Speeding up particle slowing using shortcuts to adiabaticity

John P. Bartolotta, Jarrod T. Reilly, and Murray J. Holland

Phys. Rev. A 102, 043107 – Published 16 October 2020

Abstract

We propose a method for slowing particles by laser fields that potentially has the ability to generate large forces without the associated momentum diffusion that results from the random directions of spontaneously scattered photons. In this method, time-resolved laser pulses with periodically modified detunings address an ultranarrow electronic transition to reduce the particle momentum through repeated absorption and stimulated emission cycles. We implement a shortcut to adiabaticity approach that is based on Lewis-Riesenfeld invariant theory. This affords our scheme the advantages of adiabatic transfer, where there can be an intrinsic insensitivity to the precise strength and detuning characteristics of the applied field, with the advantages of rapid transfer that are necessary for obtaining a short slowing distance. For typical parameters of a thermal oven source that generates a particle beam with a central velocity on the order of meters per second, this could result in slowing the particles to near stationary in less than a millimeter. We compare the slowing scheme to widely implemented slowing techniques that rely on radiation pressure forces and show the advantages that potentially arise when the excited-state decay rate is small. Thus, this scheme is a particularly promising candidate to slow narrow-linewidth systems that lack closed cycling transitions, such as occurs in certain molecules.

Received 14 July 2020
Accepted 17 September 2020

DOI:https://doi.org/10.1103/PhysRevA.102.043107

Physics Subject Headings (PhySH)

Atomic & molecular processes in external fields Coherent control Cooling & trapping Light-matter interaction Spontaneous emission

Atomic, Molecular & Optical

Authors & Affiliations

John P. Bartolotta , Jarrod T. Reilly , and Murray J. Holland

JILA and Department of Physics, University of Colorado, 440 UCB, Boulder, Colorado 80309, USA

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Vol. 102, Iss. 4 — October 2020

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Images

Figure 1
(a) Schematic of the normalized velocity distributions $P (v)$ of the particle beam before (solid) and after (dashed) the slowing process. We characterize the initial distribution by a Gaussian function with a standard deviation $σ_{v}$ and mean velocity $\bar{v}$ . (b) Particles exit an atom or molecule source and are collimated into the slowing region. The spatial setup of the sequentially pulsed counterpropagating lasers, which have time-dependent frequencies $ω_{1} (t)$ and $ω_{2} (t)$ , and a sample particle with velocity $v$ in the laboratory frame are displayed in the slowing region. The circular inset shows the two-level internal structure of each particle.
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Figure 2
Left: frequency diagram of an isolated subset of states in the laboratory frame. The laser frequencies $ω_{i} (\bar{p}, t)$ [see Eq. (9)] are dynamically updated according to the solution derived from the Lewis-Riesenfeld invariant shortcut method to promote quick, coherent transfer from (a) $| g, \bar{p} 〉$ to $| e, \bar{p} - ℏ k 〉$ , followed by (b) $| e, \bar{p} - ℏ k 〉$ to $| g, \bar{p} - 2 ℏ k 〉$ . Right: experimental parameters and particle dynamics over a $| \bar{p} 〉 \to | \bar{p} - 2 ℏ k 〉$ sequence of period $2 T$ . (c) Square-pulse Rabi frequency profiles $Ω_{1} (t)$ (red) and $Ω_{2} (t)$ (blue) with amplitude $Ω$ [see Eq. (28)]. (d) Lewis-Riesenfeld detuning profile $δ (t)$ [see Eq. (29)] with cutoff frequency $\pm δ_{cut}$ [see Eq. (42)] for each laser. (e) Ideal excited-state fraction $P_{e}$ dynamics. (f) Ideal average momentum $〈 \hat{p} 〉$ dynamics. Parameters are $T = 0.032 / ω_{r}, Ω = 100 ω_{r}, δ_{cut} = 250 ω_{r}, Γ = 0, \bar{p} = 100 ℏ k$ , and $β = 0.85 π / 2$ .
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Figure 3
Bloch sphere trajectories (thick curves) between the initial state $| G 〉$ and final state $| E 〉$ parametrized by the auxiliary angle $β$ [see Eq. (23)] in the free-energy interaction picture [see Eq. (30)]. We set the Rabi frequency $Ω$ to be equal for all trajectories, and the slowing periods $T$ are given by Eq. (28). The cutoff detunings for $β = \frac{1}{40} \frac{π}{2}$ and $β = \frac{1}{2} \frac{π}{2}$ are $δ_{cut} = 318 Ω$ and $δ_{cut} = 225 Ω$ , respectively [see Eq. (42)].
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Figure 4
Initial (blue) and final (orange) momentum distributions $P (p)$ of a system subject to the slowing protocol. The system is initialized in the internal ground state $| g 〉$ and with a momentum distribution of either the $100 ℏ k$ eigenstate (top row) or a Gaussian state with average momentum $〈 \hat{p} 〉 = 100 ℏ k$ and width $σ_{p} = 10 ℏ k$ (bottom row). The excited-state linewidth is $Γ = 0$ in (a) and (c), and $Γ = ω_{r}$ in (b) and (d). Further details are discussed in the text. Other parameters are $Ω = 100 ω_{r}, T = 0.032 / ω_{r}, t_{tot} = 1.6 / ω_{r}$ , and $δ_{cut} = 244 ω_{r}, β = 0.85 π / 2,$ and ${\bar{p}}_{0} = 100 ℏ k$ . Subplots (b) and (d) are averaged over 1000 trajectories.
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Figure 5
Robustness comparison of $π$ -pulse slowing (purple, plus), SWAP slowing (green, cross), and the shortcut scheme (blue, circle with cross) over a $\bar{p} \to \bar{p} - 2 ℏ k$ transfer process. Ideally, the resulting impulse $Δ p$ satisfies $Δ p / 2 ℏ k = - 1$ . The evolution is purely coherent ( $Γ = 0$ ), the momentum of the particle (which is addressed by the laser frequencies) is $\bar{p} = 2 ℏ k$ , and the Rabi frequency is $Ω = 10 ω_{r}$ for all processes. (a) Impulse $Δ p$ experienced by the particle, in units of the ideal impulse magnitude $2 ℏ k$ , as a function of the error in the Rabi frequency amplitude $ε$ [see Eq. (47)]. The shortcut scheme is the most robust protocol when $ε > 0$ . (b) $Δ p / 2 ℏ k$ as a function of the relative momentum of the particle $δ p$ with respect to the momentum $\bar{p} = 2 ℏ k$ used in the laser frequencies [see Eqs. (9) and (48)]. For this set of parameters, the shortcut and $π$ -pulse scheme are generally more robust than SWAP slowing. $π$ -pulse parameters are $T = 0.314 / ω_{r}$ and $β = π / 2$ . SWAP slowing parameters are $T = 1 / ω_{r}$ and $Δ = 50 ω_{r}$ . Shortcut parameters are $T = 0.44 / ω_{r}, δ_{cut} = 230 ω_{r}$ , and $β = 0.5 π / 2$ .
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Figure 6
Impulse $Δ p$ experienced by the particle, in units of the ideal impulse magnitude $2 ℏ k$ , as a function of the laser pulse overlap fraction $f$ . The impulse is calculated under both coherent ( $Γ = 0$ ) and dissipative ( $Γ = ω_{r}$ ) dynamics. Other parameters are $T = 0.44 / ω_{r}, δ_{cut} = 230 ω_{r}, Ω = 10 ω_{r}, \bar{p} = 2 ℏ k$ , and $β = 0.5 π / 2$ . All points are averaged over 1000 trajectories.
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