CSQ1D Formalism
CSQ1D Formalism
This page describes the complex-structure formalism implemented by NcmCSQ1D.
The formalism provides a classical parametrization of the one-dimensional harmonic oscillator in terms of a complex canonical solution encoding two linearly independent real solutions of the oscillator equations. Although often interpreted in terms of canonical quantization, adiabatic vacua, and particle production, the construction itself is classical and applies more generally to harmonic-oscillator evolution.
Dynamical System
The canonical variables \(\phi\) and \(P_\phi\) satisfy \[ \phi' = \frac{P_\phi}{m}, \qquad P_\phi' = -m\nu^2\phi, \] where \(' \equiv \mathrm{d}/\mathrm{d}t\), \(m(t)\) is an effective mass and \(\nu(t)\) the instantaneous oscillator frequency.
The pair \((\phi,P_\phi)\) forms a two-dimensional phase-space system whose general solution may be expressed using two linearly independent real solutions or, equivalently, a single complex canonical mode.
Define \[ \xi \equiv \ln(m\nu). \]
This quantity controls the adiabatic evolution of the system.
We further decompose \[ \gamma = \delta\gamma + \xi, \] so that \(\delta\gamma\) measures deviations from the instantaneous adiabatic configuration determined by \(\xi\).
Adiabatic Hierarchy
The adiabatic hierarchy is defined by \[ F_n \equiv \left( \frac{1}{2\nu} \frac{\partial}{\partial t} \right)^n \xi. \]
The quantities \(F_n\) measure successive adiabatic corrections associated with the evolution of \(m(t)\) and \(\nu(t)\) and naturally organize the adiabatic expansion.
Internal Adiabatic Variables
Rather than evolving the complex mode directly, NcmCSQ1D evolves the adiabatic variables \((\alpha,\delta\gamma)\).
The variable \(\alpha\) controls the mixing between positive- and negative-frequency components, while \(\delta\gamma\) quantifies deviations from the instantaneous adiabatic scaling.
These variables satisfy \[ \alpha' = -2\nu\sinh(\delta\gamma), \] and \[ \delta\gamma' = 2\nu \left[ -F_1 + \cosh(\delta\gamma)\tanh(\alpha) \right]. \]
This system is internally integrated by NcmCSQ1D.
Phase Decomposition
The oscillatory phase entering the mode functions is decomposed as \[ \theta(t) = \int_{t_i}^{t} \nu(t')\,\mathrm{d}t' + \delta\theta(t). \]
The first contribution corresponds to the instantaneous oscillatory phase, while \(\delta\theta\) captures the residual correction generated by the full complex-structure evolution.
The full phase obeys \[ \theta'(t) = e^{\gamma(t)-\xi(t)} \operatorname{sech}(\alpha(t)) \nu(t) = e^{\delta\gamma(t)} \operatorname{sech}(\alpha(t)) \nu(t). \]
Therefore \[ \delta\theta'(t) = \nu(t) \left[ e^{\delta\gamma(t)} \operatorname{sech}(\alpha(t)) - 1 \right]. \]
Near the adiabatic regime, where \[ \alpha \rightarrow 0, \qquad \delta\gamma \rightarrow 0, \] direct evaluation may suffer from cancellation errors. The implementation therefore evaluates the algebraically equivalent form \[ \delta\theta'(t) = \nu(t) \frac{ \operatorname{expm1}(\delta\gamma(t)) - 2\sinh^2(\alpha(t)/2) }{ \cosh(\alpha(t)) }, \]
which provides improved numerical stability.
Canonical Mode Reconstruction
The evolved variables reconstruct the complex canonical mode according to \[ \begin{pmatrix} \phi(t)\\ P_\phi(t) \end{pmatrix} = e^{-i\theta(t)} \begin{pmatrix} e^{-\gamma(t)/2} \sqrt{\frac{\cosh\alpha(t)}{2}} \\[6pt] -\dfrac{ e^{-\alpha(t)+\gamma(t)/2} (i+e^{\alpha(t)})^2 }{ 2\sqrt{2\cosh\alpha(t)} } \end{pmatrix}. \]
This complex mode encodes the full oscillator evolution and corresponds to a particular complex representation of the two-dimensional phase-space solution.
API Reference
See NcmCSQ1D for the full class reference.
For the workflow used in this page, the most relevant methods are:
- initialization and setup:
ncm_csq1d_set_init_cond_adiab,ncm_csq1d_prepare - state evaluation:
ncm_csq1d_eval_at,ncm_csq1d_eval_at_frame - phase correction:
ncm_csq1d_eval_delta_theta_at - frame conversion:
ncm_csq1d_change_frame
The state accessors are documented in NcmCSQ1DState, notably ncm_csq1d_state_get_phi_Pphi and ncm_csq1d_state_get_ag.