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Journal of Physics B Atomic Molecular and Optical Physics

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

William Goldshlag, R Su, Sehyun Park, Thomas Reboli, James Gary Eden

Published Aug 2, 2021• 3 citations
Publisher

Abstract

Abstract 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.

DOI
10.1088/1361-6455/ac19f6

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INTERFERENCE BETWEEN THE 5d5/2 – 5p3/2 AND 5p3/2 – 5s1/2 COHERENCES (386.4 AND 384.1 THz) IN Rb OBSERVED BY ULTRAFAST FOUR-WAVE MIXING SPECTROSCOPY

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\begin{wrapfigure}{r}{0pt} \includegraphics[scale=0.32]{Fig1.eps} \end{wrapfigure} Coherences at 386.2 THz and 384.1 THz, corresponding to the 5d$_{5/2}$ – 5p$_{3/2}$ and 5p$_{3/2}$ – 5s$_{1/2}$ difference frequencies, respectively, have been established in the Rb atom during pump-probe experiments involving pairs of identical ~150 fs pulses produced by a Ti:\chem{Al_2O_3} laser and a Michelson interferometer. The interference between the two coherences within the atom is observed through a parametric four-wave mixing process in Rb and detection of the signal wave intensity at 420 nm. Scanning the time delay between the pump and probe pulses over a 600 ps interval produces a sampling rate over 300 THz and allows for a spectral domain resolution of 0.05 cm$^{-1}$ to be achieved. The figure at right shows several spectra recorded near 2.1 THz, the (5d$_{5/2}$ – 5p$_{3/2}$) - (5p$_{3/2}$ – 5s$_{1/2}$) difference frequency, with varying angle between the pump and probe pulses (i.e., the phase matching angle) and varying Rb background density. A Fano interference window is clearly observed, and analysis of these and similar spectra demonstrates that the amplitude and phase of the coherently-coupled, three Rb state system can be controlled precisely.

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Probing the interactions between a mode-locked laser pulse and saturated rubidium vapor with a pump–probe technique has resulted in observations of coherences within the atom that persist longer than 150 ps. Quantum beats due to the interference between the 7S and 5D_{5/2} energy states were observed by a four-wave mixing process. The difference frequency between the 5D–5P_{3/2} and the 5P_{3/2}–5S transitions was also observed for interpulse delays up to 180 ps. The temporal evolution of the beat amplitude indicates the existence of a superfluorescent process and a scattering process detected simultaneously with the four-wave mixing process. A model was developed to explain the processes and was used to predict the range of pulse parameters for which the scattering process and four-wave mixing could be simultaneously detected.

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Dissociation of a diatomic molecule and the excited-state distribution of the nascent atomic fragments can be detected and characterized by atomic wavepacket interferometry and a coherent nonlinear optical process, such as parametric four-wave mixing (PFWM), in ultrafast pump–probe experiments. Underlying these experiments is a reliance on atom–atom interaction to alter the properties of an atomic wavepacket which, in turn, impacts the phase and amplitude of a coherent optical signal. Specifically, quantum beating in the atomic species provides a sensitive, in situ probe of molecular dissociation by detecting approaching dissociation fragments through long-range dipole–dipole interaction. The resulting influence of this interaction on the amplitude and phase of the quantum beating is observed in temporal or Fourier domains by probing the wavepacket by interferometry and PFWM with 100–150 fs laser pulses. The wavepacket thus serves as a detector of molecular dissociation fragments and the dynamics of atom–atom interactions are converted into the macroscopic domain by the PFWM signal and idler waves. Femtosecond pump–probe experiments are described in which the predissociation of electronically excited Rb2 states in the ~24 000–28 000 cm−1 interval, and the distribution of nascent atomic fragments into Rb excited states (7s, 5d, 6s, 4d and 5p) spanning an energy range >1.25 eV, have been observed in Rb vapour with atomic number densities of ~6 × 1013–3 × 1017 cm−3. Quantum beating at 18.2 THz (corresponding to the Rb 7s–5dJ (J = 5/2) energy defect of ~608 cm−1) is superimposed onto the axially phase matched PFWM signal wave generated at λS ~ 420 nm (Rb 62PJ → 52S1/2 transitions) and recovered by Fourier analysis of the signal wave intensity as the pump–probe time delay (Δt) is scanned. The dominant exit channels for Rb2 predissociation are found to be sensitive to the interval of internuclear separation R in which the molecular wavepacket, produced by the pump pulse through two-photon association of Rb–Rb collision pairs, is localized. Generating Rb2 wavepackets on 3Λu (Λ = Σ, Δ) potential surfaces at large R (~7–9 A) favours the Rb 7s and 5d dissociation channels. In contrast, producing dimer wavepackets localized in the R ~ 3–4 A region suppresses Rb (7s, 5d) generation and favours the production of Rb atomic fragments (primarily 5p) with less internal energy, but maximum velocities of ~15–20 A ps−1. Laser excitation spectroscopy on the nanosecond time scale suggests that the (3)3Δu and (7)3Σ+u states of Rb2 (correlated with Rd (5d) + Rb (6s) in the separated atom limit), and possibly a 3Σ+u state derived from the Rb (7p) + Rb (5s) asymptote, are populated by two-photon absorption of Rb–Rb ground-state collision pairs and predissociation of these levels provides the excited atomic fragments subsequently detected by atomic wavepackets. The data presented here demonstrate the observation of the molecular dissociation transient and the determination of the nascent statistical distribution of atomic product states in a manner that is unencumbered by radiative lifetime or collisional effects. A wavepacket, in tandem with the dipole–dipole interaction and a coherent nonlinear optical process, provides a new avenue for pursuing atom–atom and atom–molecular interactions over a broad range in inter-particle separations.

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Dipole-dipole interactions between excited Rb atoms at long range (approximately 300a0-2150a0) have been observed with molecular wave packets and a coherent nonlinear optical process. Fourier analysis of the parametric four-wave mixing (PFWM) signal wave intensity produced in femtosecond pump-probe experiments demonstrates the appearance of sidebands associated with the Rb 7s-5d(5/2) quantum beating frequency of approximately 18.3 THz. Calculations show that the observed sideband splittings and Fourier domain profiles result from multiple atom, dipole-dipole interactions, and ensembles comprising five or fewer Rb (7s, 6p) atoms account for virtually all of the data.

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Switching from a sequential transition to quantum beating in atomic rubidium pumped by a femtosecond laser

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We demonstrate temporal coherent control in two-photon transitions. A pair of broadband (Δλ∼20nm), ultrashort (Δt∼100fs), collinear pulses with a variable time delay excites the rubidium atoms into the 5D state from the 5S ground state where the two-photon transitions are enhanced by the intermediate level 5P. The excited atoms radiate 5μm (5D–6P) and 420nm (6P–5S) light. As a result of tuning the wavelength of the input laser, a superfluorescence at 420nm exhibits different temporal behaviors. A switching from a beating at the frequency given by the difference between the sequential atomic transitions that involve the 5P3/2 intermediate level, to a quantum beating due to two different two-photon excitation paths, 5S→5P1/2→5D and 5S→5P3/2→5D is observed. Based on the simple atom-field interaction theory, an analytic solution, which qualitatively elucidates experimental results, is obtained.

Journal of the Optical Society of America B•Feb 18, 2011•11 citations
Abstract 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.
\begin{wrapfigure}{r}{0pt} \includegraphics[scale=0.32]{Fig1.eps} \end{wrapfigure} Coherences at 386.2 THz and 384.1 THz, corresponding to the 5d5/2_{5/2}5/2​ – 5p3/2_{3/2}3/2​ and 5p3/2_{3/2}3/2​ – 5s1/2_{1/2}1/2​ difference frequencies, respectively, have been established in the Rb atom during pump-probe experiments involving pairs of identical ~150 fs pulses produced by a Ti:\chem{Al_2O_3} laser and a Michelson interferometer. The interference between the two coherences within the atom is observed through a parametric four-wave mixing process in Rb and detection of the signal wave intensity at 420 nm. Scanning the time delay between the pump and probe pulses over a 600 ps interval produces a sampling rate over 300 THz and allows for a spectral domain resolution of 0.05 cm−1^{-1}−1 to be achieved. The figure at right shows several spectra recorded near 2.1 THz, the (5d5/2_{5/2}5/2​ – 5p3/2_{3/2}3/2​) - (5p3/2_{3/2}3/2​ – 5s1/2_{1/2}1/2​) difference frequency, with varying angle between the pump and probe pulses (i.e., the phase matching angle) and varying Rb background density. A Fano interference window is clearly observed, and analysis of these and similar spectra demonstrates that the amplitude and phase of the coherently-coupled, three Rb state system can be controlled precisely.
Probing the interactions between a mode-locked laser pulse and saturated rubidium vapor with a pump–probe technique has resulted in observations of coherences within the atom that persist longer than 150 ps. Quantum beats due to the interference between the 7S and 5D_{5/2} energy states were observed by a four-wave mixing process. The difference frequency between the 5D–5P_{3/2} and the 5P_{3/2}–5S transitions was also observed for interpulse delays up to 180 ps. The temporal evolution of the beat amplitude indicates the existence of a superfluorescent process and a scattering process detected simultaneously with the four-wave mixing process. A model was developed to explain the processes and was used to predict the range of pulse parameters for which the scattering process and four-wave mixing could be simultaneously detected.
Dissociation of a diatomic molecule and the excited-state distribution of the nascent atomic fragments can be detected and characterized by atomic wavepacket interferometry and a coherent nonlinear optical process, such as parametric four-wave mixing (PFWM), in ultrafast pump–probe experiments. Underlying these experiments is a reliance on atom–atom interaction to alter the properties of an atomic wavepacket which, in turn, impacts the phase and amplitude of a coherent optical signal. Specifically, quantum beating in the atomic species provides a sensitive, in situ probe of molecular dissociation by detecting approaching dissociation fragments through long-range dipole–dipole interaction. The resulting influence of this interaction on the amplitude and phase of the quantum beating is observed in temporal or Fourier domains by probing the wavepacket by interferometry and PFWM with 100–150 fs laser pulses. The wavepacket thus serves as a detector of molecular dissociation fragments and the dynamics of atom–atom interactions are converted into the macroscopic domain by the PFWM signal and idler waves. Femtosecond pump–probe experiments are described in which the predissociation of electronically excited Rb2 states in the ~24 000–28 000 cm−1 interval, and the distribution of nascent atomic fragments into Rb excited states (7s, 5d, 6s, 4d and 5p) spanning an energy range >1.25 eV, have been observed in Rb vapour with atomic number densities of ~6 × 1013–3 × 1017 cm−3. Quantum beating at 18.2 THz (corresponding to the Rb 7s–5dJ (J = 5/2) energy defect of ~608 cm−1) is superimposed onto the axially phase matched PFWM signal wave generated at λS ~ 420 nm (Rb 62PJ → 52S1/2 transitions) and recovered by Fourier analysis of the signal wave intensity as the pump–probe time delay (Δt) is scanned. The dominant exit channels for Rb2 predissociation are found to be sensitive to the interval of internuclear separation R in which the molecular wavepacket, produced by the pump pulse through two-photon association of Rb–Rb collision pairs, is localized. Generating Rb2 wavepackets on 3Λu (Λ = Σ, Δ) potential surfaces at large R (~7–9 A) favours the Rb 7s and 5d dissociation channels. In contrast, producing dimer wavepackets localized in the R ~ 3–4 A region suppresses Rb (7s, 5d) generation and favours the production of Rb atomic fragments (primarily 5p) with less internal energy, but maximum velocities of ~15–20 A ps−1. Laser excitation spectroscopy on the nanosecond time scale suggests that the (3)3Δu and (7)3Σ+u states of Rb2 (correlated with Rd (5d) + Rb (6s) in the separated atom limit), and possibly a 3Σ+u state derived from the Rb (7p) + Rb (5s) asymptote, are populated by two-photon absorption of Rb–Rb ground-state collision pairs and predissociation of these levels provides the excited atomic fragments subsequently detected by atomic wavepackets. The data presented here demonstrate the observation of the molecular dissociation transient and the determination of the nascent statistical distribution of atomic product states in a manner that is unencumbered by radiative lifetime or collisional effects. A wavepacket, in tandem with the dipole–dipole interaction and a coherent nonlinear optical process, provides a new avenue for pursuing atom–atom and atom–molecular interactions over a broad range in inter-particle separations.
Dipole-dipole interactions between excited Rb atoms at long range (approximately 300a0-2150a0) have been observed with molecular wave packets and a coherent nonlinear optical process. Fourier analysis of the parametric four-wave mixing (PFWM) signal wave intensity produced in femtosecond pump-probe experiments demonstrates the appearance of sidebands associated with the Rb 7s-5d(5/2) quantum beating frequency of approximately 18.3 THz. Calculations show that the observed sideband splittings and Fourier domain profiles result from multiple atom, dipole-dipole interactions, and ensembles comprising five or fewer Rb (7s, 6p) atoms account for virtually all of the data.
Two processes of coupled difference-frequency axially phase-matched parametric six-wave mixing are carried out in Rb vapour by two-photon excitation using fs laser pulses, and parametric six-wave mixing signals in the infrared and near infrared regime are detected. The infrared parametric six-wave mixing signals are up-converted into the visible spectral range by sum-frequency mixing with the pump laser in a LHO 3 crystal. Moreover, quantum beating at 608 cm -1 , corresponding to the 7 s – 5d energy difference in Rb, is observed from the sum-frequency signal at 495 nm. As a result, we obtain modulated light signals in the visible, near infrared and infrared spectral ranges, and study the interference between 7s and 5d states of Rb.
We demonstrate temporal coherent control in two-photon transitions. A pair of broadband (Δλ∼20nm), ultrashort (Δt∼100fs), collinear pulses with a variable time delay excites the rubidium atoms into the 5D state from the 5S ground state where the two-photon transitions are enhanced by the intermediate level 5P. The excited atoms radiate 5μm (5D–6P) and 420nm (6P–5S) light. As a result of tuning the wavelength of the input laser, a superfluorescence at 420nm exhibits different temporal behaviors. A switching from a beating at the frequency given by the difference between the sequential atomic transitions that involve the 5P3/2 intermediate level, to a quantum beating due to two different two-photon excitation paths, 5S→5P1/2→5D and 5S→5P3/2→5D is observed. Based on the simple atom-field interaction theory, an analytic solution, which qualitatively elucidates experimental results, is obtained.