Time-Resolved RDM Movies

• Time-resolved RDMs are techniques that capture the evolution of quantum correlations by propagating reduced density matrices without requiring the full many-body wavefunction.
• They employ reconstruction strategies and dynamical purification to ensure energy, trace, and spin conservation during the simulation of complex quantum systems.
• RDM movies facilitate spatiotemporal visualization of observables in quantum quenches and relaxation studies, empowering analyses in both quantum chemistry and many-body physics.

Time-resolved reduced density matrices (RDMs), often visualized as "RDM movies," provide a dynamic, multi-particle-resolved characterization of quantum systems under time evolution. These approaches allow direct access to correlation functions and time-dependent observables without recourse to the full many-body wavefunction, facilitating the study of relaxation, thermalization, and correlation dynamics following external perturbations or quantum quenches. Recent methodological advances address both quantum chemistry (strongly correlated molecules under driving fields) and many-body physics (non-equilibrium quenched lattices), enabling the spatiotemporal visualization of correlated quantum dynamics through the propagation and extraction of RDMs (Lackner et al., 2014, Fagotti et al., 2013).

1. Mathematical Foundation: Equations of Motion for Time-Resolved RDMs

The core of time-resolved RDM analysis is the propagation of an $n$-particle reduced density matrix $D^{(n)}(t)$ under the system Hamiltonian and interactions. For $n=2$, the evolution follows from the BBGKY hierarchy: $$i\,\partial_t D_{12}(t) = [H_{12}, D_{12}] + \operatorname{Tr}_3 [W_{13} + W_{23}, D_{123}]$$ where $D_{12}$ is the 2-RDM, $H_{12}$ the two-body Hamiltonian, and $D_{123}$ the unknown three-body RDM. To close the hierarchy, reconstruction strategies approximate $D_{123}$ as a functional of $D_{12}$, defining a closed equation of motion: $$i\,\partial_t D_{12} = [H_{12}, D_{12}] + C_{12}^R[D_{12}]$$ where $C_{12}^R$ is the reconstructed collision operator. This formalism ensures trace conservation, energy conservation (if the reconstruction is contraction-consistent), and, upon careful block propagation, spin preservation for singlet states (Lackner et al., 2014).

In quantum lattice systems after quenches, time-dependent RDMs of subsystems are extracted via the computation of correlation matrices (e.g., of Majorana fermions for the transverse-field Ising model), with the full RDM encoded as a Gaussian state determined entirely by its two-point correlation matrix (Fagotti et al., 2013).

2. Reconstruction of Higher-Order RDMs

Closure of the dynamical equations for $D_{12}$ requires reconstructing the three-body RDM. Two major strategies are adopted:

• Valdemoro (V) Reconstruction:

$$D_{123}^V[D_{12}] = 9\,D_{12} \wedge D_1 - 12\,D_1 \wedge D_1 \wedge D_1$$

where $D_1$ is the traced 1-RDM and $\wedge$ denotes antisymmetrized products. This construction neglects the three-particle cumulant.

• Contraction-Consistent (C) Reconstruction:

$$D_{123}^{\rm C}[D_{12}] = D_{123}^V[D_{12}] + \Delta_{123;\perp}[D_{12}]$$

where $\Delta_{123;\perp}$ is the contraction-expressible part of the cumulant, ensuring energy conservation and spin contraction properties (Lackner et al., 2014).

$N$-representability constraints (positivity of both the 2-RDM and 2-hole RDM, i.e., D/Q conditions) are enforced via dynamical purification after each integration step, preserving physicality of the propagated object.

3. Numerical Time Propagation and Purification

Time-stepping of the coupled equations for the 2-RDM coefficients and underlying orbital basis is performed using explicit high-order integrators such as Runge–Kutta 4. At each step, the method involves:

• Reconstructing $D_{123}$ from $D_{12}$.
• Computing collision terms.
• Evolving the RDM and orbitals.
• Applying dynamical purification: diagonalizing the provisional $D_{12}$ and 2-hole RDM, correcting negative occupations iteratively until positivity is restored.

For spin-adapted singlet cases, only the $\uparrow\downarrow$ block is propagated, decomposed into symmetric/antisymmetric components and purified separately. This ensures computational tractability and the physicality of the time-evolved RDM (Lackner et al., 2014).

4. Extraction and Visualization of Observables

From the time-resolved $D_{12}(t)$, all two-body observables are available exactly. Examples include:

• Electron density: $\rho(z,t) = \sum_{\sigma} D(z\sigma;z\sigma;t)$.
• Pair density: $$\rho(z_1, z_2, t) = \sum_{\sigma_1\sigma_2} D(z_1\sigma_1, z_2\sigma_2; z_1\sigma_1, z_2\sigma_2; t)$$.
• Mean interaction energy:

$$E_{\rm int}(t) = \iint dz_1 dz_2\, \rho(z_1, z_2, t) \frac{1}{\sqrt{(z_1 - z_2)^2 + d}}$$

• Dipole moment: $d(t) = d_0 - \int z\,\rho(z, t) dz$.

For "RDM movies," a grid (e.g., $200\times200$ points for $(z_1, z_2)$) is chosen, and the pair density $\rho(z_1, z_2, t)$ is evaluated at sampled time steps, visualized as false-color maps or spectral slices. Animation with 100–200 frames over several laser cycles captures the spatiotemporal evolution of correlation features. One-dimensional cuts or momentum-space observables may also be visualized. Parameters are chosen for numerical stability and resolution: $\Delta t \approx 0.01-0.05$ a.u., basis size $r=10-50$ orbitals, and 5–10 purification iterations to achieve all eigenvalues $>-10^{-8}$ (Lackner et al., 2014).

In lattice systems, the RDM $\rho_{\ell}(t)$ for an $\ell$-site block is constructed via its time-dependent correlation matrix, enabling the tracking of local relaxation and approach to stationary ensembles (Fagotti et al., 2013).

5. RDM Movies in Quantum Quench and Relaxation Studies

For quantum quenches in integrable systems, a prototypical example is the infinite transverse-field Ising chain. The time-dependent RDM $\rho_A(t)$ of a block $A$ is reconstructed from the evolving Gaussian correlation matrix: $$\Gamma_{mn}(t) = \operatorname{Tr}[\rho(t)a_n a_m] - \delta_{mn}$$ leading to

$$\rho_\ell(t) = \frac{1}{Z} \exp\left[\frac14 \sum_{m,n=1}^{2\ell} a_m W_{mn}(t) a_n\right]$$

with $W$ related to $\Gamma$ by $\tanh(W/2) = \Gamma$. The spectrum of observables, approach to stationarity, and entropy growth are then visualized framewise, facilitating direct observation of nontrivial relaxation phenomena.

Quantitative metrics, such as the normalized Frobenius distance between the time-evolved $\rho_\ell(t)$ and the stationary GGE RDM, exhibit universal scaling laws: $$\mathcal{D}(\rho_\ell(t), \rho_\ell^{\rm GGE}) \to k(\ell) (Jt)^{-3/2}, \quad t \gg 1$$ where $k(\ell) \propto \ell^2$, revealing the power-law approach to generalized thermalization (Fagotti et al., 2013).

6. Computational Protocols and Scaling

Efficient construction of "RDM movies" is achieved via algorithms matching the physical scenario and observable complexity. For time-dependent 2-RDM propagation in molecular contexts, each step involves computational effort scaling as $\mathcal{O}(r^7)$ per collision calculation (with $r$ the number of spin orbitals), but typical molecules are tractable with $r$ up to $50$. In quantum chain models, time evolution and RDM extraction leverage fast-Fourier-transform (FFT)-like integrals and scale as $\mathcal{O}(N_t \ell^2)$ for $N_t$ time steps and block size $\ell$ (Lackner et al., 2014, Fagotti et al., 2013).

Visualization and analysis tools can include Matplotlib, ParaView, and FFmpeg for frame rendering. A practical pseudocode workflow in the quantum quench context includes single-particle data precomputation, time-dependent generation of block correlation matrices by integration, assembly of full RDMs, and optional evaluation of distance metrics.

7. Significance and Applications

Time-resolved RDM analysis ("RDM movies") delivers detailed, wavefunction-free insight into the spatiotemporal evolution of correlation and entanglement in many-body dynamics—essential for both quantum chemistry under driving fields and non-equilibrium quantum statistical physics. Observables drawn directly from RDMs offer faithful characterizations of relaxation, electronic response, and correlation build-up. The propagation methodologies, particularly with robust reconstruction and purification, allow simulation of systems with strong correlations and long-range interactions that are otherwise inaccessible to full wavefunction methods (Lackner et al., 2014, Fagotti et al., 2013).

This framework provides a universal language and computational toolbox for visualizing and quantifying quantum dynamics, enabling rigorous tests of equilibration theories and facilitating the exploration of interaction-driven phenomena in both finite and thermodynamically large systems.