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Physical Review A

Pump-coupled high-Qmicromasers with conditional measurements of atoms: Transient and steady-state entanglement of nonlocal fields

Pál Bogár, János A. Bergou, Mark S. Hillery

Published Mar 1, 1995• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.51.2396

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Fortschritte der Physik•May 1, 2000•8 citations
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.
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.
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.
Stimulated emission of two-level atoms in a high-Q cavity is considered under conditions when pumping produces correlated many-atomic states. Based on the many-body problem, a kinetic equation is obtained for the Glauber–Sudarshan probability, which describes the field in the Fokker–Planck approximation. The statistics of light in this approximation are determined by the atomic correlation functions of the order of no higher than the second. The noise of light is found in the regime of micromaser operation for two types of pumping producing the initial separable states with a classical correlation and entangled states. It is shown that the presence of the initial diatomic correlation enhances the intensity noise. The entangled state of atoms is found from which nonclassical light is generated with a steady-state phase and noise, which can be almost completely suppressed in the low-frequency spectral region.
This dissertation considers both a lossless and a dissipative micromaser in which a monoenergetic beam of two-level atoms in a coherent superposition of their upper and lower states is injected inside a single mode, high-Q cavity. In the lossless case, we find that under appropriate conditions a field initially in a mixed state will evolve to previously unknown pure states, which we call the tangent and cotangent states. In various limits, these states exhibit interesting properties such as sub-Poissonian photon statistics and squeezing, and most importantly they acquire the characteristics of "macroscopic" quantum superpositions. When dissipation of the cavity mode is incorporated into the model, we find that although the field no longer evolves to a pure state, the mixed steady-state field may still retain the properties of a macroscopic superposition under experimentally realizable damping rates. We then evaluate the experimental conditions necessary for the preparation and detection of such macroscopic superpositions.
In recent experiments it was demonstrated that number or Fock states of the one-atom maser can be generated. Two types of experiments have been performed. In the first one the Fock states are achieved in steady state via the so-called trapping states of the micromaser field. In the second experiment the number states were generated in a dynamical experiment by state reduction. In the latter case the generated field was afterwards probed by observing the dynamics of a probe atom in the field. The fact that Fock states of the micromaser field can be realised leads to new applications of the micromaser one of them is the generation of a many particle entanglement of the GHZ type.