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arXiv:atom-ph/9608001 · Physical Review Letters

Magnetic Tomography of a Cavity State

R. Walser, J. I. Cirac, P. Zoller

Published Sep 23, 1996• 48 citations
arXivHTMLPDF

Abstract

A method to determine the state of a single quantized cavity mode is proposed. By adiabatic passage, the quantum state of the field is transferred completely onto an internal Zeeman submanifold of an atom. Utilizing a method of Newton and Young [Ann. Phys. 49, 393 (1968)], we can determine this angular momentum state uniquely, by a finite number of magnetic dipole measurements with Stern-Gerlach analyzers. An example illustrates the influence of dissipation.

DOI
10.1103/physrevlett.77.2658

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CRC Press eBooks•Sep 18, 2015•0 citations
A method to determine the state of a single quantized cavity mode is proposed. By adiabatic passage, the quantum state of the field is transferred completely onto an internal Zeeman submanifold of an atom. Utilizing a method of Newton and Young [Ann. Phys. 49, 393 (1968)], we can determine this angular momentum state uniquely, by a finite number of magnetic dipole measurements with Stern-Gerlach analyzers. An example illustrates the influence of dissipation.
A complete description of the quantum state of a physical system is the fundamental knowledge necessary to statistically predict the outcome of measurements. In turning this statement around, Wolfgang Pauli raised already in 1933 the question, whether an unknown quantum state could be uniquely determined by appropriate measurements--a problem that has gained new relevance in recent years. In order to harness the prospects of quantum computing, secure communication, teleportation, and the like, the development of techniques to accurately control and measure quantum states has now become a matter of practical as well as fundamental interest. However, there is no general answer to Pauli's very basic question, and quantum state reconstruction algorithms have been developed and experimentally demonstrated only for a few systems so far. This thesis presents a novel experimental method to measure the unknown and generally mixed quantum state for an angular momentum of arbitrary magnitude. The (2F + 1) x (2F + 1) density matrix describing the quantum state is hereby completely determined from a set of Stern-Gerlach measurements with (4F + 1) different orientations of the quantization axis. This protocol is implemented for laser cooled Cesium atoms in the 6S₁/₂(F = 4) hyperfine ground state manifold, and is applied to a number of test states prepared by optical pumping and Larmor precession. A comparison of the input and the measured states shows successful reconstructions with fidelities of about 0.95.
We demonstrate a general method to measure the quantum state of an angular momentum of arbitrary magnitude. The (2F+1)x(2F+1) density matrix is completely determined from a set of Stern-Gerlach measurements with (4F+1) different orientations of the quantization axis. We implement the protocol for laser cooled Cesium atoms in the 6S(1/2)(F = 4) hyperfine ground state and apply it to a variety of test states prepared by optical pumping and Larmor precession. A comparison of input and measured states shows typical reconstruction fidelities F >/~0.95.
A scheme for the preparation of Fock states and general superposition states of the electromagnetic field in a cavity is studied in detail. The scheme uses adiabatic passage in a strongly coupled atom-cavity system to ``map'' atomic ground-state Zeeman coherence onto the cavity-mode field. We model photon-counting and homodyne measurements of the field exiting the cavity and demonstrate the possibility of generating and detecting highly nonclassical states of the field parameter values close to currently realizable experimental values. The adiabatic passage process is also reversible, enabling cavity-mode fields to be mapped onto atomic ground-state Zeeman coherence. Application of this property to the measurement of cavity fields is discussed, with particular consideration given to a possible scheme for quantum measurements of the intracavity photon number.
Abstract In 1921, Otto Stern conceived the idea for an experiment that would decide between a classical and a quantum description of atomic behavior, as epitomized by the Bohr–Sommerfeld–Debye model of the atom. This model entailed not only the quantization of the magnitude of the orbital electronic angular momentum but also of the projection of the angular momentum on an external magnetic field – the so‐called space quantization. Stern recognized that space quantization would have observable consequences: namely, that the magnetic dipole moment due to the orbital angular momentum would be space quantized as well, taking two opposite values for atoms whose only unpaired electron has just one quantum of orbital angular momentum. When acted upon by a suitable inhomogeneous magnetic field, a beam of such atoms would be split into two beams consisting of deflected atoms with opposite projections of the orbital angular momentum on the magnetic field. In contradistinction, if atoms behaved classically, the atomic beam would only broaden along the field gradient and have maximum intensity at zero deflection, i. e., where there would be a minimum or no intensity for a beam split due to space quantization. Stern anticipated that, although simple in principle, the experiment would be difficult to carry out – and invited Walther Gerlach to team up with him. Gerlach's realism and experimental skills together with his sometimes stubborn determination to make things work proved invaluable for the success of the Stern–Gerlach experiment (SGE). After a long struggle, Gerlach finally saw, on 8 February 1922, the splitting of a beam of silver atoms in a magnetic field. The absence of the concept of electron spin confused and confounded the interpretation of the SGE, as the silver atoms were, in fact, in a 2S state, with zero orbital and spin angular momentum. However, a key quantum feature whose existence the SGE was designed to test – namely space quantization of electronic angular momentum – was robust enough to transpire independent of whether the electronic angular momentum was orbital or due to spin. The SGE entails other key aspects of quantum mechanics such as quantum measurement, state preparation, coherence, and entanglement. Confronted with the outcome of the SGE, Stern noted: “I still have objections to the idea of beauty of quantum mechanics. But she is correct.”
This chapter discusses the prime cause of the magnetic moment within an individual atom. In the classical model, the angular momentum of the electrons can be used to determine the magnetic moments of the electrons by invoking the concept of electrical charge in motion. There are two contributions to the electronic magnetic moment: an orbital magnetic moment due to orbital angular momentum and a spin magnetic moment due to electron spin. The component of orbital angular momentum along the axis of a magnetic field is restricted to discrete values by quantum mechanics. Electrons have spin angular momentum, which can be represented by the spin quantum number. As the magnetic field is increased and the field splitting becomes greater than the multiple splitting, the anomalous Zeeman effect changes over to a normal Zeeman effect. The chapter looks at experimental evidence confirming the quantization of magnetic moments, specifically the Zeeman effect and the Stern-Gerlach experiment.