Mechanical rotation via optical pumping of paramagnetic impurities

Pablo R. Zangara, Alexander Wood, Marcus W. Doherty, and Carlos A. Meriles

Phys. Rev. B 100, 235410 – Published 6 December 2019

Abstract

Hybrid quantum systems exhibiting coupled optical, spin, and mechanical degrees of freedom can serve as a platform for sensing, or as a bus to mediate interactions between qubits with disparate energy scales. These systems are also creating opportunities to test foundational ideas in quantum mechanics, including direct observations of the quantum regime in macroscopic objects. Here, we make use of angular momentum conservation to study the dynamics of a pair of paramagnetic centers featuring different spin numbers in the presence of a properly tuned external magnetic field. We examine the interplay between optical excitation, spin evolution, and mechanical motion, and theoretically show that in the presence of continuous optical illumination, interspin cross relaxation must induce rigid rotation of the host crystal. The system dynamics is robust to scattering of spin-polarized phonons, a result we build on to show this form of angular momentum transfer should be observable using state-of-the-art torsional oscillators or trapped nanoparticles.

Received 4 April 2019
Revised 7 September 2019

DOI:https://doi.org/10.1103/PhysRevB.100.235410

Physics Subject Headings (PhySH)

Cooling & trapping Defects Optical pumping Phonons Quantum metrology Spin dynamics

Interdisciplinary PhysicsAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Pablo R. Zangara¹, Alexander Wood³, Marcus W. Doherty⁴, and Carlos A. Meriles^1,2,*

¹Department of Physics, CUNY-City College of New York, New York, NY 10031, USA
²CUNY-Graduate Center, New York, NY 10016, USA
³School of Physics, The University of Melbourne, Melbourne, Victoria 3010, Australia
⁴Laser Physics Centre, Research School of Physics, Australian National University, Canberra, Australian Capital Territory 2601, Australia

^*cmeriles@ccny.cuny.edu

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Issue

Vol. 100, Iss. 23 — 15 December 2019

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Figure 1
The interplay between optical spin pumping and the crystal's mechanical degrees of freedom. (a) Schematics of a coupled NV-P1 pair in a diamond crystal. The P1 center interacts with other P1s farther removed from the NV. (b) Energy level diagram of the NV and P1 spins (top and bottom respectively. At ∼51 mT the energies associated with each individual spin transition match, i.e., $δ E_{S} \approx δ E_{I}$ . (c) Starting with NV spin optical initialization, the NV-P1 pair undergoes a cycle of cross relaxation and generation of spin-polarized phonons and rigid lattice rotation. The cycle completes with P1 spin diffusion and spin-lattice relaxation accompanied by the emission of spin-polarized phonons.
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Figure 2
Modeling spin-crystal momentum conversion. (a) “Tight-binding” representation of the rotor pumping process: Upper and lower chains correspond to spin states $| 0, + 1 / 2 〉$ and $| - 1, - 1 / 2 〉$ , respectively, while chain sites indicate rotational states $| m_{L} 〉$ . Starting from a state $| m_{S}, m_{I}, m_{L} 〉$ , the system evolution is governed by the dipolar rate $Γ_{d}$ , the optical pumping rate $Γ_{o}$ , and the rotor decoherence rate $Γ_{L}$ . (b) Occupational probability of rotational states after evolution in the presence of continuous optical excitation for a fixed time interval $t = 500$ µs and various rotor depahasing rates $Γ_{L}$ ; the faint black trace indicates the population distribution assumed for $t = 0$ . The dynamics is evaluated using the Trotter-Suzuki method (see Methods); each curve shows the result after 50 averages. (c) Mean angular momentum $L_{z}$ as a function of the normalized evolution time $Γ_{d} t$ for some rotor decoherence rates $Γ_{L}$ ; the initial rotor state is that of (b). (d) Long-term evolution for the case $Γ_{L} = 0$ . At sufficiently large rotational energies, the spin-crystal momentum transfer is inefficient and the system evolves towards a pseudoterminal angular speed, whose value can be adjusted by shifting the magnetic field (shaded region of the plot). In (b), (c), and (d) we use $Γ_{d} = 0.5$ MHz, $Γ_{o} = 1$ MHz, and $ℏ / 2 J = 10$ Hz.
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Figure 3
Spin-phonon angular momentum conversion. (a) Nonresonant mechanism of spin-phonon conversion, i.e., $c | k | > ω_{0}$ . Straight (wavy) arrows indicate spin (phonon spin) change; clockwise (counter clockwise) corresponds to phonons with negative (positive) angular momentum. The initial and final states are $| i_{k} 〉 = | ... n_{k -}, n_{k +}, ... 0, + 1 / 2 〉$ and $| f_{k} 〉 = | ... n_{k -} - 1, n_{k +} + 1, ... - 1, - 1 / 2 〉$ , respectively. The upper and lower virtual states are $| g_{k, 1} 〉 = | ... n_{k -} - 1, n_{k +}, ... - 1, + 1 / 2 〉$ , and $| g_{k, 2} 〉 = | ... n_{k -}, n_{k +} + 1, ... - 1, + 1 / 2 〉$ , respectively. (b) Resonant mechanism, i.e., $c | k_{0} | = ω_{0}$ . The notation used for all kets is the same as in (a), except that $k \to k_{0}$ , and the intermediate state is $| g_{k_{0}} 〉 = | ... n_{k_{0} -} - 1, n_{k_{0} +}, ... - 1, + 1 / 2 〉$ . (c) Resonant spin-phonon conversion rate as a function of the crystal volume; $λ_{0} \equiv 2 π c / ω_{0}$ denotes the wavelength of resonant phonons, not supported by crystals of smaller size. (d) NV–P1 spin “reset” involving NV spin pumping and P1 spin diffusion into the bulk. Spin-phonon relaxation subsequently transfers the P1 polarization to the phonon bath. (e) Schematic representation of the NV–P1 spin cycle. Repeated sequences of NV spin pumping, P1-enabled spin cross relaxation, and NV–P1 spin resets simultaneously produce crystal rotation and polarization of the phonon bath spin.
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