Coherent spin manipulation in an exchange-only qubit

Initialization, two-spin coherent manipulation, and readout of a three-spin qubit are demonstrated using a few-electron triple quantum dot. The three-spin qubit is designed to allow all operations for full qubit control to be tuned via nearest-neighbor exchange interaction. Fast readout of charge states takes advantage of multiplexed reflectometry. Decoherence measured in a two-spin subspace is found to be consistent with predictions based on gate voltage noise with a uniform power spectrum. The theory of the exchange-only qubit is developed and it is shown that initialization of only two spins suffices for operation. Requirements for full multiqubit control using only exchange and electrostatic interactions are outlined.

Figure 1

(Color online) (a) Device and measurement circuit. Patterned topgates define three quantum dots and QPC charge sensors on left and right; voltages applied to gates L and R control the energy levels of the device, while voltages $V_{\text{L}}^{\text{QPC}}$ and $V_{\text{R}}^{\text{QPC}}$ tune QPC conductances $g_{\text{L}}$ and $g_{\text{R}}$. The QPCs are incorporated into resonant tank circuits comprising chip inductors $L_{\text{L}}$ and $L_{\text{R}}$ combined with parasitic capacitances $C_{\text{L}}^P$ and $C_{\text{R}}^P$; bias tees allow the QPCs to be measured both at DC and via RF reflectometry. An RF carrier, generated by combining signals at resonant frequencies $f_{\text{L}}$ and $f_{\text{R}}$, is applied to the device via a directional coupler; the reflected signal, after amplification, is demodulated by mixing with the original carrier frequencies to yield voltages $V_{\text{L}}^{\text{RF}}$ and $V_{\text{R}}^{\text{RF}}$ sensitive predominantly to left and right QPCs, respectively. Two RF switches (Minicircuits ZASWA-2-$50\text{DR}+$) allow incident and reflected signals to be blanked except during device readout, reducing backaction and preventing gate pulse coupling to the demodulation circuit. (b) Microwave transmission $S_{21}$ of the cryogenic part of the circuit as a function of frequency, measured between ports 1 and 2 in (a) using a network analyzer. As the QPCs are pinched off, separate resonances develop corresponding to reduced reflection from left and right tank circuits. Carrier frequencies $f_{\text{L}}$ and $f_{\text{R}}$ are chosen to match the two resonance frequencies. (c) and (d), QPC pinchoff measured simultaneously in reflectometry and DC conductance. (e) Reflectometry signal for the right sensor measured as a function of $V_{\text{L}}$ and $V_{\text{R}}$, showing steps corresponding to charge transitions. Electron configurations for each gate setting are indicated.

Figure 2

(Color online) An exchange-only qubit. (a) Electron spins in three adjacent quantum dots are coupled by nearest-neighbor exchange. (b), The eight states of the system can be divided into a quadruplet, $Q$, and two doublets, $D^{\prime }$ and $D$, distinguished by the multiplicity (singlet or triplet) of the rightmost pair of spins. An alternative choice, denoted $\overline{D}$ and ${\overline{D}}^{\prime }$, distinguishes the doublets according to the multiplicity of the leftmost spin pair (dashed boxes). (c), Choosing an element from each doublet as the qubit basis [highlighted in (b)], arbitrary unitary transformations are equivalent to rotations on the Bloch sphere shown, where doublet states $|D_{\pm 1/2}^{\prime }⟩$ and $|D_{\pm 1/2}⟩$ correspond to north and south poles and states $|{\overline{D}}_{\pm 1/2}^{\prime }⟩$ and $|{\overline{D}}_{\pm 1/2}⟩$ to poles of an axis tilted by $120°$. Exchange between middle and right dots drives rotations about the $D-D^{\prime }$ axis, while exchange between left and middle dots drives rotations about the $\overline{D}-{\overline{D}}^{\prime }$ axis. In combination, any rotation can be accomplished.

Figure 3

(Color online) (a) Three-electron energy levels as a function of detuning $ϵ$, showing Zeeman and exchange splitting (see Appendix for details of calculation). The case where left and right inter-dot tunnel couplings are equal is plotted; the case of strong asymmetry, corresponding to the experiment, is discussed in Appendix . Near zero detuning the device is configured in (1,1,1) with negligible exchange; increasing (decreasing) $ϵ$ lowers the energy of the $D^{\prime }$ $\left({\overline{D}}^{\prime }\right)$ doublet by exchange $J_{23}\left(J_{12}\right)$. For $ϵ>{ϵ}_{+}\left(ϵ<{ϵ}_{-}\right)$, states in doublet $D^{\prime }$ $\left({\overline{D}}^{\prime }\right)$ correspond to a predominant (1,0,2) [(2,0,1)] configuration. Doublet levels corresponding to excited charge configurations are shown as unlabeled light gray lines and play no part in spin manipulation. (b) Ground-state configuration of a triple dot as a function of gate voltages $V_{\text{L}}$ and $V_{\text{R}}$ coupled to left and right dots (Ref. 14). The detuning axis is shown.

Figure 4

(Color online) Coherent spin exchange. (a) Probability $P_{D^{\prime }}$ to return to the initial $|D_{S_z}^{\prime }⟩$ state following an exchange pulse sequence, measured as a function of $ϵ$ during the exchange pulse and pulse duration ${\tau }_{\text{E}}$. Dark and bright regions respectively indicate odd and even numbers of complete spin exchanges. (b), Points: Measured $P_{D^{\prime }}$ as a function of ${\tau }_{\text{E}}$ for values of $ϵ$ indicated by horizontal lines in (a). Lines: Fits to exponentially damped phase-shifted cosines, corresponding to coherent rotations dephased by electric fields with a white noise spectrum (see text). The fitted exchange $J_{23}\left(ϵ\right)$ for each curve is shown.

Figure 5

(Color online) Energy levels of three coupled spins, labeled as in Fig. 3a, for the case of asymmetric tunnel couplings $t_{\text{L}}⪡t_{\text{R}}$. The divergence of the doublet energy levels on the left becomes much sharper, making the effects of $J_{12}$ difficult to observe.