Group of  Nonlinear Dynamics & Complex Systems

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Research Fields —Quantum Chaos and Quantum Computer/Computing

Classical chaos refers to the sensitive dependence on initial condition  which is commonly found in nonlinear systems. In quantum mechanics, the trajectory loses its significance completely. Moreover the Schroedinger equation is a linear equation leaving no room for chaos. The correspondence  principle, on the other hand, demands that in the semi-classical regime, namely, at length scale large compared with the de Brogle wavelength, quantum mechanics continuously develops into classical mechanics. Therefore, at first glance, the name of quantum chaos seems self-contradictory. After several years debate, now people commonly accept that quantum chaos refers to the study of quantum mechanical behavior of the systems whose classical counterparts are chaotic. This field has been very active in last two decades. Three manifestation of chaos in quantum systems have been studied so far.

• Quantum Manifestation of Classical Chaos I: Energy level spacing statistics

Energy level statistics has some universal features in the semiclassical limit. It has been conjectured that level fluctuations depend only on general space-time symmetry and they are as  predicted by the Random Matrix Theory [RMT]. For instance, the energy level spacing statistics in circular billiard (representing integrable systems) is Poisson distribution, whereas that in stadium (or Sinai) billiard (representing chaotic systems with time reversal symmetry) has Wigner distribution. For a generic (mixed) system -neither complete chaotic nor complete integrable – the energy level spacing statistics is given by Berry-Robnik surmise.

Energy level spacing statistics

 

• Quantum Manifestation of Classical Chaos II: statistical properties of stationary wavefunctions

Along with eigenenergies, wavefunctions are also used to probe quantum fingerprints of classical chaos. Usually wavefunctions provide more information about the dynamics than eigenenergies. In fact, it is the space structure of wavefunction that determines the properties of spectral statistics such as level repulsion (in chaotic systems) and/or clustering (in integrable sytems) etc. So far the only proved theorem about the eigenfunctions is Shnirelman’s theorem. It agrees with the conjecture of Berry and Voros that the probability density of most eigenstates of a chaotic billiard approaches a uniform distribution. It also agrees with the Porter-Thomas distribution of RMT. Numerical studies of a large number of high-lying eigenstates of billiars have confirmed the Gaussian distribution of local wave functions.

Classical trajectories	Quantum Stationary Wave Functions
Circular billiard                        Stadium billiard	Circular billiard                             Stadium billiard

• Quantum Manifestation of Classical Chaos III: Dynamical evolution of states

One of the most important discoveries in quantum chaos is  the dynamical localization, namely the quantum

interference effects suppress the classical diffusive process (in phase space) which may take place in classical systems under external periodic perturbations. It has been shown that the dynamical localization can be mapped to the Anderson localization for electrons in 1d systems with random impurity. This fact bridges Two different field – Quantum Chaos and Solid State Physics. Dynamical localization has been confirmed in several experiments such as Rydberg atom in a microwave field and an atom moving in a modulated standing wave etc.

The group has been working in the following topics:

• Wave function structure and statistics in quantum billiard,

• Quantitative study of scars (wavefunction localization along the unstable classical periodic orbit) in far semi-classical limit.

• Energy level statistics and wave functions in mixed systems. 

• Dynamical localization in quantum billiards.

• Semi-classical propagator for chaotic quantum systems.

• Semi-classical analysis of  correlation function  in chaotic eigenstates. (g)  Quantum chaos in non-KAM  systems.

F Borgonovi, G Casati and B Li,  Phys. Rev. Lett. 77, 4744 (1996).

The definition of classical chaos – sensitive dependence on initial condition – loses its meaning in quantum mechanics, because the unitarity properties of quantum mechanics, namely, the overlap between two evolving wave functions – a natural indicator of distance between them is preserved with time, hence there is no divergence. An alternative definition of chaos – the sensitive dependence on perturbation - has been suggested recently. This new definition is meaningful both in classical and quantum mechanics. Classically, even for Small perturbation, one generically expects rapid divergence when the systems are chaotic, as the perturbation, i.e. the difference between equations of motion, soon introduces a small displacement between the trajectories. Quantum mechanically, the overlap between the wave functions begins at unity, then decays with time, and the rate of this decay – a measure of the sensitivity of quantum evolution to perturbations in the equation of motion – can be used as a signature of quantum chaos.

Recently, we have investigated the crossover of the quantum Loschmidt echo (or fidelity) from the golden rule regime to the perturbation-independent exponential decay regime by using the kicked top model

where H is a perturbed Hamiltonian from H0 which is chaotic. 

It is shown that the deviation of the perturbation independent decay of the averaged fidelity from the Lyapunov decay results from quantum fluctuations in individual fidelity, which are caused by the coherence in the initial coherent states. With an averaging procedure suppressing the quantum fluctuations effectively, the perturbation-independent decay is found to be close to the Lyapunov deccay.   Fig (right) The decay of fidelity from golden-rule decay to perturbation-independent decay. The dot line is decay with Lyapunov exponent

W. G Wang and B Li, Phys. Rev. E 66, 056208 (2002)

In classical mechanics, chaos severely limits the operation of a reversible computer. Any uncertainty in the initial conditions is magnified exponentially by chaotic dynamics, rendering the outcome of the computation unpredictable. This is why practical computational scheme are irreversible.

A quantum computer does not have this option. It relies on the reversible unitary evolution of entangled quantum mechanical states, which does not tolerate dissipation. On the other hand, the exponential gain of quantum computing is due to exponentially large size of Hilbert space which grows exponentially with the number of qubits which are the basis of quantum computers. In order to perform logical operations in quantum computers, these qubits should be coupled. As a consequence, quantum computers represent many body systems with interaction. Similar systems have been recently studied in the field of quantum chaos with applications to different many body systems such as nuclei, complex atoms, quantum dots and quantum spin glasses. It had been found that a sufficiently strong coupling leads to the emergence of quantum ergodicity and dynamical (internal) thermalization. In this regime the systems eigenstates become very complex and strongly different from the eigenstates of non interacting many body systems. At first glance, one would expect this regime to appear when the coupling is comparable with the spacings between multiparticles levels. This naïve estimate would give an absurdly strong restriction for the coupling strength and therefore a too severe limitation for the realization of quantum computers. This raises the question:

1. What limitations quantum chaos might pose on quantum computing?
2. What restrictions of quantum chaos might pose on quantum error correction.
3. Whether the suppression of quantum chaos (dynamical localization) improve the fidelity for recovery from errors of decoherence.