Tsallis Pseudo-coherent States

• Tsallis pseudo-coherent states are q-deformed quantum states obtained by substituting the standard exponential with the Tsallis q-exponential, extending the concept of coherent states.
• They feature a modified oscillator algebra and generalized uncertainty relations that bridge non-extensive thermodynamics with applications in quantum optics and gravity.
• These states offer practical insights into experimental quantum optics and quantum information, enabling the exploration of non-classical correlations and deformed commutation relations.

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 (Ferri et al., 2015, Jagannathan et al., 2019, Jizba et al., 2023).

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 $q \to 1$ (Ferri et al., 2015, Jagannathan et al., 2019). 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:

$$A(q,\alpha) = \left\{\frac{q-1}{5-q}\left(\frac{q-1}{2}\right)^{2/(1-q)} F_D \Bigl(\frac{5-q}{q-1};\tfrac1{q-1},...,z_i\Bigr)\right\}^{-1/2}$$

(Ferri et al., 2015). The construction can be equivalently performed in Fock space and for deformed oscillator algebras using the same $q$-exponential structure (Jagannathan et al., 2019).

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 $\phi_T(N) = N/[1+(q-1)(N-1)]$ (Jagannathan et al., 2019). 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 (Jagannathan et al., 2019). This algebraic deformation leads to a one-parameter ($q$) family interpolating between canonical boson oscillators ($q \to 1$) and bounded-spectrum, nonlinear oscillators ($1two-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 (Ferri et al., 2015).

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 (Ferri et al., 2015, Jizba et al., 2023).

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 $w_q(r) = e_q(-r)$ (Jagannathan et al., 2019). 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 (Jizba et al., 2023). 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 (Jizba et al., 2023).

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 (Jagannathan et al., 2019). 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 (Jizba et al., 2023). 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 (Jizba et al., 2023). 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" (Ferri et al., 2015)
• "On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential" (Jagannathan et al., 2019)
• "Coherent states for generalized uncertainty relations as Tsallis probability amplitudes: new route to non-extensive thermostatistics" (Jizba et al., 2023)