J. Semicond. > 2024, Volume 45 > Issue 10 > 101701
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REVIEWS

The exchange interaction between neighboring quantum dots: physics and applications in quantum information processing

Zheng Zhou, Yixin Li, Zhiyuan Wu, Xinping Ma, Shichang Fan and Shaoyun Huang

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 Corresponding author: Shaoyun Huang, syhuang@pku.edu.cn

DOI: 10.1088/1674-4926/24050043CSTR: 32376.14.1674-4926.24050043

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Abstract: Electron spins confined in semiconductor quantum dots (QDs) are one of potential candidates for physical implementation of scalable quantum information processing technologies. Tunnel coupling based inter exchange interaction between QDs is crucial in achieving single-qubit manipulation, two-qubit gate, quantum communication and quantum simulation. This review first provides a theoretical perspective that surveys a general framework, including the Helter−London approach, the Hund−Mulliken approach, and the Hubbard model, to describe the inter exchange interactions between semiconductor quantum dots. An electrical method to control the inter exchange interaction in a realistic device is proposed as well. Then the significant achievements of inter exchange interaction in manipulating single qubits, achieving two-qubit gates, performing quantum communication and quantum simulation are reviewed. The last part is a summary of this review.

Key words: exchange interaction, quantum dots, tunnel coupling, quantum computation



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Fig. 1.  (Color online) The exchange interaction $J$ is the energy difference between the antisymmetric orbital wave function and the symmetric orbital wave function of two electrons.

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Fig. 2.  (Color online) (a) Quartic potential as illustrated in Eq. (2) with $ \hbar\omega_0=3\text{ meV} $ and $ a=19\text{ nm} $. The potential is used to simulate the coupling of two electrons locating in two harmonic wells centered at $ \left( { - a,0} \right) $ and $(a,0)$. The effective Bohr radius of harmonic well is ${a_{\text{B}}} = \sqrt {\hbar /m{\omega _0}} $. (b) The exchange coupling strength $J$ between two spins as a function of the inter-dot spacing $d = a/{a_{\text{B}}}$ with $ \hbar\omega_0=3\text{ meV} $, $ a_{\text{B}}=19\text{ nm} $ and $c = 2.4$ [See Eq. (6)].

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Fig. 3.  (Color online) (a) Double quantum dots in GaAs/AlGaAs heterostructure. (b) Two dimensional stability diagram of double quantum dots. The purple dashed-line is the position where $\Delta \varepsilon $ is zero, the yellow dashed-arrow indicates the direction along which $\Delta \varepsilon $ increases.

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Fig. 4.  (Color online) (a) Energy of $\left| {{\psi _{\text{A}}}} \right\rangle $ and $\left| {{\psi _{\text{B}}}} \right\rangle $ versus the detuning energy $\Delta \varepsilon $. (b) The electrostatic potential $U$ of quantum dots in Fig. 3(a) along the X-axis at Y ~ $0.08{\text{ }}\mu {\text{m}}$. The barrier height ${E_{\text{B}}}$ between two quantum dots can be manipulated by the barrier gate voltage ${V_{\text{b}}}$. (c) The barrier height ${E_{\text{B}}}$ is negatively proportional to the barrier gate voltage ${V_{\text{b}}}$. The black straight line is a linear fit.

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Fig. 5.  (Color online) (a) Infinitely deep double-well model. $a$ is the width of the well, $L$ the width of the barrier between two wells, and ${E_{\text{B}}}$ the barrier height between two wells. For simplicity, the potential out of the double wells is set infinitely high due to Coulomb blockade effect. (b) The tunnel coupling ${t_{\text{c}}}$ as a function of barrier height ${E_{\text{B}}}$. The data points are calculated based on the device shown in Fig. 3(a). The curve is a fit to Eq. (22).

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Fig. 6.  (a) The exchange energy $J$ as a function of the barrier gate voltage ${V_{\text{b}}}$ as described in Eqs. (18) and (22). (b) The exchange energy $J$ as a function of detuning energy $\Delta \varepsilon $ as described in Eq. (18).

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Fig. 7.  (Color online) (a) The construction of S−T0 qubit. The exchange interaction $J$ between quantum dots, and a magnetic field gradient $\Delta B$ is required to build a S−T0 qubit with double quantum dots. (b) States with two electron spins. The states with color mapping of orange and cyan color are the computational states of S−T0 qubit. The gray color mapped states are the leakage states. (c) Two states of $\left| S \right\rangle $ and $\left| {{T_0}} \right\rangle $ form a two-level subspace. (d) Exchange coupling $J$ and a longitudinal magnetic-field gradient $\Delta B$ provide two orthogonal manipulation axes for S−T0 qubit operation.

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Fig. 8.  (Color online) (a) The construction of exchange-only qubit. Only the exchange interaction ${J_{12}}$ and ${J_{23}}$ between quantum dots are required to build the exchange-only qubit with linearly coupled triple quantum dots. (b) States with three electron spins. The states with color mapping of orange and cyan color are the computational states of exchange-only qubit. The gray color mapped states are the leakage states. (c) Two states of $\left| D \right\rangle $ and $\left| {D'} \right\rangle $ form a two-level subspace. $\left| D \right\rangle $ is the mixture of $\left| {{D_{ + 1/2}}} \right\rangle $ and $\left| {{D_{ - 1/2}}} \right\rangle $, $\left| {D'} \right\rangle $ is the mixture of $\left| {D_{ + 1/2}'} \right\rangle $ and $\left| {D_{ - 1/2}'} \right\rangle $. (d) The exchange coupling parameters of ${J_{12}}$ and ${J_{23}}$ provide two independent manipulation axes of the exchange-only qubit manipulation.

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Fig. 9.  (Color online) The spin configuration of $ \left| {D'} \right\rangle $ and $ \left| D \right\rangle $. ${S_1}$ is the spin quantum number of electron in the first quantum dot. ${S_{23}}$ is the total spin quantum number of two electrons, which makes the second and the third quantum dot singly occupied.

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Fig. 10.  (Color online) (a) The quantum circuit of SWAP gate. (b) Double quantum dots with one single electron in each dot. ${{\boldsymbol{S}}_{1}}$(${{\boldsymbol{S}}_{2}}$) is the spin of electron in left (right) quantum dot. The exchange interaction $J$ is required to obtain the SWAP gate in specific duration.

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Fig. 11.  (Color online) (a) The quantum circuit of CNOT gate. (b) Double quantum dots with one single electron in each dot. ${{\boldsymbol{S}}_1}$(${{\boldsymbol{S}}_2}$) is the spin of electron in left (right) quantum dot. The exchange interaction $J$ between two quantum dots, and a magnetic field gradient $\Delta {\boldsymbol{B}}$ are required to achieve the CNOT gate operation. (c) Schematic energy level diagram adopted from Table 5. (d) The spin of electron in the left quantum dot is flipped with absorbing a photo from an alternating magnetic field at frequency of $f_{\left| {{\psi _{\text{R}}}} \right\rangle = \left| \uparrow \right\rangle }^{\text{L}}$ when the spin of electron in the right quantum dot is $\left| \uparrow \right\rangle $.

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Fig. 12.  (Color online) (a) Spin-coherent transport through adiabatic passage. (b) Long-range coupler of spins to transfer an arbitrary spin from the left most Alice to the right most Bob[91]. (c) A representation of triple quantum dots for simulation of interaction-driven Mott metal−insulator transition. (d) Energy spectrum of low spin state and ferromagnetic state as a function of tunnel coupling strength, where $S$ is the total spin number of the three electrons.

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Table 1.   The eigenstates and the corresponding eigenenergy of the Hamiltonian (Eq. (19)).

EigenstateEigenenergyDeviation
$\left| {{\psi _{\text{A}}}} \right\rangle $${\varepsilon _{\text{M}}} + \sqrt {{{(\Delta \varepsilon )}^2}/4 + t_{\text{c}}^{\text{2}}} $$\Delta = \sqrt {{{\left( {\Delta \varepsilon } \right)}^2} + 4t_{\text{c}}^{\text{2}}} $
$\left| {{\psi _{\text{B}}}} \right\rangle $${\varepsilon _{\text{M}}} - \sqrt {{{(\Delta \varepsilon )}^2}/4 + t_{\text{c}}^{\text{2}}} $
DownLoad: CSV

Table 2.   The eigenstates and corresponding eigenenergies of Hamiltonians $ \hat H_{{\text{S}} - {{\text{T}}_{\text{0}}}}^1 $ and $ \hat H_{{\text{S}} - {{\text{T}}_{\text{0}}}}^2 $ with $\Delta B \equiv {B_2} - {B_1}$.

HamiltonianEigenstateEigenenergy
$ \hat H_{{\text{S}} - {{\text{T}}_{\text{0}}}}^{\text{1}} $$\left| { \uparrow \downarrow } \right\rangle $$ - \mu \Delta B/2$
$\left| { \downarrow \uparrow } \right\rangle $$\mu \Delta B/2$
$\left| { \uparrow \uparrow } \right\rangle $$\mu ({B_1} + {B_2})/2$
$\left| { \downarrow \downarrow } \right\rangle $$ - \mu ({B_1} + {B_2})/2$
$ \hat H_{{\text{S}} - {{\text{T}}_{\text{0}}}}^{\text{2}} $$\left| S \right\rangle $$ - J$
$\left| {{T_0}} \right\rangle $$0$
$\left| { \uparrow \uparrow } \right\rangle $$0$
$\left| { \downarrow \downarrow } \right\rangle $$0$
DownLoad: CSV

Table 3.   The eigenstates and corresponding eigenenergies of Hamiltonian $ \hat H_{{\text{E}} - {\text{O}}}^{\text{1}} $ and $ \hat H_{{\text{E}} - {\text{O}}}^{\text{2}} $[41].

Hamiltonian Eigenstate Eigenenergy
$ \hat H_{{\text{E}} - {\text{O}}}^{\text{1}} $ $\left| {{Q_{ + 3/2}}} \right\rangle $ 0
$ \left| {{Q_{ + 1/2}}} \right\rangle $ 0
$ \left| {{Q_{ - 1/2}}} \right\rangle $ 0
$ \left| {{Q_{ - 3/2}}} \right\rangle $ 0
$\left| {{{\bar D}_{ + 1/2}}} \right\rangle $ 0
$\left| {{{\bar D}_{ - 1/2}}} \right\rangle $ 0
$ \left| {{{\bar D}'}_{ + 1/2}} \right\rangle $ $ - {J_{12}}$
$ \left| {{{\bar D}'}_{ - 1/2}} \right\rangle $ $ - {J_{12}}$
$\hat H_{{\text{E}} - {\text{O}}}^{\text{2}}$ $\left| {{Q_{ + 3/2}}} \right\rangle $ 0
$ \left| {{Q_{ + 1/2}}} \right\rangle $ 0
$ \left| {{Q_{ - 1/2}}} \right\rangle $ 0
$ \left| {{Q_{ - 3/2}}} \right\rangle $ 0
$\left| {{D_{ + 1/2}}} \right\rangle $ 0
$\left| {{D_{ - 1/2}}} \right\rangle $ 0
$\left| {D_{ + 1/2}'} \right\rangle $ $ - {J_{23}}$
$\left| {D_{ - 1/2}'} \right\rangle $ $ - {J_{23}}$
DownLoad: CSV

Table 4.   The eigenstates and corresponding eigenenergies of Hamiltonian $ {\hat H_{{\text{SWAP}}}}. $

HamiltonianEigenstateEigenenergy
$ {\hat H_{{\text{SWAP}}}} $$\left| S \right\rangle $$ - J$
$\left| {{T_0}} \right\rangle $$0$
$\left| { \uparrow \uparrow } \right\rangle $$0$
$\left| { \downarrow \downarrow } \right\rangle $$0$
DownLoad: CSV

Table 5.   The eigenstates and corresponding eigenenergies of the Hamiltonian ${\hat H_{{\text{CNOT}}}}.$

HamiltonianEigenstateEigenenergy
$ {\hat H_{{\text{CNOT}}}} $$ \left| { \uparrow \uparrow } \right\rangle $$\mu ({B_1} + {B_2})/2$
$ \left| { \Downarrow \Uparrow } \right\rangle $$\left[ { - J + \sqrt {{J^2} + {{\left( {g\Delta B\mu } \right)}^2}} } \right]/2$
$ \left| { \Uparrow \Downarrow } \right\rangle $$\left[ { - J - \sqrt {{J^2} + {{\left( {g\Delta B\mu } \right)}^2}} } \right]/2$
$ \left| { \downarrow \downarrow } \right\rangle $$ - \mu ({B_1} + {B_2})/2$
DownLoad: CSV
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    Received: 28 May 2024 Revised: 16 July 2024 Online: Accepted Manuscript: 14 August 2024Uncorrected proof: 16 August 2024Published: 15 October 2024

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      Article Navigation >  Journal of Semiconductors  > 2024  >  45(10): 101701
      Zheng Zhou, Yixin Li, Zhiyuan Wu, Xinping Ma, Shichang Fan, Shaoyun Huang. The exchange interaction between neighboring quantum dots: physics and applications in quantum information processing[J]. Journal of Semiconductors, 2024, 45(10): 101701. doi: 10.1088/1674-4926/24050043 ****Z Zhou, Y X Li, Z Y Wu, X P Ma, S C Fan, and S Y Huang, The exchange interaction between neighboring quantum dots: physics and applications in quantum information processing[J]. J. Semicond., 2024, 45(10), 101701 doi: 10.1088/1674-4926/24050043
      Citation:
      Zheng Zhou, Yixin Li, Zhiyuan Wu, Xinping Ma, Shichang Fan, Shaoyun Huang. The exchange interaction between neighboring quantum dots: physics and applications in quantum information processing[J]. Journal of Semiconductors, 2024, 45(10): 101701. doi: 10.1088/1674-4926/24050043 ****
      Z Zhou, Y X Li, Z Y Wu, X P Ma, S C Fan, and S Y Huang, The exchange interaction between neighboring quantum dots: physics and applications in quantum information processing[J]. J. Semicond., 2024, 45(10), 101701 doi: 10.1088/1674-4926/24050043
      shu

      The exchange interaction between neighboring quantum dots: physics and applications in quantum information processing

      DOI: 10.1088/1674-4926/24050043
      CSTR: 32376.14.1674-4926.24050043

      • School of Electronics, Beijing Key Laboratory of Quantum Devices, Key Laboratory for the Physics and Chemistry of Nanodevices, Peking University, Beijing 100871, China

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      • Zheng Zhou got his B.S. from University of Science and Technology Beijing in 2022. Now he is a master student at Peking University under the supervision of Prof. Shaoyun Huang. His research focuses on quantum computation and semiconductor quantum devices
      • Yixin Li got her B.S. from Renmin University of China in 2020. Now she is a PhD student at Peking University under the supervision of Prof. Shaoyun Huang and Prof. Shimin Hou. Her research focuses on quantum transport properties and semiconductor quantum devices
      • Zhiyuan Wu got his B.S. from Hebei University of Technology in 2021. Now he is a Master student at Peking University under the supervision of Prof. Shaoyun Huang. His research focuses on gate defined InAs quantum dot and quantum-classical interface
      • Xinping Ma got her B.S. from JiLin University in 2022. Now she is a Master student at Peking University under the supervision of Prof. Shaoyun Huang. Her research focuses on Si/SiGe quantum dot and electrical control of Si/SiGe spin qubit
      • Shichang Fan got his B.S. from East China University of Science and Technology in 2024. Now he is a Master student at Peking University under the supervision of Prof. Shaoyun Huang. His research focuses on the preparation of germanium quantum dots and the theoretical calculation of semiconductor spin qubits
      • Shaoyun Huang received B.S. and M.S. degrees in physics from Nanjing University, Nanjing, China, in September 1997 and 2000, respectively, and Ph.D. degree in semiconductor nanoelectronics from Tokyo Institute of Technology in September 2003. He is an associate professor of School of Electronics, Peking University. He is also a key member of the Beijing key laboratory of quantum devices and the Key Laboratory for the Physics and Chemistry of Nanodevices. His research interests include semiconductor quantum dot spin qubits, solid-state quantum computations, semiconductor low dimensional nanostructure based quantum devices, low-temperature transport
      • Corresponding author: syhuang@pku.edu.cn
      • Received Date: 2024-05-28
      • Revised Date: 2024-07-16
        • Available Online: 2024-08-14

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