Chapter 8
Quantum Ratchets
In statistical mechanics, there exists a commonly accepted postulate according to
which a “small” system interacting with a “large” system (a heat bath), which stays
in the state of statistical equilibrium, eventually approaches the state of statistical
equilibrium itself. Bogolyubov, [1, 2] rigorously proved the validity of this postulate
for the first time. As the system and the heat bath, he considered an oscillator linearly
coupled to a collection of harmonic oscillators, simulating the bath in which the
single oscillator is immersed. However, without explicitly working out the statistical
properties of the fluctuations. These properties, together with a quantum mechanical
transcription of the model, has been accomplished by Magalinski ˘ i [3]. Is this model
we will unwrap in this chapter and we apply it to understand a quantum ratchet in the
quantum range. For an excellent review of the evolution of the quantum Langevin
equation, see P. Reimann [4].
8.1 The Quantum Langevin Equation
Consider an oscillator with mass m and frequency ω 0 , linearly coupled with a
set of a large number of independent harmonic oscillators with frequencies ω k
(k = 1, 2, . . . , N; N 1), simulating the bath in which the single oscillator is
immersed. The Hamiltonian of the system is given by
H =
p 2
2m
+
1
2
mω
2
0 x
2
+
1
2
k
p
2
k + ω
2
k x
2
k
+ gx
k
β k x k
(8.1)
where x and p are the corrdinate and momentum of the single oscillator, x k and p k
are those of the k − th oscillator of the medium, and gβ k is the coefficient of the
coupling with the k − th oscillator. To obtain the equation of the motion for the
single oscillator we have to solve de system of Hamilton’s equations corresponding
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_8
123
Quantum Ratchets
In statistical mechanics, there exists a commonly accepted postulate according to
which a “small” system interacting with a “large” system (a heat bath), which stays
in the state of statistical equilibrium, eventually approaches the state of statistical
equilibrium itself. Bogolyubov, [1, 2] rigorously proved the validity of this postulate
for the first time. As the system and the heat bath, he considered an oscillator linearly
coupled to a collection of harmonic oscillators, simulating the bath in which the
single oscillator is immersed. However, without explicitly working out the statistical
properties of the fluctuations. These properties, together with a quantum mechanical
transcription of the model, has been accomplished by Magalinski ˘ i [3]. Is this model
we will unwrap in this chapter and we apply it to understand a quantum ratchet in the
quantum range. For an excellent review of the evolution of the quantum Langevin
equation, see P. Reimann [4].
8.1 The Quantum Langevin Equation
Consider an oscillator with mass m and frequency ω 0 , linearly coupled with a
set of a large number of independent harmonic oscillators with frequencies ω k
(k = 1, 2, . . . , N; N 1), simulating the bath in which the single oscillator is
immersed. The Hamiltonian of the system is given by
H =
p 2
2m
+
1
2
mω
2
0 x
2
+
1
2
k
p
2
k + ω
2
k x
2
k
+ gx
k
β k x k
(8.1)
where x and p are the corrdinate and momentum of the single oscillator, x k and p k
are those of the k − th oscillator of the medium, and gβ k is the coefficient of the
coupling with the k − th oscillator. To obtain the equation of the motion for the
single oscillator we have to solve de system of Hamilton’s equations corresponding
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_8
123