NOTE: End date changed to 8/31/2020 per U. Israelsson/JPL (Ed., 8/6/19)

Method: Rb atoms are cooled to below 1 microK and laser-excited into a low-angular-momentum Rydberg level with principal quantum number n on the order of 40. Using an adiabatic atomic-state transformation method, the Rydberg atoms are then transferred into the circular state |n,n1=0,n2=0,m=n-1> (m, n1, and n2 are the magnetic and parabolic quantum numbers). An optical lattice (wavelength 1064 nm or 532 nm on the ground, 850 nm at CAL) is adiabatically ramped on. The Rydberg atoms are trapped in the lattice via the ponderomotive interaction of the Rydberg electron with the lattice light. During the hold time of the atom cloud in the lattice, the electric-quadrupole transition into the state |n+2,n1=1,n2=1,m=n-1> is driven by amplitude-modulating the optical lattice at the transition frequency. The transition is probed using state-selective electric-field ionization. The measured transition frequency allows one to extract the Rydberg constant. Key advantages of the method are that the transition is free of first-order Stark and Zeeman shifts, that QED, nuclear-penetration, hyperfine, and other shifts are small, and that the optical lattice causes only moderate trap-induced transition shifts. Theoretical work includes the modeling of the circular-state preparation using multi-level adiabatic passage and of the circular-state transition dynamics in a shallow, microwave-modulated optical lattice, in which center-of-mass wave-packet expansion and tunneling must be considered.

Relevance to NASA and relevance of microgravity: High-precision measurement of fundamental constants represents an important part of contemporary atomic-physics research. The proposed high-precision measurement of the Rydberg constant adds a new component to the NASA Fundamental Physics program, and to the Cold Atom Laboratory (CAL) research portfolio. To achieve optimal spectroscopic performance it is critical to reduce systematic lattice-induced transition shifts and related shifts. Since the most critical shifts diminish with decreasing optical-lattice depth, the lattices must be chosen as shallow as possible. Microgravity conditions are important because they enable the use of shallow lattices without the atoms falling out. The microgravity environment provided by the CAL/ its successor instrument, including its 850 nm optical-lattice laser, offers an ideal platform for the project to reach its ultimate uncertainty goal. An International Space Station (ISS)-based setup could also provide a sufficient period of measurement time to acquire good statistics in the high-precision spectra. More generally, the research could bring Rydberg atoms into space. This has far-reaching additional potential for the development of novel quantum technologies for microwave / THz / thermal-field sensing and imaging, and well as for exploration of the physics of very dilute, but strongly interacting Bose Einstein Condensates.

The research was concentrated on the investigation of ponderomotive optical lattices for Rydberg atoms and on ways how to drive Rydberg-atom transitions by microwave-modulating Rydberg-atom lattices. Several high-precision measurements of Rydberg atoms were conducted, and related theoretical studies were performed. Near the end of the performance period, the team published on tractor atom interferometry, a novel atom-interferometric method. In the following, we provide a brief timeline of the activities under this grant. The publications referenced provide details on the work performed. [Ed. Note: See Cumulative Bibliography for complete citations.]

2015: In ponderomotive spectroscopy, an amplitude-modulated optical standing wave was employed to probe Rydberg-atom transitions, utilizing a ponderomotive rather than a dipole-field interaction. Here, we engaged nonlinearities in the modulation to drive dipole-forbidden transitions up to the fifth order. We reached transition frequencies approaching the sub-THz regime. We also demonstrated magic-wavelength conditions, which resulted in symmetric spectral lines with a Fourier-limited peak at the line center. Applicability to precision measurement were discussed, which include a measurement of the Rydberg constant using circular Rydberg atoms in optical lattices. The work was published in Phys. Rev. Lett. 115, 163003 (2015).

2016 and 2017: We have conducted a study of the systematics that will affect a measurement of the Rydberg constant using circular Rydberg atoms in an intensity-modulated optical lattice. These states have long lifetimes, as well as negligible quantum electrodynamics (QED), and no nuclear-overlap corrections. Due to these advantages, the measurement can yield a Rydberg-constant value that is free of interdependencies with QED calculations and the nuclear-charge distribution. In our 2017 study, we considered circular Rydberg atoms that are trapped in an optical lattice and that are driven using our lattice-modulation technique to perform Doppler-free spectroscopy. We have optimized laser wavelengths and beam geometries to realize magic-trapping conditions. The selected transitions have no first-order Zeeman and Stark corrections, leaving only manageable second-order Zeeman and Stark shifts. For rubidium, the projected relative uncertainty of the Rydberg constant in a measurement under Earth’s gravity is 10^−11, with the main contribution to the uncertainty coming from the residual lattice shifts. In a future microgravity implementation, this could be reduced into the low 10^-12 range. The study was published in Phys. Rev. A 96, 032513 (2017) and was reported at the CPEM 2018 in Paris ( https://doi.org/10.1109/CPEM.2018.8501136 ).

2018: To better understand the physics of Rydberg atoms at high spectroscopic precision, we have measured the hyperfine structure of rubidium-85 nS1/2 Rydberg states for n = 43, 44, 45, and 46. From the splittings, the hyperfine coupling constant, AHFS, was determined to be 15.372(80) GHz. This result represented an order-of-magnitude improvement from previous measurements. The work was published in Phys. Rev. A 100, 062515 (2019).

2019 and 2020: A leading challenge in performing spectroscopy and quantum information science with circular Rydberg atoms is to prepare such atoms. While we have done this before with low-frequency electromagnetic fields, a method that dovetails much better with Rydberg atoms trapped in an optical lattice is to use the lattice itself to do the circularization. To this end, we have theoretically explored three schemes of initializing circular-state Rydberg atoms via optical couplings provided by the optical ponderomotive effect. In our first method, a radial optical trap consisting of two Laguerre-Gaussian beams of opposite winding numbers transfers orbital angular momentum to the Rydberg atom, providing a first-order coherent coupling between an F-type Rydberg state and a circular state. Next, we considered a one-dimensional ponderomotive optical lattice modulated at RF frequencies, providing quadrupole-like couplings in the hydrogenic manifold for rapid adiabatic passage through a series of intermediate Rydberg states into the circular state. In the third scheme, a two-dimensional ponderomotive optical lattice with a time-orbiting trap center induces an effective circularly polarized RF field of tunable purity for all-optical rapid adiabatic passage into the circular state. The work was published in Phys. Rev. A 101, 013434 (2020).

Over the course of the project, we have delved deeper into developing a thorough understanding of the sub-Doppler characteristics of lattice-modulation spectroscopy of Rydberg atoms. We developed and studied quantum and semiclassical models of Rydberg-atom spectroscopy in amplitude-modulated lattices. Both initial- and target-state Rydberg atoms were assumed to be trapped in the lattice. Unlike in other spectroscopic schemes, the modulation-induced ponderomotive coupling between the Rydberg states is spatially periodic and perfectly phase-locked to the lattice trapping potentials. This leads to a novel type of sub-Doppler mechanism which we have explained in detail. In our exact quantum model, we solved the time-dependent Schrödinger equation in the product space of center-of-mass (COM) momentum states and the internal-state space. We also developed a perturbative model based on the band structure in the lattice and Fermi’s golden rule, as well as a semiclassical trajectory model in which the COM was treated classically and the internal-state dynamics quantum-mechanically. In all models we obtained the spectrum of the target Rydberg-state population versus the lattice modulation frequency. Applications in Rydberg-atom spectroscopy were discussed. The work was published in Phys. Rev. A 101, 033414 (2020).

In our third 2020 study, we continued our push towards higher angular momentum states by measuring the quantum defects of G Rydberg states of rubidium. This is important in view of measuring the Rydberg constant because the quantum defects of moderately high angular momentum states, such as G and H, reveal the atoms’ core polarizability, which will be needed for Rydberg-constant measurements at the < 10^-11 level using circular atoms. We utilized two-photon, high-precision microwave spectroscopy of nG→ (n + 2)G transitions to precisely measure the high-angular-momentum G-series quantum defect of 85Rb. Samples of cold Rydberg atoms in the nG state were prepared via a three-photon optical excitation combined with controlled electric-field mixing and probed with microwave pulses. We obtained values for the G quantum defects of delta0 = 0.0039990(21) and = −0.0202(21). The work was published in Phys Rev. A 102, 062817 (2020).

2021: In view of earlier works on atom interferometry, and motivated through various NASA Fundamental Physics (FP) conferences, the team became interested in following up on earlier thoughts about novel methods in atom interferometry. We have devised a method called tractor atom interferometry (TAI). TAI employs uninterrupted 3D confinement of interfering atomic wave packets in user-controlled tractor potentials throughout the AI sequence. TAI provides guaranteed closure, suppresses wave-packet dispersion, and is robust against coherence loss from environmental platform dynamics. It allows for arbitrary user-controlled AI paths for maximal sensitivity to certain observables while suppressing contributions from forces that may not be of interest. Designer tractor trajectories can eliminate satellite rotation effects or implement arbitrary hold times. For instance, TAI can selectively maximize sensitivity to acceleration, or rotation, or the gravity-gradient etc. Using Crank-Nicolson simulation of the time-dependent Schrödinger equation, we computed the quantum evolution of scalar and spinor wave functions in several TAI sample scenarios. The work was published in Phys. Rev. A 104, 013307 (2021).

In another piece of 2021 work, we have investigated subtle yet fundamentally interesting aspects of photoionization of Rydberg atoms in optical lattices. We have developed a formalism for photoionization (PI) and potential energy curves (PECs) of Rydberg atoms in ponderomotive optical lattices and applied it to examples covering several regimes of the optical-lattice depth and atomic angular momentum. Characterization of PI in GHz-deep Rydberg-atom lattices is expected to be beneficial for optical control and quantum-state manipulation of Rydberg atoms, while data on PI in shallower lattices are useful in high-precision spectroscopy, such as measurement of the Rydberg constant, and in quantum-computing applications of lattice-confined Rydberg atoms. The work was published in the New Journal of Physics 23, 063074 (2021).

2022: The project finished up with hyperfine measurements of nP1/2 Rydberg states of Rb-85. We have measured the hyperfine structure of nP1/2 Rydberg states for n = 42, 43, 44, and 46 using cold-atom mm-wave spectroscopy. Our result on the nP1/2 hyperfine coupling constant was AHFS = 1.443(31) GHz. This measurement is expected to be useful in studies of long-range Rydberg molecules, Rydberg electrometry, and quantum simulation with dipole-dipole interactions involving nP1/2 atoms. The work was published in Phys. Rev. A 106, 052810 (2022).

2023: A patent was filed on tractor atom interferometry. See https://patents.google.com/patent/US20230056032A1/en Reference: Raithel, Georg A., and Alisher Duspayev, “Tractor Atom Interferometry.” U.S. Patent Application No. 17/891,673.

Through the lifetime of the grant, three students graduated (Kaitlin Moore, Ryan Cardman, and Andira Ramos). They all now hold positions in research and development (R&D) companies in quantum sensing (Kaitlin Moore, Ryan Cardman) and in the space industry (Andira Ramos).

Method: Rb atoms are cooled to below 1 microK and laser-excited into a low-angular-momentum Rydberg level with principal quantum number n on the order of 40. Using an adiabatic atomic-state transformation method, the Rydberg atoms are then transferred into the circular state |n,n1=0,n2=0,m=n-1> (m, n1, and n2 are the magnetic and parabolic quantum numbers). An optical lattice (wavelength 1064 nm or 532 nm on the ground, 850 nm at CAL) is adiabatically ramped on. The Rydberg atoms are trapped in the lattice via the ponderomotive interaction of the Rydberg electron with the lattice light. During the hold time of the atom cloud in the lattice, the electric-quadrupole transition into the state |n+2,n1=1,n2=1,m=n-1> is driven by amplitude-modulating the optical lattice at the transition frequency. The transition is probed using state-selective electric-field ionization. The measured transition frequency allows one to extract the Rydberg constant. Key advantages of the method are that the transition is free of first-order Stark and Zeeman shifts, that QED, nuclear-penetration, hyperfine, and other shifts are small, and that the optical lattice causes only moderate trap-induced transition shifts. Theoretical work includes the modeling of the circular-state preparation using multi-level adiabatic passage and of the circular-state transition dynamics in a shallow, microwave-modulated optical lattice, in which center-of-mass wave-packet expansion and tunneling must be considered.

Relevance to NASA and relevance of microgravity: High-precision measurement of fundamental constants represents an important part of contemporary atomic-physics research. The proposed high-precision measurement of the Rydberg constant adds a new component to the NASA Fundamental Physics program, and to the Cold Atom Laboratory (CAL) research portfolio. To achieve optimal spectroscopic performance it is critical to reduce systematic lattice-induced transition shifts and related shifts. Since the most critical shifts diminish with decreasing optical-lattice depth, the lattices must be chosen as shallow as possible. Microgravity conditions are important because they enable the use of shallow lattices without the atoms falling out. The microgravity environment provided by the CAL/ its successor instrument, including its 850 nm optical-lattice laser, offers an ideal platform for the project to reach its ultimate uncertainty goal. An International Space Station (ISS)-based setup could also provide a sufficient period of measurement time to acquire good statistics in the high-precision spectra. More generally, the research could bring Rydberg atoms into space. This has far-reaching additional potential for the development of novel quantum technologies for microwave / THz / thermal-field sensing and imaging, and well as for exploration of the physics of very dilute, but strongly interacting Bose Einstein Condensates.

Experiment

(1) The cryogenic spectroscopic setup was used to perform measurements of the hyperfine structure splittings of nS1/2 Rydberg states of 85-Rb for principal quantum numbers n=43, 44, 45, and 46. We have obtained Fourier-limited line shapes and determined the line-centers of the hyperfine levels F=2 and F=3 to within 2 kHz or below. The hyperfine splitting is given by A_HFS/(n^*)^-3, with the effective principal quantum number n^* and the hyperfine coupling constant, A_HFS. From the measured splittings, the hyperfine coupling constant A_HFS has been determined to be 15.372(84) GHz. This result is an order-of-magnitude improvement from previous measurements from other groups. We have studied and accounted for systematic uncertainty sources, such as unwanted electric and magnetic fields, van der Waals interactions, and AC shifts. A manuscript is in preparation.

(2) To allow for the above measurement, the three-photon excitation scheme was slightly modified from our earlier works. For the hyperfine measurement, we used an excitation scheme 5S F=3 to 5P3/2 F=4 (780 nm), and 5P3/2 F=4 to 5D5/2 F=* (776 nm), and 5D5/2 F=* to nP3/2, F=2 to 4 (1260 nm), where F refers to the respective hyperfine quantum numbers. We have then used microwaves to drive the nP3/2 to nS1/2 (F=2 and F=3) transitions. There, the P Rydberg state has negligible hyperfine structure, while the nS1/2 hyperfine structure is well resolved and allows us to measure A_HFS. To be able to access the nS1/2 F=2 level, the 5P3/2-5D5/2 (776 nm) was driven off-resonantly by about 100 MHz. In this way, the F=2, 3 and 4 states of nP3/2 could be populated. This, in turn, allows microwave transitions into both F-states of nS1/2, enabling measurements of the hyperfine splitting. The method also has the advantage that the 5D5/2 state does not become strongly populated, which leads to a reduction of background signal from Penning ionization of 5D5/2 atoms.

(3) Zeroing of E and B fields in the spectroscopy enclosure. We developed protocols to reduce the background electric and magnetic fields to 0.2 mV/cm and 2 mG using microwave spectroscopy of Rydberg states. Microwave transitions between Rydberg states are much narrower than optical transitions (on the order of 10 kHz vs several MHz), allowing a much improved field zeroing. This level of field zeroing was required for the hyperfine structure measurement, and it also is a prerequisite for circular-state production and spectroscopy.

Theory

(4) We have developed theoretical models for optical circularization of Rydberg atoms using three schemes based on amplitude-modulated optical lattices. In the first method, the Rydberg atoms are circularized in a single (first-order) interaction step by placing them in the field of two Laguerre-Gaussian beams with opposite winding numbers and a frequency difference that equals the transition energy between the initial, low-angular-momentum state (which can be an S, P, or D state) and the desired circular state. The transition frequency is in the order of a THz. The manuscript includes a proposal how this setup could be realized.

In the second method, the atoms are circularized by a sequence of steps akin to the standard rapid adiabatic passage method. Each individual step is a quadrupolar transition effected by an amplitude-modulated ponderomotive optical lattice that increases angular momentum by two units. Since the required modulation frequency is only on the order of a few 100 MHz, the method should be easy to implement.

The third method is an extension of the second in which two orthogonal lattices are phase-modulated in a way that leads to a time-orbiting, sub-micron-sized ponderomotive potential well with an orbiting frequency of several 100 MHz. The resultant poderomotive transitions driven in the atoms are equivalent to transitions driven by a purely circularly polarized RF field. Such fields yield the most robust results in Rydberg-atom circularization, as has been shown in Haroche’s group.

(5) Modeling of lattice modulation spectroscopy in the ultracold regime

The experimental work is accompanied by a theoretical effort lead by the co-Principal Investigator (PI) V. Malinovsky, with some associated modeling work provided by the PI. The present objective of the theory work is to model the spectroscopic line shapes of Rydberg transitions in an amplitude-modulated Rydberg-atom lattice in a manner that takes the quantization of the center-of-mass motion into account. This ability is an important ingredient to determine the line center with 1Hz uncertainty, as required in the present project. To take into account the quantization of motion of the Rydberg atoms in the optical lattice, a theoretical model has been developed in which both ground and excited wave functions are subject to the periodic potentials, representing the lattice fields, while the coupling (the effective Rabi frequency) is also periodic as function of the translational coordinate. To obtain the spectrum of the excited-state population, the time-dependent Schroedinger equation has been solved in the momentum representation. The spectra are also modeled using Fermi’s Golden Rule on transitions between thermally populated spinor Bloch states in the optical lattice. The results are all in place. Current work is still focused on finalizing two manuscripts on the theory work. The emphases are to (a) make it very clear why this is a new method of Doppler-free spectroscopy, (b) emphasize the utility and the limits when applied in very shallow lattices afforded under microgravity conditions, and (c) to discuss the portability to other spectral regions and molecular-size domains (such as molecules in x-ray lattices that are modulated at optical frequencies).

(6) A paper on three-photon electromagnetically induced transparency (EIT) in a vapor cell has been submitted. The work was conducted by Kaitlin Moore (NASA-funded), an additional National Science Foundation-funded graduate student, and two undergraduate students. This work was loosely related to the NASA-funded work in that some equipment end considerable expertise was shared.

Students

(7) Another graduate student has graduated from the project. After Kaitlin Moore graduated in 2018, this year Andira Ramos has graduated. The training of the new student (Ryan Cardman) has been completed.

Conference presentations

(8) The work was presented at the following national and international conferences (1) the annual NASA fundamental physics conference, (2) the ASGSR (American Society for Gravitational & Space Research) conference in Bethesda, Maryland, the annual Division of Atomic, Molecular and Optical Physics (DAMOP) conference of the American Physical Society (APS), and the CPEM 2018 in Paris, France (Conference on precision electromagnetic measurements). This conference was focused on fundamental constants and the redefinition of the SI (International System of Units), and had a session on the proton radius.

Method: Rb atoms are cooled to below 1 microK and laser-excited into a low-angular-momentum Rydberg level with principal quantum number n on the order of 40. Using an adiabatic atomic-state transformation method, the Rydberg atoms are then transferred into the circular state |n,n1=0,n2=0,m=n-1> (m, n1, and n2 are the magnetic and parabolic quantum numbers). An optical lattice (wavelength 1064 nm or 532 nm on the ground, 850 nm at CAL) is adiabatically ramped on. The Rydberg atoms are trapped in the lattice via the ponderomotive interaction of the Rydberg electron with the lattice light. During the hold time of the atom cloud in the lattice, the electric-quadrupole transition into the state |n+2,n1=1,n2=1,m=n-1> is driven by amplitude-modulating the optical lattice at the transition frequency. The transition is probed using state-selective electric-field ionization. The measured transition frequency allows one to extract the Rydberg constant. Key advantages of the method are that the transition is free of first-order Stark and Zeeman shifts, that QED, nuclear-penetration, hyperfine, and other shifts are small, and that the optical lattice causes only moderate trap-induced transition shifts. Theoretical work includes the modeling of the circular-state preparation using multi-level adiabatic passage and of the circular-state transition dynamics in a shallow, microwave-modulated optical lattice, in which center-of-mass wave-packet expansion and tunneling must be considered.

Relevance to NASA and relevance of microgravity: High-precision measurement of fundamental constants represents an important part of contemporary atomic-physics research. The proposed high-precision measurement of the Rydberg constant adds a new component to the NASA Fundamental Physics program, and to the Cold Atom Laboratory (CAL) research portfolio. To achieve optimal spectroscopic performance it is critical to reduce systematic lattice-induced transition shifts and related shifts. Since the most critical shifts diminish with decreasing optical-lattice depth, the lattices must be chosen as shallow as possible. Microgravity conditions are important because they enable the use of shallow lattices without the atoms falling out. The microgravity environment provided by the CAL/ its successor instrument, including its 850 nm optical-lattice laser, offers an ideal platform for the project to reach its ultimate uncertainty goal. An International Space Station (ISS)-based setup could also provide a sufficient period of measurement time to acquire good statistics in the high-precision spectra. More generally, the research could bring Rydberg atoms into space. This has far-reaching additional potential for the development of novel quantum technologies for microwave / THz / thermal-field sensing and imaging, and well as for exploration of the physics of very dilute, but strongly interacting Bose Einstein Condensates.

Progress: (1) Comprehensive study of the systematics in the pursued high-precision measurement.

Understanding the systematics is of utmost importance in any high-precision measurement (HPM). The team has been working on effort since several years. There was no rush in getting this published. It was far more important that the team was convinced the results are correct, that the list of systematics is complete, and that all systematics are thoroughly understood and well modeled. The paper was published in 2017, “Measuring the Rydberg constant using circular Rydberg atoms in an intensity-modulated optical lattice,” Andira Ramos, Kaitlin Moore, and Georg Raithel, PHYSICAL REVIEW A 96, 032513 (2017) and arXiv:1705.02682 [physics.atom-ph]. The Principal Investigator (PI) hopes that there may be useful feedback and possibly additional insights coming back from the community.

Progress (2). Quantum-defect and core polarizability measurements.

One of the systematics is associated with the ionic core polarizability of rubidium. The team is working on this and has submitted a paper titled “Measurement of Rb G-series quantum defect using two-photon nG to (n+2)G microwave spectroscopy.” Two-photon high-precision spectroscopy is engaged to precisely measure the high-angular-momentum G-series quantum defect of Rb-85. The transition frequencies are in the sub-THz regime.

Measurements of atomic transition frequencies are the cornerstone of precision spectroscopy, used in applications ranging from atomic clocks to measuring gravitational redshifts and the radius of the proton. Often, cold atoms are used in these measurements. Alkali atoms, which have a single valence electron similar to hydrogen, are easier to laser-cool than hydrogen due to a lower recoil energy and near-infrared cooling-transition wavelengths. However, in an alkali atom such as rubidium (Rb), the interaction between the ionic core of the atom and the valence electron depresses the energy levels of the valence electron below the expected hydrogenic levels (the "quantum defect"). In Rydberg-atom-based high precision spectroscopy, it is imperative to precisely quantify this quantum defect for the commonly-used alkali species. For electrons in high-angular-momentum states, the quantum defect is dominated by the polarizability of the ionic core, which may be extracted from high-angular-momentum defect measurements.

Since in the current NASA-funded project all of the above applies, the team pursues HPM of rubidium quantum defects and dipolar core polarizabilities (the quadrupolar polarizability term in not important for circular-state Rydberg atoms). In the most recent previous experimental measurement of the nG-series quantum defect, microwave spectroscopy of nD to (n+1)G [J. Lee, J. Nunkaew, and T. Gallagher, Phys. Rev. A 94 022505 (2016)] was performed, whereas in the submitted work sub-THz spectroscopy was used to measure nG to (n+2)G transitions in a field-free environment. Furthermore, the presently used two-photon transition depends only on one set of quantum defects and takes advantage of equal Lande g-factors in the lower and upper states, which is a great advantage.

In the present experiment, the samples of cold nG Rydberg atoms were prepared via a three-photon optical excitation and controlled electric-field mixing and probed with 40-microsecond long microwave interaction pulses. Results were compared with other recent measurements, performed elsewhere, and the new (our) measurement were found to be consistent and one order of magnitude more precise. The experimental procedure included a careful cancellation of stray electric fields in all three dimensions. In the submitted paper, future extensions towards precision measurements of atomic polarizabilities are discussed.

Progress (3) Modeling of lattice modulation spectroscopy in the ultracold regime.

The experimental work is accompanied by a theoretical effort lead by the co-PI V. Malinovsky, with some associated modeling work provided by the PI. The present objective of the theory work is to model the spectroscopic line shapes of Rydberg transitions in an amplitude-modulated Rydberg-atom lattice in a manner that takes the quantization of the center-of-mass motion into account. This ability is an important ingredient to determine the line center with 1Hz uncertainty, as required in the present project. To take into account the quantization of motion of the Rydberg atoms in the optical lattice, a theoretical model has been developed in which both ground and excited wave functions are subject to the periodic potentials, representing the lattice fields, while the coupling (the effective Rabi frequency) is also periodic as function of the translational coordinate. To obtain the spectrum of the excited-state population, the time-dependent Schroedinger equation has been solved in the momentum representation. The spectra are also modeled using Fermi’s Golden Rule on transitions between thermally populated spinor Bloch states in the optical lattice. The results are all in place. Current work is focused on finalizing two manuscripts on the theory work. The emphases are to (a) make it very clear why this is a new method of Doppler-free spectroscopy, (b) emphasize the utility and the limits when applied in very shallow lattices afforded under microgravity conditions, and (c) to discuss the portability to other spectral regions and molecular-size domains (such as molecules in x-ray lattices that are modulated at optical frequencies).

Progress (4) Experimental tasks.

In the last year, the team has installed a 1260-nm laser amplifier from Thorlabs and procured two laser systems from Moglabs (a 776-nm and a 1260-nm master laser). The 776-nm system has already been delivered and is being tested. The lasers have a specified linewidth of 200 kHz, when locked using the built-in Pound-Drever-Hall lock (which feeds back on an internal laser etalon and the laser current). The senior student (Kaitlin Moore) has graduated, and a new student is being trained (Ryan Cardman). To minimize the effect of the student turnover on the progress of the project, I have assigned a second senior student (half-time) to the project. The current spectroscopic work is focused on electric-field zeroing and Rydberg-atom circularization.

As a start into circular-state spectroscopy work, the team has procured a microwave frequency doubler, waveguides and a microwave horn for two-photon microwave spectroscopy of circular-state transitions of the type n to n+1 (off-resonant) to n+2. There also is continued sub-THz spectroscopy work on the nH-series quantum defect.

Progress (5). Conference submissions.

The work was presented at national and international conferences and workshops (last at the Atomic Physics workshop in Dresden, Germany). The work was presented at Division of Atomic, Molecular and Optical Physics (DAMOP) conference 2017 (K1.00082 Blackbody effects in high-precision microwave spectroscopy with circular Rydberg atoms, and Q1.00024 Progress towards measuring the Rydberg constant with circular Rydberg atoms), and will also be presented at DAMOP 2018, and the NASA research conference in La Jolla. The PI has submitted a two-page paper abstract to the CPEM 2018 (Conference on precision electromagnetic measurements). This conference will have a session focused on fundamental constants and the proton radius. The accepted conference papers will be published on Institute of Electrical and Electronics Engineers (IEEE).

Method: Rb atoms are cooled to below 1 microK and laser-excited into a low-angular-momentum Rydberg level with principal quantum number n on the order of 40. Using an adiabatic atomic-state transformation method, the Rydberg atoms are then transferred into the circular state |n,n1=0,n2=0,m=n-1> (m, n1, and n2 are the magnetic and parabolic quantum numbers). An optical lattice (wavelength 1064 nm or 532 nm on the ground, 850 nm at CAL) is adiabatically ramped on. The Rydberg atoms are trapped in the lattice via the ponderomotive interaction of the Rydberg electron with the lattice light. During the hold time of the atom cloud in the lattice, the electric-quadrupole transition into the state |n+2,n1=1,n2=1,m=n-1> is driven by amplitude-modulating the optical lattice at the transition frequency. The transition is probed using state-selective electric-field ionization. The measured transition frequency allows one to extract the Rydberg constant. Key advantages of the method are that the transition is free of first-order Stark and Zeeman shifts, that QED, nuclear-penetration, hyperfine and other shifts are small, and that the optical lattice causes only moderate trap-induced transition shifts. Theoretical work includes the modeling of the circular-state preparation using multi-level adiabatic passage and of the circular-state transition dynamics in a shallow, microwave-modulated optical lattice, in which center-of-mass wave-packet expansion and tunneling must be considered.

Relevance to NASA and relevance of microgravity: High-precision measurement of fundamental constants represents an important part of contemporary atomic-physics research. The proposed high-precision measurement of the Rydberg constant would add a new component to the Cold Atom Laboratory (CAL) research portfolio. To achieve optimal spectroscopic performance it is critical to reduce systematic lattice-induced transition shifts and related shifts. Since the most critical shifts diminish with decreasing optical-lattice depth, the lattices must be chosen as shallow as possible. Microgravity conditions are important because they enable the use of shallow lattices without the atoms falling out. The microgravity environment provided by the CAL, including its 850 nm optical-lattice laser, offers an ideal platform for the project to reach its ultimate uncertainty goal. The CAL setup may also provide an extended period of measurement time to acquire good statistics in the high-precision spectra.

The Quanta Magazine reported on August 11, 2016: “This [new physics] “would, of course, be fantastic,” said Randolf Pohl of the Max Planck Institute of Quantum Optics in Garching, Germany, who led both the 2010 experiment and the new study. “But the most realistic thing is that it’s not new physics.” ”.

And it is further reported “His [Pohl’s] personal guess is that physicists have misgauged the Rydberg constant, a factor that goes into calculating the expected differences between atomic energy levels.”

SOURCE: https://www.quantamagazine.org/20160811-new-measurement-deepens-proton-radius-puzzle/ So, as of August 2016, the proton radius puzzle has gotten worse.

This situation has reinforced the need to confirm the accuracy of the Rydberg constant. Previous precision experiments with this goal have involved low-lying states of hydrogen, limited typically by statistical uncertainties and second-order Doppler shifts. These have led to the best current available relative uncertainty for the Rydberg constant value of 5.9 x 10^-12 [Codata]. There has also been a study involving circular Rydberg states of hydrogen (relative uncertainty of 2.1 x 10^-11, by deVries and Kleppner et al.) and a proposal involving circular states of lithium, with an expected relative uncertainty of about 10^-10 (Haroche and coworkers). The main limitations for the former have been the presence of a non-zero Zeeman shift (2.6 Hz), short interaction times and a non-negligible second-order Doppler shift (1.4 Hz). The approaches involving low-lying states and circular states deal with significantly different frequency regimes: optical versus microwave. Therefore, measurements involving low-lying states of hydrogen can, in principle, have a better relative uncertainty than results for circular states (under the assumption of similar absolute uncertainty). However, circular states are insensitive to critical systematics that are limiting in spectroscopy of low-lying states. These are the nuclear penetration (the origin of the proton radius puzzle) and QED shifts.

The project “High-Precision Microwave Spectroscopy of Long-Lived Circular-State Rydberg Atoms in Microgravity” deals with measuring the Rydberg constant and other atomic constants, such as polarizabilities, using long-lived circular Rydberg atoms. The large size of the Rydberg atoms allows one to employ a fundamentally new method of spectroscopy. The method is based on harnessing a term known as the “A-square term” for the purpose of spectroscopy. The method requires one to have a modulated light field with a spatial modulation period that is on the same order as the diameter of the atom that is being probed. At the same time, the field must be periodically modulated in time, with a modulation frequency that matches the frequency of the atomic transition that is to be probed. Rydberg atoms in electro-optically modulated standing wave light fields satisfy both conditions. A manuscript, with an exhaustive list of expected systematics, is close to submission. This manuscript will present the project to the wider scientific community. Some useful new input may result from this.

In extracting the Rydberg constant, the polarization quantum defect and the core polarizability of rubidium, the atomic species used in the work, must be known better than it is currently known (by up to a factor of ten, depending which published numbers one uses). The core polarizability accounts for the fact that the atomic core (the Rb+ ion in this case) becomes electrically polarized by the Rydberg electron that orbits the Rb+ ion. The core polarization leads to Rydberg level shifts on the order of 10 kHz. The small, simplified spectroscopy chamber set up in 2015 in the current project has been used in 2016 to perform a precision measurement of the rubidium g-level quantum defects. This work has been completed and is close to submission. One graduate student has, in large parts, graduated with this result. In her work, results are compared with another recent measurement, which is by Gallagher et al. [J. Lee, J. Nunkaew, and T. Gallagher, Physical Review A 94, 022505 (2016)]. The measurements are consistent with each other, and the present one (ours) is about one order of magnitude more precise.

In the last year, the team has continued to work on a new cryogenic spectroscopic setup that is supposed to last through the entire performance period of about 3 more years. The setup features a three-dimensional optical lattice for Rydberg atom trapping and lattice modulation spectroscopy. It incorporates all ingredients necessary for preparing the above-mentioned circular Rydberg states, which will be required to reach the desired spectroscopic accuracy and precision. The circular-state production method follows pioneering work by D. Kleppner et al. at MIT [R. G. Hulet and D. Kleppner, Physical Review Letters 51, 1430 (1983)] and Haroche at ENS [P. Nussenzveig, F. Bernardot, M. Brune, J. Hare, J. M. Raimond, S. Haroche, and W. Gawlik, Phys. Rev. A 48, 3991 (1993)]. The setup is ready for cryogenic operation at 4 Kelvin, which will be required in order to suppress unwanted transitions and lifetime reductions due to black-body thermal radiation. The setup also includes the required laser-cooling infrastructure as well as control of electric and magnetic fields within the spectroscopic volume. The vacuum setup and its complicated internal electrode structure has been completed, baked out, and integrated with lasers. The setup has been fitted with a primary-MOT glass cell from Precision Glass Blowing. The primary MOT, a 2D+ beam MOT, and the secondary MOT inside the spectroscopic enclosure have been set up and are operated on a near-daily basis. The secondary MOT can also be run in molasses mode, which will be required in upcoming experiments with very well controlled magnetic fields.

The experimental work is accompanied by a theoretical effort conducted by V. Malinovsky. The present objective of the theory work is to model the spectroscopic line shapes of Rydberg transitions in an amplitude-modulated Rydberg-atom lattice in a manner that takes the quantization of the center-of-mass motion into account. This ability will be an important ingredient to determine the line center with 1Hz uncertainty. To take into account the quantization of motion of the Rydberg atoms in the optical lattice, a theoretical model is developed in which both ground and excited wave functions are subject to the periodic potentials, representing the lattice fields, while the coupling (the effective Rabi frequency) is also periodic as function of the translational coordinate. To obtain the spectrum of the excited-state population, the time-dependent Schroedinger equation has been solved in the momentum representation. Two manuscripts on the theory work are in preparation.

References

J. De Vries. A precision millimeter-wave measurement of the Rydberg frequency, PhD thesis, (2001).

J. Hare, A. Nussenzweig, C. Gabbanini, M. Weidemueller, P. Goy, M. Gross, and S. Haroche. IEEE Transactions on Instrumentation and Measurement, 42, 331 (1993).

Method: Rb atoms are cooled to below 1microK and laser-excited into a low-angular-momentum Rydberg level with principal quantum number n on the order of 40. Using an adiabatic atomic-state transformation method, the Rydberg atoms are then transferred into the circular state |n,n1=0,n2=0,m=n-1> (m, n1, and n2 are the magnetic and parabolic quantum numbers). An optical lattice (wavelength 1064nm or 532nm on the ground, 850nm at CAL) is adiabatically ramped on. The Rydberg atoms are trapped in the lattice via the ponderomotive interaction of the Rydberg electron with the lattice light. During the hold time of the atom cloud in the lattice, the electric-quadrupole transition into the state |n+2,n1=1,n2=1,m=n-1> is driven by amplitude-modulating the optical lattice at the transition frequency. The transition is probed using state-selective electric-field ionization. The measured transition frequency allows one to extract the Rydberg constant. Key advantages of the method are that the transition is free of first-order Stark and Zeeman shifts, that QED, nuclear-penetration, hyperfine and other shifts are small, and that the optical lattice causes only moderate trap-induced transition shifts. Theoretical work includes the modeling of the circular-state preparation using multi-level adiabatic passage and of the circular-state transition dynamics in a shallow, microwave-modulated optical lattice, in which center-of-mass wave-packet expansion and tunneling must be considered.

Relevance to NASA and relevance of microgravity: High-precision measurement of fundamental constants represents an important part of contemporary atomic-physics research. The proposed high-precision measurement of the Rydberg constant would add a new component to the Cold Atom Laboratory (CAL) research portfolio. To achieve optimal spectroscopic performance it is critical to reduce systematic lattice-induced transition shifts and related shifts. Since the most critical shifts diminish with decreasing optical-lattice depth, the lattices must be chosen as shallow as possible. Microgravity conditions are important because they enable the use of shallow lattices without the atoms falling out. The microgravity environment provided by the CAL, including its 850nm optical-lattice laser, offers an ideal platform for the project to reach its ultimate uncertainty goal. The CAL setup may also provide an extended period of measurement time to acquire good statistics in the high-precision spectra.

The large size of the Rydberg atoms allows one to employ a fundamentally new method of spectroscopy. The method is based on harnessing a term known as the “A-square term” for the purpose of spectroscopy. The method requires one to have a modulated light field with a spatial modulation period that is on the same order as the diameter of the atom that is being probed. At the same time, the field must be periodically modulated in time, with a modulation frequency that matches the frequency of the atomic transition that is to be probed. Rydberg atoms in electro-optically modulated standing wave light fields satisfy both conditions. It must be stressed that (1) this method of making an atomic transition has not been done before, and (2) that the method is a critical part in the the project “High-Precision Microwave Spectroscopy of Long-Lived Circular-State Rydberg Atoms in Microgravity.”

To that end, in the first year of the project the research team has tried and succeeded in demonstrating this new type of spectroscopy. The group is leading in Rydberg-atom trapping in magnetic fields and in optical lattices. An optical lattice is a laser-induced crate for atoms, made by standing waves of light, that researchers use to enhance spectroscopic resolution and to prepare the trapped atoms for quantum gates, spectroscopy, etc. The optical lattice is often compared with an egg carton that holds eggs in a way that is equivalent to how periodic laser trap holds cold atoms. The Raithel group has first characterized such traps and has now begun using them for high precision spectroscopy. The group’s effort to help solving the “proton radius puzzle” by measuring transitions between laser-trapped circular Rydberg atoms has previously been awarded with a National Institute of Standards and Technology (NIST) Precision Measurement Grant (2012-2015), and is now being continued within the Fundamental Physics Program of the NASA Cold Atom Lab. In 2014/2015 the group succeeded in measuring the 58S-59S quadrupole Rydberg transition by “A-square,” or “ponderomotive,” lattice modulation spectroscopy. The transition has a frequency of about 38GHz. This work, which was published in Nature Communications, is crucial for the present project.

To reach high relative spectroscopic resolution, it is imperative to both decrease the spectroscopic linewidth (the frequency uncertainty to which transitions can be pinpointed) and to increase the absolute frequency of that transition. In the present case, the targeted numbers are 1Hz for the uncertainty of the transition, and 200GHz for the transition frequency itself (corresponding to a relative uncertainty of 5 parts in 10^12). Since no suitable methods exist to electro-optically modulate the optical lattice 200GHz exists, it is critical to find a workaround that allows one to increase the accessible frequency range from 40GHz, the approximate limit of current electro-optic modulation technology, to 200GHz. In 2015 the research team has succeeded in identifying such a workaround. Essentially, the nonlinearity of electro-optic modulation allows one to drive all odd overtones of the modulation frequency, without the need of increasing the optical-lattice laser power. Currently, the team has shown that transitions with frequencies up to 5 times the modulation frequency can be driven. The team has already demonstrated a transition at about 95GHz. The goal of about 200GHz is therefore within close reach. This work has been published in Physical Review Letters.

In the last year, the team has designed a new cryogenic spectroscopic setup that is supposed to last through the entire performance period of about 4 more years. The setup features a three-dimensional optical lattice for Rydberg atom trapping and lattice modulation spectroscopy. It incorporates all ingredients necessary for preparing the above-mentioned circular Rydberg states, which will be required to reach the desired spectroscopic accuracy and precision. The circular-state production method follows pioneering work by Group Leader D. Kleppner et al. at MIT (Massachusetts of Technology) [8]. The setup is ready for cryogenic operation at 4 Kelvin, which will be required in order to suppress unwanted transitions and lifetime reductions due to black-body thermal radiation. The setup also includes the required laser-cooling infrastructure as well as control of electric and magnetic fields within the spectroscopic volume. The setup has proceeded well into the construction phase.

The measurement of the Rydberg constant at a precision of 5 parts in 10^12 necessitates improved knowledge of the ionic core polarizability of the utilized rubidium atoms. The core polarizability accounts for the fact that the atomic core (the Rb+ ion in this case) becomes electrically polarized by the Rydberg electron that orbits the Rb+ ion. The core polarization leads to Rydberg level shifts on the order of 10kHz. A small, simplified spectroscopy chamber, assembled from spare parts and some new components, has been set up to perform high-precision microwave spectroscopy on states with angular momenta of up to about five. Transitions between such states are ideal to determine the core polarizability. The goal of this component of the work is to increase the number of significant digits for the core polarizability from presently three to five. The five significant digits will be sufficient to eventually extract the Rydberg constant at a precision of 5 parts in 10^12.

The experimental work is accompanied by a theoretical effort conducted by V. Malinovsky. The present objective of the theory work is to model the spectroscopic line shapes of Rydberg transitions in an amplitude-modulated Rydberg-atom lattice in a manner that takes the quantization of the center-of-mass motion into account. This ability will be an important ingredient to determine the line center with 1Hz uncertainty. To take into account the quantization of motion of the Rydberg atoms in the optical lattice, a theoretical model is developed in which both ground and excited wave functions are subject to the periodic potentials, representing the lattice fields, while the coupling (the effective Rabi frequency) is also periodic as function of the translational coordinate. To obtain the spectrum of the excited-state population, the time-dependent Schroedinger equation will be solved in the momentum representation.

References:

[1] R.G. Hulet, D. Kleppner, “Rydberg atoms in circular states,” Phys. Rev. Lett. 51, 1430-3 (1983).

[2] R. Lutwak, J. Holley, P. P. Chang, S. Paine, D. Kleppner, T. Ducas, “Circular states of atomic hydrogen,” Phys. Rev. A 56, 1443-52 (1997).

[3] J. Liang, M. Gross, P. Goy, S. Haroche, “Circular Rydberg-state spectroscopy,” Phys. Rev. A 33, 4437-9 (1986).

[4] P. Nussenzveig, F. Bernardot, M Brune, J. Hare, J. M. Raimond, S. Haroche, W. Gawlik, “Preparation of high-principal-quantum-number circular states of rubidium,” Phys. Rev. A 48, 3991-4 (1993).

[5] A. Nussenzweig, J. Hare, A. M. Steinberg, L. Moi, M. Gross, S. Haroche, “A continuous beam of circular Rydberg atoms for fundamental tests and applications in metrology,” Europhys. Lett. 14, 755-60 (1991).

[6] J. Hare, A. Nussenzweig, C. Gabbanini, M. Weidemueller, P. Goy, M. Gross, S. Haroche, “Toward a Rydberg constant measurement on circular atoms,” IEEE Trans. Instrum. Meas. 42, 331-4 (1993).

[7] R. J. Brecha, G. Raithel, C. Wagner, “Circular Rydberg states with very large n,” H. Walther, Optics Comm. 102, 257-64 (1993).

[8] R. Lutwak, J. Holley, J. DeVries, D. Kleppner, T. W. Ducas, “Millimeter-wave measurement of the Rydberg frequency,” Proceedings of the Fifth Symposium on Frequency Standards and Metrology, p 259-63, (1996).

Method: Rb atoms are cooled to below 1microK and laser-excited into a low-angular-momentum Rydberg level with principal quantum number n on the order of 40. Using an adiabatic atomic-state transformation method, the Rydberg atoms are then transferred into the circular state |n,n1=0,n2=0,m=n-1> (m, n1, and n2 are the magnetic and parabolic quantum numbers). An optical lattice (wavelength 1064nm or 532nm on the ground, 850nm at CAL) is adiabatically ramped on. The Rydberg atoms are trapped in the lattice via the ponderomotive interaction of the Rydberg electron with the lattice light. During the hold time of the atom cloud in the lattice, the electric-quadrupole transition into the state |n+2,n1=1,n2=1,m=n-1> is driven by amplitude-modulating the optical lattice at the transition frequency. The transition is probed using state-selective electric-field ionization. The measured transition frequency allows one to extract the Rydberg constant. Key advantages of the method are that the transition is free of first-order Stark and Zeeman shifts, that QED, nuclear-penetration, hyperfine and other shifts are small, and that the optical lattice causes only moderate trap-induced transition shifts. Theoretical work includes the modeling of the circular-state preparation using multi-level adiabatic passage and of the circular-state transition dynamics in a shallow, microwave-modulated optical lattice, in which center-of- mass wave-packet expansion and tunneling must be considered.

Relevance to NASA and relevance of microgravity: High-precision measurement of fundamental constants represents an important part of contemporary atomic-physics research. The proposed high-precision measurement of the Rydberg constant would add a new component to CAL's research portfolio. To achieve optimal spectroscopic performance it is critical to reduce systematic lattice-induced transition shifts and related shifts. Since the most critical shifts diminish with decreasing optical-lattice depth, the lattices must be chosen as shallow as possible. Microgravity conditions are important because they enable the use of shallow lattices without the atoms falling out. The microgravity environment provided by the CAL lab, including its 850nm optical-lattice laser, offers an ideal platform for the project to reach its ultimate uncertainty goal. The CAL setup may also provide an extended period of measurement time to acquire good statistics in the high-precision spectra.