© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2205.15248 · Journal of Physics B Atomic Molecular and Optical Physics

Direct measurement of the Wigner function of atoms in an optical trap

Falk-Richard Winkelmann, Carrie Ann Weidner, Gautam Ramola, Wolfgang Alt, Dieter Meschede, Andrea Alberti

Published Aug 22, 2022• 19 citations

Abstract

Abstract We present a scheme to directly probe the Wigner function of the motional state of a neutral atom confined in an optical trap. The proposed scheme relies on the well-established fact that the Wigner function at a given point (x, p) in phase space is proportional to the expectation value of the parity operator relative to that point. In this work, we show that the expectation value of the parity operator can be directly measured using two auxiliary internal states of the atom: parity-even and parity-odd motional states are mapped to the two internal states of the atom through a Ramsey interferometry scheme. The Wigner function can thus be measured point-by-point in phase space with a single, direct measurement of the internal state population. Numerical simulations show that the scheme is robust in that it applies not only to deep, harmonic potentials but also to shallower, anharmonic traps.

DOI: 10.1088/1361-6455/ac8bb8

Related papers

Direct measurement of the Wigner function by photon counting

Konrad Banaszek, Czesław Radzewicz, K. Wódkiewicz +1 more

We report a direct measurement of the Wigner function characterizing the quantum state of a light mode. The experimental scheme is based on the representation of the Wigner function as an expectation value of a displaced photon number parity operator. This allowed us to scan the phase space point-by-point, and obtain the complete Wigner function without using any numerical reconstruction algorithms.

arXiv:quant-ph/9903027•Physical Review A•Jul 1, 1999•193 citations

Determination of the Wigner function from photon statistics

Konrad Banaszek, Czesław Radzewicz, K. Wódkiewicz +1 more

We present an experimental realisation of the direct scheme for measuring the Wigner function of a single quantized light mode. In this method, the Wigner function is determined as the expectation value of the photon number parity operator for the phase space displaced quantum state.

arXiv:quant-ph/9904089•Apr 26, 1999•2 citations

Fresnel Representation of the Wigner Function: An Operational Approach

Pavel Lougovski, E. Solano, Z. M. Zhang +3 more

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 [Phys. Rev. Lett. 76, 1796 (1996)]] and in cavity QED [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.

arXiv:quant-ph/0206083•Physical Review Letters•Jun 30, 2003•47 citations

Generalized parity and quasi-probability density functions

M. G. Benedict, Attila Czirják

By shifting the parity operator in phase space, one obtains a class of operators, which we call the Wigner operator, because its expectation value is equal to the Wigner function. Calculating the commutator of Wigner operators with different arguments, we show that the corresponding observables cannot be measured simultaneously. Introducing a new parameter, we give a trace-class generalization of this operator. The displaced number states of the harmonic oscillator constitute the natural eigenstate basis of the corresponding generalized Wigner operator. Establishing the integral representation of this operator we show that its expectation value is proportional to the Q- and P- and Wigner functions at special values of the parameter. We illustrate this connection with the coherent, number and squeezed states of the harmonic oscillator.

Journal of Physics A Mathematical and General•Aug 21, 1995•9 citations

Scheme for direct measurement of the Weyl characteristic function for the motion of a trapped ion

Shi-Biao Zheng, Xiwen Zhu, Mang Feng +1 more

We propose a scheme for the reconstruction of the motional state of an ion trapped in a one-dimensional harmonic potential. In the scheme the ion is multichromatically excited by three lasers. Then the measurement of the population of the lower internal state directly yields the Weyl characteristic function for the motional state. The scheme is easily generalized to the two-dimensional case. The scheme operates in the Lamb-Dicke limit.

Physical Review A•Aug 15, 2000•15 citations

Quantum state measurement of any arbitrary entangled field-state via Ramsey interferometry

Ashfaq H. Khosa, Ishaq Ahmad, M. Iqbal

Multipartite entangled states are the key resource and play a crucial role in latest applications of quantum mechanics. We propose a scheme for the measurement of quantum state of multimode entangled field state trapped in multiple cavities. The scheme is based on the measurement of photon statistics of the displaced entangled field state in Ramsey type set-up. In this set-up, the atoms undergo a dispersive phase shift when they pass through the off-resonant entangled field in cavities. By measuring the internal states of the atoms, the photon statistics and the Wigner function can be reconstructed.

Journal of Modern Optics•Aug 12, 2016•0 citations

Abstract We present a scheme to directly probe the Wigner function of the motional state of a neutral atom confined in an optical trap. The proposed scheme relies on the well-established fact that the Wigner function at a given point (x, p) in phase space is proportional to the expectation value of the parity operator relative to that point. In this work, we show that the expectation value of the parity operator can be directly measured using two auxiliary internal states of the atom: parity-even and parity-odd motional states are mapped to the two internal states of the atom through a Ramsey interferometry scheme. The Wigner function can thus be measured point-by-point in phase space with a single, direct measurement of the internal state population. Numerical simulations show that the scheme is robust in that it applies not only to deep, harmonic potentials but also to shallower, anharmonic traps.

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 [Phys. Rev. Lett. 76, 1796 (1996)]] and in cavity QED [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.

By shifting the parity operator in phase space, one obtains a class of operators, which we call the Wigner operator, because its expectation value is equal to the Wigner function. Calculating the commutator of Wigner operators with different arguments, we show that the corresponding observables cannot be measured simultaneously. Introducing a new parameter, we give a trace-class generalization of this operator. The displaced number states of the harmonic oscillator constitute the natural eigenstate basis of the corresponding generalized Wigner operator. Establishing the integral representation of this operator we show that its expectation value is proportional to the Q- and P- and Wigner functions at special values of the parameter. We illustrate this connection with the coherent, number and squeezed states of the harmonic oscillator.

Multipartite entangled states are the key resource and play a crucial role in latest applications of quantum mechanics. We propose a scheme for the measurement of quantum state of multimode entangled field state trapped in multiple cavities. The scheme is based on the measurement of photon statistics of the displaced entangled field state in Ramsey type set-up. In this set-up, the atoms undergo a dispersive phase shift when they pass through the off-resonant entangled field in cavities. By measuring the internal states of the atoms, the photon statistics and the Wigner function can be reconstructed.