Quantum Fisher Information: Probe to Measure Fractional Evolution

Abstract

We develop an analytical solution of fractional Schrödinger equation to explain the interaction between a two-level atom and a single electromagnetic field mode inside a cavity. Based on this solution, two different cases are discussed; the first one reflects the standard Jaynes Cummings model, while the second solution gives rise to a wave equation for a dipole interacted with a field in semi-classical sense. By controlling different parameters inside cavity, the estimation degree in terms of quantum Fisher information is investigated. The obtained results show that the precision estimation increases as the detuning decreases and number of photons inside cavity increases. Moreover robust estimation is obtained for the standard Jaynes Cummings model. To measure the amount of information, a comparative study between von Neumann entropy and quantum Fisher information show the same behavior. As a result, we conclude that quantum Fisher information may detect entanglement of the interacted atom-field system.

Appendix

The fractional derivative meaning have been established using different methods, the most popular ones are Caputo and Riemann-Liouville concepts [37, 38]. Both methods are basically defined using Riemann-Liouville integration. For any function ζ(t) of order α > 0, it is indicates as

$$ I^{\alpha} \zeta(t)=\frac{1}{{\Gamma}(\alpha)} {{\int}_{0}^{t}} (t-\alpha)^{\alpha-1} \zeta(\tau) d\tau, \qquad \alpha>0, \quad t>0, $$

(22)

where Γ(α) called Gamma Function [23] and I^α verify the following properties

$$ \begin{array}{@{}rcl@{}} I^{\alpha} I^{\beta} \zeta (t)&=& I^{\alpha+\beta} \zeta (t), \\ I^{\alpha} t^{\gamma}&=& \frac{{\Gamma}(\gamma+1)}{{\Gamma}(\alpha+\gamma+1)}. \end{array} $$

(23)

Using the Caputo notion, the fractional derivative of ζ(t) takes the following form

$$ D^{\alpha} \zeta(t)=I^{j-\alpha} D^{j} \zeta(t)= \frac{1}{{\Gamma}(j-\alpha)} {{\int}_{0}^{t}} \frac{\zeta^{(j)}(\tau)}{(t-\tau)^{\alpha-j+1}} d\tau. $$

(24)

The main advantage of Caputo’s methods is based on its possibility to solve the fractional derivative system with variable initial conditions. However, in order to improve the solution of linear fractional differential equations we investigate the stability properties. Consider the following fractional differential linear system,

$$ \frac{d^{\alpha}}{d t^{\alpha}}v(t)=M v(t), \qquad v(0)=v_{0}, $$

(25)

where $v\in \mathbb {R}^{n}$ and $M\in \mathbb {R}^{n} \times \mathbb {R}^{n}$, moreover α_i and $\frac {\partial ^{\alpha _{i}}}{\partial t^{\alpha _{i}}}$ for i = 1, ... , n are the fractional orders and Caputo fractional derivative, respectively. If α₁ = α₂,.... = α_n then the solution of (3) is improved in Ref. [23]. Whereas the general solution is then expressed in terms of Mittag-Leffler function, E_α(t^α) as,

$$ v(t)=u E_{\alpha}(\xi t^{\alpha}), $$

(26)

where ξ and u are an arbitrary constant and vector to be determined, respectively. However, E_α(s) indicates the Mittag-Leffler function of a single parameter α, it takes the following form

$$ E_{\alpha} (s)=\sum\limits_{j=0}^{\infty} \frac{s^{j}}{\Gamma (\alpha j+1)}. $$

(27)

Substituting (26) into (25) allows to the following result

$$ u\xi E_{\alpha}(\xi t^{\alpha})= M u E_{\alpha}(\xi t^{\alpha}), $$

(28)

after canceling the nonzero factor, E_α(ξt^α) gives rise to

$$ (M-\xi I_{n\times n}) u=0, $$

(29)

where I_n×n represents the identity matrix of (n × n) dimension. Therefore we conclude from (29) that ξ and u indicate the eigenvalue and eigenvector of the matrix M. Finally the general solution of the linear fractional system is then given by

$$ v(t)=C_{1} E_{\alpha} (k_{1} t^{\alpha}) u_{1}+ C_{2} E_{\alpha} (k_{2} t^{\alpha}) u_{2}+...+C_{n} E_{\alpha} (k_{n} t^{\alpha}) u_{n}, $$

(30)

where, C_i (i = 1,..n) are arbitrary constants and E_α(ξ_it^α) may be calculated from (27).