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Physical Review A

Quantum interference in two-photon excitation with squeezed and coherent fields

N. Ph. Georgiades, Eugene Simon Polzik, Harry J. Kimble

Published Jan 1, 1999• 27 citations
Publisher

Abstract

Two-photon excitation of a three-level atom in a ladder configuration $(\stackrel{\ensuremath{\rightarrow}}{1}\stackrel{\ensuremath{\rightarrow}}{2}3)$ by simultaneous illumination with fields in squeezed vacuum and coherent states results in quantum interference for the excitation process. The particular configuration considered here is one for which the signal and idler output fields of a subthreshold nondegenerate optical parametric oscillator are in resonance with the two-stepwise dipole atomic transitions $(\stackrel{\ensuremath{\rightarrow}}{1}2,\stackrel{\ensuremath{\rightarrow}}{2}3),$ while a ``reference oscillator'' field is in two-photon resonance with the quadrupole transition $(\stackrel{\ensuremath{\rightarrow}}{1}3).$ In an extension of the work of Ficek and Drummond [Phys. Rev. A 43, 6247 (1991)], a theoretical formulation based on the full quantum master equation for the problem is presented. The combined effects of quantum interference and the nonclassical character of the squeezed state are investigated, and offer the potential for a new detection strategy for quantum fluctuations of the electromagnetic field with ultrahigh frequencies (10's--100's THz). Based on the theory developed, we analyze quantum interference in excitation in several special cases relevant to experimental realizations, including the effects of a small focusing angle of the squeezing onto the atoms, and unusual population inversions. Special emphasis is given to identifying intrinsically quantum optical field effects versus classical field effects. Procedures that could distinguish between the two (i.e., classical and nonclassical) are suggested.

DOI
10.1103/physreva.59.676

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Two-photon excitation of a three-level atom in a ladder configuration (1\ensuremath→2\ensuremath→3)(\stackrel{\ensuremath{\rightarrow}}{1}\stackrel{\ensuremath{\rightarrow}}{2}3)(1\ensuremath→​2\ensuremath→​3) by simultaneous illumination with fields in squeezed vacuum and coherent states results in quantum interference for the excitation process. The particular configuration considered here is one for which the signal and idler output fields of a subthreshold nondegenerate optical parametric oscillator are in resonance with the two-stepwise dipole atomic transitions (1\ensuremath→2,2\ensuremath→3),(\stackrel{\ensuremath{\rightarrow}}{1}2,\stackrel{\ensuremath{\rightarrow}}{2}3),(1\ensuremath→​2,2\ensuremath→​3), while a ``reference oscillator'' field is in two-photon resonance with the quadrupole transition (1\ensuremath→3).(\stackrel{\ensuremath{\rightarrow}}{1}3).(1\ensuremath→​3). In an extension of the work of Ficek and Drummond [Phys. Rev. A 43, 6247 (1991)], a theoretical formulation based on the full quantum master equation for the problem is presented. The combined effects of quantum interference and the nonclassical character of the squeezed state are investigated, and offer the potential for a new detection strategy for quantum fluctuations of the electromagnetic field with ultrahigh frequencies (10's--100's THz). Based on the theory developed, we analyze quantum interference in excitation in several special cases relevant to experimental realizations, including the effects of a small focusing angle of the squeezing onto the atoms, and unusual population inversions. Special emphasis is given to identifying intrinsically quantum optical field effects versus classical field effects. Procedures that could distinguish between the two (i.e., classical and nonclassical) are suggested.
Non-classical properties and quantum interference (QI) in two-photon excitation of a three level atom (|1〉), |2〉, |3〉) in a ladder configuration, illuminated by multiple fields in non-classical (squeezed) and/or classical (coherent) states, is studied. Fundamentally new effects associated with quantum correlations in the squeezed fields and QI due to multiple excitation pathways have been observed. Theoretical studies and extrapolations of these findings have revealed possible applications which are far beyond any current capabilities, including ultrafast nonlinear mixing, ultrafast homodyne detection and frequency metrology. The atom used throughout the experiments was Cesium, which was magneto-optically trapped in a vapor cell to produce a Doppler-free sample. For the first part of the work the |1〉 → |2〉 → |3〉 transition (corresponding to the 6S1/2F = 4 → 6P3/2F' = 5 → 6D5/2F = 6 transition) was excited by using the quantum-correlated signal (Ɛs) and idler (Ɛi) output fields of a subthreshold non-degenerate optical parametric oscillator, which was tuned so that the signal and idler fields were resonant with the |1〉 → |2〉 and |2〉 → |3〉 transitions, respectively. In contrast to excitation with classical fields for which the excitation rate as a function of intensity has always an exponent greater than or equal to two, excitation with squeezed-fields has been theoretically predicted to have an exponent that approaches unity for small enough intensities. This was verified experimentally by probing the exponent down to a slope of 1.3, demonstrating for the first time a purely non-classical effect associated with the interaction of squeezed fields and atoms. In the second part excitation of the two-photon transition by three phase coherent fields Ɛ1 , Ɛ2 and Ɛ0, resonant with the dipole |1〉 → |2〉 and |2〉 → |3〉 and quadrupole |1〉 → |3〉 transitions, respectively, is studied. QI in the excited state population is observed due to two alternative excitation pathways. This is equivalent to nonlinear mixing of the three excitation fields by the atom. Realizing that in the experiment the three fields are spaced in frequency over a range of 25 THz, and extending this scheme to other energy triplets and atoms, leads to the discovery that ranges up to 100's of THz can be bridged in a single mixing step. Motivated by these results, a master equation model has been developed for the system and its properties have been extensively studied.
The level population of the hydrogen molecule is dramatically altered due to quantum interference, when the molecules are illuminated by two fields (coherent and squeezed) simultaneously. The molecules from the $B{}^{1}{\ensuremath{\Sigma}}_{u}(v=0,j=1)$ is excited stepwise to the level of same symmetry, $B{}^{1}{\ensuremath{\Sigma}}_{u}(v=20,j=1)$ by nondegenerate optical oscillator via an intermediate $E,F{}^{1}{\ensuremath{\Sigma}}_{g}(v=8,j=0),$ and directly by a two-photon transition by a CO2{\mathrm{CO}}_{2}CO2​ laser. The population of the states are also affected by the correlation developed between the two modes of the squeezed field. The correlation and the quantum interference jointly affect the molecules in such a way that the inversion of population occurs periodically. This feature can further be used for studying lasing and other important optical activities.
One of the canonical questions in quantum optics is the nature of the radiative properties of an atom when the normal vacuum fluctuations of the electromagnetic reservoir are replaced by the asymmetric, reduced fluctuations of a squeezed vacuum. While the basic radiative linewidth-narrowing effect has been known for over a decade [C. W. Gardiner, Phys. Rev. Lett. 56, 1917 (1986)], experimental realizations with operationally definable definitive manifestations of the quantum nature of the squeezed reservoir have been largely lacking from subsequent investigations. This paper presents measurements on an experimentally realized atom--squeezed-light system, in which the squeezed-light output of a subthreshold optical parametric oscillator illuminates an atom strongly coupled to a high-finesse optical resonator. Transmission of a weak probe field incident on the atom-cavity system is investigated both theoretically and experimentally. Alteration of the transmitted probe spectrum has been observed, as has a transmission modulation that depends on the phase of the squeezed field relative to a saturating coherent field (displaced squeezing). In certain parameter regimes, properties unique to the quantum nature of the squeezed light have been identified in the theoretical treatment, but complications in the experiment prevent their unequivocal measure. It is found that the observed effects of the squeezed light are dramatically reduced relative to the predictions of an idealized theory. This is quantitatively attributed to the effects of atomic beam fluctuations and a simple modeling of the atomic beam as an additional loss mechanism in the theory leads to reasonable agreement with the data.
A three-level ladder atom driven by two coherent laser fields with rather different frequencies in various kinds of reservoirs is considered. Making use of the master equation and Bloch equation, we calculate numerically and show graphically the steady-state populations versus two-photon detuning. We also discuss and compare the two-photon and one-photon processes occurring for various cases. For a squeezed vacuum, only when the carrier frequency of the squeezed vacuum equals the average of frequencies of the two driving fields and the phase matching condition is fulfilled, will there be a strong two-photon resonant absorption from the squeezed vacuum, a characteristic different from absorption of photons from a classical field; Otherwise, the two-photon ab-sorption rate from the squeezed vacuum will be greatly diminished.
Light emitted from laser-driven atoms often has squeezed quadrature fluctuations reflecting the phase dependence of the atomic source excited into a coherent superposition by the driving field. In this paper, we study the maximum squeezing in the emitted light fields from three-level atoms, paying particular attention to the role of atomic coherences in governing the optimum squeezing which is possible. Some phase dependent nonlinear optical processes have the potential for substantial quantum noise reduction at useful intensities: for example, four-wave mixing involving atomic systems has already been used to generate quadrature-squeezed light. The threoretical description of such a source is generally complicated with damping processes, coherent and incoherent pumping, and so on, all needing to be taken into account. Fully quantum treatments are available, but the basic physics of the process is often difficult to grasp in the necessarily complicated analysis. Although less important from the point of view of efficient practical sources of squeezed light, studying the generation of squeezed light in simpler systems such as two- or three-level atoms interacting with one or two modes (as in Jaynes-Cummings models) or in spontaneous emission or resonance fluorescence gives greater insight into the fundamentals of the squeezing process. In particular, the relationship between the atomic coherences and the degree of squeezing in the generated fields can be explored. We study the maximum (optimized) squeezing that can be obtained in the emitted fields from the irreversible decay from three-level atoms for any choice of initial conditions, excitations and decay process, for all quadrature components and for all possible configurations (V, \ensuremath{\Lambda}, and ``ladder'' systems).We show that for V and \ensuremath{\Lambda} systems, optimum squeezing is actually associated with a two-level state (in a suitable basis) involving a single one-photon coherence, but for ladder systems the state for maximum noise reduction is not equivalent to a two-level system and involves a single two-photon coherence and no intermediate state population. We pay particular attention to whether the total field is in a minimum uncertainty state, and the nature of the atomic state associated with maximum squeezing, especially whether it is a mixed or pure state. In the case where the initial free field has a zero amplitude at the detector and the source atoms are confined to a region small compared to the wavelength (Dicke source) we show that the squeezing in the total field is given in terms of the squeezing in the source field, and hence related to atomic populations and atomic one-photon and two-photon coherences for the case of three level atom. Previously unconsidered source-free field interferences terms are shown to be zero. The choice of quadrature phase and frequency is optimized to minimize the source field normally ordered variance. We then minimize further with respect to the choice of atomic density matrix elements subject to the constraints that the density matrix is Hermitean, positive, and has a trace equal to unity. In all cases we find that the optimum squeezing is produced when the source field is in a minimum uncertainty state.
The resonance two-photon interaction of the broadband squeezed field (without coherent displacement) with the dipole-forbidden two-level atoms via nonresonant virtual levels is investigated. It is demonstrated that the broadband squeezed field excites coherently the system of radiators (atoms, molecules) that have dipole-forbidden transitions. The two-photon Rabi frequency, in this case, is amplified due to the large number of couples of correlated modes of the broadband squeezed field involved. It should be mentioned that this coherent excitation is not specific for optical allowed dipole transitions. In the high-field limit, the quantum statistics of the photon pairs generated in the process of resonance fluorescence is analysed. As the fluorescence spectrum consists of three bands of the correlated photon pairs we analysed the second-order fluctuations in the central and sideband peaks of the triplet. It is shown that the amplitude-squared squeezing that is presented in the sidebands arises from the interaction of the spontaneously radiated photon pairs with the vacuum of the electromagnetic field.