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arXiv:2308.11095

Ramsey interferometry in three-level and five-level systems of $^{87}Rb$ Bose-Einstein condensates

Anushka Thenuwara, Andrei I. Sidorov

Published Aug 22, 2023• 0 citations
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Abstract

Our work here presents the analytical expressions for a typical Ramsey interferometric sequence for a three- and a five-level system. The analytical expressions are derived starting from the first principals of unitary time evolution operators. We focus on the three- and five-level systems because we propose a novel Ramsey interferometer created by a trapped two-state Bose-Einstein Condensate driven by dipole oscillations and gravitational sag. It involves the $^{87}Rb$ atoms in states $\vert F=2, m_F=+2 \rangle$ $(\vert +2 \rangle)$ and $\vert F=2, m_F=+1 \rangle$ $(\vert +1 \rangle)$ of the $5 ^2S_{\frac{1}{2}}$ ground state. Though the interferometer focusses on the two-levels, the experimental readouts involve all the five states in $F = 2$ hyperfine manifold. Therefore, the analytical derivation was first tested for three-levels and then expanded to five-levels. We developed the expressions for five-levels for greater analytical accuracy of the experimental scenario. This work provides a step-by-step outline for the derivation and methodology for the analytical expressions. These analytical formulae denote the population variation during Rabi and Ramsey oscillations for each state as well as the overall average for both the three- and five-level cases. The expressions are derived within the rotating wave approximation (RWA) under the equal Rabi condition. Further, by following the derivation methodology, these analytical expressions can be easily expanded for Ramsey sequences with unequal pulses, and Ramsey sequences with spin echo techniques.

DOI
10.48550/arxiv.2308.11095

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Ramsey interferometry in three-level and five-level systems of 87Rb^{87}Rb87Rb Bose-Einstein condensates
Our work here presents the analytical expressions for a typical Ramsey interferometric sequence for a three- and a five-level system. The analytical expressions are derived starting from the first principals of unitary time evolution operators. We focus on the three- and five-level systems because we propose a novel Ramsey interferometer created by a trapped two-state Bose-Einstein Condensate driven by dipole oscillations and gravitational sag. It involves the 87Rb^{87}Rb87Rb atoms in states ∣F=2,mF=+2⟩\vert F=2, m_F=+2 \rangle∣F=2,mF​=+2⟩ (∣+2⟩)(\vert +2 \rangle)(∣+2⟩) and ∣F=2,mF=+1⟩\vert F=2, m_F=+1 \rangle∣F=2,mF​=+1⟩ (∣+1⟩)(\vert +1 \rangle)(∣+1⟩) of the 52S125 ^2S_{\frac{1}{2}}52S21​​ ground state. Though the interferometer focusses on the two-levels, the experimental readouts involve all the five states in F=2F = 2F=2 hyperfine manifold. Therefore, the analytical derivation was first tested for three-levels and then expanded to five-levels. We developed the expressions for five-levels for greater analytical accuracy of the experimental scenario. This work provides a step-by-step outline for the derivation and methodology for the analytical expressions. These analytical formulae denote the population variation during Rabi and Ramsey oscillations for each state as well as the overall average for both the three- and five-level cases. The expressions are derived within the rotating wave approximation (RWA) under the equal Rabi condition. Further, by following the derivation methodology, these analytical expressions can be easily expanded for Ramsey sequences with unequal pulses, and Ramsey sequences with spin echo techniques.
We propose and explore the feasibility of a novel Ramsey interferometer created by a trapped two-state Bose-Einstein condensate (BEC) driven by dipole oscillations and gravitational sag. The BEC is formed in a pure cigar shaped compressed magnetic trap (CMT) via a dilute atom cloud of 87Rb^{87}Rb87Rb atoms in state ∣F=2,mF=+2⟩\vert F=2, m_F=+2 \rangle∣F=2,mF​=+2⟩ (∣+2⟩)(\vert +2 \rangle)(∣+2⟩) of the 52S125 ^2S_{\frac{1}{2}}52S21​​ ground state. Here, Rmasey interferometry is performed with states ∣F=2,mF=+1⟩\vert F=2, m_F=+1 \rangle∣F=2,mF​=+1⟩ (∣+1⟩)(\vert +1 \rangle)(∣+1⟩) and ∣+2⟩\vert +2 \rangle∣+2⟩. The proposed interferometer utilises the response of atoms to the harmonic oscillator trapping potential and the gravitational sag due to the variation in the mFm_FmF​ state. Briefly, the state ∣+1⟩\vert +1 \rangle∣+1⟩ experiences a shallower radial trap with a larger gravitational sag; whereas, state ∣+2⟩\vert +2 \rangle∣+2⟩ experiences a tighter radial trap with a gravitational sag which is half of state ∣+1⟩\vert +1 \rangle∣+1⟩. Due to this, a superposition between the states ∣+1⟩\vert +1 \rangle∣+1⟩ and ∣+2⟩\vert +2 \rangle∣+2⟩ experiences multipath propagation resulting in an interference pattern. This may be utilised to measure local gravitational fields and measure inter-sate scattering lengths. Here, a theoretical framework is reported which is developed via the two-level system in combination with the Gross-Pitaevskii equation (GPE). Further, the development of a simulation tool via GPELabs in MATLAB that explores the prosed interferometer is reported along with key insights and findings.
We present results on a free-space atom interferometer operating on the first order magnetically insensitive |F = 1,mF = 0) --> |F = 2,mF = 0) ground state transition of Bose-condensed (87)Rb atoms. A pulsed atom laser is output-coupled from a Bose-Einstein condensate and propagates through a sequence of two internal state beam splitters, realized via coherent Raman transitions between the two interfering states. We observe Ramsey fringes with a visibility close to 100% and determine the current and the potentially achievable interferometric phase sensitivity. This system is well suited to testing recent proposals for generating and detecting squeezed atomic states.
The Ramsey interferometer is a prime example of precise control at the quantum level. It is usually implemented using internal states of atoms, molecules or ions, for which powerful manipulation procedures are now available. Whether it is possible to control external degrees of freedom of more complex, interacting many-body systems at this level remained an open question. Here we demonstrate a two-pulse Ramsey-type interferometer for non-classical motional states of a Bose-Einstein condensate in an anharmonic trap. The control sequences used to manipulate the condensate wavefunction are obtained from optimal control theory and are directly optimized to maximize the interferometric contrast. They permit a fast manipulation of the atomic ensemble compared to the intrinsic decay processes and many-body dephasing effects. This allows us to reach an interferometric contrast of 92% in the experimental implementation.
Coherent-population trapping (CPT) is a multilevel quantum coherence phenomenon of promising applications in atomic clocks and magnetometers. In particular, multipulse CPT Ramsey interferometry is a powerful tool for improving the performance of CPT atomic clocks. Most studies on multipulse CPT Ramsey interferometry consider periodic pulse sequence and time-independent detuning. However, to further improve the accuracy and precision, one may modify the spectrum symmetry which involves a pulse sequence with time-dependent detuning or phase shift. Here we theoretically analyze the multipulse CPT Ramsey interferometry under arbitrary pulse sequences of time-dependent detuning and obtain a general analytical formula. Using our formula, we analyze the popular CPT Ramsey interferometry schemes such as two-pulse symmetric and antisymmetric spectroscopy and multipulse symmetric and antisymmetric spectroscopy. Under periodic pulses, we quantitatively obtain the influences of pulse width, pulse period, pulse number, and Rabi frequency. Furthermore, we examine the impact of CPT pulses and Rabi frequency on the light shift in both two-pulse and multipulse CPT Ramsey interference. Our theoretical results can guide the experimental design to improve the performance of atomic clocks via multipulse CPT Ramsey interferometry.
Bose-Einstein condensates present to us the opportunity to probe into the atomic interac- tions that govern a macroscopic quantum mechanical system. The degenerate hyperfine manifold in the bosonic atoms splits in the presence of an external B-field; radio-frequency induced coupling releases experimentally-attainable knowledge about the Zeeman mani- fold in 87Rb. The purpose of this study is to investigate quantum dynamics in two and three-component Bose-Einstein condensate (BEC) systems. We start with a theoretical analysis of Josephson tunneling dynamics between identical BECs trapped in a double potential well, paying particular attention to the non-linear self-trapping effect observed as a consequence of the intra-well interaction. We present a model for the non-equilibrium dynamics in a two-level system and introduce the Rabi oscillations. This is followed by a numerical and experimental investigation of Rabi oscillations in a three-level 87Rb BEC between the F = 1 hyperfine level spin states mF = {1, 0, 1}. The relation between the observed total Rabi frequency and the detuning, along with its effect on the BEC population dynamics is explored. Finally, we explore the possible suspects for the shift of the resonance at strong Rabi frequencies .
Bose-Einstein condensates are coherent matter waves, produced by cooling gaseous atomic clouds to ultra-low temperatures. For applications in atom interferometry and precision measurements, Bose-condensed sources present an intriguing alternative to thermal atoms. Although the current sensitivity achievable with interferometers using coherent atoms is not comparable to thermal beam machines (mainly due to the lower flux), there are promising ways to utilise the potential of Bose-condensed sources for atom interferometry. Among those is the low momentum width of Bose-Einstein condensates, which can generally be well controlled and is advantageous for increased interferometric sensitivities by implementing large momentum transfer beam splitters. As part of this thesis, experimental and theoretical investigations are presented to investigate the potential of Bose-Einstein condensates for such applications. We shall present the quantum projection noise limited performance of a Ramsey interferometer operating on the atomic clock transition of a freely expanding cloud of Bose-condensed rubidium 87 atoms. The results include Ramsey fringes of high visibility, not measurably affected by atomic interaction-induced dephasing effects. The achievement and detection of the quantum projection noise limit rely critically on the precision and accuracy of both the imaging setup and the coupling scheme in the interferometric beam splitters. The stabilisation of the beam splitters via an optical Sagnac interferometer is the basis for the quantum projection noise limited performance of the interferometer presented. For an increase of bandwidth and flux in atom interferometric measurements, it is advantageous to use a continuous atomic beam. A truly continuous coherent atom source has not been realised to date, and we present results on a pumping mechanism in this thesis, as a decisive step towards a continuous atom laser. By the investigation of different momentum resonances, we find that the pumping scheme relies on a Raman superradiance-like process. Finally, the thesis demonstrates two interaction measurements in rubidium. The strong mean field interactions due to the high densities in Bose-Einstein condensates are used to probe the potential of a rubidium 87 condensate with an atom laser. The measurement allows a determination of the scattering length between the two atomic states involved. In addition to this two-body scattering scheme, we present a measurement of three-body loss coefficients, extracted from loss curves in rubidium 85 Bose-Einstein condensates. The measurement provides new upper bounds on the three-body loss coefficients at the scattering lengths considered.