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Solid State Quantum Memory Using the 31P Nuclear Spin

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Solid State Quantum Memory Using the 31P Nuclear Spin Solid state quantum memory using the 31P nuclear spin John J. L. Morton,1, 2, ∗ Alexei M. Tyryshkin,3 Richard M. Brown,1 Shyam Shankar,3 Brendon W. Lovett,1 Arzhang Ardavan,2 Thomas Schenkel,4 Eugene E. Haller,4, 5 Joel W. Ager,4 and S. A. Lyon3 1Department of Materials, Oxford University, Oxford OX1 3PH, United Kingdom 2Clarendon Laboratory, Department of Physics, Oxford University, Oxford OX1 3PU, United Kingdom 3Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA 4Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley CA 94720, USA 5Materials Science Department, University of California, Berkeley, USA (Dated: October 22, 2018) The transfer of information between different physical advantage of integration with existing technologies [4] forms is a central theme in communication and computa- and the possibility of single spin detection by electrical tion, for example between processing entities and mem- measurement [13, 14, 15]. Direct measurement of the ory. Nowhere is this more crucial than in quantum com- 31P nuclear spin by NMR has only been possible at very putation [1], where great effort must be taken to pro- high doping levels (e.g. near the metal insulator tran- tect the integrity of a fragile quantum bit (qubit) [2]. sition [16]). Instead, electron-nuclear double resonance However, transfer of quantum information is particularly (ENDOR) can be used to excite both the electron and challenging, as the process must remain coherent at all nuclear spin associated with the donor site, and measure times to preserve the quantum nature of the informa- the nuclear spin via the electron [17]. This was recently tion [3]. Here we demonstrate the coherent transfer of a used to measure the nuclear spin-lattice relaxation time superposition state in an electron spin `processing' qubit T1n, which was found to follow the electron relaxation to a nuclear spin `memory' qubit, using a combination time T1e over the range 6 to 12 K with the relationship 31 of microwave and radiofrequency pulses applied to P T1n≈ 250T1e [5, 12]. The suitability of the nuclear spin donors in an isotopically pure 28Si crystal [4, 5]. The as a quantum memory element depends more critically state is left in the nuclear spin on a timescale that is on the nuclear coherence time T2n, the measurement of long compared with the electron decoherence time and which has now been made possible through the storage then coherently transferred back to the electron spin, procedure described here: by varying the storage time thus demonstrating the 31P nuclear spin as a solid-state and observing the amplitude of the recovered electron quantum memory. The overall store/readout fidelity is coherence. about 90%, attributed to imperfect rotations which can Figure 1(B) shows the coherence transfer scheme used be improved through the use of composite pulses [6]. The for the write process from a processing qubit represented coherence lifetime of the quantum memory element at by an electron spin degree of freedom, to a memory qubit 5.5 K exceeds one second. residing in a nuclear spin degree of freedom. Each π pulse Classically, transfer of information can include a copy- is equivalent to a controlled-NOT gate [18] (with some ing step, facilitating the identification and correction of additional phase which can be ignored) such that the errors. However, the no-cloning theorem limits the abil- pair of π pulses constitute a SWAP gate. The scheme as- ity to faithfully copy quantum states across different de- sumes that all pulses are on-resonance and have sufficient grees of freedom [7]; thus error correction becomes more bandwidth to completely excite an individual transition. challenging than for classical information and the trans- A read operation is performed by applying the reverse fer of information must take place directly. Experimental sequence to bring the coherent state back to the electron demonstrations of such transfer include moving a trapped spin qubit. Although the phase relationship betweeen arXiv:0803.2021v2 [quant-ph] 30 Jun 2008 ion qubit in and out of a decoherence-free subspace for the microwave and rf pulses must be constant through- storage purposes [8] and optical measurements of NV cen- out this process, any phase difference is cancelled out tres in diamond [9]. over the course of the write-read process. In practice, Nuclear spins are known to benefit from long coher- this means the microwave and rf sources need not be ence times compared to electron spins, but are slow to phase-locked, but must have high phase stability. This is manipulate and suffer from weak thermal polarisation. illustrated in calculations following the evolution of the A powerful model for quantum computation is thus one density matrix, provided in the Supplementary Material. in which electron spins are used for processing and read- Although the electron spin qubit can be prepared in out while nuclear spins are used for storage. The storage a state of high purity using experimentally accessible element can be a single, well-defined nuclear spin, or per- magnetic fields and temperatures, the small nuclear Zee- haps a bath of nearby nuclear spins [10]. 31P donors in sil- man energy results in the nuclear spin being initially in icon provide an ideal combination of long-lived spin-1/2 a highly mixed thermal state. However, for the purposes electron [11] and nuclear spins [12], with the additional of this quantum memory scheme it is not necessary to 2 A (electron echo) (nuclear echo) electron echo νrf1 4 〉 A τe1 τe1 τn τn τe2 τe2 2 〉 P nuclear π/2 π π π π coherent spin Sx Sy νμw2 π π π νμw1 superposition donor spin 1 〉 generate refocus measure E-coherence transfer to N N-coherence transfer to E νrf2 3 〉 of phase φ E l B Recovered echo intensity (a.u.) +X, φ=0˚ e φn B c t r π π 3 o +Y, φ=90˚ n s φe φx p 2 i νrf1 ν n μw1 −X, φ=180˚ e T2n =1.75 s c 1 h o −Y, φ=270˚ 0 FIG. 1: The level structure of the coupled electron and 0 0.5 1 Storage time (s) nuclear spins and scheme for the transfer of a logical Time (μs) 0 20 qubit within the two physical spin qubits. (A) The four level system may be manipulated by resonant microwave and FIG. 2: Coherent storage of an electron spin state in a radiofrequency (rf) radiation. In our experiments the logical nuclear spin state, using 31P-doped 28Si-enriched sili- electron spin `processing' qubit is represented by states j1i and con single crystal. A) An electron spin coherence is stored j2i, whose state can be transferred to a nuclear spin `memory' in the nuclear spin for 2τn ≈ 50 ms, at 7.2 K. The recovered qubit represented by states j2i and j4i. State j3i is never electron spin echo is of comparable intensity to that obtained addressed at any point and can be ignored. (B) An electron at the beginning of the sequence, even though the electron spin coherence between states j1i and j2i is transferred to the spin coherence time T2e here is about 5 ms. The lifetime of nuclear spin qubit by an rf π pulse followed by a microwave the stored state is limited instead by the nuclear decoherence π pulse. Both pulses must fully excite the transition, and time T2n, which can be measured directly by varying τn. B) be short compared with the electron and nuclear coherence The recovered echo intensity was measured a function of the times. The reverse process is used to transfer the nuclear storage time at 5.5 K while applying a dynamic decoupling se- coherence back to the electron. quence (CPMG) to the nuclear spin, yielding a T2n exceeding 1 second. perform any pre-cooling of the nuclear spin resource [34]. The above model is sufficient given a single electron- ble quantum coherence of entangled electron-nuclear spin nuclear spin pair, or a homogenous ensemble. However, states. During this period the phase δeτe, acquired be- in the experiment described here, we must consider the fore the microwave refocusing pulse, continues to unwind effects of inhomogeneous broadening across the ensem- so that when the final step of the transfer, a microwave ble of spins being manipulated. The effect of inhomo- π pulse, is applied the effect of the inhomogeneous elec- geneous broadening is to leave some electron (nuclear) tron spin packets has been completely refocused. The spins detuned from the applied microwave (rf) radiation, quantum information that was generated by the first mi- by δe (δn). In a suitable rotating reference frame, electron crowave π=2 pulse now resides entirely in the state of the (nuclear) spin coherence will thus acquire an additional nucleus. phase at a rate δe (δn), while double quantum coher- This information may be stored in the nuclear state for ences will acquire phase at a rate δe + δn. Thus, inho- some extended period so the effects of inhomogeneities on mogeneous broadening requires the application of care- the phase of the nuclear state become appreciable and a fully placed refocusing pulses to bring all spin packets preparatory rf refocusing pulse must be applied before into focus at key points during the transfer process. In the information can be recovered. During the nuclear the experiment described here, π/δe ∼ 2 µs and π/δn ∼ spin echo, the coherence is transferred back to the elec- 100 µs. tron state with a microwave π pulse followed by an rf Figure 2 shows the practical implementation of a pro- π pulse. We apply one further microwave π pulse to tocol that generates a coherent electron spin state, stores stimulate an electron spin echo representing the readout it in a state of the nuclear spin for some time, and then event.
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