Periodic Schrödinger Coherent States

• Periodic Schrödinger coherent states are quantum wavepackets defined on compact spaces such as S¹ and Tⁿ that achieve optimal localization in both position and momentum.
• They are constructed through algebraic and geometric techniques that extend canonical Gaussian coherent states to periodic settings, ensuring resolution of unity and phase-space concentration.
• Their dynamics under periodic Hamiltonians showcase stable semiclassical propagation and nonlinear modifications, with applications in spectral theory, optical lattices, and quantum-classical correspondence.

A periodic Schrödinger coherent state is a quantum wavepacket on a periodic configuration space—such as the circle $S^1$ or flat torus $T^n$—that is optimally localized in both position and momentum while exhibiting covariance under the dynamics generated by periodic Hamiltonians. These states generalize the canonical Gaussian coherent states of $\mathbb{R}^n$ to spaces with compact or discrete symmetry, supporting key roles in spectral theory, quantum dynamics in periodic/ Bloch-type media, and semiclassical analysis involving nonlinear or effective-mass corrections.

1. Algebraic and Geometric Construction

Periodic Schrödinger coherent states manifest in several settings, with a coordinated algebraic structure dictated by the topology of the underlying configuration space and the quantization of momentum.

On the unit circle $S^1$, as in (Chadzitaskos et al., 2011), the Hilbert space is $L^2(S^1,d\varphi)$ and the canonical position and momentum observables are the angle $\varphi\in[-\pi,\pi)$ and the quantized generator $\hat{P}=-i\,d/d\varphi$, respectively. The fundamental phase space is then $\mathbb{Z}\times S^1$ (momentum integer $m$, angle $\theta$). A projective Weyl representation is given by

$W(m,\theta) = e^{im\hat{Q}}\,e^{-i\theta\hat{P}}.$

The vacuum vector $|0,0\rangle$ is defined as a $2\pi$-periodic Gaussian,

$$\langle\varphi|0,0\rangle = \mathcal{A}\exp(-\tfrac12\varphi^2),$$

from which the full coherent state family

$|m,\theta\rangle = W(m,\theta)|0,0\rangle$

is generated. The analogous construction extends to the $n$-torus $T^n$, where coherent states are built by periodizing the standard $\mathbb{R}^n$ Gaussian coherent states with a translation- and momentum-label (Zanelli, 2019):

$$\varphi_h(x,\xi)(y) = \sum_{k\in\mathbb{Z}^n} \alpha_h \exp\left(\frac{i}{h}\xi\cdot(y-2\pi k)\right)\exp\left(-\frac{|y-x-2\pi k|^2}{2h}\right)$$

with labels $x,y\in T^n$, $\xi\in h\mathbb{Z}^n$.

2. Properties: Localization, Overcompleteness, and Resolution of Unity

Periodic coherent states exhibit strong phase-space localization: for $|m|\gg1$ or narrow Gaussians, the state is semiclassically concentrated at $(m,\theta)$ or $(x,\xi)$. The overlap kernel between distinct states is non-zero but rapidly decreases with phase-space separation, quantifiable via special functions (error function, Jacobi theta) (Chadzitaskos et al., 2011). Coherent states are nonorthogonal yet form an overcomplete set, resolving the identity up to normalization:

$$\sum_{m\in\mathbb{Z}}\int_0^{2\pi} |m,\theta\rangle\langle m,\theta|\, d\theta = 2\pi\mathbb{I}_{L^2(S^1)}$$

for the circle; the exact analogue holds in $L^2(T^n)$ up to $\mathcal{O}(h)$ corrections (Chadzitaskos et al., 2011, Zanelli, 2019). On $T^n$, every function admits an approximate coherent state decomposition:

$$\psi(y) = \sum_{\xi\in h\mathbb{Z}^n}\int_{T^n}\langle\psi,(x,\xi)_h\rangle (x,\xi)_h(y)dx + \mathcal{O}_{L^2}(h)$$

3. Quantum Dynamics and Semiclassical Propagation

Periodic coherent states enjoy temporal stability properties determined by the form of the Hamiltonian. For a free particle on $S^1$, evolution proceeds as

$$e^{-iHt}|m,\theta\rangle = e^{-i\frac12 m^2 t}|m,\theta+m t\rangle,$$

preserving the coherent state structure with linear evolution of the position label (Chadzitaskos et al., 2011). On the flat torus, under a general semiclassical elliptic pseudodifferential operator $P_h=\mathrm{Op}_h(b)$, coherent states track the classical Hamiltonian flow $\Phi_t(x,\xi)$ of $b$ up to $\mathcal{O}(h)$, i.e.,

$$U_h(-t)\,(x,\xi)_h = e^{iS_t(x,\xi)/h}\,(x_t,\xi_t)_h + \mathcal{O}_{L^2}(h),$$

with $S_t$ the relevant classical action (Zanelli, 2019). The entire coherent state decomposition is therefore approximately invariant under the quantum propagator.

4. Periodic Coherent States in Nonlinear and Periodic Potentials

In periodic and nonlinear environments, the structure of coherent state propagation modifies according to the Bloch-banded energy landscape and the effective nonlinearity. For nonlinear Schrödinger equations (NLSE) with periodic potentials as in (Carles et al., 2011), semiclassical initial data is prepared in the form

$$\psi_0^\varepsilon(x) = \varepsilon^{-d/4}u_0\left(\frac{x-q_0}{\sqrt\varepsilon}\right) \chi_n\left(\frac{x}{\varepsilon},p_0\right)e^{ip_0\cdot(x-q_0)/\varepsilon},$$

and evolves according to an effective envelope equation with mass tensor $M=\nabla_k^2 E_n(p(t))$,

$$i\partial_t u + \frac{1}{2}\nabla_z\cdot(M \nabla_z u) = \lambda_n |u|^{2\sigma}u,$$

plus corrections in the case of nonlocal nonlinearities (Hartree interaction). The center dynamics follow the semiclassical Bloch trajectory,

$$\dot q(t) = \nabla_k E_n(p(t)),\;\; \dot p(t) = -\nabla_q V_\mathrm{ext}(q(t)).$$

A two-scale Gaussian-Bloch structure localizes the wavepacket in phase space around the classical evolution, with error $O(\sqrt{\varepsilon})$ over time scales $O(1)$ (Carles et al., 2011).

5. Nonlinear Periodic Schrödinger Coherent States: Shape Invariance, Stability, and Breakdown

For the nonlinear Schrödinger equation with a purely quadratic confining potential, the phase-space-translated ground state

$$\psi(x,t) = \phi_0(x-q(t))\exp\left[\frac{i}{\hbar}p(t)(x-q(t)) + \frac{i}{\hbar}S(t)\right]$$

is an exact, non-spreading solution, with the shape frozen and the center following the classical Hamiltonian trajectory; this property holds only if $V''(x)=\mathrm{const}$ (Moulieras et al., 2011). Linearizing about the stationary solution yields the Bogoliubov–de Gennes spectrum. In the Thomas–Fermi regime, dipole and quadrupole mode frequencies are given by $\omega_n/\omega = \sqrt{n(n+1)/2}$, e.g., $\omega_2 = \sqrt{3}\omega$ for the quadrupole (Moulieras et al., 2011).

Perturbations from pure quadraticity ($V(x) = \frac12 m\omega^2x^2[1+\alpha x^2]$) induce splitting of the wavepacket into coherent and incoherent modes, quantifiable by a "fluidity factor" $F(\beta)$. For small anharmonicity, $F(\beta) \to 1$; as $\beta$ increases, mass evaporates into incoherent modes and coherent oscillations are damped. This signals breakdown of long-term coherent state propagation under nonlinear dynamics—an essential distinction from the linear setting (Moulieras et al., 2011).

6. Key Examples and Special Functions

The state structure and dynamics are supported by several special function representations. Overlaps of periodic coherent states employ error functions and generalizations to theta functions via Poisson summation:

$$\sum_{k\in\mathbb{Z}} e^{-\frac12(\varphi-\theta+2\pi k)^2} = \sqrt{2\pi} \vartheta_3\left(\frac{\varphi-\theta}{2}\big|\frac{i}{2\pi}\right)$$

(Chadzitaskos et al., 2011). On $T^n$, coherent states are nearly orthogonal, with overlaps decaying exponentially. The decomposition and almost-orthogonality are crucial in semiclassical spectral and microlocal analyses (Zanelli, 2019).

7. Applications and Implications

Periodic Schrödinger coherent states furnish a technical framework for analyzing quantum dynamics in finite, compact, or periodic geometries. They underpin microlocal analysis on the torus, spectral estimates for elliptic pseudodifferential operators, and quantum-classical correspondence in periodic media. In nonlinear optics and Bose-Einstein condensates in optical lattices, such states describe the regime where the wavepacket remains phase-space localized and its center obeys an effective Bloch-band or harmonic oscillator dynamics up to nonlinear corrections (Moulieras et al., 2011, Carles et al., 2011). Their deformation, breakdown, or persistence under perturbations and nonlinearities encodes information about damping, decoherence, and transport in these quantum systems.