Effective Hamiltonian for the hybrid double quantum dot qubit

Abstract

Quantum dot hybrid qubits formed from three electrons in double quantum dots represent a promising compromise between high speed and simple fabrication for solid state implementations of single-qubit and two-qubits quantum logic ports. We derive the Schrieffer–Wolff effective Hamiltonian that describes in a simple and intuitive way the qubit by combining a Hubbard-like model with a projector operator method. As a result, the Hubbard-like Hamiltonian is transformed in an equivalent expression in terms of the exchange coupling interactions between pairs of electrons. The effective Hamiltonian is exploited to derive the dynamical behavior of the system and its eigenstates on the Bloch sphere to generate qubits operation for quantum logic ports. A realistic implementation in silicon and the coupling of the qubit with a detector are discussed.

Appendices

Appendix 1: Energy levels

This appendix is devoted to the analysis of the energy levels of the hybrid qubit. The results obtained with the euristic Hamiltonian in Ref. [36] are recovered exploiting our effective Hamiltonian (14). For generality, we consider the basis with the intermediated state $|E\rangle \equiv |\downarrow \rangle |S\rangle$ in addition to the logical basis (16) previously introduced. The state $|E\rangle$ that has one electron in the left dot and two electrons in the right dot conserving the same total angular momentum $S^2$ and $S_z$ is directly involved in the physical process that leads to transitions between the two logical states. Explicit calculations of the matrix elements of the Hamiltonian in the basis $\{|0\rangle ,|1\rangle ,|E\rangle \}$ give as a result

$$\begin{aligned} H= \left( \begin{array}{lll} -\frac{3}{4}J' &{} -\frac{\sqrt{3}}{4}(J_1-J_2) &{} \frac{3}{8}(J_2-J_1+J')\\ -\frac{\sqrt{3}}{4}(J_1-J_2) &{} \frac{1}{4}J'-\frac{1}{2}(J_1+J_2) &{} -\frac{\sqrt{3}}{8}(J_1+3J_2-J')\\ \frac{3}{8}(J_2-J_1+J') &{} -\frac{\sqrt{3}}{8}(J_1+3J_2-J') &{} -\frac{3}{4}J_2-\varepsilon \end{array}\right) , \end{aligned}$$

(26)

where the detuning $\varepsilon$, proportional to the difference between the energy levels $\varepsilon _3$ and $\varepsilon _1$, is introduced.

Figure 6, in which the energy levels of the Hamiltonian (26) are represented, shows that transitions from logical state $|0\rangle$ to $|1\rangle$ can be induced by first pulsing the avoided crossing between $|0\rangle$ and $|E\rangle$ and then pulsing the avoided crossing between $|E\rangle$ and $|1\rangle$. The same argument can be applied to induce transition conversely from logical state $|1\rangle$ to $|0\rangle$.

Fig. 6

Energy levels of the Hamiltonian (14) as a function of detuning $\varepsilon$ between the two dots

Appendix 2: Dynamical evolution

Time-dependent Schrödinger equation for the hybrid qubit described by Hamiltonian (19) is here solved.

The state of the system at the initial time $t=0$ is written as a normalized superposition of the states of the logical basis $\{|0\rangle ,|1\rangle \}$ with probability amplitudes given by $a(0)$ and $b(0)$. The normalization condition $|a(0)|^2+|b(0)|^2=1$ is satisfied. Due to the conservation of the total angular momentum operator, it follows that also at a generic time instant $t$, the state of the system can be written analogously with probability amplitudes $a(t)$ and $b(t)$ depending explicitly on time

$$\begin{aligned} |\psi (0)\rangle =a(0)|0\rangle +b(0)|1\rangle \quad \Rightarrow |\psi (t)\rangle =a(t)|0\rangle +b(t)|1\rangle . \end{aligned}$$

(27)

By inserting this expression into the time-dependent Schrödinger equation $H|\psi (t)\rangle =i|\dot{\psi }(t)\rangle$ and by solving the system of two first order differential equations for the probability amplitudes $a(t)$ and $b(t)$, we finally obtain

$$\begin{aligned} \left\{ \begin{array}{ll} a(t)=c_1e^{\lambda _1t}+c_2e^{\lambda _2t}\\ b(t)=c_1\frac{\lambda _1-iA}{iC}e^{\lambda _1t} +c_2\frac{\lambda _2-iA}{iC}e^{\lambda _2t},\\ \end{array} \right. \end{aligned}$$

(28)

where

$$\begin{aligned} \lambda _{1,2}=i(\alpha \mp \beta ), \quad \alpha =\frac{A+B}{2},\quad \beta =\frac{\sqrt{(A-B)^2+4C^2}}{2} \end{aligned}$$

(29)

and

$$\begin{aligned} A=\frac{3}{4}J', \quad B=\frac{\sqrt{3}}{4}(J_1-J_2), \quad C=-\frac{1}{4}J'+\frac{1}{2}(J_1+J_2). \end{aligned}$$

(30)

Eq. (28) contains the more general form for the probability amplitudes at every time instant $t$. Once that the initial condition is fixed it is possible to extract the values for the coefficients $c_1$ and $c_2$.

In the case of the specific initial condition analyzed in Sect. 2 in which the system is prepared in the state of the logical basis corresponding to $|\psi (0)\rangle =|0\rangle$, the coefficients are

$$\begin{aligned} \left\{ \begin{array}{ll} a(0)=1\\ b(0)=0.\\ \end{array} \right. \end{aligned}$$

(31)

After straightforward calculations we get the probability amplitudes

$$\begin{aligned} \left\{ \begin{array}{ll} a(t)=\frac{e^{i\alpha t}}{\beta }\left[ \beta \cos (\beta t) +i(A-\alpha )\sin (\beta t)\right] \\ b(t)=-ie^{i\alpha t}\frac{(A-\alpha )^2-\beta ^2}{\beta C} \sin (\beta t).\\ \end{array} \right. \end{aligned}$$

(32)

Appendix 3: Eigenvalues and eigenvectors of three exchange-coupled spins in two limiting cases of interest

In this appendix eigenvectors and eigenvalues of the hybrid qubit, described by the effective Hamiltonian (14), are presented in two special cases. Two limiting conditions of interest from the practical point of view, are analyzed.

• 1. Case $J_2\gg J'\simeq J_1$ Under the condition on the exchange coupling $J_2\gg J'\simeq J_1$, that means that two electron are confined in the right dot, eigenvectors and eigenvalues in Eqs. (23) and (24) become

  $$\begin{aligned} |D_{+\frac{1}{2}}\rangle&= \frac{1}{\sqrt{6}}\left( |\uparrow \uparrow \downarrow \rangle +|\uparrow \downarrow \uparrow \rangle -2|\downarrow \uparrow \uparrow \rangle \right) \nonumber \\ |D_{-\frac{1}{2}}\rangle&= \frac{1}{\sqrt{6}}\left( |\downarrow \downarrow \uparrow \rangle +|\downarrow \uparrow \downarrow \rangle -2|\uparrow \downarrow \downarrow \rangle \right) \nonumber \\ |D'_{+\frac{1}{2}}\rangle&= \frac{1}{\sqrt{2}}\left( |\uparrow \uparrow \downarrow \rangle -|\uparrow \downarrow \uparrow \rangle \right) \nonumber \\ |D'_{-\frac{1}{2}}\rangle&= \frac{1}{\sqrt{2}}\left( |\downarrow \downarrow \uparrow \rangle -|\downarrow \uparrow \downarrow \rangle \right) \end{aligned}$$

  (33)
  $$\begin{aligned} E_{D_{S_z}}&= \frac{1}{4}J_2\;E_{D'_{S_z}}=-\frac{3}{4}J_2. \end{aligned}$$

  (34)

• 2. Case $J'\gg J_2\simeq J_1$ On the other hand, the opposite condition corresponding to two electrons confined in the left dot, that is $J'\gg J_2\simeq J_1$, gives as eigenvectors and eigenvalues

  $$\begin{aligned} |\bar{D}_{+\frac{1}{2}}\rangle&= \frac{1}{\sqrt{6}}\left( |\downarrow \uparrow \uparrow \rangle +|\uparrow \downarrow \uparrow \rangle -2|\uparrow \uparrow \downarrow \rangle \right) \nonumber \\ |\bar{D}_{-\frac{1}{2}}\rangle&= \frac{1}{\sqrt{6}}\left( |\uparrow \downarrow \downarrow \rangle +|\downarrow \uparrow \downarrow \rangle -2|\downarrow \downarrow \uparrow \rangle \right) \nonumber \\ |\bar{D}'_{+\frac{1}{2}}\rangle&= \frac{1}{\sqrt{2}}\left( |\uparrow \downarrow \uparrow \rangle -|\downarrow \uparrow \uparrow \rangle \right) \nonumber \\ |\bar{D}'_{-\frac{1}{2}}\rangle&= \frac{1}{\sqrt{2}}\left( |\downarrow \uparrow \downarrow \rangle -|\uparrow \downarrow \downarrow \rangle \right) \end{aligned}$$

  (35)
  $$\begin{aligned} E_{\bar{D}_{S_z}}&= \frac{1}{4}J'\;E_{\bar{D}'_{S_z}}=-\frac{3}{4}J'. \end{aligned}$$

  (36)