Chapter 10
Time-Periodic Quantum Systems
Abstract Quantum systems with time-periodic Hamiltonians (time period T) have
discrete time-translation invariance and their time evolution can be written in terms
of a Floquet map that connects states at times t and t + T . The overlap of quantum
nonlinear resonances induced by time-periodic driving field can cause a change in
the Floquet spectral statistics from Poisson-like to Wigner-like behavior.
Experimental evidence for the existence of Floquet states is seen in optical
lattices that undergo time-periodic modulation. Only a few Floquet states are needed
to induce large-scale oscillations in the momenta of the atoms. Another system,
the time-periodically-kicked rotor, classically undergoes a transition to chaos. It’s
Floquet states are governed by an equation similar to the tight- binding model of
solid state physics. The quantum delta-kicked rotor exhibits dynamic Anderson
localization. Time-periodic driving fields have also been used to achieve control of
quantum transitions using avoided crossings among the quantum states participating
in the process.
Experiments on microwave-driven hydrogen have provided evidence for the existence of quantum nonlinear resonances and of higher-order nonlinear resonances.
The dynamics of time-periodically driven quantum systems, with two or more
degrees of freedom, is governed by the presence of an Arnol’d web and the Arnol’d
diffusion that accompanies it.
Keywords Time-periodic Hamiltonians · Floquet theory · Floquet map ·
Quantum nonlinear resonances · Quasienergies · Quantum delta-kicked rotor ·
Wigner distribution · Dynamic Anderson localization · Arnol’d web ·
Microwave-driven hydrogen; Quantum control · Avoided crossings ·
Landau-Zener transitions
10.1 Introduction
Nonlinear dynamical systems with only one space dimension can undergo a transition to chaos if they are driven by a time-periodic force. In this chapter, we focus on
the dynamics of quantum systems with time-periodic Hamiltonians as they undergo
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_10
339
Time-Periodic Quantum Systems
Abstract Quantum systems with time-periodic Hamiltonians (time period T) have
discrete time-translation invariance and their time evolution can be written in terms
of a Floquet map that connects states at times t and t + T . The overlap of quantum
nonlinear resonances induced by time-periodic driving field can cause a change in
the Floquet spectral statistics from Poisson-like to Wigner-like behavior.
Experimental evidence for the existence of Floquet states is seen in optical
lattices that undergo time-periodic modulation. Only a few Floquet states are needed
to induce large-scale oscillations in the momenta of the atoms. Another system,
the time-periodically-kicked rotor, classically undergoes a transition to chaos. It’s
Floquet states are governed by an equation similar to the tight- binding model of
solid state physics. The quantum delta-kicked rotor exhibits dynamic Anderson
localization. Time-periodic driving fields have also been used to achieve control of
quantum transitions using avoided crossings among the quantum states participating
in the process.
Experiments on microwave-driven hydrogen have provided evidence for the existence of quantum nonlinear resonances and of higher-order nonlinear resonances.
The dynamics of time-periodically driven quantum systems, with two or more
degrees of freedom, is governed by the presence of an Arnol’d web and the Arnol’d
diffusion that accompanies it.
Keywords Time-periodic Hamiltonians · Floquet theory · Floquet map ·
Quantum nonlinear resonances · Quasienergies · Quantum delta-kicked rotor ·
Wigner distribution · Dynamic Anderson localization · Arnol’d web ·
Microwave-driven hydrogen; Quantum control · Avoided crossings ·
Landau-Zener transitions
10.1 Introduction
Nonlinear dynamical systems with only one space dimension can undergo a transition to chaos if they are driven by a time-periodic force. In this chapter, we focus on
the dynamics of quantum systems with time-periodic Hamiltonians as they undergo
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_10
339