State-Dependent Wave-Packets

• State-dependent wave-packets are quantum wave functions whose evolution is governed by internal variables such as eigenmodes, structured momentum phases, and background fields.
• They are constructed using adiabatic, WKB, and Hamiltonian methods to achieve precise control over scattering characteristics and ultrafast dynamics.
• Experimental applications include ultrafast vibrational imaging, Airy electron beams, and quantum transport, demonstrating their value in advanced quantum control.

State-dependent wave-packets, broadly, are quantum wave-packet solutions whose evolution, observable characteristics, or scattering properties are determined or controlled by underlying system “state variables” such as internal eigenmodes, structured momentum-space phases, or the dynamical environment. They play a central role in semi-classical analysis, multicomponent quantum systems, nonlinear evolution, structured-packet scattering theory, and both theoretical and experimental ultrafast dynamics. Their formal construction often leverages adiabatic, WKB, or Hamiltonian methods, with mode-polarization, structured phase engineering, or large-scale background dependencies leading to nontrivial dynamical phenomena not present in single-component, unstructured, or plane-wave constructions.

1. Formal Definitions and Classification

A state-dependent wave-packet is typically characterized by at least one of the following:

• Eigenmode (band) polarization: The wave-packet is constructed to align with a specific eigenvector of a matrix-valued or multi-band Hamiltonian. This is the cornerstone of adiabatic and non-adiabatic propagation in semi-classical matrix Schrödinger systems, where each mode evolves according to its own classical dynamics, with transition amplitudes controlled by inter-mode couplings and spectral gaps (Carles et al., 2010, Hari, 2012, Kammerer et al., 2020, Fermanian-Kammerer et al., 2020).
• Phase-structured momentum amplitudes: The packet is engineered so that its momentum-space phase (while keeping the modulus—and thus energy and momentum density—fixed) is manipulated in a nonlinear fashion (e.g., cubic phase for Airy-like structures), thus encoding state-dependence at the level of group delays or scattering observables (Saxton et al., 2020, Karlovets, 2016).
• Dynamical dependence on background fields: The packet’s propagation parameters (e.g., group velocity, wavevector evolution) are slaved to slowly-evolving classical fields (density, flow velocity) or large-scale waves, as in the case of high-frequency packets on a hydrodynamic background (Shaykin et al., 2023).
• Multimodal superpositions: The total state consists of multiple decoupled (or weakly coupled) wave-packets, each following (possibly nonlinear-dressed) evolution in its associated state variable, band, or phase-space trajectory (Carles et al., 2010, Hari, 2012, Kammerer et al., 2020).

2. Adiabatic and Non-Adiabatic Evolution in Multi-State Systems

In semiclassical systems with $N$-component wave-functions, state-dependence arises naturally upon projecting wave-packets onto the eigenbasis of the matrix Hamiltonian $H(x,p)$. For an initial packet polarized along eigenvector $\chi_j(x)$ of eigenvalue $\lambda_j(x)$,

$$\psi^\varepsilon(0,x) = \varepsilon^{-d/4} a_j^0\Bigl(\frac{x-q_0}{\sqrt\varepsilon}\Bigr) e^{i p_0 \cdot (x-q_0)/\varepsilon} \chi_j(q_0)$$

the evolution, up to corrections of $o(1)$ in the semiclassical limit, is captured by (Carles et al., 2010, Hari, 2012):

$$\psi^\varepsilon(t,x) \approx \varepsilon^{-d/4} a_j\Bigl(t, \frac{x-q_j(t)}{\sqrt\varepsilon}\Bigr) e^{i S_j(t)/\varepsilon} e^{i p_j(t) \cdot (x-q_j(t))/\varepsilon} \chi_j\bigl(q_j(t)\bigr)$$

The envelope $a_j(t,y)$ solves a nonlinear envelope equation (if nonlinearity is critical in the scaling), and the phase-space center $(q_j(t),p_j(t))$ follows the classical Hamiltonian flow for $H_j(q,p) = \frac{1}{2}|p|^2 + \lambda_j(q)$.

During avoided or true eigenvalue crossings (codimension-1 crossings), non-adiabatic transitions are realized via explicit transition operators (Landau–Zener mechanisms), creating new outgoing packets in other eigenmodes with calculable amplitudes and phase-shifts (Kammerer et al., 2020, Fermanian-Kammerer et al., 2020).

3. Control via Structured Momentum Phases

A distinct mechanism for engineering state-dependent effects is via nonlinear phase modulation in the momentum space:

• For a standard Gaussian packet: $$\tilde{\psi}_G(k) \propto \exp[-(k - k_0)^2/(4 \sigma_k^2)]e^{-ikx_0}$$
• For a structured non-Gaussian (“state-dependent”) packet: $$\tilde{\psi}_{\mathrm{NG}}(k) = \tilde{\psi}_G(k) \exp\bigl[i \alpha (k-k_0)^n\bigr]$$.

This phase structure is invisible to energy or momentum density, but produces a tunable shift in the average arrival time (group delay) upon scattering from a potential barrier (Saxton et al., 2020):

$$\Delta\tau = \frac{m}{\hbar k_T} \alpha n (k_T-k_0)^{n-1}$$

where $k_T$ is the mean transmitted momentum. By selecting $\alpha$ and $n$, one attains control over the arrival time delay, independently of the energy or momentum spread. In the $n=3$ (cubic phase, Airy) case, electron or photon Airy beams exhibit this effect experimentally via holographic phase masks or spatial light modulators.

4. Dynamics in State-Dependent or Background-Driven Environments

For wave-packets propagating in a nontrivial, slowly-varying background (e.g., density/velocity fields in fluids or BECs), the state-dependence is realized by slaving the Hamiltonian parameters to the evolving hydrodynamical variables. The packet's mean coordinate $X(t)$ and carrier wavenumber $k(t)$ solve canonical Hamiltonian equations with explicit dependence on the local fields:

$$\frac{d X}{dt} = v_g(k, \rho, u), \quad \frac{d k}{dt} = -\omega_\rho \rho_x - \omega_u u_x$$

where $v_g$ is the group velocity and $\omega$ the local dispersion relation. The background evolution is entirely independent of the packet in the leading (WKB) approximation. Exact agreement with the full nonlinear (NLS) dynamics is observed in the high-$k$ regime (Shaykin et al., 2023).

5. Scattering, Quantum Corrections, and Phase-Driven Observables

State-dependent characterization is also essential in quantum scattering of packets with nontrivial phase structure. The Wigner-function formalism provides a universal method for incorporating arbitrary packet phases (including OAM and Airy structure) and yields quantum corrections to plane-wave predictions:

• The cross section acquires dependence on the packet phases (entering via an “effective impact parameter” $b_\phi$), leading to genuinely quantum-sensitive observables such as azimuthal asymmetry in the scattering yield.
• The dependence on phases can be exploited to measure subtle features such as the $t$-dependence of the Coulomb or hadronic phase of the scattering amplitude, using asymmetric beam setups or phase-imprinted wave-packet states (Karlovets, 2016).

These effects vanish in the plane-wave limit, but are prominent for Ångström-scale beams.

6. Nonlinear Superposition and Modal Robustness

In both linear and nonlinear matrix-valued Schrödinger systems, the superposition of state-dependent wave-packets exhibits remarkable robustness:

• Different-mode superpositions propagate independently to leading order, as long as energies are non-resonant and classical trajectories remain separated. Valid even in the $L^2$-supercritical nonlinear regime (Hari, 2012, Carles et al., 2010).
• Even same-mode, distinct-ray packets (with spatially non-overlapping quasi-classical trajectories) propagate as independent objects, with nonlinear cross-terms negligible at leading order.
• Explicit error bounds in semiclassical Sobolev norms show that the modal separation and independence is preserved up to timescales determined by spectral gap and potential decay parameters.

This modal decoupling is central to the persistence of state-dependent properties and underpins the effectiveness of semiclassical adiabatic and envelope equations in complex, nonlinear, or multicomponent settings.

7. Experimental Realizations and Applications

State-dependent wave-packet engineering has been realized in several physical contexts:

• Ultrafast vibrational imaging: The time-dependent modulus-square of molecular wave-packets generated by stimulated Raman processes is measured via time-resolved Coulomb explosion, mapping kinetic energy distributions of fragment ions to bond-length probability distributions. The distinct quantum superposition content, oscillation frequencies, amplitudes, and decay times are extracted with sub-picosecond resolution (Jyde et al., 19 Nov 2024).
• Airy electron beams: Application of tailored cubic phases via holographic masks in electron microscopes allows creation of state-dependent structured beams with tunable group delays and angular properties (Saxton et al., 2020).
• Quantum transport and surface hopping: State-dependent adiabatic and non-adiabatic transport theory describes electronic wave-packet dynamics across energy crossings, with explicit formulas for transmission and reflection via Landau–Zener amplitudes (Kammerer et al., 2020, Fermanian-Kammerer et al., 2020, Rajendran et al., 2018).
• Background-controlled wave propagation: High-frequency packets on evolving hydrodynamic backgrounds in BEC or shallow-water systems serve as probes of large-scale state-dependent Hamiltonian transport (Shaykin et al., 2023).

These phenomena underscore the centrality of state-dependent wave-packets for quantum control, high-precision timing, ultrafast dynamics, and quantum scattering beyond the plane-wave regime.

---

References

• (Carles et al., 2010) A Nonlinear Adiabatic Theorem for Coherent States
• (Hari, 2012) Coherent states for systems of $L^2-$supercritical nonlinear Schrödinger equations
• (Karlovets, 2016) Scattering of wave packets with phases
• (Rajendran et al., 2018) Exact dynamics of a Gaussian wave-packet in two potential curves coupled at a point
• (Fermanian-Kammerer et al., 2020) Propagation of Wave Packets for Systems Presenting Codimension 1 Crossings
• (Saxton et al., 2020) Control of Arrival Time using Structured Wave Packets
• (Kammerer et al., 2020) Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations
• (Shaykin et al., 2023) Propagation of wave packets along large-scale background waves
• (Jyde et al., 19 Nov 2024) Time-resolved Coulomb explosion imaging of vibrational wave packets in alkali dimers on helium nanodroplets