Initial State Expansion in Quantum Systems

• Initial state expansion is the evolution of a spatially inhomogeneous quantum system into revealing ballistic propagation and bound-state confinement.
• It employs analytical tools like generalized hydrodynamics and numerical methods such as tDMRG to capture the dynamics of quasiparticle motion and entanglement growth.
• The insights inform tunable quantum transport and simulation protocols, with implications for quantum memory and engineered low-entropy states.

Initial state expansion refers to phenomena where the conditions or quantum configurations of a system at $t=0$—often imposed by a nontrivial spatial pattern, inhomogeneity, or local excitation—undergo characteristic dynamical spreading, entanglement growth, and transport evolution. The form and fate of these expansions are shaped by interactions, integrability, and physical constraints, notably in strongly correlated quantum systems such as the anisotropic spin-$\frac{1}{2}$ Heisenberg (XXZ) chain. In the context of recent work on trap-expansion dynamics, initial state expansion defines protocols where a patterned or high-energy state is “released” into a larger system, and the subsequent evolution reveals selective quasiparticle propagation, regime-dependent confinement, and hierarchies of nonequilibrium timescales.

1. Preparation of Initial States and Expansion Protocols

The typical scenario involves preparing a finite segment $A$ of a one-dimensional XXZ chain in a product state of the “$p$-Néel” form: within region $A$ (size $\ell$), one repeats unit cells comprising $p$ spins up ($\uparrow$) followed by $p$ spins down ($\downarrow$). As $p \to \infty$, the pattern approaches a domain-wall state ($\uparrow\uparrow\cdots\downarrow\downarrow\cdots$). The remainder of the chain is in the ferromagnetic vacuum (all spins down), which is an eigenstate of the XXZ Hamiltonian

$$H = \sum_{j=1}^L \left( \sigma_j^x \sigma_{j+1}^x + \sigma_j^y \sigma_{j+1}^y + \Delta(\sigma_j^z \sigma_{j+1}^z - 1) \right)$$

where $\Delta$ is the anisotropy parameter. At $t=0$, region $A$ is “suddenly” coupled to the vacuum, triggering spatial expansion dominated by the internal excitations (quasiparticles) seeded in $A$.

Once the system is released, its unitary evolution rapidly leads to the emission of quasiparticles initially present in the high-energy region. The expansion protocol is a quantum quench, analogous to trap expansion in cold atom settings, but realized here through a designed product state with spatial inhomogeneity.

2. Quasiparticle Content and Bound-State Confinement

The XXZ model’s excitations can be classified by Bethe Ansatz. Unbound magnons ($n=1$) propagate ballistically, while larger bound states ($n>1$) involve composite objects of $n$ magnons. The initial $p$-Néel state contains both magnons and bound states, with the fraction of bound states increasing with $p$.

For sufficiently strong interactions ($\Delta>1$), the expansion dynamics reveal a remarkable confinement mechanism. In the hydrodynamic regime (large $t$, $x$ with fixed $\zeta = x/t$), elastic scattering causes the “dressed” group velocity $v_n(\lambda)$ for bound states to become negative (at $\zeta=0$), trapping them within the initial region $A$. The unbound magnons ($n=1$), in contrast, escape with positive group velocity.

As $p$ increases, the system’s bound-state content changes. For $p<4$, only bound states with $n>p$ remain confined at large $\Delta$. For $p\gtrsim4$, even the dominant $n=p$ bound states are confined, matching the absence of transport in the domain-wall limit ($p\to\infty$). Eventually, as all magnons “evaporate” from $A$, bound states become liberated and start to propagate—this liberation occurs at later, parametrically large times compared to the ballistic magnon emission.

3. Space-Time Profiles and Operator Diagnostics

The fate of initial state expansion is traceable in the spatiotemporal profiles of local observables. Projection operators

$$P_{x,\uparrow}^{\otimes n} = \frac{1}{2^n}\prod_{j=0}^{n-1} (1 + \sigma_{x+j}^z)$$

measure contiguous strings of $n$ up spins starting at site $x$, serving as selective probes for bound states of size $n$. These operators reveal “fingerprints” of confinement: the expectation value $\langle P_{x,\uparrow}^{\otimes n}\rangle(t)$ remains localized to region $A$ for confined bound states, while ballistic jets (peaks in the $\zeta$ profiles) are visible only for magnons and liberating bound states.

The evolution of magnetization, higher-order correlators, and projection operator profiles, calculated through analytical generalized hydrodynamics (GHD) or numerically via algorithms like tDMRG, confirm the coexistence of ballistic transport and stationary confined components.

4. Hierarchy of Timescales and Entanglement Growth

Bound-state confinement leads to a distinct hierarchy of emission timescales: magnons (unbound $n=1$) are released first, bound states remain trapped until the density of magnons in $A$ drops to zero, after which each bound state species is liberated in time orders parametrized by the region size $\ell$.

This multistage release is reflected in the von Neumann entanglement entropy $S(t)$. Initially, the entropy grows linearly with time, proportional to the sum over velocities and Yang–Yang entropy densities for the unbound magnons: $$S(t) \sim t \int d\lambda\, |v_1(\lambda)|\, s_1^{YY}(\lambda)$$ Only after the bound states are deconfined do further stages in entropy growth appear—yielding a “tiered” structure as different quasiparticle classes begin to contribute. This dynamic hierarchy underscores the deep entanglement between initial-state structure and non-equilibrium entropy dynamics.

5. Tuning Confinement and Quantum Simulation Implications

The sensitivity of bound-state confinement to both the initial pattern parameter $p$ and model anisotropy $\Delta$ provides a tunable knob for controlling transport and localization in integrable systems. In quantum simulators—cold atoms, ion chains, or quantum computers—one can experimentally realize such initial state expansions, measure observable profiles, and exploit confinement to engineer regions of high bound-state density conducive to quantum distillation.

Confined bound states are particularly robust against decoherence and might serve as protected carriers of quantum information or facilitate preparation of low-entropy, locally ordered states. The quantitative connection between microscopic initial state design and observable hydrodynamic outcomes illustrates the utility of integrable models in benchmarking and controlling complex quantum dynamics.

6. Generalized Hydrodynamics: Analytical and Numerical Framework

The generalized hydrodynamics (GHD) framework provides the analytical backbone for describing initial state expansion dynamics. In this formalism, the evolution of local observables is governed by continuity equations for quasiparticle densities, with dressing arising from elastic interactions. GHD predicts velocities, densities, and space-time profiles for all quasiparticle types, and successfully captures both ballistic and confined behavior.

Numerical approaches, particularly time-dependent density matrix renormalization group (tDMRG), complement GHD by resolving details of entanglement growth, operator expectation profiles, and the stepwise ballistic-confinement transitions. Together, these tools enable a full characterization of initial state expansion—including finite-size effects and very long-time liberation phenomena.

7. Broader Impact and Theoretical Perspectives

The paper of initial state expansion, specifically bound-state confinement after trap expansion in integrable systems, advances understanding of transport hierarchies, dynamical decoherence, and emergent universality in quantum matter. The mechanisms uncovered elucidate how initial state engineering influences long-time dynamics and open new avenues for quantum control—potentially enhancing quantum memories or protected logic gates in future technologies.

The observed connections between initial patterning, interaction-induced dressing of eigenstates, and the emergence of nontrivial hydrodynamic regimes—encompassing ballistic jets, confined domains, and entropy “tiers”—represent an overview of microscopic, hydrodynamic, and information-theoretic principles in quantum many-body physics (Biagetti et al., 27 Feb 2024).