Interference between atomic Rb (5d 5/2–5p 3/2) and (5p 3/2–5s 1/2) coherences: observation of an exceptional point by quantum beating at ∼2.1 THz

Coupled oscillators are prevalent in nature and fundamental to fields as disparate as astrophysics, photonics, the mechanical sciences, and geophysics. Theory has identified singularities in the response of coupled oscillators, known as exceptional points (EPs), that are associated with non-Hermitian operators and lie at the transition between weak and strong coupling of the oscillator. Although several EPs have been reported or predicted to exist in nanophotonic resonators and Feshbach resonances, for example, tuning the phase of two interfering atomic or molecular coherences near an EP has not been demonstrated previously. We report the observation of an EP associated with a pair of interfering atomic coherences in Rb, oscillating at 386.3 and 384.2 THz, and confirm the theoretical prediction of an abrupt phase shift of ∼π/4 as the EP is traversed by independently varying two experimental parameters. Pairs (and trios) of coupled coherences in thermal Rb atoms are established among the 7s 1/2, 5d 5/2, 5p 3/2, and 5s 1/2 states in pump–probe experiments with <200 fs laser pulses, and observed directly in the temporal and spectral domains through the ensuing quantum beating in the ∼2–36 THz interval. Interference between the (5d 5/2–5p 3/2) and (5p 3/2–5s 1/2) coherences is mediated by the 5p 3/2 state and detected through quantum beating in the vicinity of the (5d 5/2–5p 3/2)–(5p 3/2–5s 1/2) difference frequency of 2.11 THz which is monitored by a parametric four-wave mixing process. Phase of this composite atomic oscillator is first controlled by varying the mean Rb–Rb nearest neighbor distance (⟨R⟩) in a thermalized vapor. A discontinuous transition of (0.8 ± 0.2) ∼ π/4 radians in the phase of the coupled oscillator occurs when ⟨R⟩ is varied over the ∼80–90 nm interval, a phase shift associated with the transformation of a broadband, dissipative oscillator (characterized by a Fano interference window) into a strongly-coupled system resonant at 2.1 THz.


Introduction
Physical systems described by non-Hermitian operators are characterized by singularities in parameter space at which PT symmetry is broken. Known as exceptional points (EPs) [1][2][3][4][5], such singularities have proven challenging to observe experimentally [1,6,7] and virtually all of those documented to date have been identified in coupled-photonic resonators such as photonic molecules [6,[8][9][10][11][12][13]. Classical and quantum models of coupled oscillators [1,4,6,14,15] demonstrate that: (1) EPs are situated at the transition between weak and strong coupling of the composite oscillator, and (2) their fundamental properties are captured in the spectral domain by a Fano lineshape which reflects the interference between competing resonances or exit channels [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Recently, coupled resonator models have also explored the common thread between Fano and Feshbach resonances associated with interfering decay channels [3], and have yielded generalized Fano spectral profile expressions [6,31] which provide insight into the linkages between quantum optical phenomena such as electromagnetically-induced transparency and Autler-Townes splitting [4,6,32,33]. Undoubtedly the most significant property of EPs is the predicted shift of ∼π/4 in the phase of a coupled system as an EP is approached by varying an appropriate system parameter [2]. Günther et al have noted: 'the phase-jump behavior can be used as an implicit indicator of a possible close location of an EP. . . ' [2]. In summary, despite the progress of the past decade in recognizing and analyzing EPs, few time-resolved studies of interfering atomic or molecular oscillators have appeared in the literature and the theoretically-predicted, abrupt shift in phase in the vicinity of an atomic EP has not been confirmed experimentally.
We describe here the observation of an EP associated with a pair of interfering coherences produced simultaneously in the Rb atom. Furthermore, the phase of this coupled oscillator is found to change discontinuously by (0.8 ± 0.2) ∼ π/4 radians as the EP is traversed, thereby confirming one signature of the singularity. Both pairs and trios of coupled atomic oscillators are prepared by establishing quantum coherences among at least four states of atomic Rb through one-and two-photon excitation of the atom in the vapor phase with <200 fs laser pulses. The temporal evolution of the coupled system is monitored by quantum beating in the ∼2-36 THz frequency interval and is detected optically by parametric fourwave mixing (PFWM) [34,35]. In pump-probe experiments, quantum beating is observed at multiple frequencies, including the 7s 2 S 1/2 -5d 2 D 5/2 (denoted 7s 1/2 -5d 5/2 ) atomic state difference frequency of 18.22 (5) THz. Of particular interest, however, is interference between the Rb (5d 5/2 -5p 3/2 ) and (5p 3/2 -5s 1/2 ) coherences, coupled through the intermediate 5p 3/2 state, that manifests itself as a generalized Fano lineshape [6,31] associated with the (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) resonance at 2.11 THz. Controlling the phase of this composite quantum system, denoted (5d 5/2 -5p 3/2 ): (5p 3/2 -5s 1/2 ), and its associated lineshape is first accomplished by varying the mean Rb-Rb interatomic separation ( R ) in a thermalized vapor (which is collision-free throughout the measured (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) coherence time of <30 ps), thus affirming the conclusion of Limonov et al [29] that the ability to manipulate the Fano profile '. . . provides a route toward understanding and controlling interference processes across several branches of physics'. The ability to tune the phase of the composite oscillator in this way is attributed to induceddipole interactions with background thermal Rb atoms which define the interatomic potential at R , V( R ). Increasing R from ∼80 nm to ∼90 nm results in an abrupt shift of ∼π/4 radians in the Fano phase which is in accord with the calculations of reference [2] and is attributed to induced-dipole interactions (C 3 R −3 ) with background atoms. Over the narrow region in R in which the phase transition occurs, the interference window in the Fano lineshape is suppressed, and the resonance peak at ∼2.1 THz appears, the strength of C 3 R −3 interactions increases by a factor of ∼2-3. Tuning of the coupled-oscillator phase through an EP with the laser intensity and a three coupled-coherence system, (7s 1/2 -5d 5/2 ): (5d 5/2 -5p 3/2 ): (5p 3/2 -5s 1/2 ), have also been observed. Regardless of the experimental parameter that is varied, interference structure dominating the Fourier spectrum of the non-resonant coupled oscillator collapses as the EP is traversed. Thus, the impact of the observed phase shift is to transform a broadband, dissipative oscillator into a strongly-coupled resonant system. Conducting these experiments in thermalized vapor-gas mixtures allows for the impact of atom-atom interactions on the coupled quantum coherences to also be observed. Indeed, comparison of Fano profiles near 2.1 THz in Rb vapor with those in Rb-Cs and Rb-Ar mixtures provides further support for the presence of the EP and demonstrates tuning of coupled atomic coherences through an EP by means of Rb state-specific interactions, at long range (50-100 nm), with background atoms.

Probing interfering atomic coherences by quantum beating and PFWM
Panel (a) of figure 1 is a partial energy level diagram for Rb, illustrating the optical processes responsible for generating and detecting quantum beating at 2.1 THz and other atomic state difference frequencies. Both PFWM and quantum beating are driven simultaneously by <200 fs pulses from a Ti:Al 2 O 3 laser which have sufficient bandwidth (∼20 nm FWHM) to encompass both the 7s 1/2 ←← 5s 1/2 and Optical processes responsible for producing and detecting coupled atomic coherences, and representative time-and frequencydomain data: (a) qualitative, partial energy level diagram for Rb, illustrating one-and two-photon excitation of the atom with <200 fs pulses (centered at ∼768 nm) so as to produce multiple quantum oscillators. Quantum beating is monitored in pump-probe experiments through PFWM. The vertical red arrows represent coherences established between the 5d 5/2 , 5p 3/2 , and 5s 1/2 states of Rb while one-and two-photon excitation transitions, as well as the PFWM idler waves, are indicated by black lines. For clarity, resonant excitation of the 5p 3/2 state by a single-photon transition is not shown. By varying the pump-probe delay time (Δt), Ramsey fringes associated with the signal wave intensity are observed (shown in blue) which, in the Fourier domain, yield a generalized Fano spectral profile having an interference window lying at ∼70.3 cm −1 (∼2.1 THz) which is ∼0.1 cm −1 above the resonance difference frequency for the (5d 5/2 -5p 3/2 ): (5p 3/2 -5s 1/2 ) oscillator. Also, dotted horizontal lines denote virtual states and the red shading represents twice the laser bandwidth; (b) and (c) Ramsey fringes, representative of those recorded throughout the experiments and observed by varying Δt over the 4.5-6.5 ps and 598.8-600.2 ps intervals, respectively, while monitoring the signal wave intensity. In (b), oscillations having periods of ∼55 fs and ∼475 fs, corresponding to frequencies of 18.2 THz and 2.1 THz, respectively, are evident whereas 18.2 THz quantum beating dominates at long time-delays. These data were acquired for [Rb] = 2.6 × 10 14 cm −3 . 5d 5/2 ←← 5s 1/2 two-photon transitions at 760.1 nm and 778.1 nm, respectively. Photoexciting both transitions with each pulse establishes coherent superpositions of states, and quantum beating at several atomic difference frequencies ensues. Of these, 18.2 THz (7s 1/2 -5d 5/2 ) is most prominent but single-photon excitation of Rb also produces a coherent superposition of the 5p 3/2 fine structure level and ground. Consequently, the (5d 5/2 -5p 3/2 ) and (5p 3/2 -5s 1/2 ) coherences are also produced which correspond to frequencies of 386.3 THz and 384.2 THz, respectively. As shown in figure 1(a), quantum beating is monitored by driving the PFWM process: 5s 1/2 →→ (7s 1/2 , 5d 5/2 ) → 6p 3/2 → 5s 1/2 and interference between the coherent idler waves (generated near the 7s 1/2 ,5d 5/2 -6p 3/2 transitions) modulates the PFWM signal wave (6p 3/2 → 5s 1/2 ) intensity at ∼420.2 nm (shown in blue in figure 1(a)). Accordingly, the temporal evolution of the quantum beating associated with each observed coherence is monitored by scanning the time delay between the pump and probe pulses (Δt) while recording the relative PFWM signal wave intensity, as illustrated in figure 1(a). In the Fourier domain, the temporal dynamics of the coherences driving the PFWM process are reflected by spectra near ∼2.1, 10.7, and 18.2 THz. It should be mentioned that these quantum oscillators may also be detected by six-wave parametric mixing [36]. Quantum beating at the 7s 1/2 -5d 5/2 difference frequency of Rb was first reported in reference [34] and sidebands were observed by Shen et al [35], but not resolved. As discussed in the text to follow, most of the experimentally-observed coherences appear in the Fourier domain as spectrally-narrow (i.e. high-Q) peaks. An exception is the (5d 5/2 -5p 3/2 ): (5p 3/2 -5s 1/2 ) coupled oscillator. In the vicinity of the (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) difference frequency of 2.1 THz, a Fano lineshape having the classic interference window is observed (cf figure 1(a)). The physics of this pair of coupled coherences is of interest because the 5p 3/2 state determines the coupling strength between the 386.3 and 384.2 THz oscillators, but it is the 5d 5/2 state that serves as the interface between the coherences and the PFWM process. The broad, generalized Fano profile at lower right in figure

Experimental parameters and data acquisition
In these experiments, optical coherences are established among the 5s 1/2 , 5p 3/2 , 5d 5/2 , and 7s 1/2 states of Rb through simultaneous one-and two-photon excitation of the atom with 50-200 fs pulses provided by a coherent Ti:sapphire (Ti:Al 2 O 3 ) laser system. A schematic diagram of the experimental arrangement is given in figure 2. In order to maximize the amplitude of 7s 1/2 -5d 3/2 , 5/2 quantum beating (for the purpose of having the 7s 1/2 -5d 5/2 difference frequency serve as an experimental reference), the peak of the laser spectrum was intentionally set with custom internal optics to ∼768 nm which lies slightly to the blue side of the midpoint between the 7s 1/2 ←← 5s 1/2 and 5d 5/2 ←← 5s 1/2 twophoton transitions (owing to the relative cross-sections for Δl = 0 and 2 two-photon transitions), and negative chirp was imposed upon the laser pulses (monitored by the FROG unit of figure 2). The experiments reported here employed natural abundance or isotopically-enriched Rb vapor thermalized at temperatures of 423-488 K which correspond to Rb number densities of [Rb] = 1.0-9.2 × 10 14 cm −3 . Rubidium vapor or Rb-Cs or Rb-Ar mixtures were contained in sapphire or fused silica cells with a length of 2.5 cm. Conducting these experiments in thermalized vapor/gas mixtures allows for the impact of atom-atom interactions on the coupled oscillators to be observed directly, and quantum beating to be detected through PFWM.
A Michelson interferometer having a temporal resolution of 3.3 fs (0.5 μm spatial resolution) has been developed that allows for Δt to be scanned up to 1100 ps while reducing noise in the Fourier domain by at least two orders of magnitude, relative to previous interferometers [33][34][35]. For each of the ∼200 000 steps in a typical Δt = 0-670 ps scan, for example, 20 measurements of the relative PFWM signal wave intensity (recorded with a Si photodiode) were averaged. Data were acquired in >50 scans of Δt, each of which required 17 h despite the laser pulse repetition frequency being set at 1 kHz. The data acquisition rate is constrained, not by the laser but rather by the piezoelectric driver of the interferometer's scanning translation stage and, specifically, the time necessary for the driver to settle reliably at a new position. Because the sampling frequency is 300 THz, the Nyquist frequency is 150 THz which provides more than sufficient bandwidth to observe the atomic difference frequencies of interest here. Extensive tests confirm that, given the interferometer spatial resolution and setting reproducibility, the experimental resolution in the Fourier (spectral) domain is <0.05 cm −1 and the dynamic range was measured to be >50 dB. By averaging the peak positions measured from 21 scans, the 7s 1/2 -5d 5/2 difference frequency was determined to be 18.22 (5) THz which corresponds to 607.93(7) cm −1 . It must be emphasized that the precision of this measurement (which improves slightly on the value reported by Sansonetti [37]: 607.939 ± 0.02 cm −1 ) is possible only because of the stability of the interferometer and the exceptional coherence time associated with 7s 1/2 -5d 5/2 quantum beating which is found to be >1 ns. Throughout the experiments, the pump and probe pulse energies were fixed at discrete values in the 150-350 μJ range and the phase history of both pulses was monitored continuously by frequencyresolved optical gating (FROG). No significant changes in the cross-sections and transverse intensity distributions of the pump and probe pulse laser beams were observed as the pulse energy was varied. For the maximum single-pulse energy available with the present Ti:sapphire laser system, the peak intensity at the optical cell is estimated to be ∼1.9 GW cm −2 for 200 fs pulses. Most of the experiments to be described here were conducted with single pulse energies of 300 μJ. 3 Typically, scans of the dependence of the relative signal wave (420 nm) intensity on Δt were recorded over the 0-670 ps interval, and representative data are shown in figures 1(b) and (c) for the 4.5 Δt 6.5 ps and 598.8-600.2 ps intervals, respectively. The results presented in both panels are representative of those acquired over the course of >50, 17 h scans. Several frequency components are evident in the oscillatory signal of figure 1(b), the most prominent of which has a period of ∼475 fs (corresponding to ∼2.1 THz) and is clearly modulating the amplitude of the PFWM signal. The higher frequency structure in panel (b) has a periodicity of ∼55 fs which is associated with the 7s 1/2 -5d 5/2 energy difference. Ramsey fringes for this quantum oscillator have been found to persist for >1 ns which corresponds to more than 1.8 × 10 4 oscillation periods, thereby enabling the precise measurement of the 7s 1/2 -5d 5/2 energy difference (607.93(7) cm −1 ). An example of the data routinely recorded beyond Δt = 500 ps is presented in figure 1(c). Note that although 2.1 THz oscillations dominate the waveform early ( figure 1(b)), this frequency component of the (5d 5/2 -5p 3/2 ): (5p 3/2 -5s 1/2 ) coupled quantum oscillator dephases quickly and is no longer detectable for Δt ∼ 600 ps. In fact, as will be evident in figure 3, the 2.1 THz frequency component has vanished by Δt = 100 ps. The Fourier (spectral) domain representation of temporal data such as that of figure 1 is given in figure 3(a) which identifies the observed quantum beating frequencies. The most prominent of these are the (7s 1/2 -5d 5/2 ), (8s 1/2 -6d 5/2 ), and (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) difference frequencies at 18.2, 10.7, and 2.1 THz, respectively. As mentioned previously, the latter is unique in that one state (5p 3/2 ) is common to two interacting oscillators. Both the second and third harmonics of 18.2 THz are also observed, and coupling between the 7s 1/2 -5d 5/2 and 8s 1/2 -6d 5/2 coherences is responsible for the presence of several frequency components described by (7s 1/2 -5d 5/2 ) ± nβ, where n is an integer and β = (7s 1/2 -5d 5/2 )-(8s 1/2 -6d 5/2 ) = 250.2 ± 0.1 cm −1 (7.5 THz). The locations of 4 such overtones are indicated in figure 3(a).

Generalized Fano lineshapes for the (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) oscillator
Since the dynamic range of experimental spectra (such as that of figure 3(a)) is >50 dB, spectral features lying near the noise floor and not detectable previously are now accessible. As one example, consider the spectral region lying near the base of the 18.2 THz resonance shown in the left-hand portion of figure 4. Note the symmetry of the spectrum with respect to the 7s 1/2 -5d 5/2 frequency, a characteristic of most of the observed spectra which we attribute to amplitude modulation of the 18.2 THz peak by other quantum beating frequencies. Several of the weaker features in panel (a) may be associated with a shallow electronic excited state of the Rb dimer. These, as well as the strong pairs of lines displaced by ∼1.0 and ∼2.4 cm −1 from the atomic resonance, will be discussed in detail elsewhere but, for present purposes, it must be emphasized that the quantum beating spectral features observed throughout the ∼0. 5 [31] and fitted to experimental spectra. Specifically, equation (29) of [31] describes the interference of two resonances and their interaction with a single continuum, and allows each of the observed spectra of figure 5 to be described by a single, modified Fano expression. For the T = 423 and 433 K profiles, for example, the resonance-resonance and resonance-continuum decay (damping) frequencies (γ 1 and γ 2 , respectively) extracted from the fitting procedure were found to be identical for both values of [Rb]: (γ 1 , γ 2 ) = (7, 7) cm −1 and (3.5, 3.5) cm −1 , respectively. Note also that the ∼5-8 cm −1 spectral breadths (FWHM) of the Fano profiles of figures 5(b) and 4 (panel at right) are consistent with the temporal decay of the 2.1 THz Fourier amplitude in figure 3(b). Accounting for the narrowing of  (29) of Durand et al [31]) with experimental data for each of the spectra of (a). Simulations show the resonance-resonance and resonance-continuum decay (damping) frequencies to be: (γ 1 , γ 2 ) = (7, 7), (3.5, 3.5) and (0.4, 8) cm −1 , respectively. Values of R were determined from the nearest-neighbor distributions [38,39] corresponding to each Rb number density, and the laser pump and probe pulse energies were fixed at 300 μJ throughout. the high-Q resonance at 70.3 cm −1 in figure 5(d), however, requires γ 1 to fall to 0.4 cm −1 but γ 2 increases to 8 cm −1 . The Fano phase (δ) is also extracted from the fitting of equation (29) of [31] to experiment. For the lower [Rb] values of figures 5(b) and (c), δ is found to be 3.1 radians for both spectra whereas the corresponding value for figure 5(d) is 2.7 radians. The fittings of equation (29) of [31] to the experimental spectra in figure 5, and their associated constants, are not unique but are representative of the parameter sets that best describe the data.
It is evident from figure 5 that the Fano profiles describe well the observed spectral lineshapes but do not capture the interference structure apparent on either side of the resonance at 70.3 cm −1 in figures 5(b) and (c). Undulations modulating a broadband continuum, superimposed onto the Fano lineshapes (which are Lorentzian at low Rb number densities: [Rb] <2 × 10 14 cm −3 ) and the T = 443 K spectrum, in particular, appear to arise from at least two sources, one of which is the competition between the multiple pathways by which the (5d 5/2 -5p 3/2 ) and (5p 3/2 -5s 1/2 ) coherences may be established by the pump and probe pulses. Interference with the 7s 1/2 -5d 5/2 oscillator is expected to be another factor. Note also the virtual collapse of the interference structure when R falls to ∼70 nm ( figure 5(d)). Finally, it must be emphasized that, over the interval in R of figure 5, the ∼2.1 THz dephasing time of <30 ps is approximately an order of magnitude smaller than the mean time between Rb-Rb

Transition of the Fano phase and spectral profile near an EP
The spectra of figure 5 reveal a dramatic and rapid transformation in the Fano profile over a narrow range in R (factor of 2-3 interval in [Rb]). This lineshape conversion process results (with increasing [Rb]) in a single, narrow bandwidth peak situated at the (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) resonance of 2.11 THz, lying immediately to the low-frequency side of the Fano windows illustrated in figure 1(a) and observed in panels (b) and (c) of figure 5. Although not shown in figure 5, continuing to raise [Rb] beyond 4 × 10 14 cm −3 reduces the peak Fourier amplitude at 2.1 THz but the spectral profiles are otherwise indistinguishable from figure 5(d). For these reasons, figure 6 focuses on the R ≈ 87 nm region (corresponding to [Rb] = 2.6 × 10 14 cm −3 and T = 443 K) in which the transition from a broad spectrum with one or more characteristic Fano windows to a single narrow feature occurs. In figure 6(a), the narrow resonance peak at 2.1 THz has emerged whereas the Fano interference window, which was weak in figure 5(b) and (c), has intensified sharply. In an effort to examine the influence of Rb-Rb interactions on the observations of figure 5, spectra similar to that of figure 6(a) were recorded with Rb-Ar and Rb-Cs vapor mixtures. Argon and Cs were chosen as perturbers because the former is a closedshell atom of low static polarizability (1.66a 0 3 ), whereas the corresponding value for Cs is ∼1.5 orders of magnitude larger and the excited states for this alkali atom lie outside the regions in energy accessible to the pump and probe laser pulses by one-and two-photon transitions. Figure 6(b) compares Rb-Ar and Rb-Cs spectra acquired at the same temperature (443 K) as that of panel (a). For these experiments, [Ar] was set at ∼3.2 × 10 16 cm −3 , and [Cs] is ∼5.5 × 10 14 cm −3 for saturated vapor. Clearly, neither Ar nor Cs perturbs the Fourier spectrum of the (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) coupled oscillator to the degree observed for a Rb atom-only background, but the high-frequency portion of the spectral profile is suppressed relative to that for figure 6(a). Simulations of the lineshapes of the Rb, Rb-Ar, and Rb-Cs experimental profiles with the modified Fano lineshape expressions of reference [31] yield the red curves of figures 6(a) and (b). For the T = 443 K spectrum of figure 6(a), the damping frequencies were determined to be (γ 1 , γ 2 ) = (0.2, 5.5). In contrast, the Rb-Ar and Rb-Cs experimental spectra show broadening of the interference window while sharpening the 70.3 cm −1 peak. Before leaving the data of figures 6(a) and (b), it should be emphasized that the Rb/Cs and Rb/Ar spectra of panel (b) differ significantly from each other and the Rb/Rb spectrum of figure 6(a). Although both show a suppressed response in the 70-74 cm −1 interval (relative to figures 6(a), 5(b) and (c), and 4), the effect is most pronounced for Cs and we presume that spectra similar to those of figure 6(b) offer a new diagnostic of the long-range interactions between excited Rb atoms and Cs, Ar, or other perturber species.
From simulations of spectra represented by figures 5 and 6, the Fano phase δ was extracted, and the results are given by the open circles of figure 6(c) for Rb vapor when R is between ∼55 and 120 nm. For the Rb-Ar and Rb-Cs data of figure 6(b), δ was found to be almost identical for both perturbers and is represented by a single open triangle (Δ). One concludes from figures 6(a) and (c) that both the Fano spectral profile and the phase of the composite (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) oscillator undergo a sharp transition in the range of R = 80-90 nm. Not only does the Fano phase change discontinuously near R ∼ 87 nm, but the magnitude of the phase shift is (0.8 ± 0.2) ∼ π/4 radians.

Laser intensity-dependent data and discussion
We interpret the data of figures 3(b)-6 as demonstrating the interference between coupled (5d 5/2 -5p 3/2 ) and (5p 3/2 -5s 1/2 ) atomic coherences, characterized by a phase that is controlled primarily by long-range interactions of the composite oscillator with background Rb atoms. The abrupt shift in the phase of the coupled coherences, accompanying the emergence of the sharp peak at the expected position of the (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) resonance (70.3 cm −1 ), and the appearance of the strong window at T = 443 K (figure 6(a) and (c)), are signatures of an EP. When T reaches 453 K (figure 5(d)), the interference window has broadened and weakened which suggests that the EP has been traversed. Furthermore, the (0.8 ± 0.2) ∼ π/4 radian shift in the phase of the 2.1 THz oscillator, measured when Rb is varied over the ∼80-90 nm interval, is remarkably close to the phase-jump predicted by Günther et al [2]. The ability to tune the phase of the composite quantum oscillator is attributed to induced-dipole interactions which alter the phase of the pair of coupled coherences through the interatomic potential V( R ). Over the range in R in which the interference window in the Fano lineshape is suppressed, the resonance peak at ∼2.1 THz appears, and the interference-modulated continuum vanishes, the strength of C 3 R −3 interactions [40] changes by a factor of 2-3. Since the static polarizability of a Rb (5d 5/2 ) atom is known to be a factor of ∼20 larger than its counterpart in the 5p 3/2 state, it is presumed that interactions of the Rb 5d 5/2 wavefunction with background atoms in both the ground and excited states are predominantly responsible for the observed effects. The markedly different behaviors of Rb vapor and Rb-Ar and Rb-Cs mixtures near the EP suggest that long-range interactions with background Rb(5d, 5p) atoms do, indeed, play a decisive role in approaching and moving beyond the EP.
Results similar to those of figures 5 and 6 are obtained if the laser intensity is varied. The left-hand portion of figure 7 shows Fourier spectra recorded for the same Rb number density and, hence, R value of figure 6 but the laser pulse energy (E p ) is now increased incrementally from 100 μJ to 300 μJ. One notices immediately that the spectrum is a continuum at the lowest optical field intensity (violet trace, figure 7), and the resonant peak at ∼70.3 cm −1 is first noticeable when E p reaches 150 μJ (blue). The latter spectral profile is a Fano lineshape exhibiting a null between ∼67 and 68.8 cm −1 . As E p is increased further, the interference structure discussed earlier strengthens, particularly on the high frequency side of the resonance. Above E p ∼ 200 μJ, however, the undulation-modulated continuum quickly diminishes in intensity as energy is transferred into the narrow feature at 2.11 THz. Indeed, the right-half of figure 7 illustrates the initial nonlinear growth of the 70.3 cm −1 peak amplitude when E p is increased above 200 μJ. The ordinate at right represents the fraction of the Fourier power spectrum (|F(ω)| 2 , where F(ω) is the Fourier transform of the f (Δt) experimental waveform) attributable to the 2.1 THz resonance line (denoted 'RES' in the integral of figure 7 and having a spectral width of ∼0.4 cm −1 ). When normalized to the power spectrum integrated over the ∼65-88 cm −1 interval (denoted 'TOT'), the value of the integral rises rapidly between E p = 200 μJ and ∼250 μJ but saturates thereafter. We attribute this behavior to the onset of strong coupling of the composite (5d 5/2 -5p 3/2 ): (5p 3/2 -5s 1/2 ) coupled oscillator in the vicinity of the EP. At still higher pulse energies, the background undulations and continuum strength remain unchanged.

Conclusions
In summary, an EP (singularity) associated with the response of a coupled pair of atomic coherences has been observed in thermalized Rb vapor. Quantum beating in Rb in the vicinity of the (5d 5/2 -5p 3/2 )-(5p 3/2 -5s 1/2 ) difference frequency resonance at ∼2.1 THz exhibits a generalized Fano spectral profile that is transformed by independently varying two experimental parameters, R of the background vapor or the laser intensity. In the R interval of ∼80-90 nm, the measured phase of the coupled (5d 5/2 -5p 3/2 ): (5p 3/2 -5s 1/2 ) quantum system abruptly shifts by ∼0.8 radians. The