Coherent interactions at a distance provide a powerful tool for quantum simulation and computation. The most common approach to realize an effective long-distance coupling ‘on-chip’ is to use a quantum mediator, as has been demonstrated for superconducting qubits and trapped ions. For quantum dot arrays, which combine a high degree of tunability with extremely long coherence times, the experimental demonstration of the time evolution of coherent spin–spin coupling via an intermediary system remains an important outstanding goal. Here, we use a linear triple-quantum-dot array to demonstrate a coherent time evolution of two interacting distant spins via a quantum mediator. The two outer dots are occupied with a single electron spin each, and the spins experience a superexchange interaction through the empty middle dot, which acts as mediator. Using single-shot spin readout, we measure the coherent time evolution of the spin states on the outer dots and observe a characteristic dependence of the exchange frequency as a function of the detuning between the middle and outer dots. This approach may provide a new route for scaling up spin qubit circuits using quantum dots, and aid in the simulation of materials and molecules with non-nearest-neighbour couplings such as MnO , high-temperature superconductors and DNA. The same superexchange concept can also be applied in cold atom experiments. Leon Camenzind FAM, 21.10.2016

The exchange Gate (SWAP) 𝑡 𝑁 𝜖 𝐽 ~ 𝑡 𝑁 2 /𝜖 𝜙=𝐽 𝜖 𝜏 𝐸 /ℏ Petta et al., Science 2005 𝜖→−∞, separate dot 𝜖=0 Martins et al., PRL116 (2015)

≫ Exchange Interaction 𝑡 𝑁 𝐽 𝑁 ~ 𝑡 𝑁 2 𝜖 𝑡 𝑁 : nearest-neighbour tunnel coupling 𝜖: detuning of dots

Quantum Mediator→ Superexchange «Virtual Transition»: virtual occupation of a non-magnetic intermediate state t 𝐽 𝑆𝐸 ~ 𝑡 𝑚,𝑙 2 𝑡 𝑚,𝑟 2 𝛿 2 𝜖 𝛿: detuning of intermediate quantum mediator. 𝜖: detuning of outer dots 𝐽 𝑆𝐸 See e.g.: Erich Koch, Exchange Mechanisms, Lecture Notes Kramers, H. L’interaction entre les atomes magnétogènes dans un cristal paramagnétique. Physica 1, 182–192 (1934) →𝑀 𝑛 2 𝑂

Motivation I – large range SWAP gate SWAP Operation on a network of Qbits Tunable Quantum Mediator qubit All electrical Turn on/off of J gate

Sources of for spin projection change: charge exchange with the reservoirs (control ~ 10 −5 ) hyperfine interaction 𝛿 𝐵 ⊥ ~7𝑚𝑇 vs 3.5T → 10 −6 flips per transfer spin–orbit (SO) interaction Experiment: no spin rotation due to shuttling Scarlino et al., PRL 113 (2014)

Long-range transfer using second order tunneling Detuning of Middle Dot See also

Back to Superexchange: Spin Relaxation and Measurement Scheme Readout 1 vs time B = 3.2T Readout fidelities of 95.9% and 98.0% for spin species Spins completely separated L+R are «virtual gates»: compensating crosscapacitances We have ~ 147ms for this configuration || 10ms for 90° @ 3.2T

Measurement scheme In Detunig space F Mixture | ↑0↓ 𝑎𝑛𝑑 | ↑0↑ Spins well separated J=0

Superexchange-driven spin oscillations 𝐽 𝑁 ~ 𝑡 𝑁 2 𝜖 𝐽 𝑆𝐸 ~ 𝑡 𝑚,𝑙 2 𝑡 𝑚,𝑟 2 𝛿 2 𝜖 𝜙=𝐽 𝜖 𝜏 𝐸 /ℏ From EDSR: Δ 𝐸 𝑍 ~130𝑀𝐻𝑧 𝑡 𝑚,𝑙 =8.5 𝐺𝐻𝑧= 𝑡 𝑁 𝑡 𝑚,𝑟 =11.8 𝐺ℎ𝑧

From Superexchange to nearest Neigbour exchange 𝐽 𝑁 ~ 𝑡 𝑁 2 𝜖 𝑙,𝑚 ~ 𝑡 𝑁 2 𝛿 𝐽 𝑆𝐸 ~ 𝑡 𝑚,𝑙 2 𝑡 𝑚,𝑟 2 𝛿 2 𝜖 𝛿>0 𝜙=𝐽 𝜖 𝜏 𝐸 /ℏ 𝛿<0 𝜖=−120𝜇𝑒𝑉 𝐽 𝑁 𝐽 𝑆𝐸 Δ 𝐸 𝑧 J up to 900 MHz

Conclusion Quantum Dot array with very good control + subsequentual spin read out Quantum gate between spins at distance via virtual occupation of a quantum mediator Δ 𝐸 𝑧 → 𝐽 𝑆𝐸 → 𝐽 𝑁 and good agreement with theory

M=-42mV M=-56mV

Simplified argument M=0 M=0 M ≠ 0 Erich Koch, Exchange Mechanisms, Lecture Notes

M=-412mV M=-382mV L+R are «virtual gates»: compensating crosscapacitances

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