Driven-dissipative four-mode squeezing of multilevel atoms in an optical cavity

Bhuvanesh Sundar, Diego Barberena, Ana Maria Rey, and Asier Piñeiro Orioli

Phys. Rev. A 109, 013713 – Published 17 January 2024

Abstract

We utilize multilevel atoms trapped in a driven resonant optical cavity to produce scalable multimode squeezed states for quantum sensing and metrology. While superradiance or collective dissipative emission by itself has been typically a detrimental effect for entanglement generation in optical cavities, in the presence of additional drives it can also be used as an entanglement resource. In a recent work [B. Sundar, D. Barberena, A. M. Rey, and A. Piñeiro Orioli, companion paper, Phys. Rev. Lett. 132, 033601 (2024)], we described a protocol for the dissipative generation of two-mode squeezing in the dark state of a six-level system with only one relevant polarization. There we showed that up to two quadratures can be squeezed. Here, we develop a generalized analytic treatment to calculate the squeezing in any multilevel system where atoms can collectively decay by emitting light into two polarization modes in a cavity. We show that in this more general system up to four spin squeezed quadratures can be obtained. We study how finite-size effects constrain the reachable squeezing, and analytically compute the scaling with $N$ . Our findings are readily testable in current optical cavity experiments with alkaline-earth-like atoms.

Received 23 September 2023
Accepted 11 December 2023

DOI:https://doi.org/10.1103/PhysRevA.109.013713

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Quantum metrology Squeezing of quantum noise Superradiance & subradiance

Holstein-Primakoff method

Atomic, Molecular & Optical

Authors & Affiliations

Bhuvanesh Sundar ^*, Diego Barberena , Ana Maria Rey , and Asier Piñeiro Orioli

Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA and Joint Institute for Laboratory Astrophysics, National Institute of Standards and Technology, Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

^*Present address: Rigetti Computing, Berkeley, CA 94710, USA.

Squeezing Multilevel Atoms in Dark States via Cavity Superradiance

Bhuvanesh Sundar, Diego Barberena, Ana Maria Rey, and Asier Piñeiro Orioli

Phys. Rev. Lett. 132, 033601 (2024)

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Vol. 109, Iss. 1 — January 2024

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Images

Figure 1
(a) An ensemble of atoms trapped in a deep lattice in an optical cavity, with the cavity frequency on resonance with the atomic transition from the ground states to the excited states (with spins $F_{g}$ and $F_{e}$ ), $ω_{c} = ω_{a}$ . The cavity is driven by a resonant laser, and the atoms superradiantly decay at rate $Γ$ from the excited to the ground states. (b), (c) Transitions driven by the collective atomic excitation operators. (b) $L^{\pm}$ and $R^{\pm}$ transitions due to coupling to a left or right circularly polarized photon when we choose the quantization axis to be parallel to the cavity axis. (c) $Π^{\pm}$ and $Σ^{\pm}$ transitions due to coupling to a vertically or horizontally polarized photon when we choose the quantization axis to be perpendicular to the cavity axis.
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Figure 2
(a) An ensemble of effective two-level atoms is driven by a right-circularly polarized laser with strength $Ω$ and superradiantly decay at rate $Γ$ to the ground state. The quantization axis is parallel to the cavity axis. (b) Negative of the curvature of the potential, $- d^{2} V (ζ, 0; Ψ) / d ζ^{2} |_{ζ = 0}$ . There are two critical points, at $θ_{0} = \pm π / 2$ , indicated by two dots. The thick black line marks the stable region. (c) An ensemble of eight-level atoms is driven by a $Σ$ -polarized laser, and superradiantly emits $Σ$ - and $Π$ -polarized light. The quantization axis is perpendicular to the cavity axis. (d) Regions of stability and instability vs the angle $β$ in the ground state manifold and state preparation pulse area $θ_{0}$ (see text). The system is stable to emission of both polarizations in regions marked 1, unstable to emission of $Σ$ -polarized light in region 2, unstable to emission of $Π$ -polarized light in region 3, and unstable to emission of both in region 4. Green lines and dots mark critical manifolds and points where the system crosses from a stable or unstable region to an unstable or stable region for each polarization, and cyan and red lines show the critical manifolds where emission of only one polarization crosses from stable to unstable.
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Figure 3
(a) An ensemble of effective two-level atoms [see also Fig. 2]. (b) Illustration of the steady-state squeezing in bosonic quadratures. The steady state is the coherent vacuum of ${\hat{X}}^{b}$ and ${\hat{Y}}^{b}$ , which makes it squeezed in ${\hat{X}}^{c}$ and antisqueezed in ${\hat{Y}}^{c}$ (see text for definitions of the quadratures). (c) Visualizing the collective spin squeezing on a Bloch sphere in the two-level system. The squeezing is along ${\hat{S}}^{x}$ . The visualization shows a particular example where the steady state is near $θ_{c} = π / 2$ . (d) Steady-state squeezing vs $θ_{0}$ in the two-level case, obtained from an exact numerical calculation with $N = 100$ atoms (solid line). The black dashed line plots the HP prediction, $cos θ_{0}$ , and the blue dotted line plots $Ξ_{X X}^{c}$ in the coherent state. (e) The best squeezing achievable vs $N$ has a scaling close to $N^{- 1 / 3}$ .
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Figure 4
Visualization of the evolution of the quantum noise distribution, projected onto the ${\hat{Y}}_{1}^{c} − {\hat{Y}}_{2}^{c}$ plane (see text), and the emergence of squeezing and/or antisqueezing. The noise distribution in the initial coherent state is isotropic (dashed circle), and the evolution conserves the noise along ${\hat{Y}}_{2}^{c}$ . (a) When $ϕ_{Σ} \neq 0$ [see Eq. (43)], the circular noise distribution shears into an ellipse, leading to one squeezed and one antisqueezed mode. (b) When $ϕ_{Σ} = 0$ , the circular noise distribution fattens into an ellipse if $∥ {\vec{y}}_{Σ} ∥ < ∥ {\vec{x}}_{Σ} ∥$ , leading to one antisqueezed mode but no squeezed modes (and shrinks into a narrower ellipse with one squeezed mode if $∥ {\vec{y}}_{Σ} ∥ > ∥ {\vec{x}}_{Σ} ∥$ ).
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Figure 5
Steady-state squeezing for the eight-level system shown in Fig. 2. (a), (b) Squeezing in a combination of the $\hat{X}$ and $\hat{Y}$ quadratures, respectively, when the system collectively emits only $Σ$ -polarized light. (c) Squeezing when the system collectively emits only $Π$ -polarized light. There are two squeezed modes, one in a combination of $\hat{X}$ quadratures and one in a combination of $\hat{Y}$ quadratures, and they both have the same value. Red and blue dashed lines are critical lines, and white regions are unstable. (d)–(f) Squeezing when the system collectively emits light of both polarizations. The system is squeezed only in the regions where it is stable to emission of both polarizations, and the value of the squeezing in this region is the same as in (a)–(c).
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