Fresnel Representation of the Wigner Function: An Operational Approach

We present an operational definition of the Wigner function. Our method relies on the Fresnel transform of measured Rabi oscillations and applies to motional states of trapped atoms as well as to field states in cavities. We illustrate this technique using data from recent experiments in ion traps [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)] and in cavity QED [B. Varcoe et al., Nature (London) 403, 743 (2000)]. The values of the Wigner functions of the underlying states at the origin of phase space are $W_{|0⟩}(0)=+1.75$ for the vibrational ground state and $W_{|1⟩}(0)=-1.4$ for the one-photon number state. We generalize this method to wave packets in arbitrary potentials.

Figure 1

Fresnel representation of Wigner function illustrated by the first excited energy eigenstate of a harmonic oscillator. The atomic dynamics $P_g(\tau ;\alpha )$, Eq. (1), as a function of interaction time $\tau$ and real-valued displacement $\alpha$ (top) serves as input to the truncated Fresnel representation, Eq. (4). For a finite measurement time ${\tau }_m$ the so-obtained function $W(\alpha ;{\tau }_m)$ is complex valued. The real part of $W(\alpha ;{\tau }_m)$ (bottom) approaches the correct Wigner function, $W_{|1⟩}$, already for moderate measurement times.

Figure 2

Asymptotic approach of the truncated Fresnel representation for the first excited state. Real part (black line) and imaginary part (gray line) of $W(\alpha =0;{\tau }_m)$ oscillate around their asymptotic values $-2$ and $0$, respectively, with slowly decreasing amplitudes. The asymptotic values emerge already after a measurement time of the order of $2\pi$.

Figure 3

Wigner function at the origin of phase space obtained from experimental Rabi oscillations (top) of a stored ion [20] using the truncated Fresnel representation, Eq. (4): Approach (bottom) of real (black line) and imaginary parts (gray line) of $W(\alpha =0;{\tau }_m)$ towards their asymptotic values (dashed line) yielding $W_{|0⟩}(0)=1.75+i0.05$. The ion was prepared initially in the atomic and vibrational ground state.

Figure 4

Dependence of the Wigner function $W^{(N)}(\alpha =0)$ at the origin of phase space on the cutoff $N$ of the overdetermined system, Eq. (5). The values scatter around $W_{|1⟩}(0)=-1.4$. The Rabi oscillations [23] (inset) result from a cavity field prepared in a one-photon number state.