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arXiv:1004.4739 · Physical Review A

Tomography, control, and characterization of entanglement in a three-level atomic system

S. N. Sandhya, V. Ravishankar

Published Dec 2, 2010• 8 citations
arXivHTMLPDF

Abstract

We study the quantum correlations of the radiation emitted by three-level atoms (cascade type) interacting with two driving fields. In the linear regime, and in the Weisskopf-Wigner approximation, we show that the atomic and the two-photon density matrix are equivalent to each other. This facilitates the tomography of the two-mode state to be realized by measurements on either the atomic system or the emitted fields. While, in general, one needs ${4}^{N}$ measurements for the tomography of an $N$-photon state, we show that one needs $(N+1){}^{2}\ensuremath{-}1$ observables for the tomography of photons emitted by an atomic system. Thus there is an exponential reduction in the number of observables for the reconstruction of the class of $N$-photon states emitted by atoms. We show that the driving field strengths and detunings provide the control parameters for the preparation of a specific target state. Finally, we study the characterization of entanglement of the two-photon state. We observe that a characterization of entanglement in terms of a single parameter is not possible when the system is in a mixed state; therefore, we provide a description in terms of the newly introduced probability distribution for entanglement, in various regimes of interest.

DOI
10.1103/physreva.82.062301

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We study the quantum correlations of the radiation emitted by three-level atoms (cascade type) interacting with two driving fields. In the linear regime, and in the Weisskopf-Wigner approximation, we show that the atomic and the two-photon density matrix are equivalent to each other. This facilitates the tomography of the two-mode state to be realized by measurements on either the atomic system or the emitted fields. While, in general, one needs 4N{4}^{N}4N measurements for the tomography of an NNN-photon state, we show that one needs (N+1)2\ensuremath−1(N+1){}^{2}\ensuremath{-}1(N+1)2\ensuremath−1 observables for the tomography of photons emitted by an atomic system. Thus there is an exponential reduction in the number of observables for the reconstruction of the class of NNN-photon states emitted by atoms. We show that the driving field strengths and detunings provide the control parameters for the preparation of a specific target state. Finally, we study the characterization of entanglement of the two-photon state. We observe that a characterization of entanglement in terms of a single parameter is not possible when the system is in a mixed state; therefore, we provide a description in terms of the newly introduced probability distribution for entanglement, in various regimes of interest.
Entanglement distribution in VVV and ΛΛΛ systems
We determine the two mode fock state of the radiation emitted by VVV type three level atomic systems using the method of tomography described in our previous work. Utilizing the knowledge of the atomic/two photon state we provide a complete description of the entanglement distribution in the full parameter space in the case of the VVV system. The VVV system shows weak entanglement and is predominantly in a mixed state in two dimensional subspace. Unlike the case of VVV and the ladder systems, atomic state tomography of the ΛΛΛ system requires the measurement of anti-normal ordered operators and more importantly, there is no equivalence between the atomic and the two photon density matrix. Tomography of this atomic state is not possible with existing types of measurements. However the two photon state may be determined when it is not in a dark state.
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We suggest a scheme to reconstruct a two-mode entangled state in cavity QED by using the interaction of a V-configuration three-level atom and by driving the cavity field. After the atomic interaction with the cavity fields, the probability of the atom being in its initial ground state is found to be directly related to the two-mode Wigner characteristic function. The Wigner function and the four-dimensional density-matrix elements can be obtained for the two-mode entangled field by simple transformations. We consider both the cases where two entangled modes are prepared in one cavity or in two spatially separated cavities.
Given the state of a quantum system, one can calculate the expectation value of any observable of the system. However, the inverse problem of determining the state by performing different measurements is not a trivial task. In various experimental setups it is reasonably straightforward to reconstruct the state of a quantum system employing linear tomographic technique. In this way the elements of the density matrix can be linearly related to a set of measured quantities. But since different observables of a quantum system are not commuting with each other, one often has to perform series of successive measurements of observables which cannot be done simultaneously. Simultaneous measurement of observables cost less time and energy and is more beneficial. In this paper we review the strategy of quantum state tomography with simultaneous measurement of commuting observables. This can be done by introducing an assistant system of which the state is known. We show that the interaction between the assistant and the system of interest within different frame works allows the reconstruction of the state of the system. Specifically, we consider a two-level system and reconstruct its initial state by introducing an assistant which can be either another two-level system or a single cavity mode of the electromagnetic field.
Abstract Cooperative atomic emission spectrum as a key probe of entanglement dynamics of an independent and a classically-pumped qubits pair is showed and investigated. The qubits are initially maximally-entangled while they penetrate a detuned, single-photon and coherent cavity embedded in nonlinear media. The system Hamiltonian; that is a nonlinear and a time-dependent system, is diagonalized and the time evolution unitary operator in an exact form is derived. Various characteristics are monitored while observations are theoretically documented—via tracing the field state tomography—and experimentally calibrated. Cooperation and decoherence as well as the statistics of the emitted radiation are discussed and analyzed in terms of the other phenomena. We show that, a measurement of the spectrum outputs represent a highly effective key calibration for entanglement attributes, where complete time dynamics may be predicted with a high considerable accuracy. The most striking feature realized is that, on high detuned interaction, a full matching of these phenomena outputs can be achieved regardless of the absence or presence or pumping. Once the pumping starts, a partial dephasing is encouraged, thus, the spectrum is inversely but slightly affected, nevertheless, matching remains highly closed. Within various surroundings of Kerr materials of different Kerr constants, that is an additional source of an effective dephasing, spectrum rapidly oscillates and so also entanglement that shows rapid Rabi oscillations during its fast degradation depending on the detuning width. However, on measuring the intracavity field state tomography, measurements confirm such a matching, that is, calibration is well documented.
In the first part of this thesis,we examine the preparation of a single- mode radiation field in arbitrary pure quantum states via resonant inter- action with a sequence of two-level atoms.The preparation is achieved by choosing an appropriate (in general entangled)initial state of the atomic sequence,and does neither require a final state measurement of the atoms,nor a control of the atom-field interaction.Furthermore,the method is applicable also when starting from mixed initial field states. We show how to determine the optimal initial atomic state which pre- pares the desired field state with the maximum fidelity,and prove the feasibility of our state preparation method by numerical calculations. In the second part,we demonstrate the noise-induced control of quantum jumps in a fundamental open quantum system.Here,in addition to the subsequent interaction with a flux of two-level atoms,the quantized field is also coupled to a thermal environment.Under certain experimental conditions,the photon field exhibits a bistable behavior,with quantum jumps between two metastable states.In the presence of a small peri- odic signal (i.e.,a modulation of the initial state of the two-level atoms crossing the single-mode resonator),the best synchronization of these quantum jumps with the signal is achieved at an optimal,nonvanishing temperature of the environment.This stochastic resonance e ffect can be observed in di fferent components of the atomic Bloch vector on exit from the cavity. The third part treats a speci fic problem concerning the characterization of entanglement between two quantum mechanical two-level systems. We consider the optimal decomposition of a two-qubit state into an en- tangled and a separable part,with maximal weight of the latter,and derive necessary and su fficient conditions for the optimality of the de- composition.