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arXiv:2211.13068

Frequency Measurement with Superradiant Pulses of Incoherently Pumped Calcium Atoms: Role of Quantum Measurement Backaction

Huihui Yu, Yuan Zhang, Qilong Wu, Chongxin Shan, Klaus Mølmer

Published Nov 23, 2022• 0 citations
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Abstract

A recent experiment demonstrated heterodyne detection-based frequency measurements with superradiant pulses from coherently pumped strontium atoms in an optical lattice clock system, while another experiment has analyzed the statistics of superradiant pulses from incoherently pumped calcium atoms in a similar system. In this article, we propose to perform heterodyne detection of the superradiant pulses from the calcium atoms, and analyze theoretically the corresponding atomic ensemble dynamics in terms of the rotation of a collective spin vector and the incoherent quantum jumps among superradiant Dicke states. We examine the effect of quantum measurement backaction on the emitted field and the collective spin vector dynamics, and we demonstrate that it plays an essential role in the modelling of the frequency measurements. We develop a stochastic mean field theory, which is also applicable to model frequency measurements with steady-state superradiance signals, and to explore quantum measurement effects in the dynamics of atomic ensembles.

DOI
10.48550/arxiv.2211.13068

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A recent experiment demonstrated heterodyne detection-based frequency measurements with superradiant pulses from coherently pumped strontium atoms in an optical lattice clock system, while another experiment has analyzed the statistics of superradiant pulses from incoherently pumped calcium atoms in a similar system. In this article, we propose to perform heterodyne detection of the superradiant pulses from the calcium atoms, and analyze theoretically the corresponding atomic ensemble dynamics in terms of the rotation of a collective spin vector and the incoherent quantum jumps among superradiant Dicke states. We examine the effect of quantum measurement backaction on the emitted field and the collective spin vector dynamics, and we demonstrate that it plays an essential role in the modelling of the frequency measurements. We develop a stochastic mean field theory, which is also applicable to model frequency measurements with steady-state superradiance signals, and to explore quantum measurement effects in the dynamics of atomic ensembles.
We develop a stochastic mean-field theory to describe active frequency measurements of pulsed superradiant emission, studied in a recent experiment with strontium-87 atoms trapped in an optical lattice inside an optical cavity [M. Norcia et al., Phys. Rev. X 8, 021036 (2018)PRXHAE2160-330810.1103/PhysRevX.8.021036]. Our theory reveals the intriguing dynamics of atomic ensembles with multiple transition frequencies, and it reproduces the superradiant beats signal, noisy power spectra, and frequency uncertainty in remarkable agreement with the experiments. Moreover, using longer superradiant pulses of similar strength and shortening the experimental duty cycle, we predict a short-term frequency uncertainty 7×10^{-17}/sqrt[τ/s], which makes active frequency measurements with superradiant transitions comparable with the record performance of current frequency standards [M. Schioppo et al., Nat. Photonics 11, 48 (2017)NPAHBY1749-488510.1038/nphoton.2016.231]. Our theory combines cavity quantum electrodynamics and quantum measurement theory, and it can be readily applied to explore conditional quantum dynamics and describe frequency measurements for other processes such as steady-state superradiance and superradiant Raman lasing.
In this Letter, we use quantum trajectory theory to simulate heterodyne detection of narrow bandwidth superradiant lasing from an incoherently excited atomic ensemble. To this end, we describe the system dynamics and account for stochastic measurement backaction by second-order mean-field theory. Our simulations show how heterodyne measurements break the phase symmetry, and initiate the atomic coherence with a random phase and a long temporal phase coherence. More importantly, our theory allows direct simulation of experimental procedures for extraction of spectral information which do not lend themselves to evaluation with the quantum regression theorem.
We simulate the superradiant dynamics of ensembles of atoms in the presence of collective and individual atomic decay processes. We apply the Monte-Carlo wave-function method and identify quantum jumps in a reduced Dicke state basis, which reflects the permutation symmetry of the system. While the number of density matrix elements in the Dicke representation increases polynomially with atom number, the quantum jump dynamics populates only a single Dicke state at the time and thus efficient simulations can be carried out for tens of thousands of atoms. The superradiant pulses from initially excited atoms agree quantitatively with recent experimental results of strontium atoms but rapid atom loss in these experiments does not permit steady-state superradiance. By introducing an incident flux of new atoms to maintain a large average atom number, our theoretical calculations predict lasing with a millihertz linewidth despite rapid atom number fluctuations.
Dicke superrandiance is a cooperative phenomenon which arises from the collective coupling of an ensemble of atoms to the electromagnetic radiation. Here we discuss the quantifying of quantum coherence for the Dicke model of superradiance in the mean-field approximation. We found the single-atom l1{l}_{1}l1​ norm of coherence is given by the square root of the normalized average intensity of radiation emitted by the superradiant system. This validates quantum coherence as a useful figure of merit towards the understanding of superradiance phenomenon in the mean-field approach. In particular, this result suggests probing the single-atom coherence through the radiation intensity in superradiant systems, which might be useful in experimental realizations where is unfeasible to address atoms individually. Furthermore, given the nonlinear unitary dynamics of the time-dependent single-atom state that effectively describes the system of NNN atoms, we analyze the quantum speed limit time and its interplay with the l1{l}_{1}l1​ norm of coherence. We verify the quantum coherence speeds up the evolution of the superradiant system, i.e., the more coherence stored on the single-atom state, the faster the evolution. These findings unveil the role played by quantum coherence in superradiant systems, which in turn could be of interest for communities of both condensed matter physics and quantum optics.
A recent experiment demonstrated delayed superradiance from strontium-88 atoms, which are coupled to a longitudinal mode of a cavity while being excited by laser pulses propagating along a transversal direction [Nat. Commun. 15, 1084 (2024)]. A coherent picture of the atomic ensemble dynamics in this experiment requires complementary representations of the external driving dynamics and the superradiant dynamics. To complement previous analyses, we introduce these representations by considering in-phase and out-of-phase superpositions of transverse collective spin components of two atomic sub-ensembles, and analyze the dynamics with the corresponding collective Dicke states and Bloch vectors. This approach also explains Ramsey spectroscopy experiments with the ensemble, and it may be employed to explore other phenomena, such as weak-to-strong coupling phase transitions and triggered superradiance.
Superradiance is an extensive cooperative spontaneous emission phenomenon, exhibited by some atomic collective-spin states. However, distinct initial states differ in their decay dynamics. Dicke states ∣j,m⟩|j,m\rangle∣j,m⟩ with distinct numbers of excitations n=m+jn=m+jn=m+j, driven by vacuum fluctuations, have their peak emission intensity shifted in time. Rotating Dicke states relative to the decay axis introduces an interesting parity structure that affects the pulse profile and photon correlations. Squeezed-bath prepared states undergo a squeezing-controlled crossover to the rotated Dicke states, making the emission character dependent on the amount of squeezing transferred from light to the atomic state. For semiclassical states with a macroscopic dipole moment, like the atomic coherent state, the emission intensity depends on their polarization. We present detailed results on the superradiant dynamics of a representative selection of states expanded in Dicke states to highlight the initial state as an independent dynamical control parameter. For large-NNN, we are able to predict fairly accurately the pulse profile in each case using the mean-field approximation, an approach based on the Fokker--Planck equation. We also present comparative results on the intensity correlation function, quantify the coherent and incoherent contributions of the emission, and contrast between small and large ensembles.