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Physical Review A

Pump-coupled micromasers: Entangled trapping states of nonlocal fields

Pál Bogár, János A. Bergou

Published Mar 1, 1995• 13 citations
Publisher

Abstract

A single beam of excited two-level atoms couples two micromasers in a series. It is shown that in the absence of dissipation the possible steady states of their fields are superpositions, of two-field trapping number states. Which one is realized depends upon the initial state of the fields and the interaction parameters, g'\ensuremath{\tau}' and g''\ensuremath{\tau}'', of the two cavities. A large number of these trapping states are pure quantum states, some of them showing entanglement of the two nonlocal micromaser fields of the form \ensuremath{\Vert}N,N+M〉\ifmmode\pm\else\textpm\fi{}\ensuremath{\Vert}N+M,N〉. Here, N and N+M are arbitrary trapping numbers that belong to disconnected blocks of the photon number space and M specifies the order of correlation between the fields. The time evolution of the system toward steady states is investigated numerically, mainly concentrating on the production of pure entangled trapping states of the form discussed earlier. We describe a special procedure to amplify a number state, \ensuremath{\Vert}N,N〉, into such states that is based on conditions regarding the interaction parameters. In principle, N and M can be made arbitrarily large resulting in a steady-state nonlocal quantum superposition of distinct macroscopical fields (nonlocal ``Schr\"odinger cat''). We also present a solution of the standard master equation of a damped harmonic oscillator at finite temperature and apply it to study the effect of dissipation on the production of entangled trapping states at regular and Poissonian pump statistics.It is found that although the entanglement does not survive at a steady state it can build up in the short-time transient regime when cavity losses and the number of thermal photons are not too large. In this small-loss (large ${\mathit{N}}_{\mathrm{ex}}$) regime the quantum correlation between the micromaser fields decays into a steady-state classical superposition at a rate that depends on the cavity lifetime and the difference (or order of correlation), M, between the superposed photon numbers. In the large-loss (small ${\mathit{N}}_{\mathrm{ex}}$) regime, however, no transient correlation can be produced and the photon statistics spreads out towards a vacuum for increasing losses. The system undergoes a transition from an uncorrelated to a correlated behavior when the pumping parametrized by ${\mathit{N}}_{\mathrm{ex}}$ exceeds the threshold between the large- and small-loss regimes. Thermal photons enhance the decay of the correlation and by coupling the disconnected blocks of the photon number space they populate the trapping number states of adjacent blocks. However, it is shown that the production of transient entanglement is not significantly affected by either thermal radiation or pumping fluctuations. The experimental realization of the entanglement of nonlocal micromaser fields employing these two-field trapping states of small photon numbers is shown to be feasible in the short-time transient regime using the presently available high-Q cavities and low temperatures, and could be possible on the macroscopic scale in the near future.

DOI
10.1103/physreva.51.2381

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Two lossless micromasers are coupled in a series by the common pumping beam of two-level atoms the states of which are measured conditionally after the second cavity. Pure evolutions of the two fields are studied starting from uncorrelated coherent states for the four measurement schemes denoted by a-M'M''-a, b-M'M''-b, a-M'M''-b, and b-M'M''-a indicating the state of each atom, \ensuremath{\Vert}a〉 or \ensuremath{\Vert}b〉, before and after the two maser cavities, M' and M''. It is shown that energy-preserving schemes (first two above) produce a two-dimensional set of distinct Fock states at a steady state under the envelope of the initial amplitude distribution of the fields. Since the initial fields were uncorrelated the generated ones will be uncorrelated, too. In the case of energy-transferring schemes (second two) the system makes transitions between correlated and uncorrelated regimes. Nonlocal superpositions reminiscent of the form of \ensuremath{\Vert}N,N+M〉+\ensuremath{\Vert}N+M,N〉 can be generated at an optimum number of atoms as a result of two coexisting trapping mechanisms. This is a transient entanglement since it is destroyed by the atoms to follow due to the trapping effects themselves. However, we also show that by switching from any of the two energy-transferring schemes to any of the two preserving ones the transient correlation produced by the former scheme can be frozen into a steady state by the latter one. In the absence of dissipations this combination of schemes can generate steady-state coherent superpositions of arbitrary number states of two nonlocal fields (nonlocal Schr\"odinger cats) at reasonably high detection probabilities of the conditioned atomic states.

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Coherent superpositions of quantum states are very sensitive to coupling to the environment. The fluctuations associated with this coupling tend to wash out quantum phases, so that in practice only quantum mixtures are observed at the macroscopic level. This problem has been the object of recent interest in connection with measurement theory (Schrodinger cat). Theoretical predictions indicate that the decay of quantum coherences in the so-called pointer basis is amplified by a factor proportional to a measure of the distance in Hilbert space between the states. Micromasers offer a possible experimental tool to investigate the decay of such coherences. Theoretical studies of a lossless version of the micromaster have shown the existence of trapping states | n q 〉 of the cavity mode. These number states are such that atoms injected in the upper state into the cavity experience a 2qπ interaction with the field. Cavity states of the form | ψ 〉 = Σ q C q | n q 〉 are easily shown to be dynamic steady states of the system. Number states are not a pointer basis for the micromaser, and our theoretical results will evidence the qualitative difference in the nature of coherence decay between this and previously studied cases. Because the decay times of micromaser cavities are typically of the order of hundreds of milliseconds, our results should be amenable to experimental verification.

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Solvable model of a strongly driven micromaser

Pavel Lougovski, Federico Casagrande, ALFREDO LULLI +3 more

We study the dynamics of a micromaser where the pumping atoms are strongly driven by a resonant classical field during their transit through the cavity mode. We derive a master equation for this strongly driven micromaser, involving the contributions of the unitary atom-field interactions and the dissipative effects of a thermal bath. We find analytical solutions for the temporal evolution and the steady state of this system by means of phase-space techniques, providing an unusual solvable model of an open quantum system, including pumping and decoherence. We derive closed expressions for all relevant expectation values, describing the statistics of the cavity field and the detected atomic levels. The transient regime shows the buildup of mixtures of mesoscopic fields evolving towards a super-Poissonian steady-state field that, nevertheless, yields atomic correlations that exhibit stronger nonclassical features than the conventional micromaser.

arXiv:quant-ph/0309040•Physical Review A•Feb 26, 2004•22 citations
A single beam of excited two-level atoms couples two micromasers in a series. It is shown that in the absence of dissipation the possible steady states of their fields are superpositions, of two-field trapping number states. Which one is realized depends upon the initial state of the fields and the interaction parameters, g'\ensuremath{\tau}' and g''\ensuremath{\tau}'', of the two cavities. A large number of these trapping states are pure quantum states, some of them showing entanglement of the two nonlocal micromaser fields of the form \ensuremath{\Vert}N,N+M〉\ifmmode\pm\else\textpm\fi{}\ensuremath{\Vert}N+M,N〉. Here, N and N+M are arbitrary trapping numbers that belong to disconnected blocks of the photon number space and M specifies the order of correlation between the fields. The time evolution of the system toward steady states is investigated numerically, mainly concentrating on the production of pure entangled trapping states of the form discussed earlier. We describe a special procedure to amplify a number state, \ensuremath{\Vert}N,N〉, into such states that is based on conditions regarding the interaction parameters. In principle, N and M can be made arbitrarily large resulting in a steady-state nonlocal quantum superposition of distinct macroscopical fields (nonlocal ``Schr"odinger cat''). We also present a solution of the standard master equation of a damped harmonic oscillator at finite temperature and apply it to study the effect of dissipation on the production of entangled trapping states at regular and Poissonian pump statistics.It is found that although the entanglement does not survive at a steady state it can build up in the short-time transient regime when cavity losses and the number of thermal photons are not too large. In this small-loss (large Nex{\mathit{N}}_{\mathrm{ex}}Nex​) regime the quantum correlation between the micromaser fields decays into a steady-state classical superposition at a rate that depends on the cavity lifetime and the difference (or order of correlation), M, between the superposed photon numbers. In the large-loss (small Nex{\mathit{N}}_{\mathrm{ex}}Nex​) regime, however, no transient correlation can be produced and the photon statistics spreads out towards a vacuum for increasing losses. The system undergoes a transition from an uncorrelated to a correlated behavior when the pumping parametrized by Nex{\mathit{N}}_{\mathrm{ex}}Nex​ exceeds the threshold between the large- and small-loss regimes. Thermal photons enhance the decay of the correlation and by coupling the disconnected blocks of the photon number space they populate the trapping number states of adjacent blocks. However, it is shown that the production of transient entanglement is not significantly affected by either thermal radiation or pumping fluctuations. The experimental realization of the entanglement of nonlocal micromaser fields employing these two-field trapping states of small photon numbers is shown to be feasible in the short-time transient regime using the presently available high-Q cavities and low temperatures, and could be possible on the macroscopic scale in the near future.
Two lossless micromasers are coupled in a series by the common pumping beam of two-level atoms the states of which are measured conditionally after the second cavity. Pure evolutions of the two fields are studied starting from uncorrelated coherent states for the four measurement schemes denoted by a-M'M''-a, b-M'M''-b, a-M'M''-b, and b-M'M''-a indicating the state of each atom, \ensuremath{\Vert}a〉 or \ensuremath{\Vert}b〉, before and after the two maser cavities, M' and M''. It is shown that energy-preserving schemes (first two above) produce a two-dimensional set of distinct Fock states at a steady state under the envelope of the initial amplitude distribution of the fields. Since the initial fields were uncorrelated the generated ones will be uncorrelated, too. In the case of energy-transferring schemes (second two) the system makes transitions between correlated and uncorrelated regimes. Nonlocal superpositions reminiscent of the form of \ensuremath{\Vert}N,N+M〉+\ensuremath{\Vert}N+M,N〉 can be generated at an optimum number of atoms as a result of two coexisting trapping mechanisms. This is a transient entanglement since it is destroyed by the atoms to follow due to the trapping effects themselves. However, we also show that by switching from any of the two energy-transferring schemes to any of the two preserving ones the transient correlation produced by the former scheme can be frozen into a steady state by the latter one. In the absence of dissipations this combination of schemes can generate steady-state coherent superpositions of arbitrary number states of two nonlocal fields (nonlocal Schr\"odinger cats) at reasonably high detection probabilities of the conditioned atomic states.
The authors demonstrate the effect of atomic coherence on the generic example of two lossless micromasers coupled in series by a common pump beam of excited two-level atoms. The fields are studied via conditional measurements on the final state of the atoms. They consider the two simplest sequences: in the energy preserving (transferring) scheme they require each atom to be detected in its upper (lower) state. Due to the two paths that the atoms can follow to reach the same final state correlation between the two macroscopically separated fields can arise. The authors discuss a scheme which leads to the generation of entangled trapping states of the two fields of the form, mod n,n+M)+or- mod n+M,n), starting from number states mod n, n). Starting from initial coherent states arbitrary steady-state superpositions of the two fields can be generated by switching from the energy transferring to the preserving scheme at an optimum number of atoms. In the absence of dissipations, both methods can produce steady-state coherent superpositions of arbitrary number states of two macroscopically separated fields (nonlocal 'Schrodinger-cats'). A scheme is discussed where entanglement can be transferred from the fields to two atomic beams. Finally, the authors briefly discuss the effect of injected atomic coherence on lasers.
In a micromaser where a classical field strongly drives the atoms while they cross the cavity, remarkable atom–atom correlations show up at steady state, which vanish much faster than dissipative decay. Hence we consider atom pair correlation measurements in which the detection of the first probe atom prepares a mesoscopic superposition state of the cavity field, that entangles with a second probe atom. The conditional probabilities for the latter atomic detection provide a description of the decoherence of the superposition state, occurring in an open system in the presence of pumping, driving, dissipative, and thermal effects. The decoherence rate scales as the squared interaction time, that sets the separation in phase space between the superposition components, whereas the quantum coherence is unaffected by the atomic pumping. Hence we further investigate the system when the cavity is not pumped. Starting the correlation measurements from a thermal state, we can describe the effect of temperature on decoherence. Starting from a vacuum state, the superposition states are maximally separated Schrodinger cat states, whose decoherence can thus be monitored.
Coherent superpositions of quantum states are very sensitive to coupling to the environment. The fluctuations associated with this coupling tend to wash out quantum phases, so that in practice only quantum mixtures are observed at the macroscopic level. This problem has been the object of recent interest in connection with measurement theory (Schrodinger cat). Theoretical predictions indicate that the decay of quantum coherences in the so-called pointer basis is amplified by a factor proportional to a measure of the distance in Hilbert space between the states. Micromasers offer a possible experimental tool to investigate the decay of such coherences. Theoretical studies of a lossless version of the micromaster have shown the existence of trapping states | n q 〉 of the cavity mode. These number states are such that atoms injected in the upper state into the cavity experience a 2qπ interaction with the field. Cavity states of the form | ψ 〉 = Σ q C q | n q 〉 are easily shown to be dynamic steady states of the system. Number states are not a pointer basis for the micromaser, and our theoretical results will evidence the qualitative difference in the nature of coherence decay between this and previously studied cases. Because the decay times of micromaser cavities are typically of the order of hundreds of milliseconds, our results should be amenable to experimental verification.
We present a quantum theory of the micromaser which allows for arbitrary (sub- as well as super-Poissonian) fluctuations of the pumping beam. It handles the interaction with the active atoms (gain) and cavity decay (loss) on an equal footing. In conventional reservoir theory the rate of change of the cavity field is a sum of the changes due to separate interactions with the individual reservoirs, i.e., the interactions are uncorrelated. In our approach, corrections to reservoir theory arise. They contain the commutator of the gain and loss operators. The magnitude of these additional terms is characterized by the quantity p/Nex{\mathit{N}}_{\mathrm{ex}}Nex​ where p is a parameter describing pump-beam fluctuations. The parameter is, in fact, the negative of the Mandel Q parameter of the pump beam so that p=1 corresponds to regular pumping, p=0 to Poissonian one and p0 to super-Poissonian pump beam fluctuations. Nex{\mathit{N}}_{\mathrm{ex}}Nex​ is the number of excited atoms passing through the cavity during the lifetime of the intracavity field. Thus we recover the conventional reservoir limit if p=0 and/or Nex{\mathit{N}}_{\mathrm{ex}}Nex​ is large. In all other cases the interactions with the gain and loss reservoirs are correlated. We exploit some of the consequences of the additional terms, presenting analytical as well as numerical results for the steady-state photon statistics (mean photon number and photon-number fluctuations, in particular) without resorting to the p expansion.
We study the dynamics of a micromaser where the pumping atoms are strongly driven by a resonant classical field during their transit through the cavity mode. We derive a master equation for this strongly driven micromaser, involving the contributions of the unitary atom-field interactions and the dissipative effects of a thermal bath. We find analytical solutions for the temporal evolution and the steady state of this system by means of phase-space techniques, providing an unusual solvable model of an open quantum system, including pumping and decoherence. We derive closed expressions for all relevant expectation values, describing the statistics of the cavity field and the detected atomic levels. The transient regime shows the buildup of mixtures of mesoscopic fields evolving towards a super-Poissonian steady-state field that, nevertheless, yields atomic correlations that exhibit stronger nonclassical features than the conventional micromaser.