Multi-Particle Quantum Dynamics

• Multi-particle quantum dynamics is the study of interacting quantum systems where multiple constituents exhibit collective excitations, entanglement, and emergent phenomena through advanced analytical and numerical methods.
• Key methodologies include exactly solvable models, statistical ensembles, and quantum field constructions that elucidate spectral features, operator spreading, and decoherence in many-body settings.
• Numerical and semiclassical simulation algorithms, such as Bohmian trajectories and worldline Monte Carlo, enable robust exploration of dynamic regimes and quantum feedback in complex systems.

Multi-particle quantum dynamics concerns the evolution, correlations, and observable properties of quantum systems composed of two or more interacting constituents. Such systems manifest phenomena beyond single-particle physics, including collective excitations, entanglement, emergent fragmentation, dynamical phase transitions, and unique signatures in their spectral statistics. Rigorous modeling spans exactly solvable models, field theory constructions, stochastic unravelings under measurement, advanced numerical simulations, and semiclassical approaches, all with critical technological and foundational implications.

1. Dynamical Measures and Operator Spreading in Interacting Quantum Walks

A central observable for quantifying non-classicality in multi-particle evolution is the stabilizer Rényi entropy $M_2$, measuring the non-stabilizerness (also called “magic”) generated during time evolution. Considering a one-dimensional spin-$\tfrac12$ XXZ Heisenberg chain of length $L$

$$H = J \sum_{i=1}^{L-1} \left( S_i^x S_{i+1}^x + S_i^y S_{i+1}^y + \Delta S_i^z S_{i+1}^z \right),$$

the growth of $M_2$ distinguishes qualitatively distinct dynamical regimes determined by the anisotropy parameter $\Delta$:

• Easy-plane regime ($|\Delta| < 1$): Magic is primarily generated by single-particle dynamics with linear light-cone propagation at velocity $v_F = J$, and $M_2$ grows as $M_2(t) \sim \log_2(J t)$.
• Easy-axis regime ($\Delta > 1$): Two-particle bound pairs (“doublons”) dominate, with effective group velocity $v_F^{(D)}=J/(2\Delta)\ll J$, yielding a much slower, logarithmic growth $M_2(t)\sim \log_2[J t/(2\Delta)]$.
• Isotropic point ($\Delta=1$): Both single and doublon excitations contribute, with logarithmic growth but modified prefactors and oscillatory transients (Moca et al., 28 Apr 2025).

Magic spreads strictly within the causal light-cone set by the maximal excitation group velocity, reflecting Lieb-Robinson bounds. At long times, the Pauli spectrum of the evolved state exhibits Poissonian level-spacings, independent of interaction strength or particle number, signifying universal uncorrelated structure of the magic coefficients.

2. Real-Time Observation, Decoherence Channels, and Quantum Feedback

The multi-particle quantum dynamics under real-time measurement can be modeled via stochastic Schrödinger equations for systems such as atoms in optical lattices monitored by far-off-resonant light. In the weak spatial resolution ($\sigma\gg d$) and strong coupling ($\gamma \gg J/\hbar$) regime, a key result emerges:

• For indistinguishable particles, only the center-of-mass (COM) coordinate decoheres; all relative-coherence is preserved, supporting robust multi-particle quantum correlations.
• For distinguishable particles, additional decoherence channels act on each relative coordinate, rapidly suppressing multi-particle coherence.

Decoherence rates of off-diagonal density matrix elements are governed by differences in the COM, leading to collapse, inertial, and diffusive dynamical regimes. Ballistic propagation and quantum statistical correlations persist in the inertial regime for indistinguishable fermions and bosons. This effect enables a dynamically generated decoherence-free subspace for relative motion, making continuous-time quantum feedback and many-body wave stabilization feasible by only monitoring and controlling the collective COM (Ashida et al., 2015).

3. Statistical Ensembles, Interaction Rank, and Spectral Features

The dynamical properties of multi-particle systems with $k$-body interactions can be investigated using embedded random ensemble theory—specifically, the Fermionic/Bosonic Embedded Gaussian Orthogonal Ensembles (FEGOE/BEGOE). The spectral density of the system and the associated local density of states (LDOS) interpolate from Gaussian (few-body regime, $q \approx 1$) to Wigner semi-circular (many-body regime, $q \rightarrow 0$), described by $q$-Hermite polynomials:

• Increasing interaction rank $k$ (for particle number $m$) yields decreasing $q$ and rapid crossover in survival probability from purely Gaussian decay to oscillatory Bessel-type decay. This transition manifests distinctively for fermionic and bosonic statistics due to combinatorial constraints.
• Short-time survival probabilities and global dynamical signatures are controlled by the generating function $G_q(x, t)$ associated with $q$-Hermite polynomials—the $q$ parameter is combinatorially connected to excess kurtosis of the ensemble (Vyas et al., 2018).

Such statistical frameworks underpin universality classes and crossover behavior in many-body quantum dynamics that are directly relevant for non-equilibrium experiments and quantum chaos.

4. Quantum Field Constructions for Multi-Particle Dynamics

Multi-particle quantum fields can be constructed to systematically describe bound and interacting states beyond free field Fock spaces. Two Gaussian approaches are prominent:

• Product-propagator construction: $n$-particle field operators $\phi_n(x_1,\ldots,x_n)$ with vacuum two-point functions factorizing as products of single-particle propagators.
• Center-of-mass locality construction: Operators $P_{z}(X)$, with center-of-mass $X$ and relative separations $z_j$, commuting at space-like separations of $X$, and endowed with nontrivial form factors $M(z, z')$ capturing resonance and bound-state structure.

The combination of these constructions yields interpolations between purely “free” ($\prod_j i\Delta_{m_j}$) and “bound” ($i\Delta_M(X-X')M(z;z')$) behaviors in the spectral response. Non-Gaussian, multilinear field constructions are obtainable by polarization techniques applied to nonlinear functionals of smeared free fields. These formalisms are essential for computing general multi-particle time-ordered correlations, scattering amplitudes, and bound-state spectra within a single, well-defined Hilbert space framework (Morgan, 2015).

5. Exactly Solvable Models and Quantum Integrability

Analytically tractable multi-particle models provide indispensable benchmarks and structural insights. In the $N$-body Calogero system based on $A_{N-1}$ root systems, the dynamics is solvable in the Heisenberg picture via “sinusoidal coordinates” $\eta^{(j)}$ with closed-form time evolution: $$\eta^{(j)}(t) = (2i)^{-f_j} \sum_{\ell=0}^{f_j} (-1)^{\ell} b^{(j)}_{f_j;f_j-2\ell} e^{i(f_j-2\ell)t},$$ where $b^{(j)}_{f_j;f_j-2\ell}$ are operator coefficients related to powers of Lax matrices, and $f_j$ are exponents of the root system. Associated annihilation/creation operators commute with the Hamiltonian up to frequency shifts determined by $f_j$. In the eigenbasis, action of $\eta^{(j)}$ yields finite-step recursion relations for multivariable orthogonal polynomials constituting the eigenfunctions, encoding the model’s integrability and spectral structure (0706.0768).

6. Quantum Forces, Subsystem Dynamics, and Ehrenfest Relations

Quantum analogues of classical force and pressure naturally arise in many-particle dynamics as local (boundary) operators. For a region $x>x_0$, the kinetic and interaction force operators are: $$\hat F_{\rm kin}(x_0) = -\frac{\hbar^2}{4M} \left\{ \delta''(x-x_0) + 2\partial_x \delta(x-x_0) \partial_x + \delta''(x-x_0) \right\},$$

$\hat F_v(x_0) = -[V(\rho) - \rho V'(\rho)]_{x_0},$

with generalizations to higher dimensions via the quantum stress tensor $\Pi_{ij}$. These operators enter the second time-derivative of observables of subregions through Ehrenfest’s theorem, yielding quantum analogues of Newtonian dynamics for collective or coarse-grained degrees of freedom. The formalism underpins both theoretical developments and computational implementations in nuclear, condensed matter, and cold-atom systems (Bertsch, 2020).

7. Numerical and Semiclassical Simulation Algorithms

Simulation of multi-particle quantum dynamics faces exponential scaling with particle number, prompting the development of:

• Bohmian trajectory methods: “Interacting Pilot Wave” (IPW) schemes propagate conditional one-particle wavefunctions guided by ensembles of Bohmian configurations. Quantum entanglement is realized via coupling between pilot waves and configurations; exact dynamics is recovered in the limit of an infinite ensemble of configurations (Elsayed et al., 2017).
• Worldline Monte Carlo (WLMC): Path integral representations employing ensembles of Brownian-bridge worldlines for each particle, with reweighting for interactions, allow estimation of ground-state energy and correlation functions. WLMC scales more favorably with particle number and spatial dimension than traditional Hamiltonian diagonalization for moderate $N$ and $d$ (Ahumada et al., 31 Dec 2025).
• Semiclassical methods: For systems governed by small effective Planck constant $\varepsilon$, various algorithms are available, including variationally evolving Gaussians, Hagedorn wave packets, Gaussian beam superpositions, and Wigner–Egorov approaches. These methods exploit underlying classical trajectories to achieve $\mathcal{O}(\varepsilon^2)$ accuracy in observables, significantly reducing computational complexity in high dimensions (Lasser et al., 2020).

Each of these methods accommodates different physical regimes, degrees of quantum correlation, and observables of interest, collectively providing a powerful computational arsenal for probing nontrivial quantum many-body evolution.

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References:

• "Non-stabilizerness generation in a multi-particle quantum walk" (Moca et al., 28 Apr 2025)
• "Multi-particle quantum dynamics under real-time observation" (Ashida et al., 2015)
• "Quenched many-body quantum dynamics with $k$-body interactions using $q$-Hermite polynomials" (Vyas et al., 2018)
• "Multi-particle quantum fields for bound states and interactions" (Morgan, 2015)
• "Exact Heisenberg operator solutions for multi-particle quantum mechanics" (0706.0768)
• "Force and pressure in many-particle quantum dynamics" (Bertsch, 2020)
• "Entangled Quantum Dynamics of Many-Body Systems using Bohmian Trajectories" (Elsayed et al., 2017)
• "Multi-particle quantum systems within the Worldline Monte Carlo formalism" (Ahumada et al., 31 Dec 2025)
• "Computing quantum dynamics in the semiclassical regime" (Lasser et al., 2020)