Coupling of three-spin qubits to their electric environment

We investigate the behavior of qubits consisting of three electron spins in double and triple quantum dots subject to external electric fields. Our model includes two independent bias parameters, $\varepsilon$ and ${\varepsilon }_M$, which both couple to external electromagnetic fields and can be controlled by gate voltages applied to the quantum dot structures. By varying these parameters, one can switch the qubit type by shifting the energies in the single quantum dots, thus changing the electron occupancy in each dot. Starting from the asymmetric resonant exchange qubit with a (2,0,1) and (1,0,2) charge admixture, one can smoothly cross over to the resonant exchange qubit with a detuned (1,1,1) charge configuration, and to the exchange-only qubit with the same charge configuration but equal energy levels down to the hybrid qubits with (1,2,0) and (0,2,1) charge configurations. Here, $(l,m,n)$ describes a configuration with $l$ electrons in the left dot, $m$ electrons in the center dot, and $n$ electrons in the right dot. We first focus on random electromagnetic field fluctuations, i.e., “charge noise,” at each quantum dot resulting in dephasing of the qubit, and we provide a complete map of the resulting dephasing time as a function of the bias parameters. We pay special attention to the so-called sweet spots and double sweet spots of the system, which are least susceptible to noise. In the second part, we investigate the coupling of the qubit system to the coherent quantized electromagnetic field in a superconducting strip-line cavity, and we also provide a complete map of the coupling strength as a function of the bias parameters. We analyze the asymmetric qubit-cavity coupling via $\varepsilon$ and the symmetric coupling via ${\varepsilon }_M$.

Figure 1

Schematic illustration of a three-spin qubit coupled to a noisy electric environment. The environment can affect the electron spins directly through the gate voltages $V_i$ with $i\in \{1,2,3\}$ of each quantum dot (QD) or the exchange coupling (green cloud) between the electron spins through the gate-controlled tunnel hopping ($t_l$ and $t_r$).

Figure 2

Energy landscape of the ground-state energy gap $\omega$ as a function of the detuning parameters $\varepsilon$ and ${\varepsilon }_M$ in units of the charging energy $U$. For the tunneling parameters, we used $t_l=0.022U$ and $t_r=0.015U$. Maneuvering through the ($\varepsilon ,{\varepsilon }_M$) plane, one can access various parameter regimes that allow the use of different qubit implementations in different charge configurations $(l,m,n)$, where $l$ electrons are in the left, $m$ electrons in the center, and $n$ electrons in the right QD. We further highlight the double sweet spots (DSSs) (black dots), the location of the exchange-only (EO) qubit, the resonant exchange (RX) qubit (dashed triangle), the asymmetric resonant exchange (ARX) qubit, and the left and right hybrid ($H_{l,r}$) qubit.

Figure 3

Dephasing time $T_{\phi }$ given by Eq. (13) due to longitudinal noise as a function of the detuning parameters $\varepsilon$ and ${\varepsilon }_M$. In the top row [(a) and (b)] we plot $T_{\phi }$ resulting from charge noise in the two detuning parameters $\varepsilon$ and ${\varepsilon }_M$, in the center row [(c) and (d)] we plot $T_{\phi }$ resulting from charge noise in the two tunneling parameters $t_l$ and $t_r$, and in the bottom row [(e) and (f)] we plot $T_{\phi }$ resulting from charge noise from all four parameters combined, where we choose the parameter settings identical in each column. The left column shows results for weak tunneling and strong noise, while in the right column, results for strong tunneling and weak noise are plotted. Parameters are set as follows: $t_l=0.022U,t_r=0.015U,A_q={\left({10}^{-3}U\right)}^2$, where $q=\varepsilon ,{\varepsilon }_M$ in (a) and (e), and $A_q={\left({10}^{-4}U\right)}^2$, where $q=t_l,t_r$ in (c) and (e), for the left column, and $t_l=0.22U,t_r=0.15U,A_q={\left({10}^{-5}U\right)}^2$, where $q=\varepsilon ,{\varepsilon }_M$ in (b) and (f), and $A_{t_l}=A_{t_r}={\left({10}^{-6}U\right)}^2$, where $q=t_l,t_r$ in (d) and (f), for the right column. To include a large frequency bandwidth, we globally set the ratio of the lower and higher frequency cutoff $r=5\times {10}^6$. For the scale of $T_{\phi }$, we used an explicit value of $U=1\mathrm{meV}$; note that $T_{\phi }$ scales inversely proportional with $U$. The black dots indicate the DSSs.

Figure 4

(a) Schematic illustration of a qubit implemented in a TQD coupled to the cavity and the architecture for a (b) asymmetric and (d) symmetric qubit-cavity coupling. The center conductor of the superconducting transmission line resonator is on the potential ${\mathcal{V}}_{\text{cav}}$ while the outer conductors are connected to the ground to screen off surrounding fields. The corresponding potential (green) and electric field (blue) are shown for the asymmetric (c) and symmetric (e) arrangement as a function of the position $x$.

Figure 5

The qubit-cavity coupling strength $g$ in units of the vacuum coupling strength $g_0$ as a function of the detuning parameters $\varepsilon$ and ${\varepsilon }_M$ for the asymmetric coupling (left column) and symmetric coupling (right column). The parameters are chosen as follows: top row [(a) and (b)] $t_l=t_r=0.02U$, center row [(c) and (d)] $t_l=0.022U$ and $t_r=0.015U$, and bottom row [(e) and (f)] $t_l=t_r=0.2U$. The interdot distances $a_l$ and $a_r$ and the overlaps $S_l$ and $S_r$ are set by the strength of the tunneling parameters [42, 71]. The black dots denote the DSSs.

Figure 6

Top row: The minimal vacuum coupling $g_0$ [see Eq. (36)] needed to reach strong coupling between the qubit and the cavity under the assumption that qubit dephasing is the dominant loss mechanism. Bottom row: Minimal Q-factor of the cavity [see Eq. (37)] needed for successful entanglement between two qubits in the same cavity. Parts (a) and (c) show the results for the asymmetric architecture and only noise in the asymmetric detuning parameter $\varepsilon$, while (b) and (d) show the results for the symmetric architecture and only noise in the symmetric detuning parameter ${\varepsilon }_M$. The parameters are chosen as follows: ${\omega }_{\text{ph}}=4.7\mathrm{GHz},g_0=2\pi \times 10\mathrm{MHz},t_l=0.022U,t_r=0.015U$, and $A_q=\left({10}^{-3}U\right)$, where $q=\varepsilon$ in (a) and $q={\varepsilon }_M$ in (b). The datasets for $T_{\phi }$ are obtained from Figs. 7 and 7. For the scale of $T_{\phi }$ and $g$, we used an explicit value of $U=1\mathrm{meV}$. The black dots denote the DSS.

Figure 7

Dephasing time $T_{\phi }$ due to longitudinal noise for each noisy parameter $\varepsilon ,{\varepsilon }_M,t_l$, and $t_r$ individually in the $(\varepsilon ,{\varepsilon }_M)$ plane. Each row shows the dephasing time due to a single noisy parameter (from top to bottom: $\varepsilon ,{\varepsilon }_M,t_l,t_r$), while we choose the parameter settings identical in each column. The left column contains the results for weak tunneling and strong noise, while the right column comprises the results for strong tunneling and weak noise. Parameters are set as follows; $t_l=0.022U,t_r=0.015U,A_q={\left({10}^{-3}U\right)}^2$, where $q=\varepsilon$ in (a) and $q={\varepsilon }_M$ in (c), and $A_q={\left({10}^{-4}U\right)}^2$, where $q=t_l$ in (e) and $q=t_r$ in (g), for the left column, and $t_l=0.22U,t_r=0.15U,A_q={\left({10}^{-5}U\right)}^2$, where $q=\varepsilon$ in (b) and $q={\varepsilon }_M$ in (d), and $A_q={\left({10}^{-6}U\right)}^2$, where $q=t_l$ in (f) and $q=t_r$ in (h), for the right column. For the scale of $T_{\phi }$, we used an explicit value of $U=1\mathrm{meV}$; note that $T_{\phi }$ scales inversely proportional with $U$. The black dots represent DSS.

Figure 8

Impact of transversal noise as a function of the detuning parameters $\varepsilon$ and ${\varepsilon }_M$. In this figure, $\delta {\omega }_x$ rather than the dephasing time is plotted, thus small values lead to longer coherence times of the qubit. In the top row [(a) and (b)] we consider charge noise only from the two detuning parameters $\varepsilon$ and ${\varepsilon }_M$, in the center row [(c) and (d)] we consider charge noise only from the two tunneling parameters $t_l$ and $t_r$, and in the bottom row [(e) and (f)] we consider charge noise from all four parameters simultaneously, while we choose the parameter settings identical in each column. The left column contains the results for weak tunneling and strong noise, while the right column comprises the results for strong tunneling and weak noise. The black dots indicate DSS. Parameters are set as follows: $t_l=0.022U,t_r=0.015U,\delta q={10}^{-3}U$, where $q=\varepsilon ,{\varepsilon }_M$ in (a) and (e), and $\delta q={10}^{-4}U$, where $q=t_l,t_r$ in (c) and (e), for the left column and $t_l=0.22U,t_r=0.15U,\delta q={10}^{-5}U$, where $q=\varepsilon ,{\varepsilon }_M$ in (b) and (f), and $\delta q={10}^{-6}U$, where $q=t_l,t_r$ in (d) and (f), for the right column. For the scale of $|\delta {\omega }_x|$ we used an explicit value of $U=1\mathrm{meV}$.

Figure 9

Comparison of the dephasing time $T_{\phi }$ as a function of the detuning parameters $\varepsilon$ and ${\varepsilon }_M$ for symmetric and asymmetric tunnel coupling. In the top row [(a) and (b)] we consider charge noise only from $\varepsilon$, in the center row [(c) and (d)] we consider charge noise only from ${\varepsilon }_M$, and in the bottom row [(e) and (f)] we consider charge noise from both detuning parameters, $\varepsilon$ and ${\varepsilon }_M$, simultaneously, while we choose the parameter settings identical in each column. The left column comprises the results for symmetric tunneling, while the right column repeats the results for asymmetric tunneling given in Fig. 3 and Figs. 7 and 7. The black dots denote DSS. Parameters are set as follows: $t_l=t_r=t=0.02U$ for the left column and $t_l=0.022U,t_r=0.015U$ for the right column. Furthermore, we set $A_q={\left({10}^{-3}U\right)}^2$, where $q=\varepsilon$ in (a) and (b), $q={\varepsilon }_M$ in (c) and (d), and $q=\varepsilon ,{\varepsilon }_M$ in (e) and (f). For the scale of $T_{\phi }$, we used an explicit value of $U=1\mathrm{meV}$; note that $T_{\phi }$ scales inversely proportional with $U$.