Quantum manifestations of Nekhoroshev stability

Accepted Manuscript Quantum manifestations of Nekhoroshev stability

Daniele Fontanari, Francesco Fassò, Dmitrií A. Sadovskií

PII: DOI: Reference:

S0375-9601(16)30470-4 http://dx.doi.org/10.1016/j.physleta.2016.07.047 PLA 23974

To appear in:

Physics Letters A

Received date: Revised date: Accepted date:

8 April 2016 10 June 2016 19 July 2016

Please cite this article in press as: D. Fontanari et al., Quantum manifestations of Nekhoroshev stability, Phys. Lett. A (2016), http://dx.doi.org/10.1016/j.physleta.2016.07.047

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Highlights • • • •

Basic quantum manifestations of classical Nekhoroshev theory are studied. A simple anisochronous convex system with two degrees of freedom is proposed. Zones are uncovered in the joint expectation value spectrum of quantized actions. The width of the zones is given by the Nekhoroshev resonant normal forms.

Quantum manifestations of Nekhoroshev stability$ Daniele Fontanaria , Francesco Fass`ob , Dmitri´ı A. Sadovski´ıa,∗ a D´ epartement b Universit` a

de physique, Universit´e du Littoral – Cˆote d’Opale, 59140 Dunkerque, France di Padova, Dipartimento di Matematica, Via Trieste 63, Padova 35121, Italy

Abstract We uncover quantum manifestations of classical Nekhoroshev theory of resonant dynamics using a simple quantum system of two coupled angular momenta with conserved equal magnitudes which corresponds to a perturbed classical integrable anisochronous Hamiltonian system. Keywords: anisochronous system, coupled angular momenta, Nekhoroshev theorem, joint expectation value spectrum, resonant normal form, exponentially long time stability

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1. Introduction One of the most influential results of the modern dynamical theory was, arguably, the Kolmogorov– Arnold–Moser (KAM) theorem of the 1960’s on the persistence of a large set of strongly nonresonant tori of perturbed Hamiltonian anisochronous integrable systems. This theorem provided a basis to studies that rely on integrable approximations to concrete important physical nonintegrable systems, such as the manybody systems in celestial mechanics and their quantum analogs. The KAM result allows, through interpolation and integrable approximations, extention of torus quantization (known as Einstein-Brillouin-Keller or EBK) to these systems, see for example [2–7]. In his celebrated study [8], Nekhoroshev developed a complementary approach that describes all motions, including the resonant ones occurring in the complement to the KAM tori, for finite, exponentially long times. Applications in celestial mechanics [9], particularly to analyzing the stability of the solar system, rigid body dynamics, infinite dimensional systems, and others are widely apprised. In its simplest form, the theory applies to perturbations of anisochronous systems whose Hamiltonians are, additionally, convex functions of the actions.

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$ October

2, 2016 is the 70th birthday anniversary of Nikola´ı Nekhoroshev [1]. We wish to dedicate this article to his memory. ∗ Corresponding author Email addresses: [email protected] (Daniele Fontanari), [email protected] (Francesco Fass`o), [email protected] (Dmitri´ı A. Sadovski´ı) Preprint submitted to Physics Letters

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Nekhoroshev considers all resonances up to the highest order N whose value depends on the strength of the perturbation  as 1/ c . He considers the action space of the original (unperturbed) integrable system and divides it into resonant and nonresonant zones. Locally, in each zone, either a resonant or a nonresonant normal form is constructed. These forms bound variations of the actions over times τ ∝ eN . Furthermore, within each resonant zone, we can introduce “fast” actions F and “slow” actions S . On the time scale τ, the values of F stay constant, while the values of S have a small bounded variation. The variation of the original actions is confined to a hyperplane (the “fast drift plane”) of constant F and is controlled by the normal form. Resonances are important, if not central to many quantum theories. However, with a notable exception of quantum rotors and specific model systems [10], they are usually studied in isochronous systems, such as multi-dimensional harmonic oscillators [3, 11, 12] or equivalents [13, 14], whose zero-order Hamiltonian depends linearly on actions. It may be instructive to position some of the previous studies with respect to the general Nekhoroshev approach. In particular, the perturbations of two-oscillators in [3, 11, 12] and in many similar studies are, essentially, modeling the dynamics within and near the center of a single resonance zone of the Nekhoroshev theory. When the authors of these studies describe the “capture” of the “regular” states by the resonance zone, this means that a resonant normal form can approximate the dynamics correctly and can, for a two-degrees-of-freedom system, provide the basis for the EBK quantization. The boundary of the zone, July 20, 2016

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when it exists, is observed in a Poincar´e section [12] and, typically, no analytical estimate is given for several reasons: polynomial normalization [3] is insufficient to reproduce fully the pendulum-like slow subsystem and, furthermore, the perturbation may not necessarily be chosen small. In other words, the states outside the resonance are strongly chaotic and are beyond the scope of the Nekhoroshev theory. In other studies, for example [10], we encounter the analysis of resonances in a neighborhood of a stable periodic orbit [15, 16], a “regular” neighborhood of what may otherwise be a strongly chaotic system. In these cases, the action along the orbit plays the role of the fast action F driving slow oscillations transversal to the orbit. External parameters or energy determine whether there is a resonance and so we have again a model for a single resonance zone of the Nekhoroshev theory. One can go even further and replace the dynamical fast subsystem by an external time-dependent periodic perturbation [6]. Such systems are quite popular in the study of interactions with an external coherent field [17, 18], see also [19]. Again this provides a study case of what happens inside one single resonance. For higher dimensional perturbed systems, Arnold’s diffusion can be observed within the zone [7, 20]. In the present study, we decided to consider a model autonomous anisochronous convex system with two degrees of freedom suitable for quantum manifestations of the Nekhoroshev results. In particular, we focus on the geometry of resonances, i.e., the stratification of the action space and the phase space into nonoverlapping resonant and nonresonant zones. It seems that this idea [21] has been overlooked and yet it produces interesting and enlightening results, important for other studies and concrete applications. Specifically, in comparison to the previous studies cited above, when it comes to each particular zone, we can assess its size, and we analyze the transitional dynamics and the respective quantum states near and on both sides of zone boundaries. Furthermore, we also like to emphasize that, in comparison to any previous work on quantum Nekhoroshev estimates, notably [22], anisochronicity is the most distinctive aspect of our system. The analysis in [22] was directed at the convergence of the quantized Nekhoroshev normal form in a simpler isochronous setup [23] where again the system can be considered as, essentially, one single zone. We are less concerned with such details here; we do not study exponential stability of fast actions and, for obvious reasons, we do not have Arnold’s diffusion. Our interest is primarily in the appearance of quantum zones, the resulting stratification of the quantized action space into different zones, and

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the characterization of quantum states within each zone, specifically within the resonant ones. This goes in spirit with the original Nekhoroshev theorem and reproduces more closely the situation in concrete complex quantum systems. Our approach can be considered as a general continuation of the analysis of the relations of classical and quantum dynamics [24], a popular topic in the 1980’s and 90’s which was often referred to as “quantum chaos” [25, 26]. Presently, we have a fair understanding of these relations (see for example [27–29]). Nonetheless, as our study shows clearly, new and interesting results can still be obtained in this field. We compare fully quantum results (obtained numerically) with the asymptotic stability zones predicted by the classical Nekhoroshev theory. This demonstrates most convincingly that quantum zones exist and that the classical theory is relevant. This also provides a basis to local quantum perturbation theories which we address briefly, and to the semiclassical theories.

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2. Concrete system

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The simplest possible convex integrable system has two degrees of freedom and Hamiltonian H0 = 12 (I12 + I22 ).

(1)

Its dynamics occurs on two-tori labeled by the values of actions I1 and I2 taking values in the action space A which is an open domain A ⊂ R2 . For an anisochronous system, both frequencies ωk = ∂H0 /∂Ik with k = 1, 2 vary from torus to torus. Resonances occur when the ratio of (ω1 , ω2 ) is rational, i.e., when ν1 ω1 + ν2 ω2 = 0, 74 75 76 77

with ν = (ν1 , ν2 ) ∈ Z2 .

The sets of resonant tori Λν map to curves λν in the action space A. Specifically, λν are straight lines passing through the origin (0, 0) ∈ A, a double resonance which we ignore. As a concrete system, we use two weakly coupled identical symmetric tops. Its unperturbed Hamiltonian 2 2 H0 = 12 (L1z + L2z )

(2)

is defined on the compact phase space S ×S , the product of two spheres of radius  j, and depends only on the components of the angular momenta L1 and L2 whose respective magnitudes L1 and L2 are conserved and equal  j. Removing the critical set {L1z , L2z = ± j} from S2×S2 gives the system with Hamiltonian (1) and classical actions 2

I1 = L1z 2

and

I2 = L2z .

2

(3)

Together with respective conjugate angles φ1 and φ2 , they define cylindrical coordinates on the phase space. So, for instance,  Lkx = 2 j2 − Ik2 cos φk , with k = 1, 2. The values of (I1 , I2 ) lie in the action space

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A = {|I1 |, |I2 | <  j}

(4)

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1 2 3 4

which is an open square in R2 with coordinates (I1 , I2 ). (We are not interested in what may happen near and at ∂A where angles (φ1 , φ2 ) are not defined and where the Nekhoroshev theorem does not apply immediately.) The tops are coupled through a small cubic perturbation V = V(0,0) + V(1,0) + V(0,1) , (5)

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with

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3 L1z ,

V(0,0) = V(1,0) = 2L2z (L2z − L1z ) L1x , V(0,1) = 4L1z (L2z − L1z ) L2x ,

(6a)

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(6b)

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(6c)

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and the complete Hamiltonian is

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H = H0 + V(L1 , L2 ).

(7)

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The small parameter of this system is given by the ratio of the magnitudes of V and H0 . All terms in V are of the same degree, so they scale similarly with the small parameter. Given that V and H0 are cubic and quadratic functions, respectively, this parameter equals  j. The perturbation V is chosen so that the degeneracy of the unperturbed system is completely removed. To have a nonintegrable system, we need at least two resonances and these correspond to (6b) and (6c). On the other hand, we like to minimize the number of visible zones. Specifically, because V has only two resonant harmonics (6b) and (6c), the only resonant Nekhoroshev normal forms with nontrivial contributions of order  are the ones corresponding to the resonances ν = (1, 0) and (0, 1). According  to Nekhoroshev, the width of these zones scales as  j. So the (1, 0) and (0, 1) resonant zones will be visible even at small  j. Furthermore, the only normal form with nontrivial contribution of order  2 corresponds to the resonance (1, 1) and the (1, 1) zone, whose width is predicted to scale as  j, will become visible at larger  j. As can be verified by direct averaging (see sec. 6), the terms (6b) and (6c) are among the simplest generic cubic perturbations that result in the second order (1, 1) resonance without creating (1, −1).

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3

For the values of  j that we consider, all other resonant zones will remain extremely small and indistinguishable from the nonresonant ones. Also notice that including the V(0,0) term makes sure that the nonresonant normal form has nontrivial first order term. Furthermore, to break all degeneracies, we have to remove all possible discrete symmetries, notably with respect to interchanging indices 1 and 2, and to mirror symmetries that flip individual signs of Lkz and Lkx . The most important here is to break the invariance with respect to Lkz → −Lkz , so we can use expectation values Lˆ kz  in the corresponding quantum system. This makes our example both simple and sufficient to explore the essential aspects of the Nekhoroshev theory in two degrees of freedom. Many quantum physical interpretations of our model system can be suggested. In addition to interacting spins and rotating particles, we like to mention specific perturbations of the hydrogen atom in the n-shell approximation [13, 30], and interacting resonant double degenerate vibrations of nuclei or molecules. Specifically, the linear four-atomic molecules acetylene H−C− −C−H and cyanogen N− −C−C− −N possess four possible bending deformations of their linear D∞h -symmetric equilibrium configuration involving the angle at one of the C atoms. The respective bending vibrations form two double degenerate normal modes Πg and Πu , symmetric and antisymmetric with regard to the plane σh . Other vibrations are of much higher frequencies and can be effectively averaged out. The Πg and Πu modes are in near 1:1 resonance for acetylene, and after introducing the 1:1:1:1 shell approximation and reducing the molecular axial symmetry S1 , we obtain a system on S2×S2 which may be paralleled to the perturbed Keplerian systems in [13]. For cyanogen, the resonance is 1:2 and we should have an interesting system whose compact singular space is homeomorphic to S2×S2 [14], and which may be sufficiently anisochronous. The compactness of the phase space S2×S2 facilitates quantum mechanical calculations because the required basis is finite. Specifically, the system with Hamiltonian H in (7) is quantized in the standard way using the so(3)× so(3) algebra of angular momentum operators Lˆ 1 and Lˆ 2 , and its eigenstates are found by diagonalizing H( Lˆ 1 , Lˆ 2 ) in the basis of (2 j + 1)2 spherical harmonics | j, m1 | j, m2 . In the quantum case, basic information on the behavior of I1 and I2 is given by the expectation values Lˆ 1z  and Lˆ 2z . Note that we have removed all possible symmetries, such as the invariance with respect to the change in the direction of rotation, from the Hamiltonian H in (7) for   0. This makes the quantities

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represented by Lˆ 1z and Lˆ 2z directly observable. Since all degeneracies of H are removed for   0, we can rely on the continuity of the eigenvalues of Hˆ to define the meaningful limit for  → 0 where the eigenstates become unambiguously those of Lˆ 1z and Lˆ 2z representing traveling waves exp(imϕ) on S2 .

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4. Classical Nekhoroshev analysis

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 j = 0.005

1

 j = 0.001

It is interesting to compare Fig. 1 to the Arnold web analysis in perturbed integrable systems with three degrees of freedom [31–33]. In these studies, regular dynamics on the nonresonant tori is uncovered through numerical integration of the motion after sampling on a grid of initial conditions and the tori are represented in the frequency domain Ω as a grid with gaps in place of resonances. Of course, no diffusion can occur in our system with two degrees of freedom, but we see similar resonant zone structure. For us, quantization does the sampling in the action space A which is diffeomorphic to Ω. More importantly, we can see quantum states within resonant zones and we have specific tools to study them.

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−1

L2z /( j) 0

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−1

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0 L1z /( j)

1 −1

0 L1z /( j)

1

Figure 1: Joint expectation value spectrum (dots) of Lˆ 1z and Lˆ 2z on the eigenstates of the quantum analog of the Hamiltonian in (7) and classical resonance zones (wedges) for j = 25.

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Containment of the variation of actions (I1 , I2 ) over the exponentially long time τ to the -neighborhood of their original value, as predicted by the Nekhoroshev theorem, implies that (L1z , L2z ) remain good quantum numbers for those quantum states that correlate with nonresonant tori. Such states continue to be represented by the regular lattice in the joint expectation value spectrum (Fig. 1). To understand what happens to the resonant states, we introduce slow–fast actions (called so after the values of the respective frequencies) for each resonance ν of our system, see Table 1. Note that |F| increases as we move along the resonances λν away from 0, while S varies in the orthogonal direction to λν . Since the value of the fast action F remains conserved over τ, F is a good quantum number and, since F varies along λν , we should have a regular periodic pattern in the zone around λν labeled approximately by F. That is precisely what we observe in Fig. 1, particularly for ν = (0, 1) and (1, 0), where it may also be seen that the joint expectation values cluster on the lines λν after “sliding” in the perpendicular direction towards λν . Indeed the fast drift plane [8] is in our case a line of constant F orthogonal to λν and the theory predicts that the value of S oscillates along it within the zone. As a consequence, the expectation value S  on any resonant eigenstate with given F should be close to 0, and the corresponding point (S , F) lies on the line λν .

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5. Size of resonant zones

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Table 1: Canonical slow–fast actions S 0 and F, recentered action S with value 0 on the resonance, and the threshold parameter bν for the resonances ν of our system with Hamiltonian in (7).

ν (1, 0) (0, 1) (1, 1)

S0 I1 I2 I1

F I2 −I1 I2 − I 1

S I1 I2 (I1 + I2 )/2

bν 2 4 6 2

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7

3. Quantum resonance zones

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We can now present our results. In our  = 0 limit, i.e., for the purely SO(2)×SO(2) symmetric unperturbed system, the values Lˆ 1z  and Lˆ 2z  form a regular lattice in A from (4). In the full system, as we can see in Fig. 1, this lattice is destroyed along the lines of classical resonances λν . The two order- resonances are clearly seen near axes I1 = 0 and I2 = 0, while the order- 2 resonance is visible for large values of  j along the diagonal I1 + I2 = 0. This is in total agreement with the classical Nekhoroshev theory. So it can be conjectured that the wedge-like domains without the lattice are quantum resonance zones. Within these zones we can distinguish two kinds of states, namely the strongly resonant states   for which L1z , L2z  collapse to/near the lines of resonances λν in A, and transitional or “wandering” states   who have entered the zone but whose L1z , L2z  remain visibly distant from λν . These latter states can be seen as forming “branches”.

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Further understanding of the joint expectation value spectrum within the resonant zones can be obtained 4

1

ν = (1, 0)

ν = (0, 1)

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L2z /( j) 0

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−1

21

−1

22

0 L1z /( j)

1 −1

0 L1z /( j)

1

23

Figure 2: Joint expectation value spectrum (dots) of Lˆ 1z and Lˆ 2z on the eigenstates of the resonant normal forms Hν(1) in (8) and classical ν-resonance zones (wedges) for j = 25 and  j = 0.005.

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ν = (1, 1)

1

from the local resonant normal forms Hν of the Hamiltonian H in (7). These forms are obtained by averaging with regard to the Hamiltonian flow of

the dependence on the direction of rotation is relegated to high orders, the resonant states have this degeneracy removed. We can verify the precision and the scope of the truncated local normal forms in (8) by using them as quantum Hamiltonians instead of the complete Hamiltonian (7). The resulting spectra are shown in Fig. 2. Comparing to Fig. 1, right, we can see that the respective resonant zones are reproduced perfectly, as is the lattice of the neighboring nonresonant states. Not surprisingly, the normal forms fail for resonances other than the ones they are designed to describe. To emphasize the locality of these normal forms we fade out the parts of the lattice where they do not apply.

0.5

ν · I = ν1 I1 + ν2 I2 .

Hν(1) = H0 +V(0,0) +Vν

L2z /( j) 0

For resonances (1, 0) and (0, 1), the first average gives for ν = (1, 0) and (0, 1). (8)

ν = 1 S 2 + Hcrit (, F) cos σ Hν ≈ H 2

2 3 4 5 6 7 8 9 10 11 12 13 14

−1

−1

(9)

which is parameterized by F and which is common in the Nekhoroshev theory for one-dimensional resonances [34].Here the critical energy and the maximum value of |S | attained (when σ = π) on the respective critical orbit (separatrix) are  Hcrit = bν ()F 2 and S crit = 2bν () |F| . (10) 1

−0.5

Using slow–fast variables defined in Table 1, going to the limit of small S , and dropping trivial terms that do not contain slow variables (S , σ), we come to the one-degree-of-freedom pendulum approximation to the ν-resonant normal form

They mark the transition between vibrations and rotations and define the boundaries of the resonance zone. For our system, the boundaries are straight lines forming a wedge. We can see from eq. (10) that in accordance with the classical √ theory the width of the (1, 0) and (0, 1) zones is -small. We draw these wedges in Fig. 1 using Eq. (10) and we can see that eq. (10) gives a fair estimate of the quantum zones. Resonant quantum states have energies well below Hcrit and are rotational (traveling waves) in the F component and vibrational (standing waves) in the S component. Transitional states with energies near Hcrit can be seen entering the (1, 0) and (0, 1) zones in Fig. 1. While nonresonant states are nearly degenerate in energy because

−0.5

0 L1z /( j)

0.5

1

Figure 3: Joint expectation value spectrum L1z ,L2z  (dots) of Lˆ 1z and Lˆ 2z on the eigenstates of the resonant normal form H (2) in (11) (1,1)

and the respective classical resonance zone for j = 25 and  j = 0.005.

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6. The 1:11:1 normal form To compute the normal form for the (1, 1) resonance in our system, we have to go to the second order average (2) (2) H(1,1) = H0 + V(0,0) +  2 V(1,1)

(11)

where on the assumption L√2z + L1z ≈ 0 (the resonance condition) and to order O( ) we have (2) V(1,1) ≈ 5(L2z − L1z )4 + 12(L2z − L1z )2 (L1x L2x − L1y L2y ). 30 31 32 33 34

5

Introducing slow–fast variables (Table 1), fixing F, using S centered on the resonance, and further simplifying (2) for small S , we bring H(1,1) to the form (9), and a similar analysis follows. As predicted, the width of the (1, 1)zone is -small. Similarly to the (1, 0) and (0, 1) normal

2 3 4

forms, the quantum computation with the normal form (11) shows that this form reproduces perfectly the spectrum within its zone (fig. 3). It remains to explain the regular “chess” pattern that appears in this zone.

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5

7. Quantum patterns

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6 7 8 9 10

A deeper understanding of the joint expectation value spectrum in Fig. 1 calls for a more detailed analysis of the normal forms Hν and their pendulum approximaν . While H ν do describe the two most important tions H aspects, specifically, the distinction between the resonant and the nonresonant states and the boundary of the ν becomes zone, considering Hν as a perturbation of H essential at medium-to-large |S |. Most notably, the perturbation by the V(0,0) term breaks the S → −S invariν and allows nonzero expectation values S  ance of H for large-|S | quasidegenerate doublet states of the pendulum. Pendulum systems are well researched [35] and ν by there are many ways to treat the perturbation of H Hν . It seems, however, that we should go beyond the approaches which we cited in sec. 1 and which apply near the centre of one single zone. We consider briefly a variational study in the basis formed by the eigenstates | f, s of Sˆ and Fˆ such that

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Sˆ | f, s =  s | f, s and Fˆ | f, s =  f | f, s .

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Note that | f, s can be easily selected from all states | j, m1 | j, m2  using definitions in Table 1. For any fixed ν have a tridiagonal matrix in this baf , both Hν and H sis, and a few basis functions with small |s| may suffice to describe the patterns in Fig. 1. Thus for the (1, 1) resonance, we should consider two possibilities with half-integer and integer values of s for odd | f | < 2 j and even | f | ≤ 2 j, respectively. In the half-integer case, the two most resonant states can be approximated by a 2×2 matrix of H(1,1) in the basis {| f, − 21 , | f, + 12 }   2 −1/2  π( f ) 2 3 2f + 1  α( f ) +  ,  π( f ) +1/2 4

33

28 29

12 13 14 15 16 17 18 19

where α and π are smooth functions of ( f, j). Its eigenfunctions have two nonzero S  values on the opposite sides of the resonance line λ(1,1) , see Fig. 1. On the other hand, a similar computation for the integer case using a 3×3 matrix in the basis with s = 0, ±1 gives additionally a zero S  value straight on λ(1,1) . In simple terms, the particular pattern is due to the alternation between the integer basis which includes the | f, 0 function and the half-integer basis without it.

F/( j)

-1

-0.15

-0.1

-0.05

0

S /( j)

0.05

0.1

0.15

j = 25

Figure 4: Joint expectation value spectrum near the (1, 0) resonance, cf Fig. 2. Dark dots mark exact (Lˆ 1z , Lˆ 2z ); larger red circles represent the same values computed for the normal form H(1,0) (8) in the basis of the first nine functions | f, s with |s| ≤ 4. 31

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11

 j = 0.005

(0, 0)

0

30

The situation becomes more interesting when the zone is sufficiently large to capture many states. In our system, this happens for the (1, 0) and (0, 1) resonances with several resonant states near S ≈0 which are oscillator states composed of | f, s with |s|  −1 S crit , and  can be seen as for which the S → − S symmetry of H dominating. The challenge is in reproducing the states with essentially nonzero S  near S crit , the transitional states of the hindered 1-dimensional rotor. As illustrated in Fig. 4, this can be achieved by taking a sufficiently large number of basis functions with small |s| ≈ −1 S crit .

-0.5

1

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6

8. Discussion In this letter, we uncovered basic quantum manifestations of the classical Nekhoroshev theory and thus provided the basis to the study of quantum analogs of the resonant and nonresonant motions in nearly integrable systems, and most interestingly—of the resonant chaotic dynamics inaccessible within other frameworks, such as KAM. Nekhoroshev theory is complimentary in its treatment of resonances. The insight that we get here for two degrees of freedom, notably the understanding of how different zones fit together in the phase space, can be transferred to larger systems where the resonant dynamics is not integrable and so cannot be treated by KAM and subsequently quantized using EBK. Similar work on perturbations of quantum anisochronous systems with three and more degrees of freedom becomes possible. More daringly, for the resonance zones, we can think of the “Nekhoroshev quantization” as of a third possibility, a complement to the KAM-inspired Einstein–Brillouin–Keller (EBK) torus quantization and the periodic orbit (PO) quantization (Van Vleck, Gutzwiller, Feynman), where fast actions F can be quantized within the EBK approach, while the dynamics in the slow actions S is treated using the semiclassical PO methods.

1

Acknowledgments

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