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Physical Review A

Quantum jumps in a two-level atom: Simple theories versus quantum trajectories

Howard M. Wiseman, Gilman E. S. Toombes

Published Sep 1, 1999• 31 citations
Publisher

Abstract

A strongly driven $(\ensuremath{\Omega}\ensuremath{\gg}\ensuremath{\gamma})$ two-level atom relaxes towards an equilibrium state $\ensuremath{\rho}$ which is almost completely mixed. One interpretation of this state is that it represents an ensemble average, and that an individual atom is at any time in one of the eigenstates of $\ensuremath{\rho}$. The theory of Teich and Mahler [Phys. Rev. A 45, 3300 (1992)] makes this interpretation concrete, with an individual atom jumping stochastically between the two eigenstates when a photon is emitted. The dressed-atom theory is also supposed to describe the quantum jumps of an individual atom due to photoemissions. But the two pictures are contradictory because the dressed states of the atom are almost orthogonal to the eigenstates of $\ensuremath{\rho}$. In this paper we investigate three ways of measuring the field radiated by the atom, which attempt to reproduce the simple quantum jump dynamics of the dressed state or Teich and Mahler models. These are spectral detection (using optical filters), two-state jumps (using adaptive homodyne detection), and orthogonal jumps (another adaptive homodyne scheme). We find that the three schemes closely mimic the jumps of the dressed-state model, with errors of order $\frac{3}{4}(\ensuremath{\gamma}/\ensuremath{\Omega}{)}^{2/3},$ $\frac{1}{4}(\ensuremath{\gamma}/\ensuremath{\Omega}{)}^{2}$, and $\frac{3}{4}(\ensuremath{\gamma}/\ensuremath{\Omega}{)}^{2}$, respectively. The significance of this result to the program of environmentally induced superselection is discussed.

DOI
10.1103/physreva.60.2474

Related papers

Quantum jumps in a two-level atom

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A strongly-driven ($Ω>> γ$) two level atom relaxes towards an equilibrium state rho which is almost completely mixed. One interpretation of this state is that it represents an ensemble average, and that an individual atom is at any time in one of the eigenstates of $ρ$. The theory of Teich and Mahler [Phys. Rev. A 45, 3300 (1992)] makes this interpretation concrete, with an individual atom jumping stochastically between the two eigenstates when a photon is emitted. The dressed atom theory is also supposed to describe the quantum jumps of an individual atom due to photo-emissions. But the two pictures are contradictory because the dressed states of the atom are almost orthogonal to the eigenstates of $ρ$. In this paper we investigate three ways of measuring the field radiated by the atom, which attempt to reproduce the simple quantum jump dynamics of the dressed state or Teich and Mahler models. These are: spectral detection (using optical filters), two-state jumps (using adaptive homodyne detection) and orthogonal jumps (another adaptive homodyne scheme). We find that the three schemes closely mimic the jumps of the dressed state model, with errors of order $(3/4) (γ/Ω)^{2/3}$, $(1/4) (γ/Ω)^{2}$, and $(3/4) (γ/Ω)^{2}$ respectively. The significance of this result to the program of environmentally-induced superselection is discussed.

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Quantum jumps between dressed states: A proposed cavity-QED test using feedback

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A strongly driven cavity containing a single resonant strongly coupled atom exhibits a phase bistability. The phase of the field is strongly correlated with the phase of the atomic dipole. It has been shown previously that phase-sensitive monitoring of the field emitted by the cavity would induce conditional quantum jumps between orthogonal atomic dipole states (``dressed'' states). Here we show that such monitoring can be used to fix the atom into a single dressed state. As soon as a state-changing quantum jump is inferred from the measurement of the field, the atomic state is flipped using a $\ensuremath{\pi}$ pulse. We study this feedback scheme analytically and numerically. We show that the occupation probability of the desired fixed state can be as high as $1\ensuremath{-}1/8\ensuremath{\eta}{C}_{1},$ where ${C}_{1}\ensuremath{\gg}1$ is the single-atom cooperativity and $\ensuremath{\eta}$ the detection efficiency (which does not have to be close to unity). The control of the atomic dynamics is manifest in the fluorescence spectrum. The widths of all three peaks are modified from the usual Mollow spectrum, and almost all of the area under one of the sidebands is transferred to the other sideband. This is as expected, as one of the dressed states is essentially unoccupied, and transitions out of it do not occur. In addition, the width of the central peak goes to zero. This indicates coherent scattering due to the nonzero mean atomic dipole created by the feedback.

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Recent experiments with single atomic ions isolated in electromagnetic traps have demonstrated abrupt changes in observed fluorescence intensities, which have been interpreted as evidence for quantum jumps as the atom changes energy levels. This presents the seeming paradox of discontinuously changing observables from a system whose dynamical evolution is governed by the continuous optical Bloch equations. In this paper I present an analysis of the dynamics of a model three-level atom interacting with two laser fields in terms of the dressed states of the atom-laser system. These dressed states are eigenstates of a Hamiltonian which includes laser-atom interactions, and thus they simplify the discussion of the time evolution of the combined laser-atom system. This dressed-state description is particularly useful in the regime in which both laser intensities are strong enough to saturate their respective transitions. I calculate the radiative lifetimes of the different species of dressed states and the branching ratios for decay to other dressed states, and demonstrate the consequences of these quantities for the observed fluorescence pattern. The calculated duration of bright and dark periods shows a pronounced dependence on the detunings of the lasers.

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A number of experiments on trapped single ions or atoms have been performed in recent years [1,2,3,4]. Monitoring the intensity of scattered laser light off of such systems has shown abrupt changes that have been cited as evidence of between states of the scattered ion or atom . The existence of such jumps was required by Bohr in his theory of the atom. Bohr’s quantum jumps between atomic states [5] were the first form of quantum dynamics to be postulated. He assumed that an atom remained in an atomic eigenstate until it made an instantaneous jump to another state with the emission or absorption of a photon. Since these jumps do not appear to occur in solutions of the Schrodinger equation, something similar to Bohr's idea has been added as an extra postulate in modern quantum mechanics. Stochastic quantum jump equations [6], [7],[8]were introduced as a tool for simulating the dynamics of a dissipative system with a large Hilbert space and their links with quantum measurement the or were also noted [9],[10],[11],[12],[13].This measurement interpretation is generally known as quantum trajectory theory [14].By adding filter cavities as part of the apparatus, even the quantum jumps in the dressed state model can be interpreted as approximations to measurement-induced jumps [15]. The question arises whether an explanation of these jumps can be found to result from an Colombeau solution [16]-[18](Ψe(x, t;ħ))e of the Schrodinger equation alone without additional postulates. We found exact quasi-classical asymptotic of the quantum averages

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The quantum jump method: photon statistics and macroscopic quantum jumps of two interacting atoms

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We first use the quantum method to replicate the well-known results of a single atom relaxing, whilst demonstrating the intuitive picture it provides for dissipative dynamics. By use of individual "quantum trajectories", the method allows for simulation of systems inaccessible to ensemble treatments. This is shown by replicating resonance fluorescence, allowing us to concurrently demonstrate the method's facilitation of calculating photon statistics by the creation of discrete photon streams. To analyse these, we solidify the theoretical basis for, and implement, a computational method of calculating second-order coherence functions. A process by which to model interacting two-atom systems to allow for computation with the quantum jump method is then developed. Using this, we demonstrate cooperative effects leading to greatly modified emission spectra, before investigating the decoupling of states from dissipative and coherent interactions. Here, we find the novel insight provided by the quantum jump method both births and provides the tools with which to begin an investigation into the occurrence of macroscopic jumps and the formation of macroscopic dark periods in a system of two two-level dipole-dipole coupled atoms.

arXiv:2201.11193•Jan 21, 2022•0 citations
A strongly driven (\ensuremathΩ\ensuremath≫\ensuremathγ)(\ensuremath{\Omega}\ensuremath{\gg}\ensuremath{\gamma})(\ensuremathΩ\ensuremath≫\ensuremathγ) two-level atom relaxes towards an equilibrium state \ensuremathρ\ensuremath{\rho}\ensuremathρ which is almost completely mixed. One interpretation of this state is that it represents an ensemble average, and that an individual atom is at any time in one of the eigenstates of \ensuremathρ\ensuremath{\rho}\ensuremathρ. The theory of Teich and Mahler [Phys. Rev. A 45, 3300 (1992)] makes this interpretation concrete, with an individual atom jumping stochastically between the two eigenstates when a photon is emitted. The dressed-atom theory is also supposed to describe the quantum jumps of an individual atom due to photoemissions. But the two pictures are contradictory because the dressed states of the atom are almost orthogonal to the eigenstates of \ensuremathρ\ensuremath{\rho}\ensuremathρ. In this paper we investigate three ways of measuring the field radiated by the atom, which attempt to reproduce the simple quantum jump dynamics of the dressed state or Teich and Mahler models. These are spectral detection (using optical filters), two-state jumps (using adaptive homodyne detection), and orthogonal jumps (another adaptive homodyne scheme). We find that the three schemes closely mimic the jumps of the dressed-state model, with errors of order 34(\ensuremathγ/\ensuremathΩ)2/3,\frac{3}{4}(\ensuremath{\gamma}/\ensuremath{\Omega}{)}^{2/3},43​(\ensuremathγ/\ensuremathΩ)2/3, 14(\ensuremathγ/\ensuremathΩ)2\frac{1}{4}(\ensuremath{\gamma}/\ensuremath{\Omega}{)}^{2}41​(\ensuremathγ/\ensuremathΩ)2, and 34(\ensuremathγ/\ensuremathΩ)2\frac{3}{4}(\ensuremath{\gamma}/\ensuremath{\Omega}{)}^{2}43​(\ensuremathγ/\ensuremathΩ)2, respectively. The significance of this result to the program of environmentally induced superselection is discussed.
A strongly-driven (Ω>>γΩ>> γΩ>>γ) two level atom relaxes towards an equilibrium state rho which is almost completely mixed. One interpretation of this state is that it represents an ensemble average, and that an individual atom is at any time in one of the eigenstates of ρρρ. The theory of Teich and Mahler [Phys. Rev. A 45, 3300 (1992)] makes this interpretation concrete, with an individual atom jumping stochastically between the two eigenstates when a photon is emitted. The dressed atom theory is also supposed to describe the quantum jumps of an individual atom due to photo-emissions. But the two pictures are contradictory because the dressed states of the atom are almost orthogonal to the eigenstates of ρρρ. In this paper we investigate three ways of measuring the field radiated by the atom, which attempt to reproduce the simple quantum jump dynamics of the dressed state or Teich and Mahler models. These are: spectral detection (using optical filters), two-state jumps (using adaptive homodyne detection) and orthogonal jumps (another adaptive homodyne scheme). We find that the three schemes closely mimic the jumps of the dressed state model, with errors of order (3/4)(γ/Ω)2/3(3/4) (γ/Ω)^{2/3}(3/4)(γ/Ω)2/3, (1/4)(γ/Ω)2(1/4) (γ/Ω)^{2}(1/4)(γ/Ω)2, and (3/4)(γ/Ω)2(3/4) (γ/Ω)^{2}(3/4)(γ/Ω)2 respectively. The significance of this result to the program of environmentally-induced superselection is discussed.
A strongly driven cavity containing a single resonant strongly coupled atom exhibits a phase bistability. The phase of the field is strongly correlated with the phase of the atomic dipole. It has been shown previously that phase-sensitive monitoring of the field emitted by the cavity would induce conditional quantum jumps between orthogonal atomic dipole states (``dressed'' states). Here we show that such monitoring can be used to fix the atom into a single dressed state. As soon as a state-changing quantum jump is inferred from the measurement of the field, the atomic state is flipped using a $\ensuremath{\pi}$ pulse. We study this feedback scheme analytically and numerically. We show that the occupation probability of the desired fixed state can be as high as $1\ensuremath{-}1/8\ensuremath{\eta}{C}_{1},$ where ${C}_{1}\ensuremath{\gg}1$ is the single-atom cooperativity and $\ensuremath{\eta}$ the detection efficiency (which does not have to be close to unity). The control of the atomic dynamics is manifest in the fluorescence spectrum. The widths of all three peaks are modified from the usual Mollow spectrum, and almost all of the area under one of the sidebands is transferred to the other sideband. This is as expected, as one of the dressed states is essentially unoccupied, and transitions out of it do not occur. In addition, the width of the central peak goes to zero. This indicates coherent scattering due to the nonzero mean atomic dipole created by the feedback.
Recent experiments with single atomic ions isolated in electromagnetic traps have demonstrated abrupt changes in observed fluorescence intensities, which have been interpreted as evidence for quantum jumps as the atom changes energy levels. This presents the seeming paradox of discontinuously changing observables from a system whose dynamical evolution is governed by the continuous optical Bloch equations. In this paper I present an analysis of the dynamics of a model three-level atom interacting with two laser fields in terms of the dressed states of the atom-laser system. These dressed states are eigenstates of a Hamiltonian which includes laser-atom interactions, and thus they simplify the discussion of the time evolution of the combined laser-atom system. This dressed-state description is particularly useful in the regime in which both laser intensities are strong enough to saturate their respective transitions. I calculate the radiative lifetimes of the different species of dressed states and the branching ratios for decay to other dressed states, and demonstrate the consequences of these quantities for the observed fluorescence pattern. The calculated duration of bright and dark periods shows a pronounced dependence on the detunings of the lasers.
A number of experiments on trapped single ions or atoms have been performed in recent years [1,2,3,4]. Monitoring the intensity of scattered laser light off of such systems has shown abrupt changes that have been cited as evidence of between states of the scattered ion or atom . The existence of such jumps was required by Bohr in his theory of the atom. Bohr’s quantum jumps between atomic states [5] were the first form of quantum dynamics to be postulated. He assumed that an atom remained in an atomic eigenstate until it made an instantaneous jump to another state with the emission or absorption of a photon. Since these jumps do not appear to occur in solutions of the Schrodinger equation, something similar to Bohr's idea has been added as an extra postulate in modern quantum mechanics. Stochastic quantum jump equations [6], [7],[8]were introduced as a tool for simulating the dynamics of a dissipative system with a large Hilbert space and their links with quantum measurement the or were also noted [9],[10],[11],[12],[13].This measurement interpretation is generally known as quantum trajectory theory [14].By adding filter cavities as part of the apparatus, even the quantum jumps in the dressed state model can be interpreted as approximations to measurement-induced jumps [15]. The question arises whether an explanation of these jumps can be found to result from an Colombeau solution [16]-[18](Ψe(x, t;ħ))e of the Schrodinger equation alone without additional postulates. We found exact quasi-classical asymptotic of the quantum averages
The statistics of spontaneous photon emission times in single atom resonance fluorescence are investigated through the radiative cascade of the dressed atom. The delay function which gives the distribution of the delays between two successive emissions is calculated for a coherent as well as an incoherent laser excitation. For a two-level atom, various signals that reflect the fluorescence intensity (average value, photon counting, fluctuation spectrum, etc.) are reviewed by this method. For a three-level atom, this approach is applied to the analysis of the recently observed phenomenon of intermittent fluorescence and quantum jumps.>
We first use the quantum method to replicate the well-known results of a single atom relaxing, whilst demonstrating the intuitive picture it provides for dissipative dynamics. By use of individual "quantum trajectories", the method allows for simulation of systems inaccessible to ensemble treatments. This is shown by replicating resonance fluorescence, allowing us to concurrently demonstrate the method's facilitation of calculating photon statistics by the creation of discrete photon streams. To analyse these, we solidify the theoretical basis for, and implement, a computational method of calculating second-order coherence functions. A process by which to model interacting two-atom systems to allow for computation with the quantum jump method is then developed. Using this, we demonstrate cooperative effects leading to greatly modified emission spectra, before investigating the decoupling of states from dissipative and coherent interactions. Here, we find the novel insight provided by the quantum jump method both births and provides the tools with which to begin an investigation into the occurrence of macroscopic jumps and the formation of macroscopic dark periods in a system of two two-level dipole-dipole coupled atoms.