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Homodyne Detection and Quantum State Reconstruction Homodyne Detection and Quantum State Reconstruction† Dirk–Gunnar Welsch Friedrich-Schiller-Universit¨at Jena, Theoretisch-Physikalisches Institut Max-Wien Platz 1, D-07743 Jena, Germany Werner Vogel Universit¨at Rostock, Fachbereich Physik Universit¨atsplatz 3, D-18051 Rostock, Germany Tom´aˇsOpatrn´y Palack´yUniversity, Faculty of Natural Sciences Svobody 26, 77146 Olomouc, Czech Republic arXiv:0907.1353v1 [quant-ph] 8 Jul 2009 †In Progress in Optics, Vol. XXXIX, ed. E. Wolf (Elsevier, Amsterdam, 1999), p. 63–211. 2 CONTENTS Contents 1. Introduction 4 2. Phase-sensitive measurements of light 7 2.1. Opticalhomodyning ........................... 7 2.1.1. Basicscheme ........................... 7 2.1.2. Quadrature-componentstatistics . 9 2.1.3. Multimodedetection . 13 2.1.4. Q function............................. 16 2.1.5. Probabilityoperatormeasures . 20 2.1.6. Positive P function........................ 22 2.1.7. Displaced-photon-number statistics . ... 23 2.1.8. Homodyne correlation measurements . 25 2.2. Heterodynedetection . .. .. .. 26 2.3. Parametricamplification . 26 2.4. Measurement of cavity fields . 27 3. Quantum-state reconstruction 31 3.1. Opticalhomodynetomography . 31 3.2. Density matrix in quadrature-component bases . ...... 35 3.3. DensitymatrixintheFockbasis. 37 3.3.1. Sampling of quadrature-components . 37 3.3.2. Sampling of the displaced Fock-states on a circle . ..... 43 3.3.3. Reconstructionfrompropensities . 45 3.4. Multimode density matrices . 46 3.5. Local reconstruction of P (α; s)...................... 48 3.6. Reconstruction from test atoms in cavity QED . .... 49 3.6.1. Quantum state endoscopy and related methods . 49 3.6.2. Atomicbeamdeflection . 52 3.7. Alternativeproposals . 54 3.8. Reconstruction of specific quantities . ..... 56 3.8.1. Normally ordered photonic moments . 57 3.8.2. Quantities admitting normal-order expansion . ..... 59 3.8.3. Canonicalphasestatistics . 60 3.8.4. HamiltonianandLiouvillian . 63 3.9. Processing of smeared and incomplete data . .... 64 3.9.1. Experimental inaccuracies . 64 3.9.2. Least-squaresmethod. 69 3.9.3. Maximum-entropy principle . 70 3.9.4. Bayesianinference . 72 4. Quantum states of matter systems 74 4.1. Molecularvibrations ........................... 74 4.1.1. Harmonicregime ......................... 75 4.1.2. Anharmonicvibrations . 76 4.2. Trapped-atommotion .. .. .. 78 4.2.1. Quadraturemeasurement . 78 CONTENTS 3 4.2.2. Measurement of the Jaynes–Cummings dynamics . 81 4.2.3. Entangledvibronicstates . 84 4.3. Bose–Einstein condensates . 85 4.4. Atomicmatterwaves ........................... 87 4.4.1. Transversemotion . 87 4.4.2. Longitudinalmotion . 90 4.5. Electronmotion.............................. 91 4.5.1. Electronic Rydberg wave packets . 91 4.5.2. Cyclotronstateofatrappedelectron . 92 4.5.3. Electronbeam........................... 93 4.6. Spin and angular momentum systems . 93 4.7. Crystallattices .............................. 95 A. Radiation field quantization 97 B. Quantum-state representations 98 B.1.Fockstates ................................ 98 B.2. Quadrature-componentstates . 99 B.3.Coherentstates .............................. 99 B.4. s-Parametrized phase-space functions . 101 B.5. Quantum state and quadrature components . 102 C. Photodetection 103 D. Elements of least-squares inversion 104 References 107 4 1. INTRODUCTION 1. Introduction Since the experimental demonstration of tomographic quantum-state measurement on optical fields by Smithey, Beck, Raymer and Faridani [1993] based on the theoretical work of Vogel, K., and Risken [1989] a boom in studying quantum-state reconstruction problems has been observed. Numerous workers have considered various schemes and methods for extracting all knowable information on the state of a quantum system from measurable data, and early work has been recovered. In the history of quantum theory the concept of quantum state including quantum-state measurement has been a matter of intense discussion. The state of a quantum object is commonly described by a normalized Hilbert- space vector Ψ or, more generally, by a density operator̺ ˆ= p Ψ Ψ which is a | i i i| iih i| Hermitian and non-negative valued Hilbert-space operator ofP unit-trace. The Hilbert space of the object is usually spanned up by an orthonormalized set of basic vectors A representing the eigenvectors (eigenstates) of Hermitian operators Aˆ associated with| i a complete set of simultaneously measurable observables (physical quantities) of the object.1 The eigenvalues A of these operators are the values of the observables that can be registered in a measurement. Here it must be distinguished between an individual (single) and an ensemble measurement (i.e., in principle, an infinitely large number of repeated measurements on identically prepared objects). Performing a single measurement on the object, a totally unpredictable value A is observed in general, and the state of the object has collapsed to the state A according to the von Neumann’s projection definition of a measurement (von Ne| umanni [1932]). If the same measurement is repeated immediately after the first measurement (on the same object), the result is now well predictable – the same value A as in the first measurement is observed. Obviously, owing to the first measurement the object has been prepared in the state A . Repeating the measurement many times on identically prepared object, the relative| i rate at which the result A is observed approaches the diagonal density-matrix element A ̺ˆ A as the number of measurements tend to h | | i infinity. Measuring A ̺ˆ A for all values of A the statistics of Aˆ (and any function of Aˆ) are known. Toh completely| | i describe the quantum state, i.e., to determine all the quantum-statistical properties of the object, knowledge of all density-matrix elements A ̺ˆ A′ is needed. In particular, the off-diagonal elements essentially determine the statisticsh | | i of such sets of observables Bˆ that are not compatible with Aˆ ([A,ˆ Bˆ] = 6 0) and cannot be measured simultaneously with Aˆ. Obviously, the statistics of Bˆ can also be obtained directly – similar to the statistics of Aˆ – from an ensemble measurement yielding the diagonal density-matrix elements B ̺ˆ B in the basis of h | | i the eigenvectors B of Bˆ. Now one can proceed to consider other sets of observables that are not compatible| i with Aˆ and Bˆ, and the rather old question arises of which and how many incompatible observables must be measured in order to obtain all the information contained in the density matrix of an arbitrary quantum state. Already Pauli [1933] addressed the question whether or not the wave function of a particle can be reconstructed from the position and momentum probability distributions, i.e., the diagonal density-matrix elements in the position and momentum basis. Roughly speaking, there have been two routes to collect measurable data for 1For notational convenience we write Aˆ, without further specifying the quantities belonging to the set. 5 reconstructing the quantum state of an object under consideration. In the first, which closely follows the line given above, a succession of (ensemble) measurements is made such that a set of noncommutative object observables is measured which carries the complete information about the quantum state. A typical example is the tomographic quantum-state reconstruction mentioned at the beginning. Here, the probability distributions p(x, ϕ)= x, ϕ ̺ˆ x, ϕ of the quadrature componentsx ˆ(ϕ) h | | i of a radiation-field mode for all phases ϕ within a π interval are measured, i.e., the expectation values of the quadrature-component projectors x, ϕ x, ϕ . In the | ih | second, the object is coupled to a reference system (whose quantum state is well known) such that measurement of observables of the composite system corresponds to “simultaneous” measurement of noncommutative observables of the object. In this case the number of observables (of the composite system) that must be measured in a succession of (ensemble) measurements can be reduced drastically, but at the expense of the image sharpness of the object. As a result of the additional noise introduced by the reference system only fuzzy measurements on the object can be performed, which just makes a “simultaneous” measurement of incompatible object observables feasible. A typical example is the Q function of a radiation-field mode, which is given by the diagonal density-matrix elements in the coherent-state basis, Q(α)= π−1 α ̺ˆ α . It can already be obtained from one ensemble measurement of the complexh amplitude| | i α, which corresponds to a fuzzy measurement of the ”joint” probability distribution of two canonically conjugated observablesx ˆ(ϕ) andx ˆ(ϕ + π/2) of the object. Since the sets of quantities measured via the one or the other route (or an appro- priate combination of them) carry the complete information on the quantum state of the object, they can be regarded, in a sense, as representations of the quantum state, which can be more or less close to (or far from) familiar quantum-state repre- sentations, such as the density matrix in the Fock basis or the Wigner function. In any case, the question arises of how to reconstruct from the measured data specific quantum-state representations (or specific quantum-statistical properties of the ob- ject for which a direct measurement scheme is not available). Again, there have been two typical concepts for solving the problem. In the first, equations that relate the measured quantities to the desired quantities are derived and
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