Time-Varying Coupling¶

This section provides the theoretical foundation for time-dependent cavity-molecule coupling in Cavity HOOMD.

Note

For practical implementation, see Time-Varying Coupling.

Motivation and Physical Context ¶

Why Time-Dependent Coupling?¶

In experiments, cavity coupling can be dynamically controlled through:

Cavity tuning: Piezo-electric mirrors change cavity length
Pump-probe: External laser activates coupling
Mode switching: Transition between cavity modes
Detuning: Frequency sweeps across resonance

Time-varying coupling in simulations enables studying:

Non-equilibrium dynamics
Sudden quench experiments
Pump-probe spectroscopy
Relaxation processes
Energy redistribution

Experimental Analogues ¶

Typical Experimental Protocols:

Experiment	Implementation	Timescale
Pump-probe	Laser pulse activates coupling	fs to ps
Cavity tuning	Piezo changes cavity length	µs to ms
Mode switching	Electronic control	ns to µs
Detuning sweep	Frequency scan	ms to s

Cavity HOOMD simulates the molecular response to these protocols.

Mathematical Formulation ¶

Time-Dependent Hamiltonian ¶

The Hamiltonian with time-varying coupling:

\[H(t) = H_{\text{mol}} + \sum_{\lambda} \left[ \frac{1}{2}K_\lambda \tilde{q}_{0,\lambda}^2 + g(t)\tilde{\varepsilon}_{0,\lambda}^{(0)} \tilde{q}_{0,\lambda} D_\lambda + \frac{g(t)^2(\tilde{\varepsilon}_{0,\lambda}^{(0)})^2}{2K_\lambda} D_\lambda^2 \right]\]

Where:

\(g(t)\): Time-dependent coupling function
\(\tilde{\varepsilon}_{0,\lambda}^{(0)}\): Base coupling strength (constant)
All other symbols as in Cavity Forces

Key Point: The coupling strength \(g(t)\) multiplies both the linear coupling and quadratic self-energy terms.

Modified Equations of Motion ¶

Nuclear Motion:

\[M_{nj}\ddot{R}_{nj} = F_{nj}^{(0)} - \sum_{\lambda} \left( g(t)\tilde{\varepsilon}_{0,\lambda}^{(0)}\tilde{q}_{0,\lambda} + \frac{g(t)^2(\tilde{\varepsilon}_{0,\lambda}^{(0)})^2}{K_\lambda} D_\lambda \right) \frac{\partial d_{ng,\lambda}}{\partial R_{nj}}\]

Photonic Mode Dynamics:

\[m_{0,\lambda}\ddot{\tilde{q}}_{0,\lambda} = -K_\lambda \tilde{q}_{0,\lambda} - g(t)\tilde{\varepsilon}_{0,\lambda}^{(0)} D_\lambda\]

Time Derivative Effects:

When \(g(t)\) changes, an additional “quench force” appears if we consider the full time-dependent Hamiltonian formulation. However, for practical implementations with smooth or instantaneous changes, this effect is negligible or absorbed into the redefinition of equilibrium positions.

Step Function Protocol ¶

Mathematical Definition ¶

The step function is the simplest time-varying protocol:

\[\begin{split}g(t) = \begin{cases} g_{\text{initial}} & \text{if } t < t_{\text{switch}} \\ g_{\text{final}} & \text{if } t \geq t_{\text{switch}} \end{cases}\end{split}\]

Common case: \(g_{\text{initial}} = 0\), \(g_{\text{final}} = g_0\) (coupling activation)

Implementation:

In discrete timesteps:

\[\begin{split}g(n\Delta t) = \begin{cases} g_{\text{initial}} & \text{if } n < n_{\text{switch}} \\ g_{\text{final}} & \text{if } n \geq n_{\text{switch}} \end{cases}\end{split}\]

where \(n_{\text{switch}} = \lfloor t_{\text{switch}}/\Delta t \rfloor\).

Energy Redistribution at Switch ¶

Before switch (t < t_switch, g=0):

\[E_{\text{total}}(t^-) = E_{\text{kinetic}}^{\text{mol}} + E_{\text{potential}}\]

No cavity or coupling energy.

After switch (t ≥ t_switch, g=g_0):

\[E_{\text{total}}(t^+) = E_{\text{kinetic}}^{\text{mol}} + E_{\text{potential}} + E_{\text{cavity}} + E_{\text{coupling}} + E_{\text{self}}\]

Energy Conservation:

\[E_{\text{total}}(t^+) = E_{\text{total}}(t^-)\]

Energy is conserved during the switch but redistributes among degrees of freedom.

Physical Interpretation:

Molecular kinetic/potential energy decreases
Cavity mode gains energy
Coupling and self-energy appear
Total energy unchanged (in NVE ensemble)

Cavity Particle Displacement ¶

In q=0 mode:

Cavity particles remain at origin; only forces change.

In finite-q mode:

Cavity particle position must jump to new equilibrium:

\[\vec{q}_{\text{eq}}(t^+) = -\frac{g_0}{K} \vec{D}_{\text{total}}(t_{\text{switch}})\]

Reason: The equilibrium position changes from \(\vec{q}=0\) (g=0) to \(\vec{q}=-\frac{g}{K}\vec{D}\) (g≠0).

Implementation:

At t = t_switch:

Calculate total dipole \(\vec{D}_{\text{total}}\)
Compute new equilibrium \(\vec{q}_{\text{new}}\)
Instantly move cavity particles to \(\vec{q}_{\text{new}}\)
Cavity velocity remains unchanged (momentum conservation)

Energy accounting:

The position jump conserves total energy:

\[\Delta E_{\text{cavity}} + \Delta E_{\text{coupling}} + \Delta E_{\text{self}} = 0\]

Smooth Coupling Protocols ¶

Exponential Approach ¶

Formula:

\[g(t) = g_{\text{final}} + (g_{\text{initial}} - g_{\text{final}}) e^{-(t-t_0)/\tau}\]

for \(t \geq t_0\).

Where:

\(t_0\): Start time
\(\tau\): Characteristic timescale

Properties:

Smooth transition
Asymptotically approaches \(g_{\text{final}}\)
Rate controlled by \(\tau\)

Use cases:

Gradual cavity activation
Coupling decay (lossy cavity)
Smooth detuning

Characteristic times:

\(t_{1/2} = \tau \ln(2) \approx 0.69\tau\) (half-life)
\(t_{90\%} \approx 2.3\tau\) (90% completion)

Sinusoidal Modulation ¶

Formula:

\[g(t) = g_0 + A\sin(2\pi f t + \phi)\]

Where:

\(g_0\): DC offset
\(A\): Modulation amplitude
\(f\): Modulation frequency
\(\phi\): Initial phase

Physical realization:

Oscillating cavity length
Frequency modulation
Periodic pump pulses

Floquet regime:

When \(f \sim \omega_{\text{vib}}\), the system enters the Floquet (driven) regime with:

Parametric resonances
Sideband generation
Modified selection rules

Square Wave (Periodic On-Off)¶

Formula:

\[\begin{split}g(t) = \begin{cases} g_{\text{on}} & \text{if } \mod(t, T) < T \times \text{duty} \\ g_{\text{off}} & \text{otherwise} \end{cases}\end{split}\]

Where:

\(T\): Period
duty: Duty cycle (0-1)

Example: duty=0.5 gives 50% on, 50% off.

Applications:

Pulsed excitation
Stroboscopic measurement
Time-resolved spectroscopy

Energy Conservation Theory ¶

Total Energy Functional ¶

For time-dependent g(t):

\[E_{\text{total}}(t) = \sum_{n,j} \frac{1}{2}M_{nj}\dot{R}_{nj}^2 + V(\{R\}) + \sum_{\lambda} \left[ \frac{1}{2}m_\lambda \dot{\tilde{q}}_\lambda^2 + \frac{1}{2}K_\lambda \tilde{q}_\lambda^2 + g(t)\tilde{\varepsilon}\tilde{q}_\lambda D_\lambda + \frac{g(t)^2\tilde{\varepsilon}^2}{2K_\lambda} D_\lambda^2 \right]\]

Time derivative:

\[\frac{dE_{\text{total}}}{dt} = \frac{\partial E}{\partial t}\bigg|_{\text{explicit}} + \sum_i \frac{\partial E}{\partial q_i}\dot{q}_i + \sum_i \frac{\partial E}{\partial \dot{q}_i}\ddot{q}_i\]

For Hamiltonian dynamics with time-independent H: \(\frac{dE}{dt} = 0\)

For time-dependent H: \(\frac{dE}{dt} = \frac{\partial H}{\partial t}\)

Numerical Energy Conservation ¶

In practice:

\[\frac{dE_{\text{total}}}{dt} = \frac{\partial g(t)}{\partial t} \left[ \tilde{\varepsilon}\tilde{q}_\lambda D_\lambda + \frac{g(t)\tilde{\varepsilon}^2}{K_\lambda} D_\lambda^2 \right]\]

For step function: \(\frac{\partial g}{\partial t} = 0\) except at t=t_switch where it’s a delta function.

Energy conservation check:

\[\Delta E_{\text{rel}} = \frac{|E(t) - E(t_0)|}{|E(t_0)|} < \epsilon_{\text{tol}}\]

Typically \(\epsilon_{\text{tol}} \approx 10^{-4}\) for good simulations.