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Few-Body Systems

A Quantum Version of Wigner’s Transition State Theory

Roman Schubert, Holger Waalkens, Stephen Wiggins

Published Feb 23, 2009• 5 citations
Publisher

Abstract

A quantum version of a recent realization of Wigner’s transition state theory in phase space is presented. The theory developed builds on a quantum normal form which locally decouples the quantum dynamics near the transition state to any desired order in $${\hbar}$$ . This leads to an explicit algorithm to compute cumulative quantum reaction rates and the associated Gamov–Siegert resonances with high accuracy. This algorithm is very efficient since, as opposed to other approaches, it requires no quantum time propagation.

DOI
10.1007/s00601-009-0037-4

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A quantum version of a recent realization of Wigner’s transition state theory in phase space is presented. The theory developed builds on a quantum normal form which locally decouples the quantum dynamics near the transition state to any desired order in ℏ{\hbar}ℏ . This leads to an explicit algorithm to compute cumulative quantum reaction rates and the associated Gamov–Siegert resonances with high accuracy. This algorithm is very efficient since, as opposed to other approaches, it requires no quantum time propagation.
Bose-Einstein condensates are bosonic many-body systems in which the quantum mechanical ground state is macroscopically occupied. If the interparticle interaction between the single bosons is attractive, this ground state is metastable and separated from the collapse by a barrier. A sufficient thermal excitation of the condensate can enable the crossing of this barrier and induce its collapse. In this thesis, the corresponding decay mechanism of the thermally induced coherent collapse of Bose-Einstein condensates with additional long-range interaction is investigated for the first time. For the description of the condensate’s dynamics a variational approach to the quantum wave function is used, which allows for the precise determination of the system’s transition state that governs the reaction dynamics. In order to quantitatively calculate the decay rate and lifetime of the condensate, respectively, a novel transition state theory for quantum wave packets is developed. This is based on a variational approach as well as normal form expansions of the dynamical equations and the energy functional in the vicinity of the transition state. The dynamical equations obtained from the variational approach induce a symplectic structure onto the space of variational parameters. Furthermore, the normal form expansions serve as a tool to locally construct canonical coordinates on this symplectic manifold. For this purpose, the dynamical equations as well as the energy functional are expanded and diagonalized. The eigenvectors of the linearized equations of motion thereby define a symplectic basis of the system. Afterwards, the transformations of the higher-order terms are carried out in two steps via nonlinear near-identity transformations: In the first step, the nonresonant terms of the expansions are eliminated and in the second step the remaining resonant terms are transformed in such a way that they fulfill canonical equations. The combination of these methods finally allows for the application of the well-established classical transition state theory. The procedure is valid for a wide variety of trial wave functions, for arbitrary dimensions of the variational space, in arbitrary order of the expansions, and it is independent of the precise form of the Hamilton operator. In addition, a procedure to handle bifurcations in the transition state is developed using a uniform rate formula in connection with an appropriate normal form of the potential energy surface. In contrast to the usual treatment which breaks down at the point of the bifurcation, the new method allows to cross the bifurcation smoothly, because it is valid on both sides of the bifurcation as well as arbitrarily close to it. At the same time, one can calculate the flux over the whole configuration of saddles which emerge in the bifurcation. The applicability of the newly developed transition state theory for wave packets is demonstrated for a cubic potential well. The thermal decay rates of a point particle and a frozen Gaussian wave packet, respectively, are calculated within the classical and quantum normal form approaches as well as the variational one. It is shown that the variational approach is capable to perfectly reproduce both the classical and quantum normal form results of the point particle in different limits. In the limit of a narrow trial wave function it agrees perfectly with the results using the classical normal form. The results with a broader trial wave function go even beyond the classical approach, i.e. they coincide with those of the quantum normal form. In the field of Bose-Einstein condensates with long-range interaction the application of transition state theory for quantum wave packets is extended to the field of nonlinear Schrödinger systems. For condensates with attractive 1/r-interaction, the thermal decay rates are calculated for different numbers of coupled Gaussian wave functions as well as different normal form orders. Significant corrections of the decay rate are obtained with increasing order of the normal forms. In the low-temperature regime convergence is observed in eighth order of the expansions. Finally, the thermally induced coherent collapse of dipolar condensates is investigated. These exhibit the feature of a biconcave ground state density distribution for certain values of the physical parameters. It is shown that such condensates reveal the astonishing effect of a symmetry-breaking collapse dynamics. This effect is directly related to the occurrence of the structured ground state, and it depends on the geometry of the external trapping potential. In case of a “conventional” density distribution, the collapse possesses an s-wave symmetry, while it has a d-wave symmetry in case of a blood-cell shaped ground state. The thermally induced coherent collapse plays a major role close to the stability threshold where the energy barrier is small and the decay rates are high. In case of Cr atoms, dipolar condensate have been first realized with, the region of interest is on the order of a Bohr radius above the critical scattering length. There, the lifetime of the condensate can be reduced to a millisecond. For the elements Dy and Er which have also been condensed in the laboratory, the effects become even stronger, i.e. the parameter region of importance becomes larger, the energy barriers become smaller, and the decay rates are higher.
Out-of-Time-Order Correlators (OTOCs) quantify quantum information scrambling, but their connection to localized phase-space structures, such as chemical transition states, requires formal development. We derive a leading-order semiclassical expansion for the local microcanonical OTOC in systems with an index-1 saddle point, expressing the scrambling rate as a coherent sum over unstable periodic orbits on the Normally Hyperbolic Invariant Manifold (NHIM). Valid in the semiclassical limit and the intermediate-time regime before the Ehrenfest time, our derivation utilizes the Normal Form theory of the transition state, which transforms the Hamiltonian near the saddle into an integrable (though generally non-separable) form dependent on conserved actions. We outline the derivation of the microcanonical trace, the semiclassical propagator for integrable systems, the factorization of the stability matrix, and the Schur complement reduction of the stationary phase approximation. Our result extends periodic-orbit trace methods to scrambling observables, yielding a local instability exponent Λ(J) governing the leading semiclassical growth window. As a special case, when the observation time coincides with the intrinsic periods of the contributing orbits, the trace sum reduces to an effective 1.5Λ scaling, resulting from the competition between local hyperbolic growth and wavepacket dilution. This simplified form is conditional; the full expansion retains a coherent sum over orbit periods. Finally, we discuss how the dependence of the instability on transverse actions establishes a theoretical mechanism for mode-selective control of scrambling, and outline a numerical evaluation strategy to test these predictions.
Abstract Out-of-time-order correlators (OTOCs) quantify quantum information scrambling, but their connection to localized phase-space structures—such as chemical transition states—has remained largely unexplored. We derive a formal leading-order semiclassical expansion for the microcanonical OTOC in systems with an index-1 saddle point, expressing its oscillatory, geometry-resolving contribution as a coherent sum over resonant invariant tori of the normally hyperbolic invariant manifold (NHIM), each a family of periodic orbits, transversely unstable in the full phase space. Although formally subleading in ℏ to the smooth Weyl contribution, this orbit sum carries the localized exponent 2 Λ γ , compared with the smooth exponent Λ max , and it is the only contribution that resolves the periodic-orbit geometry of the transition state: within the local normal-form description, the NHIM is the one structure on the energy shell whose trajectories never escape the saddle neighborhood. Valid in the semiclassical limit and the intermediate-time regime before the Ehrenfest time, our derivation utilizes the normal form theory of the transition state, which transforms the Hamiltonian near the saddle into an integrable (though generally non-separable) form dependent on conserved actions. We outline the derivation of the microcanonical trace, the semiclassical propagator for integrable systems, the factorization of the stability matrix, and the explicit Schur complement reduction of the stationary phase approximation. Our result extends periodic-orbit trace methods to scrambling observables, recovering a local instability exponent Λ ( J ) governing the leading semiclassical growth window. Finally, we discuss how the dependence of the instability on transverse actions suggests a mechanism for mode-selective control of scrambling, and outline a numerical evaluation strategy to test these predictions.