Landau–Zener transitions in spin qubit encoded in three quantum dots

Quantum Inf Process (2017) 16:10 DOI 10.1007/s11128-016-1480-z Landau–Zener transitions in spin qubit encoded in three quantum dots Jakub Łuczak1 · Bogdan R. Bułka1 Received: 14 July 2016 / Accepted: 2 November 2016 / Published online: 10 December 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We study generation and dynamics of an exchange spin qubit encoded in three coherently coupled quantum dots with three electrons. For two geometries of the system, a linear and a triangular one, the creation and coherent control of the qubit states are performed by the Landau–Zener transitions. In the triangular case, both the qubit states are equivalent and can be easily generated for particular symmetries of the system. If one of the dots is smaller than the others, one can observe Rabi oscillations that can be used for coherent manipulation of the qubit states. The linear system is easier to fabricate; however, then the qubit states are not equivalent, making qubit operations more difficult to control. Keywords Exchange qubits · Quantum computation · Landau–Zener transition · Quantum dots · Spin qubit dynamics 1 Introduction Recent progress in the experimental realization of the semiconductor quantum dots (QDs) gives the tool to perform compatible and fully scalable systems needed in the quantum computations. In contrast to charge qubits, spin qubits are characterized by long decoherence times necessary in the quantum computation [1]. To encode the qubit in the single electron spin in QD, one needs to apply a magnetic field which removes the spin degeneracy. Control of the spin qubit can be performed by the electron spin resonance (ESR) [2,3]. The readout of the final qubit state can be done using transport B Jakub Łuczak 1 Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, 60-179 Poznan, Poland 123 10 Page 2 of 16 J. Łuczak, B. R. Bułka measurements in the Pauli spin blockade regime with an auxiliary QD or with a quantum point contact (QPC) [4]. There are proposals [5–14] to build the qubit in a two-spin system in doublequantum dots (2QD). The qubit logical subspace is defined by a singlet (S) and one of the triplets (T S Z ), which correspond to the north and the south pole of the Bloch sphere, respectively. Applying an external magnetic field, one removes degeneration between the triplets, and the information is stored in the S − T +1 subspace. The preparation and manipulation of the qubit can be done by fast electrical pulses (in a nanosecond scale) which change an exchange interaction between the spins [6]. The control of the qubit is performed by the Landau–Zener (L–Z) transition [15–17] through an anticrossing point in a non-adiabatic regime. The anticrossing comes from mixing between the singlet and triplet states due to nuclear hyperfine fields [6–8], a spin–orbit coupling [18], or an inhomogeneous magnetic field [12]. The mixing is needed for proper functioning of the S − T +1 qubit; however, it can cause some unwanted decoherence processes. The L–Z method can be used to implement the universal quantum gates with high fidelity [19,20] and to measure the S − T +1 splitting when the spin–orbit coupling and hyperfine interaction compete with each other [21]. The 2QD system with two spins allows also to encode the qubit in the S − T 0 subspace [6,11–13]. In the external magnetic field, the qubit state T 0 is the excited state; therefore, one needs to pass through the S − T +1 anticrossing quick enough to remain in the qubit state S. The mixing of S and T 0 states is induced in an inhomogeneous magnetic field, and the initialization can be performed with high fidelity [13]. Moreover, both the qubit states have S Z = 0; therefore, they are unaffected by noises in an uniform magnetic field. DiVincenzo et al. [22] proposed an exchange-only qubit in three-spin system in a triple-quantum-dot (TQD) device. An advantage of the proposal is encoding the qubit in the doublet states with the same spin z-component (S Z ). It was pointed out [23,24] that the doublet subspace is immune to the decoherence processes. In the system, the full unitary operations of the qubit states are done by purely electrical control of the exchange interactions between the spins. Recently, TQD in a linear geometry has been investigated, both theoretically [25–27] and experimentally [28–31]. Another proposition is a resonant exchange qubit [32,33], where the manipulation is done by applying an rf gate-voltage pulses to one of the gates. If the oscillation frequency is matched to the exchange interaction, one can observe the nutations between the qubit states. The DiVincenzo scheme is not limited only to the linear TQD system. Shi et al. [34] proposed an electrically controlled hybrid qubit encoded in 2QD with many levels and three spins. Recent theoretical studies [35–37] showed advantages encoding of the qubit on TQD with a triangular geometry. In this case, both the doublet states are equivalent and can be easily controlled by changing the TQD symmetry. Single-qubit operations, the readout and the decoherence related to external electrodes as well as leakage processes were studied as well [37]. The triangular TQD devices were fabricated in the three lateral quantum dots by the atomic force microscope [38–40], the electronbeam lithography [41], and in the vertical quantum dots [42]. Similar structures can be found in molecular magnets [43,44] which exhibit rich quantum dynamics. 123 Landau–Zener transitions in spin qubit encoded in three.. Page 3 of 16 10 In this paper, we would like to show how one can encode the spin qubit and study its dynamics by means of the L–Z transitions for different symmetries of the TQD system. We model the system within the Hubbard Hamiltonian where the symmetry is fully electrically controlled by the local gate potentials applied to the quantum dots. Two geometries of TQD will be taken into consideration, the linear and the triangular one, for which one expects significant differences in qubit generation and its dynamics. The linear case is related to the experimental papers [29,30] where a qubit state was initialized in the doublet subspace by an adiabatic passage. They observed a coherent rotation between the qubit states when an exchange pulse applied to the system induced the L–Z transition. We would like to extend the investigation on dynamic generation of the qubit states and study conditions for qubit encoding. The main purpose is to study the triangular TQD where one expects that both the qubit states can be easily generated by the L–Z transition. We will examine different symmetries of the system to generate any qubit state on the Bloch sphere. Next, the evolution of the qubit states with time-dependent gate potentials will be analyzed. We expect that the qubit states could be degenerated for a special condition (when a pseudo-magnetic field vanishes). The L–Z pa (...truncated)