Many-body singlets by dynamic spin polarization

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Many-body singlets by dynamic spin polarization

Wang Yao

Phys. Rev. B 83, 201308(R) – Published 23 May 2011

Abstract

We show that dynamic spin polarization by collective raising and lowering operators can drive a spin ensemble from arbitrary initial state to many-body singlets, the zero-collective-spin states with large-scale entanglement. For an ensemble of $N$ arbitrary spins, both the variance of the collective spin and the number of unentangled spins can be reduced to $O (1)$ [versus the typical value of $O (N)$ ], and in the steady state many-body singlets are occupied with a population of $~$ $20 %$ independent of the ensemble size. We discuss a potential implementation in a mesoscopic ensemble of nuclear spins using dynamic nuclear spin polarization by an electron. The result is of twofold significance for spin quantum technology: (1) a cleaner surrounding and less quantum noise for the electron spin and (2) a resource of entanglement for nuclear-spin-based quantum information processing. The scheme can also be applied to other spin systems where collective raising and lowering operations are available.

Received 31 March 2011

DOI:https://doi.org/10.1103/PhysRevB.83.201308

Authors & Affiliations

Wang Yao

Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China

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Issue

Vol. 83, Iss. 20 — 15 May 2011

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Figure 1
(a) A spin ensemble partitioned into subsets $A$ and $B$ ( $C$ and $D$ ) by the solid (dashed) line. (b) An example of how the operator ${ĵ}_{A}^{+} - {ĵ}_{B}^{+}$ couples various basis states $| J, M, j_{A}, j_{B} 〉$ (solid arrows). The absolute value squared of the transition matrix elements are indicated with the thickness of the arrows. The transitions related to the spin coherent states $| J, - J 〉$ are highlighted. Hollow vertical arrows show the coupling by ${Ĵ}^{-}$ . (c) Schematics of the population transfer rates between multiplets under the condition that each multiplet is initialized on the spin coherent state $| J, - J 〉$ when the ${ĵ}_{A}^{+} - {ĵ}_{B}^{+}$ operator is applied. The transfer rate of ${J, α_{J}^{k}} \leftarrow {J, α_{J}^{k^{'}}}$ is identical to the rate of the backward transfer ${J, α_{J}^{k^{'}}} \to {J, α_{J}^{k}}$ . The rate of ${J + 1, α_{J + 1}^{q}} \to {J, α_{J}^{k}}$ is by a factor of $(J + 1) (2 J + 1)$ larger than that of the backward transfer ${J, α_{J}^{k}} \to {J + 1, α_{J + 1}^{q}}$ .Reuse & Permissions
Figure 2
Simulation of squeezing control with the operators ${Ĵ}^{-}$ , ${ĵ}_{A}^{+} - {ĵ}_{B}^{+}$ , and ${ĵ}_{C}^{+} - {ĵ}_{D}^{+}$ (see text). The system is initially in the completely mixed state in the subspace with ${j_{A \cap D} = 7 / 2, j_{B \cap D} = 7 / 2, j_{B \cap C} = 7 / 2, j_{A \cap C} = 7 / 2}$ . The polarization rates are $Λ_{h} = 2000 Λ_{o}$ . (a) $P (J)$ gives the integrated probability of finding the system with total collective spin $J$ at various time. At $t = 0.5 Λ_{o}^{- 1}$ , MBSs are occupied with the population $P (0) = 0.21$ , and $〈 {\overset{\land}{J}}^{2} 〉 = 2.44$ . (b) Population distribution on various MBSs, triplets and quintets at $t = 1.5 Λ_{o}^{- 1}$ . The sum of populations on all other multiplets is less than $2 %$ .Reuse & Permissions
Figure 3
(a) Effects of spin decoherence. Black curves: 8 spin- $\frac{1}{2}$ , $Λ_{h} / Λ_{o} = 40$ ; red (dark gray) curves: 12 spin- $\frac{1}{2}$ , $Λ_{h} / Λ_{o} = 60$ ; green (light gray) curves: 8 spin- $1$ , $Λ_{h} / Λ_{o} = 200$ . $γ_{n} = 0$ for solid curves, and $γ_{n} = 0.1 Λ_{o}$ for dashed curves. The spins are initially in the maximally mixed states. (b) Squeezing of 8 spin- $\frac{1}{2}$ particles using different collective operators. Red (dark gray) curves are squeezing by ${Ĵ}^{-}$ and only one inhomogeneous operator: ${ĵ}_{A}^{+} - {ĵ}_{B}^{+}$ (solid); $0.7 ({ĵ}_{A}^{+} - {ĵ}_{B}^{+}) + 0.3 ({ĵ}_{C}^{+} - {ĵ}_{D}^{+})$ (dashed); $\sum_{n} \exp (i \frac{n}{4} π) {Î}_{n}^{+}$ (dash-dotted). $Λ_{h} / Λ_{o} = 50$ . Black curves are squeezing by ${Ĵ}^{-}$ and two inhomogeneous operators: ${ĵ}_{A}^{+} - {ĵ}_{B}^{+}$ and ${ĵ}_{C}^{+} - {ĵ}_{D}^{+}$ (solid); $0.7 ({ĵ}_{A}^{+} - {ĵ}_{B}^{+}) + 0.3 ({ĵ}_{C}^{+} - {ĵ}_{D}^{+})$ and $0.7 ({ĵ}_{E}^{+} - {ĵ}_{F}^{+}) + 0.3 ({ĵ}_{G}^{+} - {ĵ}_{H}^{+})$ (dashed). $Λ_{h} / Λ_{o} = 100$ . ${A | B} = {1234 | 5678}$ , ${C | D} = {1278 | 3456}$ , ${E | F} = {1357 | 2468}$ , and ${G | H} = {1458 | 2376}$ give different bipartition of the eight spins.Reuse & Permissions
Figure 4
(a) Schematics of an electron in a nuclear spin bath. The first coordination shell has four nuclear spins in green (light gray) color and the second shell has eight nuclear spins in blue (dark gray) color. (b) dc hyperfine coupling coefficients at the various lattice sites. (c) ac hyperfine coupling coefficients with ac electric field in the $x$ direction. The heights of the bar give the magnitude.Reuse & Permissions

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