Probing phase-space noncommutativity through quantum beating, missing information, and the thermodynamic limit

In this work we examine the effect of phase-space noncommutativity on some typically quantum properties such as quantum beating, quantum information, and decoherence. To exemplify these issues we consider the two-dimensional noncommutative quantum harmonic oscillator whose component behavior we monitor in time. This procedure allows us to determine how the noncommutative parameters are related to the missing information quantified by the linear quantum entropy and by the mutual information between the relevant Hilbert space coordinates. Particular questions concerning the thermodynamic limit of some relevant properties are also discussed in order to evidence the effects of noncommutativity. Finally, through an analogy with the Zeeman effect, we identify how some aspects of the axial symmetry of the problem suggest the possibility of decoupling the noncommutative quantum perturbations from unperturbed commutative well-known solutions.

Figure 1

Time evolution of the phase-space coordinates, $[{\mathit{Q}}_1\left(t\right),{\Pi }_1\left(t\right)]$ and $[{\mathit{Q}}_2\left(t\right),{\Pi }_2\left(t\right)]$, for the NC harmonic oscillator. The first plot line refers to the phase-space elliptical orbits of commutative harmonic oscillators (as if one had set $\epsilon =0$ in the NC map). From the second to the forth plot lines one has set arbitrary values for $\epsilon$, $\epsilon =1/4,1/10$, and $1/100$, respectively. The precession motion describes close orbits for these cases. One has used a BlueGreenYellow (GrayLevel) scale in order to denote the time scale $\tau$, varying from 0 [blue (dark gray)] to $2\pi /\Omega$ [yellow (light gray)], such that orbits start and finish at $(x,{\pi }_x)=(y,{\pi }_y)=(1/\sqrt{2},0)$. The last pair of plots corresponds to $\epsilon =1/2\pi$, which does not result into a closed orbit since it is not a rational number. By convenience, one has set $\alpha =\beta$, that is, equivalent to $m\omega =\hslash =1$.

Figure 2

Quantum beating for the NC harmonic oscillator described by the traced-out state vector ${\tilde{\rho }}_{n_x,n_y}^{\left(1\right)}({\mathit{Q}}_1,{\Pi }_1;t)$ in the ${\mathit{Q}}_1$-${\Pi }_1$ plane, with $n_y=2$ and $n_x=1$ (first column), 2 (second column), and 5 (third column). At time $\tau =0$ the Wigner function is assumed to be centered at the origin $(0,0)$. One has considered time intervals such that $\tau =k\pi {\left(8\epsilon \Omega \right)}^{-1}$, with $k=0,1,2,3$, and 4 in order to reproduce its time evolution through the sequence of five plots. Notice that the prepared quantum state in ${\mathit{Q}}_1$-${\Pi }_1$ is projected into the traced-out quantum state, by reproducing its wave pattern at $\tau =\pi /\left(2\epsilon \Omega \right)$. It corresponds to the quantum beating effect with frequency ${\omega }_{\mathrm{beat}}=2\gamma =2\epsilon \Omega$. The contour plot follows a BlueGreenYellow scale from yellow (light gray) which corresponds to 1, to blue (dark gray) which corresponds to 0.

Figure 3

Quantum beating for the NC harmonic oscillator described by the traced-out state vector ${\tilde{\rho }}_{n_x,n_y}^{\left(1\right)}({\mathit{Q}}_1,{\Pi }_1;t)$ in the ${\mathit{Q}}_1$-${\Pi }_1$ plane, with $n_y=1$ and $n_x=1$ (first column), 2 (second column), and 5 (third column). The resulting Wigner function in the ${\mathit{Q}}_1$-${\Pi }_1$ plane is supposed to be centered at $({\mathit{Q}}_1,{\Pi }_1)=(1,0)$ at time $\tau =0$. One has considered time intervals such that $\tau =k\pi {\left(8\epsilon \Omega \right)}^{-1}$, with $k=0,1,2,3$, and 4. Once again one notices a quantum beating effect with frequency ${\omega }_{\mathrm{beat}}=2\gamma =2\epsilon \Omega$. In addition to the analogous effects described in Fig. 2, the NC parameter $\epsilon$ introduces the precession motion that follows the phase-space maps like those in Fig. 1.

Figure 4

(a) Mutual information $I_{12}$ for the NC harmonic oscillator described by the state vector ${\rho }_{n_x,n_y}^W$ for several values of the quantum numbers $n_x$ and $n_y$. (b) Approximated decoherence profile for tiny values of the NC parameter $\epsilon$, depicted for increasing values of the mutual information $I_{12}$.

Figure 5

Probability distribution (quantum orbital) distortion for the NC harmonic oscillator state vectors with radial symmetry in the ${\mathit{Q}}_1$-${\mathit{Q}}_2$ plane. For higher temperatures, i.e., for $k_{B}T\gg \hslash \Omega$ ($\sigma \ll 1$), the NC effects are erased, and the quantum behavior of the thermalized state vector ${\rho }_{th}^W$ prevails over the tiny NC quantum corrections. For very low temperatures, i.e., for $k_{B}T\ll \hslash \gamma$ ($\sigma \gg 1$), the wave function collapses into the classical limit. The usual pattern of the quantum orbitals is maximally modified by the NC element at intermediate scales, namely $1\lesssim {\sigma }_{\max }\lesssim 4$. The plots correspond to $\epsilon =0.1$ (first column) and $0.5$ (second column) with $\sigma$ assuming the values of $0.1,0.2,0.5,1,2,5$, and 10 (decreasing temperature scale). The right-side GrayLevel plots correspond to the amplification of monochrome negative image [from white which corresponds to 0 (no distortion) to black which corresponds to 1 (maximal distortion)] of the distortion effects on the left-side plots.

Figure 6

(a) Linear entropy (thin red lines) $S_{12}\left(\sigma \right)$, and mutual information (thick black lines) $I_{12}\left(\sigma \right)$, for the isotropic 2D NC harmonic oscillator, and (b) the corresponding missing information given in terms of $\Delta S_{12}\left(\sigma \right)$ and $\Delta I_{12}\left(\sigma \right)$. At high temperatures, $k_{B}T\gg \hslash \gamma$ ($\sigma \ll 1$), and the NC effects are erased. At very low temperatures, $k_{B}T\ll \hslash \gamma$ ($\sigma \gg 1$), and the wave function collapses into the classical limit. The maximal values of $\Delta S_{12}\left(\sigma \right)$ and $\Delta I_{12}\left(\sigma \right)$ are obtained at intermediate values of $\sigma$ (see the correspondence in Fig. 7). One has considered $\epsilon =0$ (dotted lines), $0.1$ (solid lines), and $0.5$ (dashed lines).

Figure 7

Numerical solution for the $\sigma$ parameter that maximizes the missing information (from Fig. 6) as a function of the NC parameter $\epsilon =\gamma /\Omega$. The results are for the maximal values of $\Delta S_{12}\left(\sigma \right)$ (thin red line), and $\Delta I_{12}\left(\sigma \right)$ (thick black line).

Figure 8

(a) Thermodynamic variables. Boltzmann entropy (thin red lines) $S_k\left(\sigma \right)/k_B$, and heat capacity (thick black lines) $C_v\left(\sigma \right)/k_B$, for the statistical mixture ${\rho }_{th}^W$. It reproduces the thermodynamic limit of the isotropic 2D NC harmonic oscillator quantum behavior. (b) Distortions due to the NC effects are quantified through $\Delta S_k\left(\sigma \right)/k_B$ and $\Delta C_v\left(\sigma \right)/k_B$. One notices that $C_v\left(\sigma \right)/k_B$ exhibits an anomalous behavior that inverts the sign of $\Delta C_v\left(\sigma \right)/k_B$ at intermediate scales of $\sigma$. The plots are for $\epsilon =0$ (dotted lines), $0.1$ (solid lines), and $0.5$ (dashed lines).