Dynamical evolution of quantum oscillators towards equilibrium

Several investigations have been carried out on the statistical processes leading to equilibration phenomena in the linearly coupled oscillator system (see for e.g., G. Klein and I. Prigogine, 19

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The terminology ‘equilibrium’ is used here in a rather non-technical sense, to imply stationary long time behaviour

Cited in the paper.

Even when the individual pure state of the whole system has zero entropy, subsystems are in a mixed state, exhibiting non-zero entropy, thus revealing ’objective’ randomness. This is in stark contrast with classical statistical mechanics, where the probability distributions that define entropy reflect our ’subjective’ lack of knowledge

Cited in the paper.

The 2 ​ N × 2 ​ N 2N\times 2N matrix 𝒮 = Λ 1 4 ​ σ ⊕ Λ − 1 4 ​ σ {\cal S}~=~\Lambda^{\frac{1}{4}}\sigma~\oplus~\Lambda^{-\frac{1}{4}}\sigma is a symplectic transformation belonging to Sp ( 2 ​ n CLOSE (2n ,R) as can be explicitly verified via the defining relation 𝒮 ​ Γ ​ 𝒮 T = Γ {\cal S}\Gamma{\cal S}^{T}=\Gamma

Cited in the paper.

The 2 × 2 2\times 2 symplectic transformation corresponding to the unitary operation U ^ ​ ( S ⁡ ( ϕ ) ) = e − i ​ ϕ 2 ​ ( P ^ 2 + Q ^ 2 ) \hat{U}(S(\phi))=e^{\frac{-i\phi}{2}(\hat{P}^{2}+\hat{Q}^{2})} on the single oscillator Hilbert space is given by [ 8 ] S ⁡ ( ϕ ) = ( cos ⁡ ϕ sin ⁡ ϕ − sin ⁡ ϕ cos ⁡ ϕ ) . S(\phi)~=~\left(\begin{array}[]{cc}\cos\phi&\sin\phi\\ -\sin\phi&\cos\phi\end{array}\right). Employing this in Eq. ( 6

Cited in the paper.

The ‘system’ could be isolated oscillators spread throughout the whole quantum state. It is for the sake of mathematical convenience that they are grouped next to each other

Cited in the paper.

P. Reimann, Phys. Rev. Lett. 101

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N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. Rev. E 79

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Closest in time.