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arXiv:0902.2249 · New Journal of Physics

Monitoring the wave function by time continuous position measurement

Thomas Konrad, Andreas Rothe, Francesco Petruccione, Lajos Diósi

Published Apr 21, 2010• 17 citations
arXivHTMLPDF

Abstract

Motivated by the technical requirements of quantum information processing and nanotechnology, the control of individual quantum systems such as single atoms, ions or even photons has become a highly desirable aim.The monitoring of quantum systems-a direct test and in many cases a prerequisite of their control-has been investigated so far only for certain observables such as the position or momentum of quantum particles.Here, we describe a method to monitor in real time the complete state of a quantum particle with unknown initial state moving in a known potential.The method is based on successively updating an estimate by the results of a continuous position measurement.We demonstrate by numerical simulations that even in a chaotic potential tracking the wave function of a particle is possible, and we show with an example that the monitoring scheme appears to be robust against sudden random perturbations.

DOI
10.1088/1367-2630/12/4/043038

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Motivated by the technical requirements of quantum information processing and nanotechnology, the control of individual quantum systems such as single atoms, ions or even photons has become a highly desirable aim.The monitoring of quantum systems-a direct test and in many cases a prerequisite of their control-has been investigated so far only for certain observables such as the position or momentum of quantum particles.Here, we describe a method to monitor in real time the complete state of a quantum particle with unknown initial state moving in a known potential.The method is based on successively updating an estimate by the results of a continuous position measurement.We demonstrate by numerical simulations that even in a chaotic potential tracking the wave function of a particle is possible, and we show with an example that the monitoring scheme appears to be robust against sudden random perturbations.
We formulate a model of a quantum particle continuously monitored by detectors measuring simultaneously its position and momentum. We implement the postulate of wave-function collapse by assuming that upon detection the particle is found in one of the meters' states chosen as a discrete subset of coherent states. The dynamics, as observed by the meters, is thus a random sequence of jumps between coherent states. We generate such trajectories using the quantum Monte Carlo wave-function method. For sparsely distributed detectors, we use methods from renewal theory of stochastic processes to obtain some semianalytic results. In particular, the different regimes of dynamics of the free particle are identified and quantitatively discussed: from stroboscopic motion in the case of low interrogation frequency, to delayed dynamics reminiscent of the Zeno effect if monitoring is frequent. For a semicontinuous spatial distribution of meters the emergence of classical trajectories is shown. Their statistical properties are discussed and compared to other detection schemes in which the operation on the system due to measurement corresponds to ``spatial filtering'' of the wave function.
One of the most fundamental principles of quantum mechanics is the collapse of the wave function upon measurement. Here, we present a novel approach called Noise vs Noise that enables continuous tracking of a quantum particle (specifically, an electron in a double-slit experiment) without causing wave function collapse. Instead of directly interacting with the particle, we introduce a well-defined reference noise field into the medium. The electron perturbs this noise through its electric field, and by analyzing the noise distortion, we extract information about the electron’s position and velocity. Numerical simulations demonstrate that the wave function remains intact (correlation = 1.0000) while achieving positional accuracy on the order of 10−8 m and velocity estimates within the correct order of magnitude (106 m/s). The uncertainty product ∆x · ∆p remains above the Heisenberg limit (ħ/2), confirming that the method re- spects quantum mechanical principles while offering a gentle measurement paradigm. This work opens new possibilities for quantum sensing, continuous monitoring, and fundamental tests of quantum mechanics.
For reliable and consistent quantum information processing carried out on a quantum network, the network structure must be fully known and a desired initial state has to be prepared on it. In this paper, we consider a spin network with only a single node accessible and provide continuous-measurement-based methods to fulfill the above two requirements. More specifically, the measurement results continuously obtained are used to update the estimate of the network structure, which converges to the true structure with high probability; for the latter problem, we further employ an adaptive measurement scheme to deterministically drive the state to a desired target state for network initialization.
Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. We use two different estimation methods: maximum likelihood and compressed sensing. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, 133{}^{133}133Cs. We apply this protocol to the case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3⊕F=4F=3 \oplus F=4F=3⊕F=4), controlled by microwaves and radio-frequency magnetic fields. We present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. We show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities >95% for low complexity quantum states, and >92% for arbitrary random states, which is a formidable accomplishment for a space of this size.
Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. We use two different estimation methods: maximum likelihood and compressed sensing. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, 133{}^{133}133Cs. We apply this protocol to the case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3⊕F=4F=3 \oplus F=4F=3⊕F=4), controlled by microwaves and radio-frequency magnetic fields. We present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. We show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities >95% for low complexity quantum states, and >92% for arbitrary random states, which is a formidable accomplishment for a space of this size.
We investigate the determination of a Hamiltonian parameter in a quantum system undergoing continuous measurement. We demonstrate a computationally rapid method to estimate an unknown and possibly time-dependent parameter, where we maximize the likelihood of the observed stochastic readout. By dealing directly with the raw measurement record rather than the quantum-state trajectories, the estimation can be performed while the data are being acquired, permitting continuous tracking of the parameter during slow drifts in real time. Furthermore, we incorporate realistic nonidealities, such as decoherence processes and measurement inefficiency. As an example, we focus on estimating the value of the Rabi frequency of a continuously measured qubit and compare maximum likelihood estimation to a simpler fast Fourier transform. Using this example, we discuss how the quality of the estimation depends on both the strength and the duration of the measurement; we also discuss the trade-off between the accuracy of the estimate and the sensitivity to drift as the estimation duration is varied.