Observation of Four-body Ring-exchange Interactionsand Anyonic Fractional Statistics

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan

Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, 69120, Heidelberg, GermanyUniversity of Science and Technology of China, 230026, Hefei, China

Funding

Reference:

* E. Knill, Nature 434, 39 (2005)§ A. Y. Kitaev, Ann. Phys. 303, 2 (2003).† H. Büchler, et. al. Phys. Rev. Lett. 95, 040402 (2005).‡ B. Paredes and I. Bloch, Phys. Rev. A. 77, 023603 (2008).¶ H.-N. Dai, et. al. Nat. Phys. (2016).

arXiv : 1602.05709

Experimental Setup

𝑉 𝑥 = 𝑉𝑥𝑠𝐶𝑜𝑠2 2𝑘𝑥 + 𝑉𝑥𝑙𝐶𝑜𝑠2 𝑘𝑥

𝑉 𝑦 = 𝑉𝑦𝑠𝐶𝑜𝑠2 2𝑘𝑥 + 𝑉𝑦𝑙𝐶𝑜𝑠2 𝑘𝑥

𝑉 𝑧 = 𝑉𝑧𝐶𝑜𝑠2 2𝑘𝑧𝑥

• Single layer of atoms

• in-situ imaging (N.A.=0.48)

87Rb𝜆𝑠 = 767 𝑛𝑚𝜆𝑙 = 1534 𝑛𝑚𝑑 = 4 𝜇𝑚

↑ = 𝐹 = 2, 𝑚𝐹 = −2 ↓ = 𝐹 = 1, 𝑚𝐹 = −1

• 2D superlattices

• Four-site plaquette model

Δ Δ

• The Four-site Plaquette Model

• “Site-Resolved” MW addressing

• State Readout

• Ring-Exchange Evolution

• Experiment vs Theory

Suppress lower order interactions

𝐻𝑅 = −𝐽□ 𝑆1+ 𝑆2

− 𝑆3+ 𝑆4

− + H. c. − 𝐽+ 𝑗,𝑘 𝑆𝑗𝑧 𝑆𝑘

𝑧

+ 𝑗 𝛥𝑗 𝑆𝑗𝑧

• Four-body Ring-Exchange

• the bare tunnelings (1st order )

• the superexchange processes (2nd order)

The 4th order dynamics becomes dominant

and

by strong interactions 𝑈 ≫ 𝐽

by gradients‡ 𝛥𝑥(𝑦) ≫ 4𝐽2/𝑈

• Coherent evolution in a two-level subspace

Ring-exchange dynamics

“effective shot-range gradient”

well confirmed with a microscopic analysis of the generalized BHM

• State initialization

• Spin-dependent superlattices¶

𝑁𝑧 = 𝑆1𝑧 − 𝑆2

𝑧 + 𝑆3𝑧 − 𝑆4

𝑧 /2= 𝑁↑ − 𝑁↓ / 𝑁↑ + 𝑁↓

Measuring the oscillations of 𝑁𝑧

deriving the major frequency

𝑉𝑥𝑙 = 𝑉𝑦𝑙 = 10 𝐸𝑟,

𝑉𝑥𝑠 = 19.2 1 𝐸𝑟, 𝑉𝑦𝑠 = 18.2 1 𝐸𝑟,

𝛥𝑥 = 115 1 Hz, 𝛥𝑦 = 145 1 Hz,

𝐽𝑥/𝑈 = 0.064, 𝐽𝑦/𝑈 = 0.075

𝑓 = 2.9 1 Hz

Resolving the Ring-Exchange Dynamics

Summary

• Prepare, address and manipulate the sites on isolated four-site optical plaquettes

• Observe the four-spin ring-exchange dynamics

• Simulate a minimal instance of the Toric code model

• Observe the fractional statistics between the anyonic excitations

Outlook

• Study topological liquid (Z2 symmetry)

• Study Many-body entangled system

Motivation

To make quantum information processing technology a reality, scientists have to solve the crucial problem of decoherence and systematic errors in real quantum systems*. The Kitaev toric model§ of fault-tolerant quantum computation by anyons, a sort of topological quasiparticles being neither bosons nor fermions, promises a way towards efficient quantum computing.

The ring-exchange interaction†, which is the key ingredient to the toric code model and many different models in condensed matter‡, remains notoriously difficult to implement in experiment due to its nature of the fourth-order spin interaction. It is usually greatly suppressed compared to the lower order processes, such as superexchange interactions.

• Fault-torrelant Quantum Computing

𝐻 = 𝑣 𝐴𝑣 + 𝑝

𝐵𝑝

𝐴𝑣 = − 𝜎1𝑥 𝜎2

𝑥 𝜎3𝑥 𝜎4

𝑥 𝐵𝑝 = − 𝜎1𝑧 𝜎2

𝑧 𝜎3𝑧 𝜎4

𝑧

𝐻𝑇 = −𝐽□ 𝜎1𝑥 𝜎2

𝑥 𝜎3𝑥 𝜎4

𝑥 − 𝐽+ 𝑗,𝑘 𝜎𝑗𝑧 𝜎𝑘

𝑧

Observing the Anyonic Fractional Statistics

• “e –particle”

• “m –particle”

• Anyon in two dimensional space

for Fermions or Bosons:

for Anyons:

𝑒𝑖𝜙 2= 1

𝑒𝑖𝜙 2≠ 1

Exchange positions of two particles two times,

• States of Toric code model

𝜎1𝑥 □

𝜎1𝑧 □

• Ground state □

• Braiding anyons

• Anyonic interferometer

𝛥𝜑 = 1.00 3 𝜋

𝐻𝑇 = −𝐽□ 𝜎1𝑥 𝜎2

𝑥 𝜎3𝑥 𝜎4

𝑥 − 𝐽+ 𝑗,𝑘 𝜎𝑗𝑧 𝜎𝑘

𝑧