Phase Space Approach to Quantum Dynamics
P. LEBŒUF Division de Physique Théorique* Institut de Physique Nucléaire 91406 Orsay Cedex, France
: We replace the Schrôdinger equation for the time propagation of states of a quantised two-dimensional spherical phase space by the dynamics of a system of N particles lying in phase space. This is done through factorisation formulae of analytic function theory arising in coherent-state representation, the "particles" being the zeros of the quantum state. For linear Hamiltonians, like a spin in a uniform magnetic field, the motion of the particles is classical. However, non-linear terms induce interactions between the particles. Their time propagation is studied and we show that, contrary to integrable systems, for chaotic maps they tend to fill, as their classical counterpart, the whole phase space.
IPNO/TH 91-20 ' Mars 1991
" Unité de recherche des Universités de Paris XI et Paris VI associée au CNRS 1 Introduction
In classical mechanics, when an integrable system is perturbed some invariant tori are broken and replaced by small layers where chaotic motion takes place. This process is amplified as the perturbation increases, the way this transition occurs being described by the KAM theorem. We do not have an equivalent quantum mechanical description, and our understanding of the mathematical and physical implications of the system being classically chaotic is still rudimentary [I]. Recently, P. Leboeuf and A. Voros [2] proposed a representation of quantum states particularly well adapted to semiclassical analysis. For compact phase spaces such as the two-dimensional torus or the sphere
2 The SU(2) Coherent-State Representation
We recall first some basic formulae arising in the context of SU(2) coherent-state rep- resentation. Consider a Hamiltonian H = H(Sz,S+,S~,t) written in terms of the generators of the SU(2) group, satisfying the algebra
[S2 ; S±\ —
[S. ; S+] = -2hSz. (2.1)
Since H commutes with S2, the Hilbert space Ti is (2s + 1 )-dimensional and spanned by the usual discrete basis
S. \m•} = mti \m) m = —s, • • •, s (2.2)
S2|m) = s(s + l)fc2|m),
while the phase space P is a two-dimensional sphere (The Riemann sphere). We hence- forth take h (2.3)
so that the radius of the sphere is fixed to one. With this convention, the phase space can then be labelled by polar and azimutal angles (B,(j>). Moreover, the semiclassical limit corresponds now to s —> oo. An alternative representation of Ti in terms of analytic functions is obtained through the spin coherent-states [3]
|r) =e«+/*| _5), Z = COt^e* (2.4)
having the norm (z\z) = (l + zz)2'. The complex variable z spanning V corresponds to a stereographic projection of the Riemann sphere onto the plane by the north pole. The state I 2) is a minimum-uncertainty packet on V centered at z = (6,tj)) with a width O(hlt2) in both directions, thus coherent-states are the most "classical" quantum-states of angular momentum. They have the property
<•!->= (.?„) •*" (2-5)
so that the coherent-state decomposition of a state 11/> ) = Sl3 am\m) oîH
am (2.6)
is a polynomial of degree 2s. It has therefore JV = 2s complex zeros {zk}^... N on the sphere. We now exploit this analytic structure: the standard factorisation formula W = cfc-4 (2-7)
where C is a normalisation constant, provides a one-to-one mapping between states | if) )
of H and configurations {zk}k=l...N °f zeros in V. The spin quantum state is therefore uniquely determined (up to a constant factor) by a set {z*} of N — Is points on V. This representation was already known to E. Majorana [4] in 1932. As he has suggested, we may associate to each zero zj. corresponding to a point P on the Riemann sphere a unit vector OP, 0 being the center of the sphere. The quantum state of the spin s particle (which classically corresponds to a single arrow pointing in a definite direction) is then "composed" of a set of Is unit vectors (spins) pointing in arbitrary directions |O.Pjb} . As we show below, a coherent-state is a very particular state for which the 2s spins point in the same direction. In spite of being particularly simple for spin systems, a similar parametrisation by the zeros can be applied to other spaces of entire functions; in particular, this has being done in [2] for the case of the two-dimensional toric phase space. Moreover, an analog factorisation structure has also being exploited in the context of the quantum Hall effect [5,6] and also in two-dimensional electrons in a periodic magnetic field [7]. Some relevant features of this exact parametrisation are [2]
(i) the number JV = 2s of zeros is controlled by the value of fi. (or viceversa, cf.Eq.(2.3)), and tends to infinity in the semiclassical limit (i.e., a thermodynamic limit); (ii) each zero constitutes a topologica) defect of ^>(~), since doing a small closed path around a (non-degenerate) zero changes its phase by 2TT;
(iii) the Husimi distribution
(z|z) (1 + zz)2'
has the same zeros as V'(2)- Thus, this positive-definite real function defined on V carries the full quantum information of the state of the system through its zeros.
2 Since ( z \ z0 ) = (1 + zoz) ' [3], a wave-packet | z0 ) has a particularly simple repre- 2 sentation: it is defined by placing the 2s zeros at the single point z = — 2C/|zo| in V, the opposite point to z0 on the sphere. Finally, in this representation, the generators (2.1) have the following realization by differential operators
( z I S21 ip ) = h(zd, — s)ip(z)
{z\S+\ip) =hz(2s -zdz)ii>(z) (2.9)
3 The Equations of Motion
Our purpose now is to replace the Schrodinger equation ihdt \rp) = H \i/>} for the time evolution of a state | ip(t) ) of Ti. by an equivalent equation for the time evolution of the zeros. The coherent-state representation of the Schrodinger equation can be written
ihdMz,t) = H(z,dz)i>{z,t). (3.1)
Given the Hamiltonian if as a function of the generators SZ,S+,S-, the symbol
H(z,d:) defined as {z\H\ijf) = H(z,dz)tl>{z) is easily obtained using Eqs.(2.9). We
will assume that it has being normal ordered, i.e., all the dz have being placed to the right of the z variables. If at a time i the state has a zero at a certain phase space point z — Zk, then the condition ip(zk + Szk,t + St) = 0 implies
Sz _ d 4> k t N (3.2)
Using Eq.(3.1) we get
(3.3) where the function computed on the right hand side must be evaluated at z = Zfc(£), the position of the zero at time t. These equations determine the dynamics of each of the JV zeros. However, it is not fully written in terms of them, since we still use the
function ip(z) to calculate zk.
Let us now assume a polynomial expansion for the symbol H(z,dz)
n n where dz means the nth derivative with respect to z and fn{z) are arbitrary functions depending on the functional form of H. The higher meaningful order n in Eq.(3.4) is
n* = 2s since d"il>(z) = 0 for n > n*. The operator dz can be interpreted as a destruction of one zero since it lowers the degree of the polynomial by one (further evaluation of
the function at z = zk annihilates the fcth zero). Conversely, multiplication by (z — z0) creates a zero at z = ZQ. Then
(3.5)
But from the factorisation formula Eq.(2.7)
N n! (3.6) E ~ Zk3)-•-(zk -zj,n_,
we get a formula allowing the elimination of i/'(2) m Eq.(3.3). Each term on the right hand side of Eq.(3.6) can be interpreted as an n-body interaction between the zeros. For an order n, the /:th zero interacts simultaneously with n — 1 of the remaining JV — 1 zeros. There are ( j different possibilities to choose the n — 1 zeros, and all these possibilities are taken into account by the sum in Eq.(3.6). Then the coefficient fn{zk) in Eq.(3.5) can be seen as a position-dependent "charge". However, this charge depends only on the position of the it h zero, and not on the position of the remaining zeros. This asymmetry reflects the fact that in general Eqs.(3.3) cannot be written as a derivative of a potential i*. = d2tV(zx, • • •, ZJV) since d:j zjt y^ d,kZj. The existence or not of n-body interactions among the zeros is determined by the operator Eq.(3.4), which in turn is fixed by the functional dependence of the Hamiltonian on the generators of the group. If this dependence is simply iinear (i.e., a spin in a uniform magnetic field), then Eqs.(2.9) and (3.6) show that only n = 1 terms would appear, and there will be no interactions between the zeros. However, the coefficient fi(zk) in Eq.(3.5) will pi wide an "external field", and the motion of a given zero will depend only on its own position. The simplest non-trivial case are quadratic terms on the generators, implying two- body interactions
= 2 Y1 . (3.7) -* -j This kind of term, as the general «-body interaction Eq.(3.6), produces strong short- distance correlations among the zeros. They are responsible in non-linear Hamiltonians for the spreading of a wave-packet, since, as we already pointed out, a wave-packet
|s0) = IO0, tj>0 ) centered at (90, point, the opposite one to z0 on the sphere. For that configuration, interaction terms like (3.7) are singular and play a dominant role in the short-time dynamics. The final forni for the equations of motion is (3.8) The equilibrium configurations for the zeros - the eigenstates - will be determined by the conditions 4 = 0, k = \,---,N. (3.9) This will be in general an algebraic system of Af - 2s non-linear coupled equations for I the 2s variables {z^}. 4 Two Examples We now consider two specific examples. If the Hamiltonian H does not depend on time, then we have a one-din.ensional integrable system. In order to have a classically chaotic dynamics, we must consider time-dependent Hamiltonians. The simplest case is a periodic sequence of delta pulses (kicks). Then stroboscopic observation of the system leads, at the classical level, to a symplectic map of V onto itself. Quantum mechanically, in the Schrodinger picture the one-step map is realised by a unitary operator U such that a state | ip ) of H is transformed, after a period T of the pulse, onto | V' ) = U \4') [8]. Our quantum approach is different, and much more resemblant to the classical picture. From Eqs.(3.8) we will construct a one-step map for the time evolution of the zeros, and study their phase space motion. The first example deals with a system which, in spite of being periodically kicked, is classically integrable. The second one has a regular to chaotic transition at the classical level. 4.1 The Linear Case Consider a simple linear Hamiltonian on the generators H=pS: + pSx £ 8(t-n) (4.1) n= — integrated over a period T=I give the following discrete map of the sphere onto itself . (4.3) Its physical interpretation is as follows: between two pulses, the constant magnetic field in the z direction produces a uniform precession of the spin by an angle p around the ; axis. Then acts the magnetic field in the x direction, producing a sudden rotation of the spin by an angle /x. around the x axis. The effective motion is just a rotation around a third axis er specified by the values of the parameters /i and p. We now consider the quantum dynamics i.e., a map for the zeros. Using equations (2.9), the Hamiltonian (4.1) gives, with Sx = (S+ + 5_)/2 V 6(t-n) (4.4) so that Eqs.(3.8) become OO zk = ipzk - i-(zl - 1) Yl 6(t-n), k = !,•••, N. (4.5) 1 n=-oo As mentioned before, due to the linearity of the Hamiltonian on the generators of the group we get a velocity Z^ for the fcth zero that depends on its position but not on the position of the other zeros. These equations are easily integrated to get the one-step map. We denote t'n' the time just after the nth kick, and t" the time just before the (n.+1 )th kick. In the interval 2 which is just, in stereographic projection, a rotation by ^i around x of the fcth zero from its position just before the kick z = z["'. We have thus arrived to the conclusion that the motion of each individual zero coincides with the motion of a classical spin s particle having the same initial conditions. The quantum dynamics can then be viewed, in this simple case, as a collection of N non-interacting point particles in phase space all of them undergoing a classical motion [9,4]. n+1) The stationary condition zk = 4 applied to Eq.(4.6) determines the two points arising from the intersection of the effective axis of rotation er with the sphere, for all k. These are obviously invariant under the dynamics. The zeros being indistinguishable, we have JV + 1 = 2s + 1 different possibilities for the location of the N zeros on the two stable points. The coincidence of the quantum motion of the zeros with the classical one is not just due to the integrability of the system, but to the linear dependence of H on the generators. For non-linear integrable systems, they will be different. Analogously, in the latter case the stationary distributions of zeros will not generically be single points but will tend, in the semiclassical regime, to form lines with a typical distance of order h between the zeros [2]. 4.2 A Quadratic Kicked Spin We now consider the Hamiltonian H = ÏSl + pS, £ 6(t-n) (4.7) which has a nonlinear dependence on the generators of the group. This kind of peri- odically kicked spin system has being consider before by several authors [10,11]. From Eqs.(4.2) we obtain now the classical map (n> (4.8) whose structure is similar to the linear case Eq.(4.3). However, the rotation around the z axis is done now by an angle proportional to the ^-component of the spin S. At /i = 0, the classical motion is integrable (a simple precession around the z axis with energy E = p/2cos2 0). As /i increases, the system undergoes a regular-to-chaotic transition. In Fig. 1 we show this transition for p = 4TT and 0 < [i < 1. The variables used to label V are ((£, cos#), which are canonical conjugates. Quantum mechanically the map is realised by the unitary transformation | ip )( "+" = U \4>) " , U = exp(—ifiSx/h)ex.p(-ipSl/(2h)). The corresponding equations of n. ^ti for the zeros, obtained from (3.8), are (4.9) The kick term is the same as in 4.1. However, the quadratic dependence of the Hamil- tonian on Sz induces two-body interactions between the zeros, and their motion is now coupled. As before, to get the one-step map we must integrate Eqs.(4.9) between f(") . The motion between kickb (l) klN (4.10) corresponds classically to a uniform precession of the spin around the z axis provided by the term 5? in Eq.(4.7). The quantum motion, described by Eqs.(4.10), is much more complicated. We were not able to solve these equations analytically to obtain an explicit form for the map between two kicks, except for the simple case s = 1 (2 zeros), where we get for the position of the two zeros just before the (n + l)th kick (n) iph/2 ! 2 M z[% = (c e ± v/(»»)) + (CW) (e'*-l)) /2, C and R<"> being the "center n { n) n) n) n of mass" and "relative" coordinates at time <<">, C< ' = z 2 + z[ and #<"> = z[ - z[ \ respectively. For an arbitrary s > 1, Eqs.(4.10) were solved numerically. We denote the change in the position of the fcth zero during that period of time by z[ —> zj^'. The kick is then easy to integrate. It is just a rotation of each zero around the x axis by an angle /x (cf.Eq.(4.6)). The total map can thus be written " UI + D - (# - De-" Stationary configurations of zeros of this kicked spin system have being shown in [2] as a function of the parameter p. We now show some global properties of the map (4.11) as the parameter fi changes from 0 to 1. Since we want to emphasize the classical underlying structures, we have chosen large values of s and as initial configuration for the zeros a coherent-state wave-packet | z0 ) peaked at some point z0 of V. The zeros are then concentrated at the opposite point (0«)0«) °n the sphere. In practice, to avoid singularities in Eq.(4.10), we have placed the zeros over a small circle centered at (<£,, 8t). Other initial conditions, like a uniform distribution of zeros over V, will mix several classically different structures, unless the 10 system is completely chaotic. As in classical mechanics, we follow the trajectory of the zeros and we plot the successive positions (4 } >n - I'2' " " "• The result obtained in the asymptotic limit t = n —> oo gives an idea of the regions of phase space explored by the quantum state during its time evolution, and of the way this exploration is done. Although in a strict sense the distribution Eq.(2.8) cannot, be interpreted as a phase space probability density since the coherent-states | z ) do not form an orthogonal set, in this "complementary" picture of quantum mechanics regions of high density of zeros in V are regions where the system has a low "probability" to be found there. This probabilistic interpretation becoming more and more correct as we approach the classical limit s —> oo where coherent-states become orthogonal. Fig. 2a shows the regions of phase space explored by the zeros in the case fi = 0 and 2s = 60 up to a time t = 150. In all the figures, the initial configuration of zeros is indicated by a star. The zeros describe in this case a very regular pattern, a series of horizontal lines (like tk° classical orbits); this structure structure is more apparent in a horizontal band around the initial latitude, becoming more fuzzy as we approach the poles. A vertical alignement also exists, but we have observed it occurs only for special values of p, such as 4TT. This is also true for the si. all diagonal bands over which the zeros do not enter, as well as the vertical Une at 4> = $*• For other values of p the zeros tend to fill uniformly the horizontal lines. The total number of lines is of order '2s i.e., one line per zero. The same global structure as in Fig. 2a was observed for other initial points ($»,cos0*). The motion of the zeros is completely different when the underlying classical dynam- ics is fully chaotic. In Fig. 2b we show the same plot as in Fig. 2a, with the same initial condition, but for fi — 1 (cf Fig. Id). Here the zeros tend to fill the whole phase space in a apparently disordered way, and the overall picture obtained is quite resemblant to the classical one (the total number of points is the same in both figures). The question of the ergodicityoi this distribution, i.e., if the zeros in their time evolution cover or not the whole phase space is an important question which, however, cannot be answered by numerical methods. The latter is also true classically. For other positions of the initial coherent-state packet | Z0 ) the same pattern of Fig. 2b was observed, except when the 11 wave-packet was concentrated over an unstable periodic orbit, like the period-one orbit located at ( 5 Conclusion 1. The covering of phase space by the zeros as some boundary conditions are changed was recently exploited to interpret the localisation properties of eigenstates of quantised classically chaotic maps [12]. In that context, zeros of delocalised states (having a Chern index different from zero) were shown to cover the whole phase space. Semiclassically, those states are associated to chaotic regions of phase space. On the contrary, zeros of eigenstates associated to regular regions (localised states 12 having zero Chern index) do not cover the entire phase space. The present study seems to confirm, but now on dynamical grounds, such a scheme. 2. We have chosen the case of the SU(2) group for technical simplicity, but other groups can be treated in the same way. Eq.(3.3) is general, but the factorisation formula Eq.(2.7) and the differential representation of generators are obviously not, so that the final form of the equations of motion will depend on the group. 3. In the semiclassical limit, and at least for sufficiently short times, the phase space location of zeros at time / = n can be interpreted as the interference of different branches of the classical action (a sort of anti-Stokes line, see ref.[2] for details). When the initial state is, as in section 4.2, a coherent-state wave packet | Z0 ) , what we have studied are just the zeros of the propagator in the coherent-state n n representation ( z \ U | zQ ) as a function of time t — n, since \ij>) = U \z0) => n ip(z,t — n) — { z I U I Z0 ). Adachi [13] has interpreted the location of zeros of n ( z I U \z0) in terms of "folding" of wave functions in phase space, the basic mech- anism, together with "streching", generating a chaotic dynamics at the classical level. 4. For multidimensional systems, the basic structure Eq.(2.7) is lost since we run now into the theory of analytic functions of several variables. I sketch here a possible way to overcome the problem by considering a quantum analog of the classical Poincaré section construction. Consider for definiteness a system of two roupled spins S1 and S2. Then the analytic coherent state representation | zxz2 ) — I Zy ) ® I Z2 ) of any vector of Hilbert space +ra + IKs11Z2)= Uz2| SI 1 = E ^1Tn2Zj "*', m-2 = -S2,- • -,S2 mi ——3 13 There are (2s2 + 1) of such functions labelled by the index m2, and each of them has 2sj zeros on the plane spanned by Z1. The set of (2s2 + 1) x 2^1 zeros clearly completely determines (up to proportionality) the quantum state. Moreover, z can Sm->( i) be considered as a quantum "Poincaré Section" of the four dimen- sional i>{zi,z2) through the section plane m2 — constant. The classical analog of this procedure is to fix the projection of the spin s2 over the vertical axis, i.e. the plane cos O2 — m2/s2 = constant. (p2 being determined by energy conservation, we get a surface of section in the variables (^mJ1), to be compared to Sj712(Z1). Eq.(5.1) is the coherent-state representation of a one-dimensional system (cf (2.6)). However, an eventual transition to chaos will not be produced here via a time dependence on H but through the coupling to the other spin. Acknowledgements I would like to thank especially O. Bohigas and A. Voros for many useful discus- sions and for their constant encouragement, and to J. Kurchan and M. Saraceno for useful and illuminating discussions. References [1] for a review of the status of the subject, see for example Giannoni M J, Voros A and Zinn-Ju&tin J eds 1991, Proceedings of Les Houches Summer School, Session LII, (Amsterdam: Elsevier Science Publishers) [2] Leboeuf P and Voros A 1990, J. Phys. A: Math. Gen. 23, 1765-1774 [3] Klauder J R and Skagerstam B 1985, Coherent States (Singapore: World Scientific) [4] Majorana E 1932, Nuovo Cimento 9, 43-50 [5] Haldane F D M and Rezayi E H 1985, Phys. Rev. B 31, 2529-2531 [6] Arovas D P, Bhatt R N, Haldane F D M, Littlewood P B and Ranimai R 1988, Phys. Rev. Lett. 60, 619-622 14 [7] Dubrovin B A and Novikov S P 1980, Sov. Phys. JETP 52, 511-516 [8] Berry M V, Balazs N L, Tabor M and Voros A 1979, Ann. Phys. (NY) 122, 26-63 (9| For the special case of a coherent-state wave-packet, the quantum motion will correspond, for all values of s and for all times, to a single point (a 2s- degenerate zero) propagating classically in phase space. This is the spin analog of the non-spreading of a wave-packet for the harmonic oscillator potential. [10] Haake F, Kus M and Scharf R 1987, Z. Phys. B 65, 381-395 [11] Nakamura K, Okazaki Y ; id Bishop A R 1986, Phys. Rev. Lett. 57, 5-8 [12] Leboeuf P, Kurchan J, Feingold M and Arovas D P 1990, Phys. Rev. Lett. 65, 3076-3079 [13] Adachi S 1989, Ann. Phys. (NY) 195, 45-93 15 CAPTIONS TO FIGURES Figure 1: The classical map Eq.(4.8) for p -= Air and (a) /t = 0, (b) M — 0.1, (c) M = 0.25, (d) M = I- Figure 2: The quantum map Eq.(4.11) for p = 47f, JV = 2s = 60 and (a) p = 0, (b) M = I- AU the zero? are at t — 0 concentrated on the star, defining thus as initial state a coherent-state wave packet. Figure 3: The same as in Fig. 2 but for /t = 0.25 and two different positions (a) and (b) for the initial wave packet. 16 I 'si Ul :'l;y":.' '.-'I tu D vu .-.•.t.'- .,,';•.: ;l; ; ; : •.'. 'I ',I '.I •J ;' CO • • • -• I I ! :f : ai *!•*" '•- : ; i • i i"!. ; ; '. ". ; "•-'-" 'i Z -•"":".- • ! i '. i • " : V ; ; : '^"•* -•* • .C. V !' . '• i '. • '• '• • i ' î Ï • - •* v;;W • :;.*!•,•. !lit » • • Si : ;.; : i •!••'* • • ; • • • i1 % •. :; '•'•'•,'•' " i »'•'.' *•' ^ .' : * • ' • -,; ". .- . ; i - : • » I • ' .1 • ' Ï •=.:iii:§ :ft.V?| ^itî? 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