384
10 Time-Periodic Quantum Systems
that this dramatic change in the behavior of the quantum system is a quantum
manifestation of the transition from the Nekhoroshev regime to the Chirikov regime
in the periodically driven optical lattice. Indeed, it appears that Arnold diffusion
gives rise to Floquet eigenstates consisting of large numbers of entangled energy
states of the nondriven quantum system and leads to energy instability of such
systems. The energy instability we observe in the driven 2d optical lattice, appears
to be a universal property of any periodically driven material system with two or
more degrees of freedom.
10.9 Quantum Control
Laser radiation provides a means to control intra-molecular processes in a robust
manner. The process is called STIRAP (stimulated Raman adiabatic passage).
STIRAP involves the application of short laser pulses with carefully chosen carrier
frequencies to a molecular system for the purpose of exciting the molecules in a
controlled manner. This technique causes the coherent change of an entire molecular
population between targeted molecular states (Hioe 1983; Oreg et al. 1984; Shore
1990; Bergmann et al. 1998).
In the remainder of this section, we will use Floquet theory to model the exact
dynamics underlying STIRAP for the case of a particle confined to a square well
potential, but driven by two laser pulses of different frequency (Na and Reichl 2004).
10.9.1 The Model (Classical Dynamics)
The classical Hamiltonian for the model of STIRAP that we consider here is
H = p
2
+ U f (t)xcos(ω f t) + U s (t)xcos(ω s t), for |x| < 1.
(10.108)
All parameters in Eq. (10.108) are dimensionless. The amplitude U f (t) of the first
pulse (in time) and the amplitude U s (t) of the second pulse, have Gaussian time
dependence of the form,
U f (t) = U o exp(−β(t − t f )
2 ) and U s (t) = U o exp(−β(t − t s )
2 ),
(10.109)
where t f < t s . We can control the duration of each pulse by adjusting the parameter
β and we can control the amount of overlap of the two pulses by changing t f and t s ,
which give the peak times of the first and second pulses, respectively. For simplicity,
we assume that the maximum amplitude U o and the width β of the two pulses are the
same. A schematic picture of the variation in time of the amplitudes of the two pulses
is displayed in Fig. 10.19. The pulse sequence ends at the total pulse duration time
10 Time-Periodic Quantum Systems
that this dramatic change in the behavior of the quantum system is a quantum
manifestation of the transition from the Nekhoroshev regime to the Chirikov regime
in the periodically driven optical lattice. Indeed, it appears that Arnold diffusion
gives rise to Floquet eigenstates consisting of large numbers of entangled energy
states of the nondriven quantum system and leads to energy instability of such
systems. The energy instability we observe in the driven 2d optical lattice, appears
to be a universal property of any periodically driven material system with two or
more degrees of freedom.
10.9 Quantum Control
Laser radiation provides a means to control intra-molecular processes in a robust
manner. The process is called STIRAP (stimulated Raman adiabatic passage).
STIRAP involves the application of short laser pulses with carefully chosen carrier
frequencies to a molecular system for the purpose of exciting the molecules in a
controlled manner. This technique causes the coherent change of an entire molecular
population between targeted molecular states (Hioe 1983; Oreg et al. 1984; Shore
1990; Bergmann et al. 1998).
In the remainder of this section, we will use Floquet theory to model the exact
dynamics underlying STIRAP for the case of a particle confined to a square well
potential, but driven by two laser pulses of different frequency (Na and Reichl 2004).
10.9.1 The Model (Classical Dynamics)
The classical Hamiltonian for the model of STIRAP that we consider here is
H = p
2
+ U f (t)xcos(ω f t) + U s (t)xcos(ω s t), for |x| < 1.
(10.108)
All parameters in Eq. (10.108) are dimensionless. The amplitude U f (t) of the first
pulse (in time) and the amplitude U s (t) of the second pulse, have Gaussian time
dependence of the form,
U f (t) = U o exp(−β(t − t f )
2 ) and U s (t) = U o exp(−β(t − t s )
2 ),
(10.109)
where t f < t s . We can control the duration of each pulse by adjusting the parameter
β and we can control the amount of overlap of the two pulses by changing t f and t s ,
which give the peak times of the first and second pulses, respectively. For simplicity,
we assume that the maximum amplitude U o and the width β of the two pulses are the
same. A schematic picture of the variation in time of the amplitudes of the two pulses
is displayed in Fig. 10.19. The pulse sequence ends at the total pulse duration time