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Nonlocal entanglement and noise between spin qubits induced by Majorana bound states
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Nonlocal entanglement and noise between spin qubits induced by Majorana bound states
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Sha-Sha Ke , Hai-Feng Lü
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, Hua-Jun Yang , Yong Guo , Huai-Wu Zhang
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State Key Laboratory of Electronic Thin Films and Integrated Devices and School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China b Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China c Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People’s Republic of China
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Article history: Received 6 August 2014 Accepted 12 August 2014 Available online xxxx Communicated by R. Wu
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We propose a scheme to create nonlocal entanglement between two spatially separated electron spin qubits by coupling them with a pair of Majorana bound states (MBSs). The spin qubits are based on the spins of electrons confined in quantum dots. It is shown that spin entanglement between two dots could be generated by the nonlocality of MBSs. We also demonstrate that in the transport regime, the current noise cross correlation can serve as a good indicator of spin entanglement. The Majorana-dot coupling not only induces an indirect interaction between qubits, but also produces spin localization in the strong coupling limit. These two competing effects lead to a nonmonotonic dependence of current cross-correlation and entanglement on the Majorana-qubit coupling strength. © 2014 Published by Elsevier B.V.
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Topological quantum computation is to encode quantum information in nonlocal degrees of freedom obeying non-Abelian statistics [1–4]. These nonlocal degrees of freedom immune to decoherence caused by local perturbations and can be manipulated by braiding operations. The solid-state Majorana fermions (zero modes) are considered as one of the most promising candidates as this kind of non-Abelian particles [5–12]. Two well-separated Majorana fermions can define a nonlocal fermion level, and its occupied or unoccupied states are usually used to encode a qubit. It has been proposed that Majorana bound states (MBSs) could be achieved at the ends of a semiconductor nanowire with strong spin–orbit interaction in the proximity of an s-wave superconductor [13–17]. Recently, several experiments may have already provided evidence for MBSs in such a system [18–22]. Due to its long coherence time and efficient implementation of qubit gates, spin qubit defined in quantum dot is a promising candidate for quantum information processing [23–25]. For two spin qubits based on quantum dots, their exchange interaction could be modulated locally through gated control of the interdot tunnel coupling [25]. However, the range of the exchange interaction between two spin qubits is limited by their distance. It is attractive to design quantum systems which can facilitate coupling between spatially separated qubits. Recently, MBS is considered as a suitable choice to induce indirect coupling between conventional qubits [6–9,26–28]. By choosing the quantization axis for dot spin, MBS could be controlled to couple the dot electron with certain spin http://dx.doi.org/10.1016/j.physleta.2014.08.015 0375-9601/© 2014 Published by Elsevier B.V.
direction [26]. It has been suggested to transfer quantum information between topological and conventional qubit to combine their respective advantages [26–28]. It is interesting to generate indirect qubit interaction and entanglement by using the nonlocal nature of MBSs. In this paper, we investigate the Majorana-modulated current noise cross correlations (CCs) and dynamical spin entanglement between two electron spin qubits defined in quantum dots. Different from previous schemes to transfer information between topological qubit and conventional qubit [26,27], in this work, the nonlocality of MBSs is used not to define a topological qubit but to generate entanglement in conventional spin qubits. Only one pair of MBSs is involved to generate entanglement between two spatially separated spin qubits. Furthermore, we demonstrate that the measurable CCs could serve as the sensitive indicators of spin entanglement between two qubits coupled by MBSs. In many previous studies [23–25], generation of spin entanglement based on two quantum dots requires: (i) two dots are very closed (about 100 nm) and (ii) there exists an exchange interaction between two spins (like the form J S 1 S 2 ). However, it is a challenge to control one spin state separately in such a small region. An important advantage of the setup proposed here is that the two spins nested in two dots could be well separated, where the nonlocal entanglement is produced by a pair of MBSs. The scheme view of the setup is illustrated in Fig. 1. Two quantum dots are connected with a pair of MBSs supported by
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Fig. 1. (a) schematic representation of two spin qubits defined in quantum dots coupled with two ends of a semiconductor nanowire with strong spin–orbit interaction. The nanowire is in contact with an s-wave superconductor. Under large enough Zeeman field, the nanowire is driven into the topological superconducting phase and a pair of Majorana bound states appear in its ends. Each dot is connected with the source (S) and drain (D) lead. (b) The coherence between different states of Majorana-spin qubits system. The green line represents the Majorana-spin interaction, while the black line means the intradot spin-flip processes.
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a one-dimensional topological nanowire in its ends. Under proper magnetic field, the semiconductor nanowire resembles a spinless topological superconductor when adjacent to an s-wave superconductor due to the proximity effect [13–15]. At the ends of the wire, a pair of MBSs emerge. The spin qubit is based on a quantum dot with infinite intradot Coulomb repulsion energy. Each dot is connected to one MBS and two normal metal leads. Experimentally, the magnetic field is order of 0.1–1 T, which also leads a Zeeman splitting in the dots [18,19]. The Hamiltonian for the Majorana-dot part is given by
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H0 =
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σ
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†
†
U i ni ↑ ni ↓ + t 0 d1↑ d1↓ + d2↑ d2↓ + H.c.
is the Fermi distribution function and l l = E l − E l is the Bohr frequency of the transition from |l to |l with E l the eigenenergy of |l and μi α the chemical potential in reservoir i. In the wide-band limit approximation, the coupling between the i-th dot level and its reservoir is denoted by Γi α = 2π |t i α |2 νi α with νi α the density of states near the Fermi surface of reservoir i α . The current I i α flowing through dot i is calculated by
I iα = e [ˆ i α ρ ]l ,
(4)
i ωt
I α (t ) I β (0) − I α I β ,
dte
−∞
reservoirs, and t i α is the tunneling amplitude. Two Majorana fermions can be combined to form an ordinary fermion operator: η1 = f + f † , η2 = i ( f † − f ), where f † creates a nonlocal fermion and f † f = 0, 1 counts the occupation of the corresponding state [26]. In the new representation, the Hamiltonian in the central region becomes
H0 =
† 1σ d 1σ d 1σ
σ
† 2σ d 2σ d 2σ +
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M f f −
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+ λ1 f † d1↓ + f d1↓ + λ2 f † d2↓ − f d2↓ + h.c. † † + U i ni ↑ ni ↓ + t 0 d1↑ d1↓ + d2↑ d2↓ + H.c. . i
(2)
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ζ1 −
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ζ4 },
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C (ρ ) = max{0,
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with Tˆ (±ω) = (∓i ωI − W)−1 and I the unit matrix. To quantify the entanglement, we adopt Wootters’ concurrence for a general state of two qubits [34]. For the general state ρ of two qubits, its spin-flipped state is defined as ρ˜ = (σ y ⊗ σ y )ρ ∗ (σ y ⊗ σ y ), where σ y is the y component of Pauli matrix. Having defined the matrices ρ and ρ˜ , the concurrence that measures the degree of entanglement is defined as [34–36]
+ H.c.), where i αkσ is the electron energy in the reservoir i, α = S , D represent the source (S) and drain (D)
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are described by the Hamiltonian H T = † i αkσ (t i α c i αkσ d i σ
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(= η We choose the quantization axis of the dot spin such that MBSs only couples to the spin-down dot electron with strength of λi . The electron reservoirs and their coupling to the qubits
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I α (t ) I β (0) ω = Γˆα Tˆ (ω)Γˆβ ρ (0) + Γˆβ Tˆ (−ω)Γˆα ρ (0) k
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where · · · represents the thermodynamic average. Furthermore, the current–current correlation function of currents I α and I β in ω-space can be expressed as
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† )| l |d
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U i is the intradot Coulomb interaction in dot i, and di σ (di σ ) is the annihilation (creation) operator of electron. Here M is the cou† pling strength between the Majorana fermions η1 (= η1 ) and η2
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[ f (l l + μi α i σ |l| + f (l l − μi α i σ |l| ] for l = l and N ω / k W ll = − l =l W l l , with l, l ∈ {|o i , |e i }. Here f (ω) = [1 + e B T ]−1
)| l |d
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† 2 ).
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ˆ i α is the matrix form of the current operator. where The noise CC of two tunneling currents can be written by the Fourier transform of the current–current correlation function [32,33]
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where i σ = i + σ z is the spin-resolved energy level in quantum dot i (i = 1, 2 and σ = ↑, ↓), z is the Zeeman splitting in the dot,
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dρ (t )/dt = Wρ (t ),
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The Hamiltonian H 0 can be solved in the space spanned by eighteen basis states |n1 n2 p , where ni = 0, ↑, ↓ is the electronic state in dot i (where U i → ∞ is assumed to prohibit double occupation). The parity p with the values of even (e) or odd (o) denotes the state of the MBSs. For the two qubits and MBSs, the energy eigenvalues and eigenstates are given in two closed subspaces of definite parity. The parity in the MBS-dot hybrid system can be varied by one-particle sequential tunneling between quantum dots and reservoirs. At relatively high temperature and weak dot-lead coupling regime, the current is mainly determined by the lowest-order (sequential) tunneling events. We study the tunneling properties in the sequential tunneling regime with the help of the rate equation method in the diagonalized representation [29–31]. For weak coupling between the dots and reservoirs, we could construct the density matrix ρ with the eigenstates of the hybrid system of MBSs and dots. Therefore we can write down the rate equations for the population of the system eigenstates. The time evolution of density matrix ρ (t ) is given by the rate equations
(6)
where ζi s are the eigenvalues of the non-Hermitian matrix ρ ρ˜ in decreasing order. It is noted that the stationary density matrix in present case corresponds to a transport situation and the concurrence needs to be generalized to nonequilibrium case. Following Refs. [35] and [36], we take Γi S Γi D to ensure both qubits always occupied with one single electron. By tracing out the MBSs part, the density matrix ρ of two spin qubits is obtained.
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Fig. 2. (a) Concurrence, (b) current noise CC S 1S ,2S , (c) population of the Bell states, and (d) the current I i S as a function of Majorana energy splitting as taken as: i = −5.0, z = 2.0, λ0 = 20.0, t 0 = 2.0, Γi S = 10.0, Γi D = 0.2, and k B T = 5.0 (in unit of μeV).
M . Other parameters
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In the calculation, we assume that the two dots are symmetrically coupled to MBSs, i.e., λi = λ0 . The dot-lead coupling strengths are taken as Γi S = 10 μeV and Γi D = 0.2 μeV. A large enough bias voltage is taken so that the electrons flow from the source to the drain lead. Firstly we discuss the effect of the Majorana energy splitting M on the entanglement and transport properties, and the numerical results is demonstrated in Fig. 2. Experimentally, the splitting M has an oscillatory dependence on the Zeeman field and the chemical potential in the nanowire, and exponentially decays as a function of the wire length [22,37]. In the limit of M = 0, there is no overlap between the Majorana’s wave functions. Interaction of dot i and MBS in one side changes the parity of MBS, but does not affect the Majorana-dot interaction in another side. In this case, there is no communication between two MBSs, corresponding to the zero C and S 1S ,2S [28,38]. In the presence of M , there exists an energy difference for odd or even parity of MBSs. This means that the Majorana-dot interaction in one side makes a parity change of MBSs, which could affect the interaction in the other side. Therefore the spin states in both sides are correlated. As shown in Fig. 2a, the concurrence C indicates a nonmonotonic dependence on M . As M increases further and becomes much larger than other energy scales of Majorana-dot Hamiltonian, the large detuning strongly suppresses the Majorana-dot interaction and the correlation between two spin qubits. Therefore, both C and S 1S ,2S decrease for large enough M . The population of four Bell states of two qubits is helpful to analyze the property of their entanglement. For two spin qubits, their four Bell states are defined as Ψ T + = √1 (|↑↑ + |↓↓), Ψ T − = 2
√1
(|↑↑ − |↓↓), Ψ T 0 = √1 (↑↓ + |↓↑), and Ψ S0 = √1 (|↑↓ − 2 2 |↓↑). The population differences of P T + − P T − and P T 0 − P S0 2
measure the coherence of two qubits [36]. Dependence of the population of four Bell states on the Majorana energy splitting M is shown in Fig. 2c. For M = 0, the two dots are occupied by a spin-down electron due to the strong Majorana-dot interaction. As a function of M , the Majorana-dot interaction correlates both sides of MBSs and makes a population imbalance of P T + − P T − and P T 0 − P S0 . For strong enough M , the large energy detuning of the Majorana-dot interaction favors to weaken P T + − P T − and P T 0 − P S0 , and thus suppresses the concurrence. Different from the current CC, the current flowing from the source leads is determined by the dot-lead coupling strength Γ S and the population
of empty state in the dot. As M increases from zero energy, the population of the Bell states Ψ T + and Ψ T − decreases, thus facilitates electron tunneling into the dot. When M is much larger than the inner interactions t 0 and λ0 , the Majorana-dot interaction is suppressed and the two dots are always occupied by one electron most of the time, thus leading to the suppression of the current. Although the current partly reflects the population information of two qubits, the population change of four Bell states cannot be read only through the current. As shown in Fig. 2, the current CC could exhibit the competition between inner interactions, i.e., Majorana-dot interaction and intradot spin-flip scattering. One can controllably address the states of two spin qubits by modulating the Majorana-dot interaction. Before discussing the nonlocality transfer from MBSs to spin qubits, we shall emphasize the importance of intradot spin flip processes in the entanglement generation. In practice, spin scattering mechanisms in quantum dot can originate from spin–orbit interactions or hyperfine interaction. Here MBS only interacts with one type spin in the dot. If there is no spin coherence in each dot, there is no entanglement between two spin qubits. Fig. 3 illustrates the effect of intradot spin-flip processes on the current CC and concurrence. It is shown in Fig. 3 that S 1S ,2S and C equal to zero for t 0 = 0. It should be noted that even a small spin-flip scattering can change the CC S 1S ,2S properties dramatically and produce considerable C . In particular, both S 1S ,2S and C show a nonmonotonic behavior as a function of spin-flip scattering t 0 . The suppression and enhancement of S 1S ,2S and C are due to the interplay of Majorana-induced correlation and spin-flip scattering in the dots. For strong enough t 0 , the spin imbalance induced by Majorana-dot interaction is strongly suppressed, leading to the suppression of C and S 1S ,2S . Therefore, retaining proper spin-flip strength is a necessary condition for entanglement generation. For spin-flip scattering t 0 = 0, the population of Bell states Ψ T + and Ψ T − are dominant due to the strong Majorana-dot interaction. As a function of t 0 , the concurrence C and current CC S 1S ,2S indicate a double peak behavior, where the peaks lie in two sides of λ0 . When t 0 is much smaller than the interdot hopping λ0 , the Majorana-dot interaction make a population imbalance of four Bell states. The coupling strength t 0 and λ0 determines the time scale of qubit coherence and Majorana-dot interaction. For strong enough λ0 , the dwell time of spin-down electron staying in the dots is much longer than the time of spin-flip process. Therefore, for λ0 t 0 , the electrons in both qubits are freezed on the
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Fig. 3. (a) Concurrence, (b) current noise CC S 1S ,2S , (c) population of the Bell states, and (d) the current I i S as a function of intradot spin-flip strength t 0 . Other parameters as taken as those in Fig. 2 and M = 20.0.
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Fig. 4. (a) Concurrence and (b) current noise CC S 1S ,2S as a function of Majorana-dot coupling strength λ0 . Other parameters as taken as those in Fig. 3.
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spin-down state, which suppresses the entanglement between two qubits. On the contrary, for λ0 t 0 , the dot electron experiences spin-flip transition many times before it interacts with MBS. Interplay between two competing effects produces a nonmonotonic dependence of concurrence on λ0 . Both concurrence peaks appear at the maximum population difference of four Bell states. As shown in Fig. 3d, the current does not contain the information of the competition between two effects, which only indicates one resonance peak as a function of t 0 . It is interesting that the Majoranadot interaction always produces a positive population imbalance of P T 0 − P S0 , while P T + − P T − experiences two times of sign reversal as a function of t 0 . In the previous studies [36], such a double peak dependence of entanglement on the coupling strength was demonstrated, in a system of two charge qubits coupled with a common bosonic environment [36]. Fig. 4 presents the concurrence C and current noise CCs S 1S ,2S between two qubits as a function of Majorana-dot interaction λ0
for different spin-flip strengths t 0 . As illustrated in Fig. 4, there is a strong analogy between the concurrence and current CC, at least qualitatively. This means that the current CCs can be considered as a measure to the entanglement between the two spin qubits. In the absence of spin-flip process that t 0 = 0, there is no entanglement between two spin qubits and the current CC is also rather small. The spin flip scattering favors to make the population of two spin states equivalent, while the Majorana-dot interaction facilitates the spin imbalance in the dots and results in the spin-down states dominant. When t 0 is smaller than other system parameters, both the current CC S 1S ,2S and concurrence C indicate a single resonance peak structure as a function of λ0 . When t 0 is comparable to M , there are double resonance peaks appears in S 1S ,2S and C , as a result of coherence-modulated correlation and spin localization. Here both S 1S ,2S and C indicates a sensitive dependence on the competition between the Majorana-dot interaction λ0 and the spin-flip strength t 0 . Therefore, for the device proposed here, the current CCs could directly reflect the change of concurrence. Finally we discuss experimental systems where our scheme can be verified. It is important to avoid involving doubly occupied state in dot and quasiparticle excitations of topological superconductor. This requires that the Majorana-dot interaction λ0 should be much smaller than the Coulomb interaction in the dot and the superconductor gap. In the experiment, typical values of the induced superconductor gap is in the order of 0.1–1 meV [18–22], the intradot Coulomb interaction is in the order of 1–10 meV. The Majorana-dot interaction is tunable electrically and we can make λ0 ∼ 1–100 μeV. For a nanowire with its length L = 2 μm, M ranges between 0–200 μeV [37]. Compared to previous schemes [26,27], the present device only involves single pair of MBSs and it is need not to maintain the spin degeneracy in two dots. Therefore, the device proposed here is more accessible in the experiments. Compared to the previous proposal for coupling spin qubits via a superconductor [39], the main difference is that here the coupling between qubits is induced by the in-gap MBSs. Due to their topological nature, the MBSs are expected to induce more stable nonlocal coupling between conventional spin qubits. It is also noted that here only one pair of MBSs is used to generate nonlocal correlation and entanglement between two spin qubits and the Majorana energy splitting M is indispensable to induce nonlocal correlation. In the limit of M = 0, there is no communication between two spin qubits. Differently, a finite Majorana energy splitting M is not essential in the proposal where two pair MBSs are
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involved [26]. For the non-split MBSs case [26], the strong intradot Coulomb interaction can also modulate the spin states in the dot and produces the nonlocal coupling even for M = 0. In summary, we propose an experimental setup for generating nonlocal entanglement between two spin qubits defined in quantum dots by coupled them with single pair of MBSs. It is shown that the both Majorana-qubit interaction and intradot spin-flip processes could induce the population imbalance of four Bell states and the spin localization in two qubits. The competition between two effects results in monmonotonic dependence of the concurrence and current noise CCs on inner interactions. To create the spin entanglement, it is essential to maintain proper spin-flip scattering strength. More importantly, it is shown that the current CCs can be used as a possible entanglement indicator.
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Acknowledgements
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This project was supported by the Foundation for Innovative Research Groups of the NSFC under Grant (No. 61021061), the National Basic Research Program of China (2011CB606405), the NSFC (No. 11004022, 11174168, 61006081) and the Fundamental Research Funds for the Central Universities (No. ZYGX2012J052). References [1] A. Kitaev, Ann. Phys. (N.Y.) 303 (2003) 2. [2] C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. 80 (2008) 1083. [3] D.A. Ivanov, Phys. Rev. Lett. 86 (2001) 268. [4] A. Stern, Nature (London) 464 (2010) 187. [5] E. Majorana, Nuovo Cimento 14 (1937) 171. [6] G. Moore, N. Read, Nucl. Phys. B 360 (1991) 362. [7] J.D. Sau, R.M. Lutchyn, S. Tewari, S. Das Sarma, Phys. Rev. Lett. 104 (2010) 040502. [8] L. Fu, Phys. Rev. Lett. 104 (2010) 056402.
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Nonlocal entanglement and noise between spin qubits induced by Majorana bound states
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Sha-Sha Ke , Hai-Feng Lü
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, Hua-Jun Yang , Yong Guo , Huai-Wu Zhang
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State Key Laboratory of Electronic Thin Films and Integrated Devices and School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China b Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China c Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People’s Republic of China
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Article history: Received 6 August 2014 Accepted 12 August 2014 Available online xxxx Communicated by R. Wu
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We propose a scheme to create nonlocal entanglement between two spatially separated electron spin qubits by coupling them with a pair of Majorana bound states (MBSs). The spin qubits are based on the spins of electrons confined in quantum dots. It is shown that spin entanglement between two dots could be generated by the nonlocality of MBSs. We also demonstrate that in the transport regime, the current noise cross correlation can serve as a good indicator of spin entanglement. The Majorana-dot coupling not only induces an indirect interaction between qubits, but also produces spin localization in the strong coupling limit. These two competing effects lead to a nonmonotonic dependence of current cross-correlation and entanglement on the Majorana-qubit coupling strength. © 2014 Published by Elsevier B.V.
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Topological quantum computation is to encode quantum information in nonlocal degrees of freedom obeying non-Abelian statistics [1–4]. These nonlocal degrees of freedom immune to decoherence caused by local perturbations and can be manipulated by braiding operations. The solid-state Majorana fermions (zero modes) are considered as one of the most promising candidates as this kind of non-Abelian particles [5–12]. Two well-separated Majorana fermions can define a nonlocal fermion level, and its occupied or unoccupied states are usually used to encode a qubit. It has been proposed that Majorana bound states (MBSs) could be achieved at the ends of a semiconductor nanowire with strong spin–orbit interaction in the proximity of an s-wave superconductor [13–17]. Recently, several experiments may have already provided evidence for MBSs in such a system [18–22]. Due to its long coherence time and efficient implementation of qubit gates, spin qubit defined in quantum dot is a promising candidate for quantum information processing [23–25]. For two spin qubits based on quantum dots, their exchange interaction could be modulated locally through gated control of the interdot tunnel coupling [25]. However, the range of the exchange interaction between two spin qubits is limited by their distance. It is attractive to design quantum systems which can facilitate coupling between spatially separated qubits. Recently, MBS is considered as a suitable choice to induce indirect coupling between conventional qubits [6–9,26–28]. By choosing the quantization axis for dot spin, MBS could be controlled to couple the dot electron with certain spin http://dx.doi.org/10.1016/j.physleta.2014.08.015 0375-9601/© 2014 Published by Elsevier B.V.
direction [26]. It has been suggested to transfer quantum information between topological and conventional qubit to combine their respective advantages [26–28]. It is interesting to generate indirect qubit interaction and entanglement by using the nonlocal nature of MBSs. In this paper, we investigate the Majorana-modulated current noise cross correlations (CCs) and dynamical spin entanglement between two electron spin qubits defined in quantum dots. Different from previous schemes to transfer information between topological qubit and conventional qubit [26,27], in this work, the nonlocality of MBSs is used not to define a topological qubit but to generate entanglement in conventional spin qubits. Only one pair of MBSs is involved to generate entanglement between two spatially separated spin qubits. Furthermore, we demonstrate that the measurable CCs could serve as the sensitive indicators of spin entanglement between two qubits coupled by MBSs. In many previous studies [23–25], generation of spin entanglement based on two quantum dots requires: (i) two dots are very closed (about 100 nm) and (ii) there exists an exchange interaction between two spins (like the form J S 1 S 2 ). However, it is a challenge to control one spin state separately in such a small region. An important advantage of the setup proposed here is that the two spins nested in two dots could be well separated, where the nonlocal entanglement is produced by a pair of MBSs. The scheme view of the setup is illustrated in Fig. 1. Two quantum dots are connected with a pair of MBSs supported by
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Fig. 1. (a) schematic representation of two spin qubits defined in quantum dots coupled with two ends of a semiconductor nanowire with strong spin–orbit interaction. The nanowire is in contact with an s-wave superconductor. Under large enough Zeeman field, the nanowire is driven into the topological superconducting phase and a pair of Majorana bound states appear in its ends. Each dot is connected with the source (S) and drain (D) lead. (b) The coherence between different states of Majorana-spin qubits system. The green line represents the Majorana-spin interaction, while the black line means the intradot spin-flip processes.
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a one-dimensional topological nanowire in its ends. Under proper magnetic field, the semiconductor nanowire resembles a spinless topological superconductor when adjacent to an s-wave superconductor due to the proximity effect [13–15]. At the ends of the wire, a pair of MBSs emerge. The spin qubit is based on a quantum dot with infinite intradot Coulomb repulsion energy. Each dot is connected to one MBS and two normal metal leads. Experimentally, the magnetic field is order of 0.1–1 T, which also leads a Zeeman splitting in the dots [18,19]. The Hamiltonian for the Majorana-dot part is given by
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H0 =
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σ
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†
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U i ni ↑ ni ↓ + t 0 d1↑ d1↓ + d2↑ d2↓ + H.c.
is the Fermi distribution function and l l = E l − E l is the Bohr frequency of the transition from |l to |l with E l the eigenenergy of |l and μi α the chemical potential in reservoir i. In the wide-band limit approximation, the coupling between the i-th dot level and its reservoir is denoted by Γi α = 2π |t i α |2 νi α with νi α the density of states near the Fermi surface of reservoir i α . The current I i α flowing through dot i is calculated by
I iα = e [ˆ i α ρ ]l ,
(4)
i ωt
I α (t ) I β (0) − I α I β ,
dte
−∞
reservoirs, and t i α is the tunneling amplitude. Two Majorana fermions can be combined to form an ordinary fermion operator: η1 = f + f † , η2 = i ( f † − f ), where f † creates a nonlocal fermion and f † f = 0, 1 counts the occupation of the corresponding state [26]. In the new representation, the Hamiltonian in the central region becomes
H0 =
† 1σ d 1σ d 1σ
σ
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+
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(2)
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with Tˆ (±ω) = (∓i ωI − W)−1 and I the unit matrix. To quantify the entanglement, we adopt Wootters’ concurrence for a general state of two qubits [34]. For the general state ρ of two qubits, its spin-flipped state is defined as ρ˜ = (σ y ⊗ σ y )ρ ∗ (σ y ⊗ σ y ), where σ y is the y component of Pauli matrix. Having defined the matrices ρ and ρ˜ , the concurrence that measures the degree of entanglement is defined as [34–36]
+ H.c.), where i αkσ is the electron energy in the reservoir i, α = S , D represent the source (S) and drain (D)
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are described by the Hamiltonian H T = † i αkσ (t i α c i αkσ d i σ
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(= η We choose the quantization axis of the dot spin such that MBSs only couples to the spin-down dot electron with strength of λi . The electron reservoirs and their coupling to the qubits
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I α (t ) I β (0) ω = Γˆα Tˆ (ω)Γˆβ ρ (0) + Γˆβ Tˆ (−ω)Γˆα ρ (0) k
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where · · · represents the thermodynamic average. Furthermore, the current–current correlation function of currents I α and I β in ω-space can be expressed as
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† )| l |d
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[ f (l l + μi α i σ |l| + f (l l − μi α i σ |l| ] for l = l and N ω / k W ll = − l =l W l l , with l, l ∈ {|o i , |e i }. Here f (ω) = [1 + e B T ]−1
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† 2 ).
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ˆ i α is the matrix form of the current operator. where The noise CC of two tunneling currents can be written by the Fourier transform of the current–current correlation function [32,33]
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where i σ = i + σ z is the spin-resolved energy level in quantum dot i (i = 1, 2 and σ = ↑, ↓), z is the Zeeman splitting in the dot,
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dρ (t )/dt = Wρ (t ),
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The Hamiltonian H 0 can be solved in the space spanned by eighteen basis states |n1 n2 p , where ni = 0, ↑, ↓ is the electronic state in dot i (where U i → ∞ is assumed to prohibit double occupation). The parity p with the values of even (e) or odd (o) denotes the state of the MBSs. For the two qubits and MBSs, the energy eigenvalues and eigenstates are given in two closed subspaces of definite parity. The parity in the MBS-dot hybrid system can be varied by one-particle sequential tunneling between quantum dots and reservoirs. At relatively high temperature and weak dot-lead coupling regime, the current is mainly determined by the lowest-order (sequential) tunneling events. We study the tunneling properties in the sequential tunneling regime with the help of the rate equation method in the diagonalized representation [29–31]. For weak coupling between the dots and reservoirs, we could construct the density matrix ρ with the eigenstates of the hybrid system of MBSs and dots. Therefore we can write down the rate equations for the population of the system eigenstates. The time evolution of density matrix ρ (t ) is given by the rate equations
(6)
where ζi s are the eigenvalues of the non-Hermitian matrix ρ ρ˜ in decreasing order. It is noted that the stationary density matrix in present case corresponds to a transport situation and the concurrence needs to be generalized to nonequilibrium case. Following Refs. [35] and [36], we take Γi S Γi D to ensure both qubits always occupied with one single electron. By tracing out the MBSs part, the density matrix ρ of two spin qubits is obtained.
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Fig. 2. (a) Concurrence, (b) current noise CC S 1S ,2S , (c) population of the Bell states, and (d) the current I i S as a function of Majorana energy splitting as taken as: i = −5.0, z = 2.0, λ0 = 20.0, t 0 = 2.0, Γi S = 10.0, Γi D = 0.2, and k B T = 5.0 (in unit of μeV).
M . Other parameters
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In the calculation, we assume that the two dots are symmetrically coupled to MBSs, i.e., λi = λ0 . The dot-lead coupling strengths are taken as Γi S = 10 μeV and Γi D = 0.2 μeV. A large enough bias voltage is taken so that the electrons flow from the source to the drain lead. Firstly we discuss the effect of the Majorana energy splitting M on the entanglement and transport properties, and the numerical results is demonstrated in Fig. 2. Experimentally, the splitting M has an oscillatory dependence on the Zeeman field and the chemical potential in the nanowire, and exponentially decays as a function of the wire length [22,37]. In the limit of M = 0, there is no overlap between the Majorana’s wave functions. Interaction of dot i and MBS in one side changes the parity of MBS, but does not affect the Majorana-dot interaction in another side. In this case, there is no communication between two MBSs, corresponding to the zero C and S 1S ,2S [28,38]. In the presence of M , there exists an energy difference for odd or even parity of MBSs. This means that the Majorana-dot interaction in one side makes a parity change of MBSs, which could affect the interaction in the other side. Therefore the spin states in both sides are correlated. As shown in Fig. 2a, the concurrence C indicates a nonmonotonic dependence on M . As M increases further and becomes much larger than other energy scales of Majorana-dot Hamiltonian, the large detuning strongly suppresses the Majorana-dot interaction and the correlation between two spin qubits. Therefore, both C and S 1S ,2S decrease for large enough M . The population of four Bell states of two qubits is helpful to analyze the property of their entanglement. For two spin qubits, their four Bell states are defined as Ψ T + = √1 (|↑↑ + |↓↓), Ψ T − = 2
√1
(|↑↑ − |↓↓), Ψ T 0 = √1 (↑↓ + |↓↑), and Ψ S0 = √1 (|↑↓ − 2 2 |↓↑). The population differences of P T + − P T − and P T 0 − P S0 2
measure the coherence of two qubits [36]. Dependence of the population of four Bell states on the Majorana energy splitting M is shown in Fig. 2c. For M = 0, the two dots are occupied by a spin-down electron due to the strong Majorana-dot interaction. As a function of M , the Majorana-dot interaction correlates both sides of MBSs and makes a population imbalance of P T + − P T − and P T 0 − P S0 . For strong enough M , the large energy detuning of the Majorana-dot interaction favors to weaken P T + − P T − and P T 0 − P S0 , and thus suppresses the concurrence. Different from the current CC, the current flowing from the source leads is determined by the dot-lead coupling strength Γ S and the population
of empty state in the dot. As M increases from zero energy, the population of the Bell states Ψ T + and Ψ T − decreases, thus facilitates electron tunneling into the dot. When M is much larger than the inner interactions t 0 and λ0 , the Majorana-dot interaction is suppressed and the two dots are always occupied by one electron most of the time, thus leading to the suppression of the current. Although the current partly reflects the population information of two qubits, the population change of four Bell states cannot be read only through the current. As shown in Fig. 2, the current CC could exhibit the competition between inner interactions, i.e., Majorana-dot interaction and intradot spin-flip scattering. One can controllably address the states of two spin qubits by modulating the Majorana-dot interaction. Before discussing the nonlocality transfer from MBSs to spin qubits, we shall emphasize the importance of intradot spin flip processes in the entanglement generation. In practice, spin scattering mechanisms in quantum dot can originate from spin–orbit interactions or hyperfine interaction. Here MBS only interacts with one type spin in the dot. If there is no spin coherence in each dot, there is no entanglement between two spin qubits. Fig. 3 illustrates the effect of intradot spin-flip processes on the current CC and concurrence. It is shown in Fig. 3 that S 1S ,2S and C equal to zero for t 0 = 0. It should be noted that even a small spin-flip scattering can change the CC S 1S ,2S properties dramatically and produce considerable C . In particular, both S 1S ,2S and C show a nonmonotonic behavior as a function of spin-flip scattering t 0 . The suppression and enhancement of S 1S ,2S and C are due to the interplay of Majorana-induced correlation and spin-flip scattering in the dots. For strong enough t 0 , the spin imbalance induced by Majorana-dot interaction is strongly suppressed, leading to the suppression of C and S 1S ,2S . Therefore, retaining proper spin-flip strength is a necessary condition for entanglement generation. For spin-flip scattering t 0 = 0, the population of Bell states Ψ T + and Ψ T − are dominant due to the strong Majorana-dot interaction. As a function of t 0 , the concurrence C and current CC S 1S ,2S indicate a double peak behavior, where the peaks lie in two sides of λ0 . When t 0 is much smaller than the interdot hopping λ0 , the Majorana-dot interaction make a population imbalance of four Bell states. The coupling strength t 0 and λ0 determines the time scale of qubit coherence and Majorana-dot interaction. For strong enough λ0 , the dwell time of spin-down electron staying in the dots is much longer than the time of spin-flip process. Therefore, for λ0 t 0 , the electrons in both qubits are freezed on the
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Fig. 4. (a) Concurrence and (b) current noise CC S 1S ,2S as a function of Majorana-dot coupling strength λ0 . Other parameters as taken as those in Fig. 3.
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spin-down state, which suppresses the entanglement between two qubits. On the contrary, for λ0 t 0 , the dot electron experiences spin-flip transition many times before it interacts with MBS. Interplay between two competing effects produces a nonmonotonic dependence of concurrence on λ0 . Both concurrence peaks appear at the maximum population difference of four Bell states. As shown in Fig. 3d, the current does not contain the information of the competition between two effects, which only indicates one resonance peak as a function of t 0 . It is interesting that the Majoranadot interaction always produces a positive population imbalance of P T 0 − P S0 , while P T + − P T − experiences two times of sign reversal as a function of t 0 . In the previous studies [36], such a double peak dependence of entanglement on the coupling strength was demonstrated, in a system of two charge qubits coupled with a common bosonic environment [36]. Fig. 4 presents the concurrence C and current noise CCs S 1S ,2S between two qubits as a function of Majorana-dot interaction λ0
for different spin-flip strengths t 0 . As illustrated in Fig. 4, there is a strong analogy between the concurrence and current CC, at least qualitatively. This means that the current CCs can be considered as a measure to the entanglement between the two spin qubits. In the absence of spin-flip process that t 0 = 0, there is no entanglement between two spin qubits and the current CC is also rather small. The spin flip scattering favors to make the population of two spin states equivalent, while the Majorana-dot interaction facilitates the spin imbalance in the dots and results in the spin-down states dominant. When t 0 is smaller than other system parameters, both the current CC S 1S ,2S and concurrence C indicate a single resonance peak structure as a function of λ0 . When t 0 is comparable to M , there are double resonance peaks appears in S 1S ,2S and C , as a result of coherence-modulated correlation and spin localization. Here both S 1S ,2S and C indicates a sensitive dependence on the competition between the Majorana-dot interaction λ0 and the spin-flip strength t 0 . Therefore, for the device proposed here, the current CCs could directly reflect the change of concurrence. Finally we discuss experimental systems where our scheme can be verified. It is important to avoid involving doubly occupied state in dot and quasiparticle excitations of topological superconductor. This requires that the Majorana-dot interaction λ0 should be much smaller than the Coulomb interaction in the dot and the superconductor gap. In the experiment, typical values of the induced superconductor gap is in the order of 0.1–1 meV [18–22], the intradot Coulomb interaction is in the order of 1–10 meV. The Majorana-dot interaction is tunable electrically and we can make λ0 ∼ 1–100 μeV. For a nanowire with its length L = 2 μm, M ranges between 0–200 μeV [37]. Compared to previous schemes [26,27], the present device only involves single pair of MBSs and it is need not to maintain the spin degeneracy in two dots. Therefore, the device proposed here is more accessible in the experiments. Compared to the previous proposal for coupling spin qubits via a superconductor [39], the main difference is that here the coupling between qubits is induced by the in-gap MBSs. Due to their topological nature, the MBSs are expected to induce more stable nonlocal coupling between conventional spin qubits. It is also noted that here only one pair of MBSs is used to generate nonlocal correlation and entanglement between two spin qubits and the Majorana energy splitting M is indispensable to induce nonlocal correlation. In the limit of M = 0, there is no communication between two spin qubits. Differently, a finite Majorana energy splitting M is not essential in the proposal where two pair MBSs are
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involved [26]. For the non-split MBSs case [26], the strong intradot Coulomb interaction can also modulate the spin states in the dot and produces the nonlocal coupling even for M = 0. In summary, we propose an experimental setup for generating nonlocal entanglement between two spin qubits defined in quantum dots by coupled them with single pair of MBSs. It is shown that the both Majorana-qubit interaction and intradot spin-flip processes could induce the population imbalance of four Bell states and the spin localization in two qubits. The competition between two effects results in monmonotonic dependence of the concurrence and current noise CCs on inner interactions. To create the spin entanglement, it is essential to maintain proper spin-flip scattering strength. More importantly, it is shown that the current CCs can be used as a possible entanglement indicator.
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Acknowledgements
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This project was supported by the Foundation for Innovative Research Groups of the NSFC under Grant (No. 61021061), the National Basic Research Program of China (2011CB606405), the NSFC (No. 11004022, 11174168, 61006081) and the Fundamental Research Funds for the Central Universities (No. ZYGX2012J052). References [1] A. Kitaev, Ann. Phys. (N.Y.) 303 (2003) 2. [2] C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. 80 (2008) 1083. [3] D.A. Ivanov, Phys. Rev. Lett. 86 (2001) 268. [4] A. Stern, Nature (London) 464 (2010) 187. [5] E. Majorana, Nuovo Cimento 14 (1937) 171. [6] G. Moore, N. Read, Nucl. Phys. B 360 (1991) 362. [7] J.D. Sau, R.M. Lutchyn, S. Tewari, S. Das Sarma, Phys. Rev. Lett. 104 (2010) 040502. [8] L. Fu, Phys. Rev. Lett. 104 (2010) 056402.
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