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.
Summary form only given. Determining the unknown state of a quantum system from measurements of observables, a question first posed by Pauli, is a nontrivial task, which does not always permit a solution. The few examples where this inverse problem was experimentally solved include light fields, molecular vibrations, the motion of trapped ions and free atomic waves. We describe a method to reconstruct the complete density matrix of a generally mixed spin state of trapped neutral atoms. Compared to spin- 1/2 systems, the reconstruction of the spin density matrix for a large angular momentum is a much more challenging task. In our system with cesium atoms in the hyperfine ground state, a reconstruction of the 9/spl times/9 density matrix requires 80 real numbers to be experimentally determined. This can be achieved by measuring the 2F+1=9 populations along 4F+1=17 carefully chosen orientations of a Stern-Gerlach apparatus. A numerical simulation of the reconstruction shows this method to be robust against expected experimental uncertainties. We report on an experiment in progress to implement this measurement protocol in a standard laser cooling and trapping arrangement of neutral cesium atoms.
As in quantum mechanics the operators of different Cartesian components of the angular momentum do not commute these observables cannot simultaneously have sharp values. In this paper we present a simple method to calculate the probability distributions for the measurements of angular momentum components other than ĵz in eigenstates of ĵz with an arbitrary value of j, where ℏ2j(j+1) is the eigenvalue of ĵ2. Surprisingly this method is not found in textbooks, where usually the case j=12 is presented in discussions of Stern–Gerlach experiments. For larger values of j, the probability distributions have a comb-like structure. A detailed discussion of the quasiclassical limit j≫1 is presented for the measurement of ĵx. It yields rather surprising results.
Quantum states of a spin
21 (a qubit) are parametrized by the space
CP1∼S2, the Bloch sphere. A spin
j for a generic
j (a
2j+1-state system) is represented instead by a point of a larger space,
CP2j. Here we study the state of a single angular momentum/spin in the limit,
j→∞. The special class of states $ | j, {\mathbf n}\rangle \in {\mathbf {CP}}^{2j} $, with spin oriented towards definite spatial directions
n∈S2, i.e., $({\mathbf J}\cdot {\mathbf n} ) \, | j, {\mathbf n}\rangle = j\, |j, {\mathbf n}\rangle $, are found to behave as classical angular momenta,
jn, in this limit. Vice versa, general spin states in
CP2j do not become classical, even at large
j. We discuss these questions, by analysing the Stern-Gerlach processes, the angular-momentum composition rule, and the rotation matrix. Our observations help to clarify better how classical mechanics emerges from quantum mechanics in this context (e.g., with unique trajectories for a particle carrying a large spin), and to make the widespread idea that large spins somehow become classical, more precise.
The quantum states of a spin 12 (a qubit) are parametrized by the space CP1∼S2, the Bloch sphere. A spin j for a generic j (a 2j+1-state system) is represented instead by a point in a larger space, CP2j. Here we study the state of a single angular momentum/spin in the limit j→∞. A special class of states, |j,n⟩∈CP2j, with spin oriented towards definite spatial directions, n∈S2, i.e., (J^·n)|j,n⟩=j|j,n⟩, are found to behave as classical angular momenta, jn, in this limit. Vice versa, general spin states in CP2j do not become classical, even at a large j. We study these questions by analyzing the Stern–Gerlach processes, the angular momentum composition rule, and the rotation matrix. Our observations help to better clarify how classical mechanics emerges from quantum mechanics in this context (e.g., with the unique trajectories of a particle carrying a large spin in an inhomogeneous magnetic field) and to make the widespread idea that large spins somehow become classical more precise.