We present a method for reconstructing the average evolution of the photon number distribution of a field decaying in a high- Q cavity. It applies an iterative maximum likelihood state reconstruction algorithm to the diagonal elements of the field density operator. It is based on quantum non-demolition measurements carried out with atoms crossing the cavity one by one. A small set of successively detected atoms defines a positive operator valued measure (POVM). The reconstruction is performed by applying this POVM to a large ensemble of field realizations. An optimal POVM based on the detection of a minimal number of atoms is shown to be sufficient to ensure an unambiguous convergence of the reconstruction. The cavity crossing time of this minimal number of atoms must be much shorter than the lifetime of the largest photon number present in the field. We apply the method to monitor the evolution of number states prepared by quantum feedback in a recent experiment. The method could also be useful in circuit QED experiments.
We present a method for reconstructing the average evolution of the photon number distribution of a field decaying in a high- Q cavity. It applies an iterative maximum likelihood state reconstruction algorithm to the diagonal elements of the field density operator. It is based on quantum non-demolition measurements carried out with atoms crossing the cavity one by one. A small set of successively detected atoms defines a positive operator valued measure (POVM). The reconstruction is performed by applying this POVM to a large ensemble of field realizations. An optimal POVM based on the detection of a minimal number of atoms is shown to be sufficient to ensure an unambiguous convergence of the reconstruction. The cavity crossing time of this minimal number of atoms must be much shorter than the lifetime of the largest photon number present in the field. We apply the method to monitor the evolution of number states prepared by quantum feedback in a recent experiment. The method could also be useful in circuit QED experiments.
We suggest an iterative, maximum-likelihood–based, method to reconstruct the photon number distribution of the steady-state cavity field of a micromaser starting from the statistics of the atoms leaving the cavity after the interaction. The scheme is based on measuring the atomic populations of probe atoms for different interaction times and works effectively using a small number of atoms and a limited sampling of the interaction times. The method has been tested by numerically simulated experiments showing that it may be reliably used in any micromaser regime, leading to high-fidelity reconstructions for single-peaked distributions as well as for double-peaked ones and for trapping states.
Abstract In this paper, we introduce an analytical framework for the reconstruction of quantum states. The reconstruction of an unknown quantum state requires the information of a complete set of observables, obtained through experimental measurements of Hermitian operators usually defined as positive-operator-valued measures (POVMs). The scheme involves a single-qubit unambiguous state discrimination POVM, which can be generalized to perform n-qubit measurements. We also use maximum likelihood estimation as a method in the reconstruction of the density matrix from experimental data and show that the expected value of the cleaner is independent of the parameter of the density operator.
The decay of the off-diagonal elements of the density matrix for the field that accompanies the quantum non-demolition measurement, recently proposed by Brune et al (Phys. Rev. Lett., vol.65, p.976, 1990) of the number of photons present in a microwave cavity, is studied. Averaging over the time of flight of the atoms, a simple estimate of the number of atoms needed for a precise measurement, is found. It agrees well with the result obtained by Brune et al by numerically simulating the measurement, and, moreover, allows a determination, in a systematic manner, of the optimum values for the relevant physical parameters.
The relaxation of a quantum field stored in a high-Q superconducting cavity is monitored by nonresonant Rydberg atoms. The field, subjected to repetitive quantum nondemolition photon counting, undergoes jumps between photon number states. We select ensembles of field realizations evolving from a given Fock state and reconstruct the subsequent evolution of their photon number distributions. We realize in this way a tomography of the photon number relaxation process yielding all the jump rates between Fock states. The damping rates of the n photon states (0 < or = n < or = 7) are found to increase linearly with n. The results are in excellent agreement with theory including a small thermal contribution.
We present a method for measuring quantum states encoded in the temporal modes of photons. The basis for the multilevel quantum states is defined by the use of modes propagating in a dispersive medium, which is a fiber in this case. The propagation and time-resolved single-photon detection allow us to define a positive-operator valued measure (POVM). The POVM depends on the amount of dispersion and the characteristics of a detector. This framework is numerically tested by performing quantum state tomography on a large number of states for a set of realistic experimental settings. Finally, the average fidelity between the expected and reconstructed states is computed for qubits, qutrits, and entangled qubits.
We present a numerical algorithm for the maximum-likelihood estimation of the density matrix, and apply it to the homodyne tomography of a single-mode radiation field. The algorithm is based on a specific form of the Gauss decomposition for positive definite Hermitian matrices. Results from Monte Carlo simulated experiments are presented. PACS: 42.50.Dv, 03.65.Bz Over past several years a great deal of interest has been paid to the reconstruction of the complete quantum state from experimental data [1]. Theoretical research in this field, apart from designing new measurement schemes, explores also the problem of extracting the optimal amount of information from data available in a realistic experimental setup. In this communication, we present a numerical algorithm for reconstructing the density matrix using the maximum-likelihood method [2, 3]. We discuss this algorithm in the context of quantum homodyne tomography [4, 5], which is actually the most successful method for quantum state reconstruction, and, in fact, the unique technique that has been so far experimentally implemented to measure nonclassical states of light [4, 6]. The realistic homodyne measurement of the quadrature is described by the positive operator-valued measure