Coherent control of electron-nuclear states of rare-earth ions in crystals using radio-frequency and microwave radiation

We have demonstrated electron-electron and electron-nuclear spin manipulations of Gd3+ ion in CaWO4 crystal. The results suggest that the studied system is perspective for multiqubit implementation in quantum computing.

It consists of six resonance lines corresponding to various S z1 → S z2 transitions between the states of the ground spin-7/2 multiplet. Containing 8 lowestenergy spin states (S z = -7/2÷7/2), Gd 3+ ion can be viewed as model three-qubit system. Single-qubit manipulations in this system have been demonstrated in our previous work [3]. There, mw pulse excited one of the six transitions (chosen by appropriately tuned magnetic field), and the resulting Rabi oscillations were recorded in the time domain of the mw pulse.
Since the impurity gadolinium ions contain odd 155 Gd and 157 Gd isotopes with nuclear spin I = 3/2 (natural abundance 14.8% and 15.6%, respectively), one can use pulsed ENDOR in order to manipulate both the electron and nuclear spin states of these isotopes simultaneously. The total basis of 8•4 = 32 electro-nuclear states would expand the possible quantum algorithm to 5-qubit operations. In order to demonstrate the control of both the electron and nuclear spin states simultaneously, we used Davies ENDOR pulse sequence [4]: an initial mw π pulse is followed by the radiofrequency (rf) π pulse, then the remaining electron state is probed by the mw two-pulse spin-echo sequence. If the rf pulse frequency matches the nuclear transition, the final echo intensity changes. An example of ENDOR spectrum corresponding to 5/2→3/2 electronic transition near B = 244 G swept in the range 0-50 MHz of the rf pulse is shown in Fig. 2 Fig. 1 are orientation-dependent [5], one can choose an appropriate direction of B with respect to the crystallographic axes, so that two electronic transitions lie within the 800 MHz bandwidth of the two-frequency DEER equipment. EPR spectrum in certain orientation (13 degrees between B and c axis) is shown in Fig. 3. There, two high-field transitions, 5/2→3/2 and 7/2→5/2, involving three electronic levels, are drawn together, the corresponding resonance fields differ by ≈ 100 G. Then, the following two-frequency pulse sequence has been applied: π/2(ν 1 ) -π/2(ν 1 ) -π(ν 2 )π/2(ν 1 ), where the indices in brackets denote the mw pulse frequency. There, ν 1 = 9.74 GHz corresponded to the resonant transition 5/2→3/2 at 7210 G, while ν 2 was swept in the range 9.3-10 GHz. When ν 2 did not match any transition, a usual three-pulse stimulated echo for 5/2→3/2 transition was recorded. However, when ν 2 matched either 5/2→3/2 or 7/2→5/2 transition energy, the third (flipping) π pulse inversed the populations of the two involved spin states, so that the echo intensity was decreased.

7000
7200 7400 Magnetic field, G 7/2 5/2 5/2 3/2 Fig. 3. A part of pulsed EPR spectrum of Gd:CaWO 4 when B is directed at an angle of 13 degrees with respect to crystal c axis 9.4 9.5 9.6 9.7 9.8 9.9 mw frequency, GHz The coherence loss associated with the third pulse resulted in two dips in the echo intensity: at ν 2 = ν 1 (original 5/2→3/2 transition), and at ν 2 = 9.49 GHz (7/2→5/2 transition), see Fig. 4. An estimated magnitude of the observed hole burnt is ~ 10% of the one at ν 2 = ν 1 , this figure is attributed to the inhomogeneous broadening of the resonance lines. In order to measure the corresponding coherence time of the three-state ensemble (a qutrit), we have recorded the transient nutation of the 7/2→5/2 coherence. There, the duration of the third ν 2 pulse resonant with 7/2→5/2 transition was varied in the range 0-4 μs, while the duration of the π(ν 2 ) pulse was ≈ 180 ns. The resultant echo intensity vs. the pulse duration is shown in Fig.  5.
Transient pulse duration, s Finally, we have demonstrated electron-electron and electron-nuclear spin manipulations of Gd 3+ ion in CaWO 4 crystal. The results suggest that the studied system is perspective for multiqubit implementation in quantum computing.
This work was financially supported by the Russian Science Foundation (Project no. 17-72-20053).