .(Re B1Re C)21(Im B2Im C)2, we find

E

C

exp~ 2A u z u 2 1Bz 2 1Cz¯2 1Dz1Ez¯ ! d 2 z

5

p

AA 2 24BC

exp

~ D1E ! 2 ~ A 2 24BC ! 2 ~ A ~ D2E ! 12 ~ EB2CD !! 2 . 4 ~ A2B2C !~ A 2 24BC !

Applying this formula twice to the above integral gives the desired result.

j

III. QUANTUM KICK MAPS

We provide here a procedure for quantizing another class of automorphisms of the torus described in Sec. I. Exponentiating ~4!, we have for the classical maps on the torus e 2 p ix 8 5e 2 p i ~ x1p 8 ! ,

e 2 p i p 8 5e 2 p i ~ p1 e f ~ x !! .

~10!

In fact, we let ~10! also give the quantum evolution, that is, we find a quantum propagator such that the one-step dynamics is given by ~4! exactly, with x and p replaced by the Toeplitz operators xˆ and pˆ . However, it is not immediately clear that such a map will give a well-defined automorphism of the quantum algebra A on the torus. We now demonstrate this. Lemma 2: Let f be a real valued function satisfying (5). Then the sum ( kPZ˜f k U k , where

S

D

2ilk ˜f 5 12e fk k ilk

~11!

defines a bounded self-adjoint operator. Proof: This is an immediate consequence of the estimate

(k i˜f k U k i < (k u˜f ku < (k u f ku ,`. The second inequality follows from ~19!, while the last bound on the Fourier coefficients is by assumption. j We now define the quantum map corresponding to ~10!, U 8 5e 2il/2V 8 U,

S

V 8 5V exp 2 p i e

( ˜f k U k

kPZ

D

,

~12!

where l54 p 2 \. Downloaded 22 Apr 2008 to 128.103.60.225. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp

J. Math. Phys., Vol. 39, No. 4, April 1998

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1839

In the following we use the standard functional calculus for self-adjoint operators to define functions of the operators xˆ and pˆ . Lemma 3: With the above definitions, U 8 5exp~ 2 p i ~ xˆ 1 pˆ 8 !! ,

V 8 5exp 2p i ~ pˆ 1 e f ~ xˆ !! .

~13!

Proof: Using Trotter’s formula ~see, e.g., Ref. 11!, for T an integer, exp 2p i ~ pˆ 1 e f ~ xˆ !! 5 lim ~ exp@ 2 p ipˆ /T # exp@ 2 p i e f ~ xˆ ! /T # ! T ,

~14!

T→`

where the limit is meant in the strong operator topology. Using e 22 p i pˆ /T xˆ e 2 p i pˆ /T 5xˆ 22 p \/T, we see that the right-hand side of ~14! reduces to T21

2pie T

exp~ 2 p ipˆ ! lim exp T→`

5exp~ 2 p ipˆ ! lim exp T→`

S

S

( (k

j50

2pie T

5exp~ 2 p ipˆ ! exp 2 p i e

f k exp@ 2 p ik ~ xˆ 22 p j\/T !#

(k

f k exp@ 2 p ikxˆ #

(k ˜f k exp~ 2 p ikxˆ !

D

12exp~ 24 p 2 \ik ! 12exp~ 4 p 2 \ik/T !

D

5V 8 ,

where ˜f k is defined in ~11!. We also find exp~ 2 p i ~ xˆ 1pˆ 8 !! 5exp~ 2 p ipˆ 8 ! exp~ 2 p ixˆ ! exp~ 22 p 2 @ xˆ ,pˆ 8 # ! 5V 8 Ue 2il/25U 8 , and the claim follows. j Theorem 4: Let f be a real valued function satisfying (5). (1) There exists a unitary operator (the quantum propagator) which implements (12), i.e., U 8 5K † UK,

V 8 5K † VK.

Explicitly, this operator is given by K5e 2i pˆ

2 /2\ i e u xˆ ! /\ ~

e

~15!

,

where u is a real differentiable function on the unit circle defined (up to a real constant) by du(x)/dx5 f (x). (2) The map U→U 8 , V→V 8 defined above extends to an automorphism of the von Neumann algebra of observables on the quantized torus. We call this automorphism the quantized kick map. (3) The quantum dynamics has a well-defined classical limit in the norm topology. For s PC(T2 ), we have i K 21 T \ ~ s ! K2T \ ~ s + k !i →0, as \→0,

where k (x, p)5(p 8 ,x 8 ) is defined in Eq. (4). Proof: ~1! This is a straightforward calculation. Observe for K as in ~15!, K † VK5exp~ 2 p iK † pˆ K ! 5exp@ 2 p i $ exp~ 2i e u ~ xˆ ! /\ ! pˆ exp~ i e u ~ xˆ ! /\ ! % #

F S

5exp 2 p i pˆ 1 e

du ~ xˆ ! dx

DG

5V 8 ,

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J. Math. Phys., Vol. 39, No. 4, April 1998

Lesniewski, Rubin, and Salwen

where in the last step we have used Lemma 3. Likewise, K † UK5exp@ 2 p i ~ exp~ 2i e u ~ xˆ ! /\ ! exp~ ipˆ 2 /2\ ! xˆ exp~ 2ipˆ 2 /2\ ! exp~ i e u ~ xˆ ! /\ !!# 5exp@ 2 p i ~ pˆ 8 1xˆ !# 5U 8 . ~2! It follows easily from part ~1! that U 8 and V 8 satisfy ~7!. Since U and V can be expressed in terms of U 8 and V 8 , namely,

U5e il V 8 † U 8 ,

S

V5V 8 exp 22 p i e

( ˜f k~ e il V 8 † U 8 ! k

kPZ

D

,

this implies that U 8 and V 8 generate A\ . The map is thus an automorphism of the algebra A\ . ~3! As the proof of ~3! is somewhat involved, we divide it into steps. Step 1. We first calculate the terms in ~3! for the case of the pure harmonic

s mn ~ x,p ! ªexp~ 2 p i ~ mx1n p !! . Applying the Toeplitz quantization to the symbol s mn and using the Baker–Hausdorff– Campbell relation in the last step, we get

T \ ~ s mn ! 5T \ ~ exp~ & p i ~ m2ni !¯z ! exp~ & p i ~ m1ni ! z !! 5exp~ p i ~ m2ni !~ xˆ 1ipˆ !! exp~ p i ~ m1ni !~ xˆ 2ipˆ !! 5V n U m exp~ 2 p 2 \ ~ n 2 1m 2 12inm !! .

~16!

Thus, using ~14! once again, we find

K † T \ ~ s mn ! K5 ~ V 8 ! n ~ U 8 ! m exp~ 2 p 2 \ ~ n 2 1m 2 12inm !! 5 ~ V 8 ! n1m U m exp~ 2ilm/2! exp~ 2il @ m ~ m21 !# /2! exp~ 2 p 2 \ ~ n 2 1m 2 12inm !! 5V n1m U m exp~ 2l ~ n 2 1m 2 12inm12im 2 ! /4!

S

3exp 2 p i e

(k ~ m1n !˜f k~ m1n ! U k

D

~17!

,

where, for any integer n,

S

D

˜f ~ n ! 5 12exp~ 2ilkn ! f . k k ilkn

~18!

Expanding the right-hand side of ~17!, we have Downloaded 22 Apr 2008 to 128.103.60.225. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp

J. Math. Phys., Vol. 39, No. 4, April 1998

Lesniewski, Rubin, and Salwen

1841

K † T \ ~ s mn ! K5V m1n U m exp~ 2l ~ n 2 1m 2 12inm12im 2 ! /4! `

3

~ 2 p i e ! l ~ m1n ! l l!

(

l50

l

( ) ˜f k ~ m1n ! U k . j

j

k 1 ,...,k l j51

We want to compare this last expression with T \ ~ s mn + k ~ x, p !! 5T \ ~ exp~ 2 p i @ n ~ p1 e f ~ x !! 1m ~ x1p1 e f ~ x !!# !!

S

`

5T \ exp 2p i ~~ m1n ! p1mx !

(

l50

D

@ 2 p i ~ m1n ! e ~ ( kPZ f k e 2 p ikx !# l . l!

Using ~16! to expand the right-hand side of the equation above, we find

S

T \ ~ s mn + k ~ x, p !! 5V m1n U m exp 2 `

3

~ 2 p i e ! l ~ m1n ! l l!

(

l50

l @~ m1n ! 2 12i ~ m 2 1mn !# 4

S

3exp 2

D

l

( )

k 1 ,...,k l j51

f k jU k j

D

l @~ m1k 1 1¯1k l ! 2 12i ~ m1n !~ k 1 1¯1k l !# . 4

Step 2. Next we prove ~3! for the case of a pure harmonic. Since ˜f k j (n) approaches f k as \ →0, it is clear that term by term ~in powers of V p U q ! the difference K † T \ ( s mn )K and T \ ( s mn + k ) vanishes as \→0. To show the entire sum vanishes requires a little more work. We shall see that this indeed happens provided the force function f satisfies our assumption. To this end, we define the following norm of f : i f i2 ª

(k ~ 11 u k u ! 2u f ku ,`.

Notice first of all the difference

I S

i K † T \ ~ s mn ! K2T \ ~ s mn + k !i < exp 2 `

3

(

l50

S

l 2 ~ n 1m 2 12inm12im 2 ! 4

~ 2 p i e ! l ~ m1n ! l l! k 1 ,...,k l

3

(

l50

l

( j51 ) ˜f k ~ m1n ! U k

2exp 2 `

D

l @~ m1n ! 2 12i ~ m 2 1mn !# 4

~ 2 p i e ! l ~ m1n ! l l! k 1 ,...,k l

3exp2

j

j

D

l

( j51 ) f k Uk

j

j

I

l @~ m1k 1 1¯1k l ! 2 12i ~ m1n !~ k 1 1¯1k l !# , 4

where we have used the fact that i U i 5 i V i 51. We next use a simple argument to eliminate the exponentials preceding the sums in both terms on the right-hand side of the above expression. We see that Downloaded 22 Apr 2008 to 128.103.60.225. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp

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J. Math. Phys., Vol. 39, No. 4, April 1998

i K † T \ ~ s mn ! K2T \ ~ s mn + k !i