Intelligent states for a number-operator–annihilation-operator uncertainty relation

Recently a new uncertainty relation was found as an alternative to a number-phase uncertainty relation for a harmonic oscillator. In this paper we determine numerically, via the discrete-variable-representation method known from quantum chemistry, the exact states that saturate this new uncertainty relation. We analyze the physical properties of the states and compare them to the intelligent states of the Pegg-Barnett uncertainty relation. We find that for a given set of expectation values of the physical parameters, which are the particle number and the two quadratures, the two kinds of intelligent states are equivalent. The intelligent states are the eigenstates corresponding to the lowest eigenvalue of a Hermitian operator, which, when interpreted as a Hamiltonian of a physical sytem, describes a nonlinear driven harmonic oscillator, for example, a Duffing oscillator for a certain parameter range. Hence, our results can be interpreted as the determination of the ground state of such physical systems in an explicit analytic form as well. As the Pegg-Barnett intelligent states we use are expressed in terms of a coherent-state superposition facilitating experimental synthesis, we relate the states determined here to experimentally feasible ones.

Figure 1

The mean value of the number operator $\langle N\rangle$ as a function of the parameters ${\lambda }_X$ and ${\lambda }_Y$ for ${\lambda }_N=-100$.

Figure 2

The mean value of the quadrature operator $\langle X\rangle$ as a function of the parameters ${\lambda }_X$ and ${\lambda }_Y$ for ${\lambda }_N=-1$.

Figure 3

The mean value of the quadrature operator $\langle X\rangle$ as a function of the parameters ${\lambda }_N$ and ${\lambda }_Y$ for ${\lambda }_X=-1$.

Figure 4

The mean value of the quadrature operator $\langle Y\rangle$ as a function of the parameters ${\lambda }_N$ and ${\lambda }_Y$ for ${\lambda }_X=-1$.

Figure 5

The difference $\Delta \varepsilon$ between the left-hand side and the right-hand side of Eq. (1) as a function of the mean value of the number operator $\langle N\rangle$ and the quadrature operator $\langle X\rangle$ for (a) ${\lambda }_X=[-20,-40,-60,-80,-100]$ from bottom to top and (b) ${\lambda }_X=[0,-0.1,-0.5,-1,-6,-10]$ from right to left.

Figure 6

The mean value of the number operator $\langle N\rangle$ as a function of the mean value of the quadrature operator $\langle X\rangle$ for ${\lambda }_X=[0,-0.1,-0.5,-1,-6,-100]$ from left to right.

Figure 7

The mean value of the number operator $\langle N\rangle$ as a function of the mean value of the quadrature operator $\langle X\rangle$ for $\delta =1,2,3,4,5$ and a series of ${\alpha }_0$ parameter values. (For $\delta =1,3,5$ only the case ${\alpha }_0=\sqrt{\delta }$ is plotted while for $\delta =4$, ${\alpha }_0=[0.1,0.5,1,1.5,1.7,1.9,2,2.1,2.3,2.5,5,10,20]$, and for $\delta =2$, ${\alpha }_0=[0.1,0.5,1,1.2,\sqrt{2},1.6,1.8,2,5,10,15]$ are plotted. In these latter cases the curve belonging to a lower ${\alpha }_0$ runs always below the one with a greater ${\alpha }_0$.)

Figure 8

The difference $\Delta \varepsilon$ between the left-hand side and the right-hand side of Eq. (1), as a function of the mean value of the number operator $\langle N\rangle$ for various values of $u$.

Figure 9

The difference $\Delta \varepsilon$ between the left-hand side and the right-hand side of Eq. (1) as a function of the mean value of the number operator $\langle N\rangle$ for various values of ${\lambda }_X$.