---
title: Tsallis Pseudo-coherent States
url: https://www.emergentmind.com/topics/tsallis-pseudo-coherent-states
type: topic
---

# Tsallis Pseudo-coherent States

Tsallis pseudo-coherent states constitute a class of quantum states that generalize the classical concept of Glauber coherent states by incorporating the non-additive statistical structure inherent to Tsallis’ formulation of non-extensive thermostatistics. These states arise naturally when the exponential function in the harmonic oscillator’s coherent state wavefunction is replaced by the Tsallis $q$-exponential, resulting in a deformation of both the state algebra and associated statistical properties. Tsallis pseudo-coherent states establish a nontrivial link between nonextensive statistical mechanics, generalized uncertainty relations, and the theory of deformed oscillators, with implications spanning quantum optics, information theory, and quantum gravity [1511.08720][1911.02428][2308.12368].

## 1. Mathematical Foundation: The $q$-Exponential and State Construction

Central to the Tsallis framework is the $q$-exponential,
$$
e_q(x) = \bigl[1+(1-q)x\bigr]^{1/(1-q)}, \qquad q\in\mathbb{R},
$$
which reduces to the ordinary exponential as $q \to 1$ [1511.08720][1911.02428]. The corresponding pseudo-coherent state, directly analogous to the standard harmonic oscillator coherent state but deformed by $q$, is given in the coordinate representation by
$$
\psi_{\alpha q}(x) = A(q,\alpha)\left[1+\frac{q-1}{2}\bigl(x^2 - 2\sqrt{2}\alpha x + |\alpha|^2 + \alpha^2\bigr)\right]^{1/(1-q)},
$$
with normalization factor $A(q,\alpha)$ determined through Lauricella hypergeometric functions:
$$
A(q,\alpha) = \left\{\frac{q-1}{5-q}\left(\frac{q-1}{2}\right)^{2/(1-q)} F_D \Bigl(\frac{5-q}{q-1};\tfrac1{q-1},...,z_i\Bigr)\right\}^{-1/2}
$$
[1511.08720]. The construction can be equivalently performed in Fock space and for deformed oscillator algebras using the same $q$-exponential structure [1911.02428].

## 2. Algebraic Structure and Oscillator Deformation

The operator algebra underlying Tsallis pseudo-coherent states is a specific $q$-deformation of the bosonic oscillator, with creation and annihilation operators $(a, a^\dagger)$ obeying
$$
[a,a^\dagger] = \phi_T(N+1) - \phi_T(N)
$$
where $\phi_T(N) = N/[1+(q-1)(N-1)]$ [1911.02428]. In the Bargmann representation, the deformed annihilation operator $a$ corresponds to a $q$-deformed derivative, acting on $q$-exponential eigenfunctions,
$$
D_{T,q} e_q(kx) = k e_q(kx).
$$
The coherent states are eigenstates of this annihilation operator, with the number-basis representation
$$
|z\rangle_T = \mathcal{N}_T(z) \sum_{n=0}^\infty \frac{z^n}{\sqrt{[n]_{q-1}!}} |n\rangle,
$$
$\mathcal{N}_T(z)$ ensuring proper normalization [1911.02428]. This algebraic deformation leads to a one-parameter ($q$) family interpolating between canonical boson oscillators ($q \to 1$) and bounded-spectrum, nonlinear oscillators ($1<q<2$), culminating in two-level systems for $q\to2$.

## 3. Quantum Statistical and Uncertainty Properties

Tsallis pseudo-coherent states have probability distributions in both position and momentum spaces parameterized by $q$:
- Position space:
  $$
  P_q(x) = |\psi_{\alpha q}(x)|^2 = A(q,\alpha)^2 \left[1+\frac{q-1}{2}(x^2-2\sqrt{2} \mathrm{Re}\,\alpha\,x + |\alpha|^2 + \mathrm{Re}(\alpha^2))\right]^{2/(1-q)}
  $$
- Momentum space distributions can be written via confluent hypergeometric (or Fox–Wright) functions [1511.08720].

Expectation values such as $\langle x \rangle_q$, $\langle x^2 \rangle_q$, $\langle p \rangle_q$, and $\langle p^2 \rangle_q$ are expressible in closed form through Lauricella functions and $\Gamma$-functions. The quantum uncertainties,
$$
(\Delta x)_q = \sqrt{\langle x^2\rangle_q - \langle x\rangle_q^2}, \qquad
(\Delta p)_q = \sqrt{\langle p^2\rangle_q - \langle p\rangle_q^2}
$$
may be computed for each $q$. The uncertainty product $(\Delta x)_q (\Delta p)_q$ is minimized at $1/2$ for $q=1$ (ordinary Glauber state), and increases monotonically with $|q-1|$, demonstrating that the $q$-states are no longer minimum-uncertainty packets except in the extensive limit [1511.08720][2308.12368].

## 4. Overcompleteness, Resolution of Identity, and Fock Space

Tsallis pseudo-coherent states provide an overcomplete basis of the Hilbert space for all admissible $q$, with an exact resolution of the identity:
$$
\mathbf{1} = \int d^2\alpha\, |\alpha,q\rangle\, w(q,\alpha)\, \langle\alpha,q|, \qquad w(q,\alpha) \to 1/\pi \text{ as } q\to1.
$$
In Fock space, the number basis expansion involves $q$-deformed factorials, and the resolution of the identity uses a $q$-modified weight function $w_q(r) = e_q(-r)$ [1911.02428]. These overcompleteness properties allow the same function space completeness as in standard coherent-state quantization.

## 5. Connection to Generalized Uncertainty Principles and Entropy-Power Relations

Tsallis pseudo-coherent states emerge as the extremal states for generalized uncertainty principles (GUP), saturating both the (quadratic) GUP and the Tsallis entropy-power uncertainty relation (EPUR):
$$
M_{q/2}^T(|\psi|^2)\; M_{q'/2}^T(|\widetilde\psi|^2) \geq \frac{\hbar^2}{4},
$$
where $M_q^T$ denotes the Tsallis entropy power [2308.12368]. The non-extensivity parameter $q$ is monotonically related to the GUP deformation parameter $\beta$,
$$
q = 1 + \frac{\beta \gamma \hbar}{m_p^2 + \beta \gamma \hbar},
$$
implying that deformations away from canonical quantum mechanics (i.e., $\beta \neq 0$) naturally induce $q$-pseudo-coherence [2308.12368].

## 6. Special Limits and Physical Interpretations

For $q=2$, Tsallis pseudo-coherent states become the well-known phase-coherent (harmonious/pseudothermal) states with a geometric photon-number distribution and minimum generalized phase–number uncertainty [1911.02428]. As $q\to1$, all expressions reduce to their standard (extensive) quantum counterparts. The finite-band spectrum for $1 < q < 2$ may be of interest for modeling systems with bounded excitations, while $q=2$ (two-level) has applications in coding and in quantum optical phase references.

These states have physical applications as robust pointer states in GUP-modified quantum theory, and admit interpretations in emergent gravity, Loop Quantum Gravity microstate counting, and modifications of fundamental commutators in DSR theories [2308.12368]. *This suggests* that Tsallis pseudo-coherent states unify several domains where classical notions of additivity and linearity break down.

## 7. Experimental Accessibility and Broader Impact

The structure of Tsallis pseudo-coherent states is, in principle, experimentally accessible in quantum optics, where $q$-deformed probability distributions can be probed in photonic or mesoscopic systems. Potential consequences include corrections to quantum noise and coherence in optics and interferometry, especially in regimes where non-extensive statistics become significant [2308.12368]. The formalism opens routes to explore quantum information-theoretic measures, generalized entropies, and non-classical correlations beyond the extensive regime.

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**References**

- "New mathematics for the non additive Tsallis' scenario" [1511.08720]
- "On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential" [1911.02428]
- "Coherent states for generalized uncertainty relations as Tsallis probability amplitudes: new route to non-extensive thermostatistics" [2308.12368]

Source: https://www.emergentmind.com/topics/tsallis-pseudo-coherent-states