(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

This paper presents a review of the current state of the art in the research field of cold and ultracold molecules. It serves as an introduction to the focus issue of New Journal of Physics on Cold and Ultracold Molecules and describes new prospects for fundamental research and technological development. Cold and ultracold molecules may revolutionize physical chemistry and few-body physics, provide techniques for probing new states of quantum matter, allow for precision measurements of both fundamental and applied interest, and enable quantum simulations of condensed-matter phenomena. Ultracold molecules offer promising applications such as new platforms for quantum computing, precise control of molecular dynamics, nanolithography and Bose-enhanced chemistry. The discussion is based on recent experimental and theoretical work and concludes with a summary of anticipated future directions and open questions in this rapidly expanding research field.

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Cold and ultracold molecules: science, technology and applications

This Review discusses the emerging field of mid-infrared frequency comb generation, including technologies based on novel laser gain media, nonlinear frequency conversion and microresonators, as well as the applications of these combs in precision spectroscopy and direct frequency comb spectroscopy. Laser frequency combs are coherent light sources that emit a broad spectrum of discrete, evenly spaced narrow lines whose absolute frequency can be measured to within the accuracy of an atomic clock. Their development in the near-infrared and visible domains has revolutionized frequency metrology while also providing numerous unexpected opportunities in other fields such as astronomy and attosecond science. Researchers are now exploring how to extend frequency comb techniques to the mid-infrared spectral region. Versatile mid-infrared frequency comb generators based on novel laser gain media, nonlinear frequency conversion or microresonators promise to significantly expand the applications of frequency combs. In particular, novel approaches to molecular spectroscopy in the 'fingerprint region', with dramatically improved precision, sensitivity, recording time and/or spectral bandwidth may lead to new discoveries in the various fields relevant to molecular science.

Mid-infrared frequency combs

The control of the broadband frequency comb emitted from a mode-locked femtosecond laser has permitted a wide range of scientific and technological advances--ranging from the counting of optical cycles for next-generation atomic clocks to measurements of phase-sensitive high-field processes. A unique advantage of the stabilized frequency comb is that it provides, in a single laser beam, about a million optical modes with very narrow linewidths and absolute frequency positions known to better than one part in 10(15) (ref. 5). One important application of this vast array of highly coherent optical fields is precision spectroscopy, in which a large number of modes can be used to map internal atomic energy structure and dynamics. However, an efficient means of simultaneously identifying, addressing and measuring the amplitude or relative phase of individual modes has not existed. Here we use a high-resolution disperser to separate the individual modes of a stabilized frequency comb into a two-dimensional array in the image plane of the spectrometer. We illustrate the power of this technique for high-resolution spectral fingerprinting of molecular iodine vapour, acquiring in a few milliseconds absorption images covering over 6 THz of bandwidth with high frequency resolution. Our technique for direct and parallel accessing of stabilized frequency comb modes could find application in high-bandwidth spread-spectrum communications with increased security, high-resolution coherent quantum control, and arbitrary optical waveform synthesis with control at the optical radian level.

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Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb

A laser frequency comb is a broad spectrum composed of equidistant narrow lines. Initially invented for frequency metrology, such combs enable new approaches to spectroscopy over broad spectral bandwidths, of particular relevance to molecules. The performance of existing spectrometers — such as crossed dispersers employing, for example, virtual imaging phase array etalons, or Michelson-based Fourier transform interferometers — can be dramatically enhanced with optical frequency combs. A new class of instruments, such as dual-comb spectrometers without moving parts, enables fast and accurate measurements over broad spectral ranges. The direct self-calibration of the frequency scale of the spectra within the accuracy of an atomic clock and the negligible contribution of the instrumental line-shape will enable determinations of all spectral parameters with high accuracy for stringent comparisons with theories in atomic and molecular physics. Chip-scale frequency comb spectrometers promise integrated devices for real-time sensing in analytical chemistry and biomedicine. This Review gives a summary of the developments in the emerging and rapidly advancing field of atomic and molecular broadband spectroscopy with frequency combs. Frequency comb spectroscopy is a recent field of research that has blossomed in the past five years. This Review discusses developments in the emerging and rapidly advancing field of atomic and molecular broadband spectroscopy with frequency combs.

Frequency comb spectroscopy

Ultrashort laser pulses have thus far been used in two distinct modes. In the time domain, the pulses have allowed probing and manipulation of dynamics on a subpicosecond time scale. More recently, phase stabilization has produced optical frequency combs with absolute frequency reference across a broad bandwidth. Here we combine these two applications in a spectroscopic study of rubidium atoms. A wide-bandwidth, phase-stabilized femtosecond laser is used to monitor the real-time dynamic evolution of population transfer. Coherent pulse accumulation and quantum interference effects are observed and well modeled by theory. At the same time, the narrow linewidth of individual comb lines permits a precise and efficient determination of the global energy-level structure, providing a direct connection among the optical, terahertz, and radio-frequency domains. The mechanical action of the optical frequency comb on the atomic sample is explored and controlled, leading to precision spectroscopy with an appreciable reduction in systematic errors.

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United Time-Frequency Spectroscopy for Dynamics and Global Structure

We summarize recent developments in direct frequency-comb spectroscopy that allowed high-resolution, broad-bandwidth measurements of multiple atomic and molecular resonances using only a phase-stabilized femtosecond laser, opening the way for merging precision spectroscopy with coherent control.

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Direct frequency comb spectroscopy

A phase-stabilized femtosecond laser comb is directly used for high-resolution spectroscopy and absolute optical frequency measurements of one- and two-photon transitions in laser-cooled $^{87}\mathrm{Rb}$ atoms. Absolute atomic transition frequencies, such as the $5{S}_{1/2}$ $F=2\ensuremath{\rightarrow}7{S}_{1/2}$ ${F}^{\ensuremath{'}\ensuremath{'}}=2$ two-photon resonance measured at 788 794 768 921 (44) kHz, are determined without a priori knowledge about their values. Detailed dynamics of population transfer driven by a sequence of pulses are uncovered and taken into account for the measurement of the $5P$ states via resonantly enhanced two-photon transitions.

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Direct frequency comb measurements of absolute optical frequencies and population transfer dynamics.

We present a general and highly efficient scheme for performing narrow-band Raman transitions between molecular vibrational levels using a coherent train of weak pump-dump pairs of shaped ultrashort pulses. The use of weak pulses permits an analytic description within the framework of coherent control in the perturbative regime, while coherent accumulation of many pulse pairs enables near unity transfer efficiency with a high spectral selectivity, thus forming a powerful combination of pump-dump control schemes and the precision of the frequency comb. Simulations verify the feasibility and robustness of this concept, with the aim to form deeply bound, ultracold molecules.

Precise control of molecular dynamics with a femtosecond frequency comb.

We demonstrate high resolution coherent control of cold atomic rubidium utilizing spectral phase manipulation of a femtosecond optical frequency comb. Transient coherent accumulation is directly manifested by the enhancement of signal amplitude and spectral resolution via the pulse number. The combination of frequency comb technology and spectral phase manipulation enables coherent control techniques to enter a new regime with natural linewidth resolution.

Matthew C. Stowe

Papers

United Time-Frequency Spectroscopy for Dynamics and Global Structure

Direct frequency comb spectroscopy

Direct frequency comb measurements of absolute optical frequencies and population transfer dynamics.

Precise control of molecular dynamics with a femtosecond frequency comb.

High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb.