Number-operator–annihilation-operator uncertainty as an alternative for the number-phase uncertainty relation

We consider a number-operator–annihilation-operator uncertainty as a well-behaved alternative to the number-phase uncertainty relation, and examine its properties. We find a formulation in which the bound on the product of uncertainties depends on the expectation value of the particle number. Thus, while the bound is not a constant, it is a quantity that can easily be controlled in many systems. The uncertainty relation is approximately saturated by number-phase intelligent states. This allows us to define amplitude squeezing, connecting coherent states to Fock states, without a reference to a phase operator. We propose several setups for an experimental verification.

Figure 1

Numerical test of the inequality (2). $F$ refers to the Fock state $|n=25\rangle$, $C$ to the coherent state $|\alpha =5\rangle$, and $Z$ to the point that saturates inequality (2) for $(\Delta a){}^2=0.$ Solid line: Boundary of the region defined by Eq. (2) for $\langle N\rangle =25$. All points below this line correspond to aphysical $(\Delta a){}^2-(\Delta N){}^2$ values. Dashed line: Points corresponding to squeezed coherent states. The equation of the curve is given in Eq. (23). Inset: Solid line: Boundary of the region defined by Eq. (2). Dashed line: Points corresponding to states with a Gaussian wave vector Eq. (16). Circles: States corresponding to photon-added coherent states, Eq. (27).

Figure 2

Numerical test of the inequality Eq. (2). The difference between the left- and right-hand sides of Eq. (2) divided by the right-hand side is shown for states Eq. (16) with a Gaussian wave vector for (from left to right) $N_0=100,$ 25, 10, and $5.$

Figure 3

Distance of the points corresponding to states with a Gaussian wave vector Eq. (16) from the curve corresponding to states that saturate Eq. (2) for (from left to right) $N_0=100,$ 25, 10, and $5.$ $\Delta$ and $N_0$ are the parameters of the state Eq. (16).