Generating Majorana qubit coherence in Majorana Aharonov-Bohm interferometer
GGenerating Majorana qubit coherence in Majorana Aharonov-Bohm interferometer
Fei-Lei Xiong, ∗ Hon-Lam Lai, ∗ and Wei-Min Zhang
1, 2, † Department of Physics and Center for Quantum Information Science,National Cheng Kung University, Tainan 70101, Taiwan Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
We propose an Aharonov-Bohm interferometer consisted of two topological superconductingchains (TSCs) to generate coherence of Majorana qubits, each qubit is made of two Majoranazero modes (MZMs) with the definite fermion parity. We obtain the generalized exact master equa-tion as well as its solution and study the real-time dynamics of the MZM qubit states under variousoperations. We demonstrate that by tuning the magnetic flux, the decoherence rates can be modi-fied significantly, and dissipationless MZMs can be generated. By applying the bias voltage to theleads, one can manipulate MZM qubit coherence and generate a nearly pure superposition state ofMajorana qubit. Moreover, parity flipping between MZM qubits with different fermion parities canbe realized by controlling the coupling between the leads and the TSCs through gate voltages.
I. INTRODUCTION
Topological quantum computation has been widely in-vestigated as a promising candidate for realizing fault-tolerant quantum computation due to its robustnessagainst decoherence . The protection against decoher-ence during the computation process relies on highly-degenerate ground states of the qubit space, which isrealized by spatially separated Majorana zero modes(MZMs) . Theories have predicted that under certainphysical conditions, MZMs can exist at the ends of 1Deffective spinless p -wave superconductors , which can begenerated by contacting a conventional s -wave super-conductor to topological insulators , magnetic atomchains , or semiconductors with strong spin-orbit in-teraction . After nearly a decade of effort, exper-imentalists have recently observed some signatures ofMZMs in the proximitized nanowire systems .Because of the non-Abelian exchange property ofMZMs, braidings among them correspond to nontrivialunitary transformations, which plays a central role in thescheme of topological quantum computation . In the lit-erature, there are mainly two kinds of methods to realizethe braiding operations, either by changing the phys-ical parameters of the system adiabatically or byperforming projective measurements systematically .For instance, MZMs can be moved along nanowires bytuning the chemical potential and braiding operationscan be performed in T-junction structures . Otherproposals include performing braidings by controlling thecouplings between different MZMs , or by fusions ofdifferent MZMs in T-junction nanowires . As for themeasurement-based braiding method, effective braidingsof two MZMs are done by measuring the joint fermionparity of the MZM pair rather than exchanging theirspatial positions. This kind of measurements, as wellas the error correction code, can be realized by couplingquantum dots to MZMs , or by using Aharonov-Bohminterferometers .However, in realistic situations, the Majorana qubitsare unavoidably coupled to external controlling gates un- Super-gate
Magnetic field 𝜇 𝐿 𝜇 𝑅 𝛾 𝛼𝐿 𝛾 𝛼𝑅 𝛾 𝛽𝐿 𝛾 𝛽𝑅 Lead L
Lead R 𝜇 𝛼 𝜇 𝛽 Controlling-gateNanowire
FIG. 1: (Color Online) The schematic picture of the pro-posed Majorana AB interferometer. The two copper-coloredwires in the picture are the proximitized TSCs. By tuningthe chemical potential µ α and µ β with the super-gates, twoMZMs can be formed at the ends of each TSC (See the redcircles in the figure). The leads L and R are two electrodes.In the central region between the TSCs, we apply a magneticflux threading into the central region of the interferometer.The controlling gates are used for controlling the tunnelingamplitudes. der qubit operations . As a consequence, the topo-logical protection against decoherence can be destroyedby, for instance, charge fluctuations of the controllinggates . In this paper, we propose a new schemeother than braidings to manipulate the qubit states ofMZMs, where noise effects are taken into account. Ourdevice mainly consists of an Aharonov-Bohm (AB) inter-ferometer, which is constructed by connecting two TSCswith two metal leads (See Fig. 1). The TSCs are tunedto the topological phase through the super-gates so thatfour MZMs are formed at their ends. Two leads with tun-able bias voltage are coupled to the left and right endsof the TSCs, with the coupling strengths being also ad-justable through the controlling gates. The magnetic fluxthreading into the central region of the interferometer canaffect the interference pattern of the interferometer.In this device, unlike the braiding operations, thefermion parity of the MZM states can be intentionally a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b switched by letting electrons tunnel into and out of theTSCs from the leads. In addition, electrons transportcoherently from one end of a TSC to the other throughthe MZM pairs, the two paths formed by the TSCs in-terfere with each other. Quantum coherence can there-fore be generated and controlled by tuning the lead-TSCcouplings through gate voltages or by the applied mag-netic flux. Through the exact master equation involvingpairing interaction , we shall study the real-timedynamics of this Majorana AB interferometer under vari-ous operations. We discover that under the magnetic fluxcontrolling, dissipationless MZM modes can be formedbetween the leads and the TSCs. Also, intended MZMqubit states with different parities can be prepared byapplying a bias to the leads. Finally, parity flipping be-tween the MZM qubits with different fermion parity canbe done by controlling the coupling strengths betweenthe leads and the TSCs through gate voltages.Our paper is organized as follows. In Sec. II, wepropose the model of the Majorana AB interferometer,which includes two TSCs contacting with left and rightleads. The magnetic flux threading into the central re-gion of the interferometer. We construct the Hamiltonianof the electron tunnelings in this superconducting AB in-terferometer incorporating the magnetic flux. In Sec. III,we derive the exact master equation for the MZMs local-ized at the ends of two TSCs. We show that the damp-ing of the MZMs and the couplings induced by the leadsare explicitly related to the generalized non-equilibriumGreen functions for the MZMs. Furthermore, the densitymatrix of the four MZMs (two MZM qubits with differentparity respectively) can be obtained as the solution to ourexact master equation. In Sec. IV, we discuss Majoranaqubit state evolution under various kinds of operations.We show that dissipationless MZM modes will be formedby controlling the magnetic flux. Moreover, MZM qubitcoherence can be generated and controlled when a biasis applied between the two leads. The MZM qubit stateparity can also be flipped by tuning the couplings be-tween the MZMs and the leads. Finally, the conclusionsare summarized in Sec. V. II. THE MAJORANA AB INTERFEROMETERAND ITS MODELING
The Majorana AB interferometer we propose isschematically plotted in Fig. 1. The nanowires labeled α and β are two 1D spinless p -wave superconductor chains,which can be realized by, for example, strong spin-orbitinteracting nanowires proximitized by conventional s -wave superconductors. In this paper, we model them as N -site Kitaev chains and in the absence of the magnetic flux, they are described by the Hamiltonians H α ( β ) = N − (cid:88) j =1 (cid:2) − wa † α ( β ) ,j a α ( β ) ,j +1 +∆ e iφ α ( β ) a α ( β ) ,j a α ( β ) ,j +1 + h.c. (cid:3) − N (cid:88) j =1 µ α ( β ) (cid:2) a † α ( β ) ,j a α ( β ) ,j − (cid:3) . (1)Here, a α,j ( a † α,j ) denotes the annihilation (creation) op-erator of site- j in chain α (similarly for β ). The hop-ping amplitude w is real-valued and ∆ e iφ α ( β ) , with ∆and φ α ( β ) being real numbers, are the superconductinggap in chain α and β . Two leads, which are labeled L andR and modelled by the free electron gas Hamiltonians H L ( R ) = (cid:88) k (cid:15) L ( R ) k c † L ( R ) k c L ( R ) k , (2)are coupled to the TSCs through the tunneling Hamilto-nian H I = (cid:88) k (cid:0) λ αLk ( t ) c † Lk a α, + λ βLk ( t ) c † Lk a β, + λ αRk ( t ) c † Rk a α,N + λ βRk ( t ) c † Rk a β,N + h.c. (cid:1) . (3)Here, (cid:15) L ( R ) k is the single-particle energy of mode- k in lead L(R), with c L ( R ) k and c † L ( R ) k being the cor-responding annihilation and creation operators respec-tively. Moreover, λ αLk ( t ), λ βLk ( t ), λ αRk ( t ), and λ βRk ( t )stand for the coupling strengths between the modes inthe leads and the ends of the TSCs. They are controlledby tuning the controlling gates in Fig. 1 and can in gen-eral be time-dependent. In the following, if not specified,we would omit the time-dependence of the λ ’s.In addition, we apply the magnetic flux threading intothe central region of the interferometer Φ = (cid:72) A ( r ) · d r ,where A ( r ) stands for the vector potential at position r .The operators a α,j and a β,j in the Hamiltonians H α and H β should be converted according to Peierls substitution a α ( β ) ,j → e − iφ α ( β ) ,j a α ( β ) ,j , (4)and the phase functions satisfy the relation φ α ( β ) ,j +1 − φ α ( β ) ,j = e (cid:126) (cid:90) α ( β ) ,j +1 α ( β ) ,j A ( r ) · d r . (5)Therefore, the Hamiltonians of the TSCs α and β withmagnetic flux threading into the central region of theinterferometer can be written as H α ( β ) = (cid:88) j (cid:2) − we − i [ φ α ( β ) ,j +1 − φ α ( β ) ,j ] a † α ( β ) ,j a α ( β ) ,j +1 +∆ e iφ α ( β ) e − i [ φ α ( β ) ,j +1 + φ α ( β ) ,j ] a α ( β ) ,j a α ( β ) ,j +1 + h.c. (cid:3) − (cid:88) j µ α ( β ) (cid:2) a † α ( β ) ,j a α ( β ) ,j − (cid:3) . (6)Apply the substitutions that˜ a α ( β ) ,j = e iφ α ( β ) / e − iφ α ( β ) ,j a α ( β ) ,j , (7)the Hamiltonians H α , H β and H I can be expressed interms of the operators ˜ a α,j and ˜ a β,j , i.e., H α ( β ) = (cid:88) j (cid:104) − w ˜ a † α ( β ) ,j ˜ a α ( β ) ,j +1 + ∆˜ a α ( β ) ,j ˜ a α ( β ) ,j +1 + h.c. (cid:105) − (cid:88) j µ α ( β ) (cid:104) ˜ a † α ( β ) ,j ˜ a α ( β ) ,j − (cid:105) , (8) H I = (cid:88) k ( λ αLk e i ˜ φ αL c † Lk ˜ a α, + λ βLk e i ˜ φ βL c † Lk ˜ a β, + λ αRk e i ˜ φ αL c † Rk ˜ a α,N + λ βRk e i ˜ φ βR c † Rk ˜ a β,N + h.c. ) . (9)where ˜ φ α ( β ) L = φ α ( β ) , − φ α ( β ) / φ α ( β ) R = φ α ( β ) ,N − φ α ( β ) / µ α and µ β , the hop-ping amplitude w , and the pairing parameter ∆ are tunedso that two MZMs can be generated at the ends of eachnanowire, which we denote as γ αL , γ αR , γ βL and γ βR , re-spectively (See the red circles in the ends of two TSCs inFig. 1). To illustrate this property, we consider an idealparameter setting that µ α = µ β = 0 and ∆ = w , then theMZMs possess the explicit form, γ αL = ˜ a α, +˜ a † α, , γ αR = − i ˜ a α,N + i ˜ a † α,N , (10a) γ βL = ˜ a β, +˜ a † β, , γ βR = − i ˜ a β,N + i ˜ a † β,N , (10b)and the annihilation operators of the zero-energy quasi-particle excitations are b α, = 12 ( γ αR + iγ αL ) , b β, = 12 ( γ βR + iγ βL ) . (11)In this work, we consider the case that the bias betweenthe chains and the leads is much smaller than the su-perconducting gap and the excitation of quasiparticles inthe continuous bands of the TSCs is negligible . As aconsequence, in the interaction Hamiltonian (9), the com-ponents of the field operators that involved with the non-zero energy bogoliubons in the TSCs can be neglected,Then the interaction Hamiltonian is reduced to H I = 12 (cid:88) k ( λ αLk e i ˜ φ αL c † Lk γ αL + λ βLk e i ˜ φ βL c † Lk γ βL + iλ αRk e i ˜ φ αR c † Rk γ αR + iλ βRk e i ˜ φ βR c † Rk γ βR + h.c. ) . (12)In Eq. (6), the pairing phases at the left and right ends ofchain- α and the left and right ends of chain- β are − φ αL , − φ αR , − φ βL and − φ βR , respectively. Following theconvention in Feynman’s dealing with the pairing phasesin the Josephson junctions , the phases satisfy˜ φ αL − ˜ φ βL = − φ − π ΦΦ , (13a)˜ φ αR − ˜ φ βR = − φ π ΦΦ , (13b) where Φ = h/e is the flux quantum with h standing forthe Planck constant and e standing for the elementarycharge, φ = φ α − φ β is the initial pairing phase differencebetween the two TSCs.Although our modeling of the Majorana AB interfer-ometer is based on modeling the TSCs as Kitaev chainsunder special physical conditions, it is applicable to moregeneral cases. Generally speaking, when the effective 1Dspinless p -wave superconductors are in the topologicalphase and the excitation of quasiparticles in the contin-uous band is negligible, the Hamiltonians (1) character-izing the TSCs can be written as H α = i(cid:15) α γ αL γ αR and H β = i(cid:15) β γ βL γ βR , where the γ ’s are the Majorana op-erators with wave packets localized near the ends of theTSCs . The energy (cid:15) α ( β ) ∼ γ α ( β ) L and γ α ( β ) R , whichis exponentially suppressed by the length of the TSCs. Ifthe TSCs are long enough, the wave packets of the γ ’scan be seen as localized and the energy values (cid:15) α ( β ) canbe treated as zero. The pairing phase differences betweenthe ends of the TSCs can also apply to the relations inEq. (13). As a consequence, Eq. (12) and Eq. (13) to-gether describe the interaction between the TSCs and theleads, except for the fact that the Majorana operators areno longer in the form of Eq. (10). Thus, the leads andthe TSCs together form an Aharonov-Bohm interferencering. Particles exchange and interfere through the MZMsin the TSCs. The dynamics of the system is influencedby the magnetic flux Φ and the time-dependent tunnel-ing amplitudes, through which we can generate coherencebetween the two TSCs and manipulate the MZM qubitstates. III. THE EXACT MASTER EQUATION ANDTHE DENSITY MATRIXA. The exact master equation
We treat the MZMs as the principal system, and thetwo leads as the environment. Suppose that the total sys-tem is initially in a product state ρ tot (0) = ρ (0) ⊗ ρ L (0) ⊗ ρ R (0), where ρ (0) is the state of the principal system and ρ L (0) ( ρ R (0)) is the state of lead L (R). Without loss ofgenerality, we also assume that ρ L (0) ( ρ R (0)) is the ther-mal equilibrium state associated to temperature T L ( T R )and chemical potential µ L ( µ R ). By taking advantageof the path integral approach in the coherent state rep-resentation, states of the system can be found to evolveaccording to the exact master equation ˙ ρ ( t ) = − i (cid:126) [ ρ ( t ) , ˜ H L ( t )+ ˜ H R ( t )]+ (cid:88) i,j Γ Lij ( t )2 (cid:2) γ iL ρ ( t ) γ jL − { ρ ( t ) , γ jL γ iL } (cid:3) + (cid:88) i,j Γ Rij ( t )2 (cid:2) γ iR ρ ( t ) γ jR − { ρ ( t ) , γ jR γ iR } (cid:3) . (14)In the formula,˜ H L ( t ) = i U L U − L ] αβ − [ ˙ U L U − L ] βα ) γ αL γ βL (15)is the environment-induced renormalized Hamiltonian forthe left-side MZMs; Γ Lij ( t ) ( i, j = α, β ) characterize thedecoherence rates of the MZMs in the left side and canbe explicitly written in terms of the generalized non-equilibrium Green functions U L and V L ,Γ Lij ( t ) = [ ˙ V L − ( ˙ U L U − L V L + h.c. )] ij . (16)In Eqs. (15)-(16), U L and V L are short for the retardedGreen’s function U L ( t, t ) and the correlation function V L ( t ) involving pairing interactions . U L ( t, t ) satisfiesthe integro-differential equation ∂ t U L ( t, t ) + 2 (cid:90) tt dτ g L ( t, τ ) U L ( τ, t ) = 0 , (17)with the initial condition that U L ( t , t ) = I ( I is a2 × g L ( t, τ ) is given by g L ( t, τ ) = (cid:90) d(cid:15) π (cid:2) e − i(cid:15) ( t − τ ) J eL + e i(cid:15) ( t − τ ) J hL (cid:3) , (18)where J eL and J hL are short for the electron spec-tral density function J eL ( (cid:15), t, τ ) and the hole spec-tral density function J hL ( (cid:15), t, τ ), respectively; and J hL ( (cid:15), t, τ ) = J e ∗ L ( (cid:15), t, τ ). Define that ( J L ) ij = π (cid:80) k δ ( (cid:15) − (cid:15) Lk ) λ ∗ iLk ( t ) λ jLk ( τ ), where i and j are either α or β , thenthe complete expression of J eL is J eL = [ J L ] αα [ J L ] αβ e − iδ L / [ J L ] βα e iδ L / [ J L ] ββ , (19)where δ L = ˜ φ αL − ˜ φ βL . Note that the cross couplingbetween α and β is dependent on δ L which has beendefined in Eq. (13). V L ( t ) can be written in terms of theretarded Green’s function U L that V L ( t ) = 2 (cid:90) tt dτ (cid:90) tt dτ U L ( t, τ )˜ g L ( τ , τ ) U L ( τ , t ) , (20)where the system-environment correlation ˜ g L ( τ , τ ) sat-isfies˜ g L ( τ , τ ) = (cid:90) d(cid:15) π (cid:104) f eL ( (cid:15) ) e − i(cid:15) ( τ − τ ) J eL + f hL ( (cid:15) ) e i(cid:15) ( τ − τ ) J hL (cid:105) . (21)Here, f eL ( (cid:15) ) = e βL ( (cid:15) − µL ) +1 and f hL ( (cid:15) ) = 1 − f eL ( (cid:15) ) are theinitial particle number distribution of electrons and holesin lead L respectively. All the relations and conventionsare similar for the right-side MZMs, with only the index L being replaced by R .As shown in Eq. (14), couplings between the TSCs andthe leads induce interactions among the MZMs as well as the dissipation of them. Specifically, the MZMs γ αL and γ βL ( γ αR and γ βR ) in the left (right) side are coupledto each other through the renormalized Hamiltonian ˜ H L ( ˜ H R ), and dissipate to lead L ( R ) through the dissipa-tion coefficients Γ L ( R ) ij . All the MZM dynamics can becaptured by the Majorana correlation function matrix M ( t ) = (cid:104) iγ αL γ βL (cid:105) (cid:104) iγ αL γ αR (cid:105) (cid:104) iγ αL γ βR (cid:105)(cid:104)− iγ αL γ βL (cid:105) (cid:104) iγ αR γ βL (cid:105) (cid:104) iγ βL γ βR (cid:105)(cid:104)− iγ αL γ αR (cid:105) (cid:104)− iγ αR γ βL (cid:105) (cid:104) iγ αR γ βR (cid:105)(cid:104)− iγ αL γ βR (cid:105) (cid:104)− iγ βL γ βR (cid:105) (cid:104)− iγ αR γ βR (cid:105) , (22)which can be obtained in terms of non-equilibrium Greenfunctions, explicitly, M ( t ) = U M ( t ) U T − i V − V T ) , (23)where the superscript T denotes the matrix transpose.Also, the expectation value of the fermion parity for theMZM states can be written as¯ P ( t ) = − (cid:10) γ αL γ αR γ βL γ βR (cid:11) = ¯ P ( t ) det( U )+ 14 (cid:0) V Lαβ − V ∗ Lαβ (cid:1)(cid:0) V Rαβ − V ∗ Rαβ (cid:1) . (24)Note that in Eqs. (22)-(24), we have omitted the time-dependence of the Majorana operators. The Green func-tions U and V can be explicitly expressed as U ( t, t ) = U L U R and V ( t ) = V L V R respectively. B. Exact dynamics of the density matrix
The MZM density matrix in the AB interferometercan be obtained by solving the master equation. Inthe following, the basis {| (cid:105) , b † ,α | (cid:105) , b † ,β | (cid:105) , b † ,α b † ,β | (cid:105)} is used, which consists of two Majorana qubit ba-sis with different parities, the even parity qubit ba-sis {| (cid:105) , b † ,α b † ,β | (cid:105)} and the old parity qubit ba-sis { b † ,α | (cid:105) , b † ,β | (cid:105)} , where the operator b † ,α ( β ) = ( γ α ( β ) R − iγ α ( β ) L ) creates a zero-energy Bogoliubon inTSC α ( β ). This basis corresponds to the zero-energyBogoliubon occupation in α , β or both TSCs. We con-sider the case that initially the two nanowires are notcorrelated, i.e., the system initial state reads ρ ( t ) = ρ ( t ) 0 0 00 ρ αα ( t ) 0 00 0 ρ ββ ( t ) 00 0 0 ρ dd ( t ) , (25)where the subscripts 0, α , β , and d correspond to thestates | (cid:105) , b † ,α | (cid:105) , b † ,β | (cid:105) and b † ,α b † ,β | (cid:105) , respectively.Because there cannot exist coherence between differentparity eigenstates of fermions, the density matrix of thetwo MZM qubits will always possess the form ρ ( t ) = ρ ( t ) 0 0 ρ d ( t )0 ρ αα ( t ) ρ αβ ( t ) 00 ρ βα ( t ) ρ ββ ( t ) 0 ρ d ( t ) 0 0 ρ dd ( t ) . (26)At arbitrary time t , the relation between the densitymatrix elements and the Majorana correlation functionsreads ρ ( t ) = 14 (1+ M ( t )+ M ( t )+ ¯ P ( t )) , (27a) ρ αα ( t ) = 14 (1+ M ( t ) − M ( t ) − ¯ P ( t )) , (27b) ρ ββ ( t ) = 14 (1 − M ( t )+ M ( t ) − ¯ P ( t )) , (27c) ρ dd ( t ) = 14 (1 − M ( t ) − M ( t )+ ¯ P ( t )) , (27d) ρ d ( t ) = 14 [ − M ( t )+ M ( t )+ i ( M ( t ) − M ( t ))] , (27e) ρ αβ ( t ) = 14 [ − M ( t ) − M ( t ) − i ( M ( t )+ M ( t ))] . (27f)where M ij ( i, j = 1 , , ,
4) is the element of the matrix M ( t ). By substituting Eqs. (23) and (24) into Eq. (27),one can obtain the complete solution to the two MZMqubit density matrix at arbitrary time t , which is ex-pressed in terms of the initial condition of the MZMstates and the non-equilibrium Green’s functions U ( t, t )and V ( t ).Initially, the two MZM qubit density matrix is diagonaland no coherence exists. After the TSC system is cou-pled to the leads, the off-diagonal matrix elements ρ d ( t )or ρ αβ ( t ) would, in general, become finite values, i.e., onecan generate coherence in each MZM qubit state. More-over, both the dynamical process and the final state canbe manipulated by tuning the magnetic flux and the cou-pling strengths. In the following section, we shall discussthe cases of various parameter settings. We shall demon-strate that by tuning the magnetic flux Φ, the bias µ L and µ R , and the coupling strengths λ ’s, the MZM qubitstates can be modified significantly. IV. DYNAMICS OF THE MZMS WITHVARIOUS PARAMETER SETTINGS
In this section, we shall study how the coherence dy-namics of the MZM qubits are varied under differentparameter settings. For clarity, in the following anal-ysis, we set the original pairing phase difference φ of the TSCs to be zero ( φ = 0) in absence of the mag-netic flux [see Eq. (13)], and the initial state of the sys-tem as ρ ( t ) = | (cid:105)(cid:104) | . Firstly, we investigate the gen-eral dynamics of two MZM qubits with different pari-ties. For simplicity, we avoid the complicated tunnel-ing effects due to the structure of the leads and simplytake the wide-band limit of the spectral density func-tions: [ J L/R ] αα = [ J L/R ] αβ = [ J L/R ] βα = [ J L/R ] ββ = Γ with Γ standing for a constant. The matrix elements ofthe retarded Green’s functions are then explicitly givenby [ U L/R ] αα ( t, t ) = [ U L/R ] ββ ( t, t )= 12 (cid:104) e − Γ (1+ y )( t − t ) + e − Γ (1 − y )( t − t ) (cid:105) , (28a)[ U L/R ] αβ ( t, t ) = [ U L/R ] βα ( t, t )= 12 (cid:104) e − Γ (1+ y )( t − t ) − e − Γ (1 − y )( t − t ) (cid:105) , (28b)where y = cos [ π Φ / Φ ]. Note that U ( t, t ) and thus theMZM qubit density matrix ρ ( t ) show 2Φ -periodicity asa function of the magnetic flux Φ. The diagonal elementsof U ( t, t ) describe the decays of the MZM qubits. FromEq. (28a), one can see that the decay of MZM qubit statesconsist of two parts with different decay times, namely[Γ (1 + y )] − and [Γ (1 − y )] − respectively. Therefore,if y (cid:54) = ±
1, i.e., Φ / Φ (cid:54) = n ( n stands for an integer),the MZM qubits will inevitably decay away. Further-more, it is obvious from Eq. (28a) that for y = ±
1, i.e.,Φ / Φ = n , there exist dissipationless modes and part ofthe MZM qubit states will not decay. For the parametersconsidered above, when Φ / Φ is an even integer, the sys-tem generates two dissipationless MZM modes reading ( γ αL − γ βL ) and ( γ αR − γ βR ), while for Φ / Φ being anodd integer, the system forms two dissipationless modesreading ( γ αL + γ βL ) and ( γ αR + γ βR ).On the other hand, the off-diagonal elements of U ( t, t )characterize the correlations between MZMs γ αL ( R ) and γ βL ( R ) , and hence relate to the MZM qubit coherence.One can see from Eq. (28b) that [ U L/R ] αβ increasesfrom zero initially, implying that the correlations be-tween MZMs are building up. If there are no dissipation-less modes, these build-up MZM correlations will eventu-ally vanish. When dissipationless MZM modes exist, theMZM correlation functions [ U L/R ] αβ will reach a steadyvalue of 1 / / Φ being odd) or − / / Φ being even). To study the MZM qubit dynamics, theevolution of the density matrix elements of the MZMstates is shown in Fig. 2. In the case of zero bias, i.e. µ L = µ R = 0, the MZM qubit will eventually decayto a maximally mixed state if there is no dissipationlessMZM mode. As mentioned above, the qubit coherence(described by the off-diagonal elements of the densitymatrix) grows from zero initially and fades away as theMZMs decay. Explicitly, Re[ ρ αβ ] quickly grows from zeroto 0 .
25 within t ∼ / Γ (see Fig. 2b), then decreases ata decay rate depending on the magnetic flux Φ / Φ (seeEq. (28b)). On the other hand, the dissipationless MZM ��� ��� ��� ��� - �������������������� ��� ��� ��� ��� - �������������������� ρ ρ αα ρ ββ ρ dd Re [ ρ d ] Im [ ρ d ] Re [ ρ αβ ] Im [ ρ αβ ] ��� ��� ��� ��� - �������������������� ��� ��� ��� ��� - �������������������� ! = 0.1/Γ ( ! = 1/Γ ( ! = 2/Γ ( ! = 10/Γ ( Φ/Φ ( Φ/Φ ( �� FIG. 2: (Color Online) The density matrix elements of theMZM states with varying magnetic flux at time (a) (top left) t = 0 . / Γ , (b) (top right) t = 1 / Γ , (c) (bottom left) t = 2 / Γand (d) (bottom right) t = 10 / Γ . The other parametersof the device are T L = T R = 0 . (cid:126) Γ /k B , µ L = µ R = 0, and J L / Γ = J R / Γ = , and the initial state of the systemis ρ ( t ) = | (cid:105)(cid:104) | . It is noteworthy that since the evolution of ρ αα and ρ ββ are precisely the same, their curves coincide witheach other. mode, which exists when Φ / Φ = 0, 1 or 2, will preservepart of the initial qubit state information and keep theMZM qubits away from a maximally mixed state. Notethat two different dissipationless MZM modes are formedat Φ / Φ = 0 and Φ / Φ = 1 (see Re[ ρ αβ ] in Fig. 2), show-ing the 2Φ -periodicity of the MZM qubit states.Next, the bias µ L and µ R can be tuned so that twoqubit steady states will not become a maximally mixedstate. In this case, apart from the mere damping of theMZMs, electrons and holes can be pumped into or outof the two TSCs from the leads. As a result, the evenand odd parity qubit states are not equally occupied.Furthermore, MZM qubit coherence with definite paritycan also be generated by applying bias (See the curvescorresponding to Im[ ρ αβ ] and Im[ ρ d ] in Fig. 3). If thebias µ L and µ R are large enough, MZM qubits can evolveto a state with almost definite parity and perfect coher-ence. For instance, in the case of Φ / Φ = , a largeanti-symmetric bias (e.g. µ L = − µ R = 10 (cid:126) Γ ) leads thesystem to the almost pure qubit state with odd parity,namely, ( | α (cid:105) + i | β (cid:105) ) / √
2. While in the case of Φ / Φ = ,a large symmetric bias (e.g. µ L = µ R = 10 (cid:126) Γ ) leads thesystem to to the almost pure qubit state with even parity,namely, ( | (cid:105) + i | d (cid:105) ) / √ µ L = − µ R = 10 (cid:126) Γ is applied, the dominated parity is flippedwhen the cross-coupling strength [ J L ] αβ / Γ changes from ��� ��� ��� ��� - ���� - �������������������� ρ ρ αα ρ ββ ρ dd Re [ ρ d ] Im [ ρ d ] Re [ ρ αβ ] Im [ ρ αβ ] ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� Φ/Φ
Φ/Φ
Φ/Φ $ = 1/Γ $ = 2/Γ $ = 10/Γ ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - ��������������������
12 ⟩|/ + 1 ⟩|212 ⟩|0 + 1 ⟩|3 = −4 = Γ = −4 = 10Γ = 4 = 10Γ FIG. 3: (Color Online) The MZM qubit state evolutionwith various magnetic flux and bias voltages. The com-mon conditions for the device are T L = T R = 0 . (cid:126) Γ /k B , J L / Γ = J R / Γ = , and the initial states of the sys-tem are all set to be ρ ( t ) = | (cid:105)(cid:104) | . It is noteworthy that thecurves of ρ αα and ρ ββ coincide with each other. The left, mid-dle and right columns show the MZM density matrix elementsat time t = 1 / Γ , 2 / Γ and 10 / Γ , respectively. While therows show the MZM density matrix elements with the appliedbias µ L = − µ R = (cid:126) Γ (top row), µ L = − µ R = 10 (cid:126) Γ (middlerow) and µ L = µ R = 10 (cid:126) Γ (bottom row) respectively. Thedashed arrows are marked to show the state ( | α (cid:105) + i | β (cid:105) ) / √ | (cid:105) + i | d (cid:105) ) / √ positive to negative (see Fig. 4). We demonstrate indetail this parity-flip dynamics in Fig. 5, in which thecross-coupling strength of the left-hand side [ J L ] αβ / Γ is tuned so that it changes from 1 to − J L ] αβ / Γ is tuned within a very short time ( ∼ / Γ ), the MZMswill relax directly to the even parity state ( | (cid:105) + i | d (cid:105) ) / √ J L ] αβ / Γ be-comes a little longer ( ∼ / Γ ), as shown in Fig. 5b, theMZMs will relax partially to the odd parity state but then“turn” its relaxation to the even parity state. This is be-cause the coupling changes so fast that the MZM statescannot reach full relaxation. Finally, when the changingtime of [ J L ] αβ / Γ is long enough ( ∼ / Γ ), the MZMsrelaxes from the initial state | (cid:105)(cid:104) | to the odd parity state( | α (cid:105) + i | β (cid:105) ) / √
2, then relaxes again to the odd parity state,which is a parity flip between two MZM qubit states [seeFig. 5c].
V. CONCLUSION
In this paper, we propose a Majorana Aharonov-Bohminterferometer to control the MZM states. In this device, ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ρ ρ αα ρ ββ ρ dd Re [ ρ d ] Im [ ρ d ] Re [ ρ αβ ] Im [ ρ αβ ] ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ( Φ/Φ ( Φ/Φ ( ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - �������������������� ��� ��� ��� ��� - ���� - ��������������������
12 ⟩|0 + - ⟩|. /
0( 12 = 0.5 Γ ( /
0( 12 = −0.5 Γ ( /
0( 12 = − Γ ( ( ( ( FIG. 4: (Color Online) The MZM qubit state evolution withvarious magnetic flux and cross-coupling strength. The com-mon conditions for the device are set to be T L = T R =0 . (cid:126) Γ /k B , µ L = − µ R = 10 (cid:126) Γ and the initial states of thesystem are all set as ρ ( t ) = | (cid:105)(cid:104) | . It is noteworthy that theevolution of ρ αα and ρ ββ are precisely the same so their curvescoincide with each other. The left, middle and right columnsshow the MZM density matrix elements at time 1 / Γ , 2 / Γ and 10Γ respectively. While the rows show the MZM densitymatrix elements with cross-coupling strength [ J L ] αβ = 0 . (top row), [ J L ] αβ = − . (middle row) and [ J L ] αβ = − Γ (bottom row) respectively. The dashed arrows are marked toshow the state ( | (cid:105) + i | d (cid:105) ) / √ electrons and holes transport from one lead to anotherthrough the rectangular ring formed by the four spatiallyseparated MZMs. Through this transport process, thequbit states of the MZMs evolves and manifests variousfeatures such that their state evolution can be tuned bysetting the parameters of the interferometer.With path-integral approach in the coherent state rep-resentation, we obtain the exact master equation of thetwo MZM qubits, one qubit has the even fermion parityand the other has old parity. The effects of the leads onthe system are clearly revealed in the structure of themaster equation. Formally, the MZMs in the left andright evolve independently, which are respectively influ-enced by the leads on the left and right side. However,because the renormalized Hamiltonian and damping co-efficients all depend on the global quantity, i.e., the totalmagnetic flux Φ, the qubit state evolution of the MZMsactually involves interference effect. Note that the den-sity matrix of two MZM qubits shows 2Φ -periodicity ofthe magnetic flux.It is shown that by tuning the magnetic flux, the biasvoltage of the leads, and the TSC-lead coupling strength,the interference property of the MZM qubit states canbe modified significantly. The two decoherence ratesΓ (1 + y ) and Γ (1 − y ) can be changed by tuning the magnetic flux, and dissipationless modes can be formed !/Γ $ & '$ () � � � � � �� - ��� - ������������ � � � � � �� - ��� - ������������ � � � � � �� - ��� - ������������ & '$ () & '$ () FIG. 5: (Color Online) The cross coupling strength [ J L ] αβ /J (red dotted line) is tuned at different rates. The density ma-trix elements are plotted with the same line patterns as in theprevious figures. (a) (top) The MZMs relaxes directly to theeven parity state. (b) (middle) The MZMs relax partially tothe odd parity state then “turns” its relaxation to the evenparity state. (c) (bottom) The MZMs relaxes fully to the oddparity state then relaxes to the even parity state. for certain values of the magnetic flux. By setting biasamong the leads and the TSCs, MZM qubit states canbe drawn away from approaching the maximally mixedstate. The fermion parity of the MZM qubit can be po-larized and the MZM qubit coherence can also be gener-ated. The parity of the target state can be controlledby setting the bias voltages in a particular configura-tion, or by tuning the TSC-lead coupling through thecontrolling gates. If the bias is large enough, the statecan evolve to a nearly pure coherent MZM qubit statewithin the same parity. Moreover, the switch betweendifferent parity qubit states can be realized by changingthe cross-coupling strength from positive (negative) tonegative (positive) at suitable rates. Acknowledgments
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