Direct measurement of the spatial Wigner function with area-integrated detection

Mukamel, Eran A.; Banaszek, Konrad; Walmsley, Ian A.; Dorrer, C.

doi:10.1364/ol.28.001317

Cited by 57 publications

(48 citation statements)

References 16 publications

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“…The Wigner function can be directly measured by displacing the system in phase space and then measuring the parity operator [8]. Equivalently, the integral of the interference between a pair of rotated and displaced replicas of the system will give the Wigner function [9]. The Husimi Q-function can be directly measured by an eight-port homodyne apparatus or by projection on the harmonic oscillator ground state [10].…”

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confidence: 99%

Procedure for Direct Measurement of General Quantum States Using Weak Measurement

2012

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Recent work [J.S. Lundeen et al. Nature, 474, 188 (2011)] directly measured the wavefunction by weakly measuring a variable followed by a normal (i.e. 'strong') measurement of the complementary variable. We generalize this method to mixed states by considering the weak measurement of various products of these observables, thereby providing the density matrix an operational definition in terms of a procedure for its direct measurement. The method only requires measurements in two bases and can be performed 'in situ', determining the quantum state without destroying it.PACS numbers: 03.65. Ta, 03.65.Wj, 42.50.Dv, The wavefunction Ψ is at the heart of quantum mechanics, yet its nature has been debated since its inception. It is typically relegated to being a calculational device for predicting measurement outcomes. Recently, Lundeen et al. proposed a simple and general operational definition of the wavefunction based on a method for its direct measurement: "it is the average result of a weak measurement of a variable followed by a strong measurement of the complementary variable [1,2]." By 'direct' it is meant that a value proportional to the wavefunction appears straight on the measurement apparatus itself without further complicated calculations or fitting. The 'wavefunction' referred to here is a special case of a general quantum state, known as a 'pure state.' The general case is represented by the density operator ρ, which can describe both pure and 'mixed' states. The latter incorporates both the effects of classical randomness (e.g., noise) and entanglement with other systems (e.g., decoherence). The density operator plays an important role in quantum statistics, quantum information, and the study of decoherence. Because of its generality and because it follows naturally from classical concepts of probability and measures, some consider ρ, rather than Ψ, to be the fundamental quantum state description. In this letter, we propose two methods to directly measure general quantum states, one of which directly gives the matrix elements of ρ.The standard method for experimentally determining the density operator is Quantum State Tomography [3]. In it, one makes a diverse set of measurements on an ensemble of identical systems and then determines the quantum state that is most compatible with the measurement results. An alternative is our direct measurement method, which may have advantages over tomography, such as simplicity, versatility, and directness. A quantitative comparison of measures such as the signal to noise ratio, resolution, and fidelity, has not been undertaken but some limitations of the direct method have been identified in [4]. As compared to tomography, which works with mixed states, the most significant limitation of the direct measurement of the wavefunction is that it has only been shown to work with pure states.Previous works have developed direct methods to measure quasi-probability distributions, such as the Wigner function [5], Husimi Q-function [6], and the GlauberSudarshan P-function [7]...

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Procedure for Direct Measurement of General Quantum States Using Weak Measurement

2012

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“…The commutator and the anti-commutator of the position and the momentum operators are constructed with single-photon quantum interference. For an initial Gaussian wave function ψ(x), we find that applying the commutator leaves the state unchanged whereas applying the anti-commutator results a Wigner function with negativity, starkly different from the Wigner function of the initial wave function [19]. Finally, we discuss how the proposed scheme can be applied to matter-wave interferometry to directly observe the non-commutativity of the position and the momentum operators for a particle with mass or a macroscopic quantum state of matter.…”

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confidence: 82%

“…The spatial Wigner function W (x, p) can be reconstructed by employing, e.g., an areaintegrated detection scheme [19]. In Fig.…”

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Scheme for directly observing the noncommutativity of the position and the momentum operators with interference

Lee

Kim

et al. 2012

Phys. Rev. A

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Although non-commutativity of a certain set of quantum operators (e.g., creation/annihilation operators and Pauli spin operators) has been shown experimentally in recent years, the commutation relation for the position and the momentum operators has not been directly demonstrated to date. In this paper, we propose and analyze an experimental scheme for directly observing the non-commutativity of the position and the momentum operators using single-photon quantum interference. While the scheme is studied for the single-photon state as the input quantum state, the analysis applies equally to matter-wave interference, allowing a direct test of the position-momentum commutation relation with a massive particle.

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“…We completed a detailed study of the threshold gate and memory failure rates necessary for fault-tolerant quantum computation [4]. More efficient and noise-tolerant syndrome extraction raised the threshold for fault-tolerance to the 10 -3 level for certain computer architectures.…”

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confidence: 99%

Research in Atomic, Ionic and Photonic Systems for Scalable Deterministic Quantum Logic

Banaszek¹,

Foot²,

Lucas³

et al. 2005

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The "pushing gate" that we intend to use to entangle ions was thoroughly studied theoretically (milestone 1 complete). We proposed a solution to a hidden weakness in the original proposal which led to extreme sensitivity to, for example, laser intensity noise. We demonstrate that fast gates of good fidelity are possible without ground-state cooling [1].We implemented and quantitatively studied photo-ionization ion loading and optimized the loading process so that we could trap and laser-cool all the naturally-occurring isotopes of calcium (including the 0.004%-abundant 46 Ca). We can load single ions or small crystals "on demand", from a background vapour pressure 4-5 orders of magnitude lower than with electron bombardment, and with negligible drift in static electric fields between loads; this gives increased environmental stability for the ion-qubits and should reduce deposition on electrodes to a negligible level. Most importantly for the implementation of qubits, we can reliably load small, pure, crystals of the 0.14%-abundant 43 Ca+ ion (milestone 2 complete). The ability to do this without an enriched isotopic source was beyond our expectations and represents a significant experimental simplification. Isotope-selection of specific even isotopes (e.g. We began the design process for our next-generation multiple trap (start of milestone 3), with studies of possible electrode structures. We designed and had built a high-power solid-state violet laser system in order to test its suitability for implementing the "pushing gate" (start of milestone 4).

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Direct measurement of the spatial Wigner function with area-integrated detection

Cited by 57 publications

References 16 publications

Procedure for Direct Measurement of General Quantum States Using Weak Measurement

Procedure for Direct Measurement of General Quantum States Using Weak Measurement

Scheme for directly observing the noncommutativity of the position and the momentum operators with interference

Research in Atomic, Ionic and Photonic Systems for Scalable Deterministic Quantum Logic

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