Manipulation with Andreev states in spin active mesoscopic Josephson junctions
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Manipulation with Andreev states in spin active mesoscopic Josephson junctions
J. Michelsen, V.S. Shumeiko, and G. Wendin
Department of Microtechnology and Nanoscience, MC2Chalmers University of Technology,SE-41296 Gothenburg, Sweden (Dated: October 24, 2018)We investigate manipulation with Andreev bound states in Josephson quantum point contactswith magnetic scattering. Rabi oscillations in the two-level Andreev subsystems are excited byresonant driving the direction of magnetic moment of the scatterer, and by modulating the su-perconducting phase difference across the contact. The Andreev level dynamics is manifested bytemporal oscillation of the Josephson current, accompanied, in the case of magnetic manipulation,also by oscillation of the Andreev states spin polarization. The interlevel transitions obey a selec-tion rule that forbids manipulations in a certain region of external parameters, and results fromspecific properties of Andreev bound states in magnetic contacts: 4 π -periodicity with respect to thesuperconducting phase, and strong spontaneous spin polarization. PACS numbers: 74.50.+r, 74.45.+c, 71.70.Ej
I. INTRODUCTION
Recent advances in the development and experimentalinvestigation of nanowire based Josephson junctions attract new attention to rich physics of mesoscopicJosephson effect. One of particularly interesting ques-tions concerns the possibility to employ Josephson quan-tum point contacts for quantum information processing.Such contacts contain a small number of generic two-levelsystems - Andreev bound levels, whose quantum statescan be selectively manipulated and measured.
By mod-ulating the phase difference across the junction one is ableto induce the Rabi oscillation in the Andreev two-levelsystem, and therefore to prepare arbitrary superpositionof the Andreev states. A measurement of induced oscil-lation of the Josephson current allows for the Andreevlevel readout. Thus the pair of Andreev bound levelsbelonging to the same conducting mode may serve as aquantum bit.
An interesting possibility to involve a spin degreeof freedom in the contact quantum dynamics, andto use it for qubit application has been investigatedby Chtchelkatchev and Nazarov. They considered aJosephson quantum point contact with spin-orbit inter-action and showed how to manipulate with the spin ofthe Andreev state. The properties of Andreev boundstates in spin-active mesoscopic junctions and equilib-rium Josephson effect have been extensively studied inrecent literature; non-stationary as-pects of interaction of individual magnetic scatterers withJosephson current have also been discussed.
In this paper, we investigate the methods of manip-ulation with the Andreev states in Josephson quantumpoint contacts containing a magnetic scatterer, e.g. mag-netic nanoparticle situating between the superconductingelectrodes.
We investigate two manipulation meth-ods: (i) time variation of the superconducting phaseacross the contact, and (ii) time variation of the direc- tion of magnetic moment of the scatterer. We find thatin both the cases the Josephson current exhibits Rabioscillation under the resonant drive within a certain in-terval of biasing superconducting phase. In the case ofmagnetic manipulation, the effect may only exist if theAndreev states are initially spin polarized, then the cur-rent oscillation is accompanied by oscillation of the An-dreev states spin polarization. The phase interval, wherethe Rabi oscillation can be excited, decreases with in-creasing strength of the magnetic scatterer, and eventu-ally disappears at large enough strength; in particular inthe π -junction regime, the Rabi oscillation is completelyforbidden. This selection rule results from specific prop-erties of the bound Andreev states in magnetic junctionsas we will show. FIG. 1: Sketch of a magnetic Josephson point contact: su-perconducting reservoirs are connected by a nanowire ofthe length L smaller than the coherence length, magneticnanoparticle creates local classical magnetic field. For a static scatter the spin rotation symmetry aroundthe direction of its magnetic moment is preserved. Thisallows for the contact description in terms of a two-component Nambu spinor, similar to non-magneticjunctions, thus avoiding a double counting problem.Within such an approach, the two bound Andreev lev-els per conducting modes are only relevant, giving com-plete quantitative description of the stationary Joseph-son effect as well as the resonant two-level transitionsand non-stationary current response under the phasemanipulation. Consideration of the spin conjugatedNambu spinor gives a completely equivalent physical de-scription in terms of a reciprocal pair of Andreev boundstates; both the pictures mirror each other.Time variation of the direction of the magnetic mo-ment of the scatterer leads to a violation of the spinrotation symmetry, and induces coupling between thespin conjugated Nambu spinors. This results in a uni-tary rotation in the extended space of the four Andreevbound states. It turns out, however, that this rotationsplits into two equivalent rotations in invariant two-levelsubspaces, which mirror each other. Thus the contactresponse can also in this case be explained in terms ofthe two-level Rabi dynamics. For the two different waysof magnetic manipulation considered - instant switching,and small-amplitude resonant oscillation of the directionof the magnetic moment of the scatterer, the Andreevtwo-level dynamics has a physical meaning of precession,and nutation of the spin polarization of the Andreev lev-els, respectively. Thus the spin polarization of Andreevstates is required in order to observe a non-trivial dynam-ical response. Such a possibility naturally exists, as wewill show, in the contact under consideration: the equi-librium Andreev states exhibit strong spin polarization,up to the maximum values ± / II. CONTACT DESCRIPTION
Consider one-mode quantum point contact with super-conducting electrodes connected by a normally conduct-ing nanowire as shown in Fig. 1. The left and rightelectrodes (L,R) are described with the BCS Hamilto-nian, H S = Z L , R dx X σ = ↑ , ↓ ˆ ψ † σ ( x ) (cid:18) p m − µ ( x, t ) (cid:19) ˆ ψ σ ( x )+∆ ∗ ( x, t ) ˆ ψ ↑ ( x ) ˆ ψ ↓ ( x ) + ∆( x, t ) ˆ ψ †↓ ( x ) ˆ ψ †↑ ( x ) . (1)Aiming to investigate the effect of time variation ofthe superconducting phase difference ϕ across the junc-tion, we consider the order parameter having the form,∆( x, t ) = ∆ e iϕ ( x,t ) / , ϕ ( x, t ) = ϕ ( t ) sgn x , and the elec-trochemical potential, µ ( x, t ), having the form, µ ( x, t ) = E F − ~ ∂ t ϕ/
2, which provides the electro-neutrality con-dition within the electrodes. To explore the spin properties of the Andreev states,we assume that the contact nanowire contains a mag-netic scatterer, e.g., magnetic nanoparticle (Fig. 1). Weassume for simplicity that the magnetic field H ( x, t ) in-duced by the scatterer is localized within the nanowire ona distance l smaller than the distance L between the elec-trodes, thus not affecting the superconductivity within the electrodes; furthermore, we will treat this magneticfield as a given external parameter neglecting the backaction effect from the current. Then the Hamiltonian ofthe normal region of the junction has the form, H N = Z L/ − L/ dx X σσ ′ ˆ ψ † σ ( x ) (cid:20)(cid:18) p m − E F + U ( x ) (cid:19) δ σσ ′ + 12 µ B σ σσ ′ H ( x, t ) (cid:21) ˆ ψ σ ′ ( x ) , (2)where U ( x ) is the scalar potential of the scatterer. Weassume a symmetric with respect to x = 0 spatial dis-tributions of the scalar potential and the magnetic field,and also a fixed direction of the magnetic field, which willvary with time during the manipulation.In the stationary case, the Hamiltonian (2) preservesthe spin rotational symmetry around direction of themagnetic field. Choosing the spin quantization axisalong this direction, we describe the electron propaga-tion through the normal region of the junction with atransfer matrix, T e , T e = ˆ d − (cid:18) e iσ z ( β/ i ˆ r − i ˆ r e − iσ z ( β/ (cid:19) , (3)where ˆ d = diag(d ↑ , d ↓ ) and ˆ r = diag(r ↑ , r ↓ ). Contact de-scription in terms of the spin active scattering matrix hasbeen extensively discussed in literature. The impu-rity scalar potential produces spatially symmetric scat-tering with transmission amplitudes, d ↑ , d ↓ , and reflec-tion amplitudes, r ↑ , r ↓ , which may be different for dif-ferent spin orientations (spin selection). The scatteringphase shift β between the opposite spin orientations isinduced by the Zeeman effect, β = µ B Hl ~ v F . (4)The physical observables of interest, Josephson cur-rent and spin polarization, are described with a singleelectron density matrix ρ ( x, x ′ , t ) associated with a two-component Nambu field Ψ( x, t ),Ψ( x, t ) = ˆ ψ ↑ ( x, t )ˆ ψ †↓ ( x, t ) ! , ρ ( x, x ′ , t ) = h Ψ( x, t )Ψ † ( x ′ , t ) i (5)(here the angle brackets indicate statistical averaging).Alternative description is given by a density matrix,˜ ρ ( x, x ′ , t ) associated with the spin conjugated Nambufield, ˜Ψ( x, t ),˜Ψ( x, t ) = ˆ ψ ↓ ( x, t ) − ˆ ψ †↑ ( x, t ) ! , ˜ ρ ( x, x ′ , t ) = h ˜Ψ( x, t ) ˜Ψ † ( x ′ , t ) i . (6)The two Nambu fields are connected via a fundamentalsymmetry relation imposed by the singlet nature of theBCS pairing, ˜Ψ( x, t ) = iσ y (cid:0) Ψ † (cid:1) T . (7)In what follows we will use these two reciprocal rep-resentations, φ -representation and ˜ φ -representation, re-spectively (see Appendix B). This will allow us to avoidthe redundancy, introduced by Eq. (7), of a commonlyused four-component Nambu formalism, and to explicitly show that the stationary Josephson ef-fect as well as non-stationary response to the phase ma-nipulation can be fully understood within the frameworkof the two bound Andreev states per conducting modeassociated with either of these two reciprocal representa-tions. III. ANDREEV STATE SUBSYSTEM
In this section we outline the properties of the sta-tionary Andreev states, which are of importance for thediscussion of non-stationary effects. The details of deriva-tions are presented in Appendix A.Within the two-component Nambu formalism, thequasiparticle states in the stationary contact are givenby the Bogoliubov-deGennes equation, hφ = Eφ, h = ( p / m − E F ) σ z + ∆ σ x , (8)supplemented with the boundary condition at the con-tact, which for a short contact is given by the transfermatrix, T = exp( iσ z ϕ/ T e , Eq. (A3). The energy spec-trum of the bound states consists of the two levels, ac-cording to Eq. (A9), E s = θ s ∆ cos( sη − β/ , s = ± , (9)where θ s = sgn[sin( sη − β/ η is de-fined through equation, sin η = √ D sin( ϕ/ D is thecontact transparency. This energy spectrum is asymmet-ric with respect to the chemical potential, E = 0, and is4 π -periodic with respect to the phase difference as shownon Fig. 2, the spectral branches cross at ϕ = 2 πn . Onthe figure are also shown the energy levels of a recipro-cal Andreev level pair, ˜ E s = − E s , whose wave functions,˜ φ s = ( iσ y ) φ ∗ s , are associated with the spin conjugatedNambu field, Eq. (6) ( ˜ φ -representation, see Appendix B).The reason for the 4 π -periodicity and the level cross-ings is the symmetry of the problem which is reflected inthe property, Eq. (A12). Introducing the parity operator P that permutes the wave functions at the left and rightsides of the junction, P φ ( x <
0) = φ ( x > ˆΛ ˆ φ s ( x ) = sθ s ˆ φ s ( x ) , ˆΛ = P (cid:18) − σ x σ x (cid:19) . (10) π π ϕ ∆ - ∆ E E + E - E + E - θ + θ − ϕ E - E + E - E + FIG. 2: Andreev level energy spectrum for sin( β/ < √ D ( D =0.9, β =0.5), upper panel; θ -functions, shown on the lowerpanel, define the spectrum discontinuity points at 2 πn ± ϕ .Vertical dotted lines separate regions of weak and strong Zee-man effect. According to this equation, the Andreev level wave func-tions are simultaneously the eigen functions of the sym-metry operator ˆΛ with eigenvalues sθ s . Furthermore,the energy branches, E − and E + , in the neighboringphase intervals in Fig. 2, correspond to different eigenvalues of the operator Λ. The properties of the An-dreev states are therefore qualitatively different withinthe phase intervals, where θ s = θ − s , and θ s = − θ − s .To emphasize the difference we will refer to the for-mer ones as the regions of strong Zeeman effect (ZE),sin β/ > √ D sin ϕ/
2, and the latter ones as the regionsof weak ZE, sin β/ < √ D sin ϕ/
2. These regions are sep-arated by the points where the energy levels touch thecontinuum, ϕ = 2 πn ± ϕ , ϕ = 2 arcsin(1 / √ D sin β/ β/ > √ D theweak ZE regions disappear, while strong ZE regionsspread over the whole superconducting phase axis. Thebound energy levels, E ± ( ϕ ), depart from the continuumforming ”cigars”, see Fig. 3 , they belong to the orthogo-nal eigen subspaces of the operator ˆΛ at all phases. The π -contact is realized in this regime, at β = π , when thereciprocal cigars coincide and situate symmetrically withrespect to E = 0.A qualitative difference between the regions of strongand weak ZE is illustrated by the properties of Josephsoncurrent. Starting with a general expression for the chargecurrent through the density matrix, I ( t ) = e ~ mi ( ∂ x − ∂ x ′ ) [ δ ( x − x ′ ) − Tr ρ ( x, x ′ , t ) ] x = x ′ =0 , (11)we truncate it to the Andreev level subspace, I A ( t ) = X ss ′ I ss ′ (cid:18) δ s ′ s − ρ s ′ s ( t ) (cid:19) . (12) π πϕ - ∆ ∆ E E - E + E + E - FIG. 3: Andreev level energy spectrum for sin( β/ > √ D ( D =0.1, β = 2 . Here ρ ss ′ ( t ) = h φ s | ρ ( t ) | φ s ′ i = Z dxdx ′ Tr (cid:2) ρ ( x, x ′ , t ) φ s ′ ( x ′ ) φ † s ( x ) (cid:3) , (13)is the density matrix in the Andreev level representation,Eqs. (A11), (A12), the trace refers to the electron-holespace. The current matrix I ss ′ reads, I ss ′ = 2 e ~ (cid:20)(cid:18) ∂ ϕ E + ∂ ϕ E − (cid:19) + (1 − θ s θ − s ) √ RD sin ϕ p ζ + ζ − ε (cid:18) (cid:19) . (14)The diagonal elements give the expectation values ofthe currents of Andreev states, while the off-diagonalpart describes the current quantum fluctuation. The off-diagonal part is finite in the weak ZE regions, where θ s θ − s = −
1, while it is zero in the strong ZE regions.This implies that the current quantum fluctuation is fullysuppressed in the strong ZE regions: here the currentmatrix commutes with the Andreev level Hamiltonian,which is diagonal in this representation h ss ′ = δ ss ′ E s .In the equilibrium, ρ ss ′ = δ ss ′ n F ( − E s ), and the An-dreev current in Eq. (12) takes the form, I A = − e ∆ ~ X s ∂ ϕ E s tanh( E s / T ) , (15)One can confirm by direct calculation that the currentof the continuum states vanishes, thus Eq. (15) repre-sents the total Josephson current. This equation coin-cides with the result obtained using the four-componentNambu formalism, and it can be also ob-tained by working with the reciprocal Nambu represen-tation using equation (B9).Spin polarization of the Andreev states plays an im-portant role in magnetic manipulation. Asymmetry ofthe Andreev spectrum with respect to the zero energy together with the spectrum discontinuities result in a pe-culiar phase dependence of the spin polarization of An-dreev levels. The z -component of electronic spin densityin the contact is given by equation, S ( x, t ) = 12 [ δ ( x − x ′ ) − Tr ρ ( x, x ′ , t ) ] x = x ′ . (16)Truncating this equation to the Andreev level subspace,and integrating over x using normalization condition forthe bound state wave functions, we find the spin polar-ization of the Andreev level pair, S A = (1 / − f + − f − ) , (17)where f s = ρ ss are the population numbers.Thus the spin polarization of the Andreev levels is en-tirely determined by their (generally non-equilibrium)population numbers: For empty Andreev level pair, f ± = 0, the spin polarization is S = 1 /
2, while for fullypopulated levels, f ± = 1, it is S = − /
2. For the sin-gle particle occupation of the level pair, f + + f − = 1,the spin polarization is zero, S = 0. In non-magneticcontacts the equilibrium spin polarization of Andreevlevels is always zero, S = 0, by virtue of the identity, n F ( − E + ) + n F ( − E − ) = 1, that holds due to the spec-trum symmetry, E + = − E − . In magnetic contacts, thespin polarization sharply varies in the ϕ − β parameterplane, see Fig. 4. At zero temperature, f s = θ ( E s ), thespin polarization is zero in the regions bound by the lines,sin( ϕ/
2) = sin( β/ / √ D and sin( ϕ/
2) = cos( β/ / √ D ;the first region corresponds to a weak ZE. Outside theseregions the Andreev levels are strongly polarized, S =1 /
2. At small contact transparencies,
D < /
2, the abovementioned lines do not overlap, while at
D > / S = − /
2, around the point, ϕ = π , β = π/
2, see Fig. 4.This island grows with increasing transparency eventu-ally touching the lines, β = 0 and β = π at D = 1. FIG. 4: Equilibrium spin polarization of Andreev levels atzero temperature and at
D > /
2; the central region with S = − / D = 1 / ϕ = π, β = π/
2, and growswith D reaching the lines β = 0 and β = π at D = 1. In contrast to the charge density, the spin density inEq. (16) obeys the conservation equation, ∂ t S ( x, t ) + ∂ x I S ( x, t ) = 0. The consequence of this is the zero spincurrent of the Andreev states, I SA = 0: under the sta-tionary condition the partial spin current of the Andreevstate must be constant in space, and being proportionalto the bound wave function it vanishes at the infinity;thus it is identically equal to zero. IV. PHASE MANIPULATION
Now we turn to discussion of the Andreev level dy-namics under the time dependent phase. Similar to non-magnetic junctions this dynamics involve only the boundlevels belonging to the same Nambu representation, as shown on Fig. 5. The frequency of the phase timevariation must be small compared to the distance tothe gap edges to prevent the level-continuum transitions, ω ≪ ∆ − | E s | .Time evolution of the contact density matrix, ρ ( t ), isgoverned by the Liouville equation, i ~ ∂ t ρ = [ h, ρ ], withthe Hamiltonian of Eq. (8), supplemented with the non-stationary boundary condition, T = exp( iσ z ϕ ( t ) / T e .Now we truncate the density matrix using the instanta-neous Andreev eigenfunctions, φ s [ ϕ ( t )] (cf. Eq. (13)), ρ ss ′ ( t ) = h φ s ( t ) | ρ ( t ) | φ s ′ ( t ) i . (18)This density matrix obeys the Liouville equation i ~ ∂ t ρ =[ H, ρ ] with a truncated Hamiltonian, which in this basisis given by equation, H ss ′ = h φ s | h | φ s ′ i−h φ s | i ~ ∂ t | φ s ′ i = E s δ ss ′ − ~ ˙ ϕ h φ s | i∂ ϕ | φ s ′ i , (19)where E s [ ϕ ( t )] and φ s [ ϕ ( t )] are given by Eqs. (A9) and(A11), (A12) respectively. π πϕ - ∆ ∆ E π π E + E - E + E - FIG. 5: Interlevel transitions induced by time oscillation ofthe phase difference, shadow regions indicate forbidden re-gions; transitions in ˜ φ representation (reduced-intensity lines)are equivalent to the transitions in φ -representation (full-intensity lines); D =0.9, β =0.5. Inset: transition matrix el-ement as function of ϕ . The matrix element, h φ s | i∂ ϕ | φ s ′ i , is found to be zerofor s ′ = s , while for the interlevel transitions, s ′ = − s , itreads, h φ s | i∂ ϕ | φ − s i = is (1 − θ s θ − s ) Λ( ϕ )2 , Λ( ϕ ) = √ RD ϕ p ζ + ζ − ε ( ζ + + ζ − ) . (20)From this we find that in the weak ZE regions, θ s θ − s = −
1, the non-stationary Andreev level Hamiltonian hasthe form, H ( t ) = (cid:18) E + E − (cid:19) + ~ ˙ ϕ Λ( ϕ ) (cid:18) − ii (cid:19) , √ D sin ϕ > sin β . (21)This equation provides generalization to a magnetic junc-tion of the Hamiltonian derived in Ref. [6]: the level cou-pling coincides with the one in non-magnetic junctionwhen β = 0, and remains finite when β = 0 but onlyinside the weak ZE region, decreasing towards the edgesof this region, see inset in Fig. 5. In the strong ZE regionthe matrix element is identically zero ( θ s θ − s = +1), andthe Hamiltonian is diagonal, H ( t ) = (cid:18) E + E − (cid:19) , √ D sin ϕ < sin β . (22)Thus we conclude that no operation with Andreev levelsis possible in the strong ZE regime.Eq. (21) is convenient for calculation of Rabi oscillationin the Andreev level system under the resonant driving, ϕ ( t ) = ϕ + δ sin ωt , ω = ( E + − E − ) / ~ . Inserting this inEq. (21), we get H = ~ ω σ z + ~ Λ δω cos ωt σ y . (23)Assuming small amplitude of the phase oscillation, δ ≪
1, and using rotating wave approximation, we find thetime dependent density matrix in the rotating frame, ρ ( t ) = ρ (0) − f + (0) − f − (0)2 (1 − cos Λ δωt − sin Λ δωt σ x ) , (24)where ρ (0) = diag ( f + (0) , f − (0)). Rabi oscillation of theAndreev levels generate a time dependent Josephson cur-rent, I ( t ) = I (0) − e ~ ( f + (0) − f − (0)) sin Λ δωt X s ∂ ϕ E s . (25)The Rabi oscillation, Eq. (24), and the time oscillationof the Josephson current, Eq. (25), vanish if the Andreevlevels are initially fully spin polarized, S (0) = ± /
2, since f + (0) = f − (0) in this case. Thus an additional require-ment for the phase manipulation is to bias the contact inthe region outside the negative-spin island on Fig. 4. V. SPIN MANIPULATION
Now we proceed with the discussion of the spin ma-nipulation. We consider the two ways of driving An-dreev level spin, presented in Fig. 6: (i) rapid change ofthe direction of the magnetic moment of the scatterer (dcpulsing), and (ii) harmonic oscillation with resonance fre-quency of the magnetic moment direction (rf-pulsing). Inboth cases the spin rotation symmetry is violated, there-fore the junction dynamics cannot be described with onlyone Nambu pseudo-spinor, but involves both the spinconjugated Nambu pseudo-spinors. The interlevel tran-sitions in this case physically describe a rotation of theAndreev level spin.
FIG. 6: Sketch of manipulation with magnetic field; a) instantswitching of direction of magnetic field; b) small-amplitudeoscillation of magnetic field with resonant frequency (electronspin resonance)
A. dc pulsing
Suppose the Andreev levels are initially prepared in astationary state with non-zero spin, which points alongthe applied magnetic field ( z -axis). Such states were dis-cussed in previous sections. Let us now suppose that themagnetic field is rapidly rotated by angle θ around y -axis,as shown on Fig. 6a. Such a manipulation is describedby rotation of the electronic T -matrix, in Eq. (3), T e → U T e U † , U = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) , (26)and it mixes the Nambu pseudo-spinors Ψ and ˜Ψ. Todescribe the effect of this manipulation, we introduce theextended four-component Nambu space, (Ψ , ˜Ψ) T , andthe corresponding single particle density matrix,Π( x, x ′ , t ) = (cid:18) h Ψ( x, t )Ψ † ( x ′ , t ) i h Ψ( x, t ) ˜Ψ † ( x ′ , t ) ih ˜Ψ( x, t )Ψ † ( x ′ , t ) i h ˜Ψ( x, t ) ˜Ψ † ( x ′ , t ) i (cid:19) . (27) This density matrix operates in the Hilbert spacespanned by the extended eigen basis,Φ ν ( x ) = (cid:18) φ ν ( x )0 (cid:19) , Φ ˜ ν ( x ) = (cid:18) φ ν ( x ) . (cid:19) . (28)The transformation U induces rotation of the extendedbasis Φ α ( x ) → U Φ α ( x ) , α ∈ { ν, ˜ ν } . (29)The eigen energies, however, remain the same since sucha rotation just corresponds to changing the spin quanti-zation axis.The wave functions, Eq. (28), form a complete set inthe extended space, and thus any operator can be ex-pressed through them, A ( x, x ′ ) = X α,β Φ α ( x )Φ † β ( x ′ ) A αβ . (30)In particular, for a stationary system with spin rotationalinvariance we have for the density matrix,Π( x, x ′ ) = X α Φ α ( x )Φ † α ( x ′ ) f α , (31)and the Hamiltonian, H ( x ) = X α Φ α ( x )Φ † α ( x ) E α . (32)However, one has to remember that this description is re-dundant, and rigorous constraints hold on the occupationnumbers, ˜ f ν = 1 − f ν , and eigen energies, ˜ E ν = − E ν .We now write down the Hamiltonian after the mag-netic field rotation on the form, H ( x ) = X α U Φ α ( x ) E α Φ † α ( x ) U † , t > . (33)In the initial basis, this Hamiltonian is represented withthe matrix, H αβ = X µ h Φ α | U | Φ µ i E µ h Φ µ | U † | Φ β i , t > , (34)or explicitly, H νν ′ = cos θ E ν δ νν ′ − sin θ X µ = ν,ν ′ h φ ν | ˜ φ µ i E µ h ˜ φ µ | φ ν ′ i H ˜ ν ˜ ν ′ = − cos θ E ν δ νν ′ + sin θ X µ = ν,ν ′ h ˜ φ ν | φ µ i E µ h φ µ | ˜ φ ν ′ i H ν ˜ ν ′ = − sin θ h φ ν | ˜ φ ν ′ i E ν + E ν ′ . (35)Here the orthogonality relations were used, h φ ν | φ µ i = h ˜ φ ν | ˜ φ µ i = δ µν , and h φ ν | ˜ φ ν i = 0, Eq. (B5).At this point we restrict ourselves to the Andreev levelsubspace, and present the truncated Hamiltonian on theform (using the symmetries E s = − ˜ E s and h φ + | ˜ φ − i = −h φ − | ˜ φ + i ), H (4) = E + W V − E + W − V − V − E − W V E − W , (36)where E , W = (cid:18) cos θ − sin θ | M | (cid:19) E + ∓ E − ,V = − M sin θ E + + E − , (37)and the interlevel matrix element, M = h φ + | ˜ φ − i = ( θ s − θ − s ) cos β p ζ + ζ − ζ + + ζ − . (38)The matrix element (38) equals to zero in the strongZE region ( θ s = θ − s ), thus the manipulation does notproduce any effect there, which is similar to the phasemanipulation, see inset in Fig. 7.The Hamiltonian (36) has a block-diagonal form de-scribing identical rotations in the two orthogonal sub-spaces, spanned with the eigen vectors ( φ + , ˜ φ − ), and( φ − , ˜ φ + ). Thus the problem reduces to solving for twophysically equivalent two-level systems. Choosing thesubspace ( φ + , ˜ φ − ), we have the two-level Hamiltonian, H (2) = (cid:18) W VV − W (cid:19) . (39)Introducing projection operators on the eigen subspaces, H (2) = X λ = ± λ ~ Ω P λ , P λ = 12 (cid:18) λ σ z W + σ x V ~ Ω (cid:19) , ~ Ω = p W + V = ( E + + E − )2 (cid:18) cos θ θ | M | (cid:19) , (40)where λ ~ Ω are the eigen energies, we have for the timeevolution of the two-level density matrix,Π (2) ( t ) = e − i Ht ~ Π (2) (0) e i Ht ~ = X λλ ′ e i ( λ ′ − λ )Ω t P λ Π (2) (0) P λ ′ . (41)Assuming the initial density matrix to be stationary (notnecessarily equilibrium), and expressing it through thelevel occupation numbers of the φ -representation,Π (2) (0) = (cid:18) f + (0) 00 1 − f − (0) (cid:19) , (42) we obtain,Π (2) ( t ) = Π (2) (0) + 2 S A (0) (cid:18) | a ( t ) | b ( t ) b ( t ) ∗ −| a ( t ) | (cid:19) ,a ( t ) = − i V ~ Ω sin Ω t, b ( t ) = a ( t ) (cid:20) cos Ω t + i W ~ Ω sin Ω t (cid:21) , (43)where S A (0) = (1 / − f + (0) − f − (0)] is the initial spin-polarization of the Andreev levels as given by Eq. (17).Thus no rotation is induced for spin unpolarized An-dreev levels. Furthermore, the frequency of the rotationis proportional to the level splitting, ~ Ω ∝ E + − ˜ E − = E + + E − , i.e. to the magnetic field.The time-evolution of the occupation numbers f s ( t ) ofthe Andreev levels in the φ -representation is extractedfrom Eq. (43), f s ( t ) = f s (0) + 2 S A (0) V ( ~ Ω) sin Ω t. (44)This relation illustrates the non-unitary evolution of theAndreev levels in this representation, f + ( t ) + f − ( t ) =const.Eq. (44) allows us to obtain the time dependence ofthe spin polarization of the Andreev levels, S A ( t ) = 12 (1 − f + ( t ) − f − ( t )) = S A (0) (cid:18) − V ( ~ Ω) sin Ω t (cid:19) . (45)To calculate the Josephson current, we use the expres-sion through the current matrix in the φ -representation,Eqs. (12), (14), I ( t ) = X ss ′ I ss ′ (cid:18) δ s ′ s − ρ s ′ s ( t ) (cid:19) . (46)The diagonal elements of the density matrix, ρ ss , aregiven by Eq. (44). On the other hand, the off-diagonalelements, ρ ss ′ , equal zero because the spin manipulationdoes not induce transitions between the eigen states ofthe same (either φ - or ˜ φ -) representation. Therefore, I ( t ) = 2 e ~ X s ∂ ϕ E s f s ( t )= I (0) + 4 e ~ S A (0) V ( ~ Ω) sin Ω t X s ∂ ϕ E s . (47)Summarizing, the conditions for the observation of thenon-stationary contact response, biasing in a weak ZE re-gion with a finite spin polarization, can be only fulfilled inthe central island region on Fig. 4 with negative polariza-tion. This constraint can be relaxed by pumping initiallevel populations away from equilibrium as suggested inRef. .Concluding this section we note that the dc pulsingof the magnetic field does not allow one to reach every π πϕ - ∆ ∆ E π π E + E - E + E - FIG. 7: Interlevel transitions induced by magnetic manip-ulation; shadow regions indicate forbidden regions; transi-tions between the levels ( E − , ˜ E + )(reduced-intensity lines) areequivalent to the transitions ( E + , ˜ E − ) (full-intensity lines); D =0.9, β =0.5. Inset: transition matrix element as functionof ϕ . point on the Bloch sphere: this is due to the fact that U isnot an invariant operation on the Andreev level subspaceof the extended Nambu space, which physically means aleakage to the continuum (similar effect exists also forthe phase manipulation with dc pulses ). However, forsmall rotation angles and not close to the edges of theweak ZE region, the matrix element M is close to unity,and the leakage is small; it can be further reduced byusing rapid adiabatic change of the magnetic field, i.e.rapid on the time scale of the Andreev level splitting butslow on the time scale of the distance of the Andreevlevels to the continuum. This shortcoming does not existfor the resonant rf-pulsing. B. rf-pulsing
Now let us consider a time-dependent rotation of themagnetic field, T → U ( θ ( t )) T U † ( θ ( t )) . where θ ( t ), as before, is the angle of rotation around the y -axis, see Fig. 6b. We can now define the instantaneouseigenstates U ( t )Φ α ( x ) , (48)satisfying the instantaneous boundary condition, Eq. (3),with T e → U T e U † . The time-dependent Hamiltonian canthen be written similarly to Eq. (33), H ( x, t ) = X α U ( t )Φ α ( x ) E α Φ † α ( x ) U † ( t ) . (49)Since the energy eigenvalues do not depend on thedirection of the quantization axis they remain time-independent. Now similarly to Eq. (31) we can expand the density matrix in terms of these instantaneous eigen-functions,Π( x, x ′ , t ) = X α,β U ( t )Φ α ( x )Φ † β ( x ′ ) U † ( t )Π αβ ( t ) . (50)The matrix Π αβ ( t ) satisfies the Liouville equation withthe Hamiltonian, H αβ ( t ) = Z dx Φ † α ( x ) U † ( t ) [ H ( x ) − i ~ ∂ t ] U ( t )Φ β ( x )= E α δ αβ − i ~ h Φ α | U † ( t ) ∂ t U ( t ) | Φ β i . (51)Inserting Eq. (48) we get, i ~ U † ( t ) ∂ t U ( t ) = − ~ ∂ t θ (cid:18) − ii (cid:19) . (52)Truncating to the Andreev level sub-space we have theHamiltonian, H = E + − ig E − ig − ig − E + ig − E − , g = ~ ∂ t θM. (53)This Hamiltonian can again be presented in a block-diagonal form describing two equivalent two-level sys-tems. Choosing the subspace spanned by ( φ + , ˜ φ − ) anddriving the magnetic field at exact resonance, ω = E + − ˜ E − = E + + E − , with small amplitude, θ ( t ) = θ sin ωt , θ ≪
1, we have the two-level Hamiltonian, H = (cid:18) E + − i ~ Ω r cos ωti ~ Ω r cos ωt − E − (cid:19) , Ω r = 12 θ ωM, (54)from which we obtain the Rabi oscillation of the popula-tion numbers of the φ -representation, f + ( t ) = f + (0) cos Ω r t − f − (0)] sin Ω r t , [1 − f − ( t )] = [1 − f − (0)] cos Ω r t f + (0) sin Ω r t , (55)or introducing explicitly the the Andreev level spin, f s ( t ) = f s (0) + 2 S A (0) sin Ω r t . (56)This equation again illustrates the non-unitary evolutionof the Andreev levels in the φ -representation. The timeevolution of the spin polarization, and the Josephson cur-rent then become, respectively, S A ( t ) = S A (0) (cid:18) − Ω r t (cid:19) ,I ( t ) = I (0) + 4 e ~ S A (0) sin Ω r t X s ∂ ϕ E s . (57) VI. DISCUSSION
Manipulations with the Andreev levels generatestrongly non-equilibrium states whose lifetime is re-stricted by relaxation processes. Let us qualitativelydiscuss the relaxation mechanisms relevant for the non-equilibrium states induced by manipulation methods dis-cussed.The phase manipulation affects the difference of thepopulations of the Andreev levels belonging to the sameNambu representation while keeping the total populationthe Andreev level pair unchanged. At zero temperature,and for relatively small frequency of the qubit rotationcompared to the superconducting gap, the states of thecontinuum spectrum are either empty or fully occupied,and therefore the exchange between the continuum andthe Andreev levels is exponentially weak. Therefore therelaxation predominantly occurs within the Andreev levelsystem. In the strong ZE regions, the interlevel relax-ation caused by interaction with electromagnetic envi-ronment should be suppressed due to the vanishing tran-sition matrix element, Eq. (20), i.e. for the same reasonthat prevents the phase manipulation. One may expectto prolong lifetime of the excited states by taking ad-vantage of this property and shifting adiabatically thephase bias into the strong ZE region after the manipu-lation has been performed. Such an operation, however,requires a passage through the one of the singular points, ϕ = 2 πn ± ϕ , where the Andreev levels touch the con-tinuum; at this points the quantum state escapes in thecontinuum and quantum information is lost.The absence of the interlevel relaxation in the strongZE regions has interesting implications for the obser-vation of 4 π -periodicity of the Andreev level spectrumdiscussed in Section III. The equilibrium Josephson cur-rent, Eq. (15), is 2 π -periodic and does not reveal the4 π -periodicity property of the Andreev states. This maychange under non-equilibrium condition, when a smallvoltage is applied to the junction. In this case, supercon-ducting phase becomes time dependent, ϕ = 2 eV / ~ , andthe Andreev levels adiabatically move along the ϕ -axiskeeping constant level population during long time (lim-ited by a weak level-continuum quasiparticle exchange).If the magnetic effect is weak while contact is trans-parent, sin( β/ < √ D , the levels touch the contin-uum every Josephson cycle, and the level population willbe periodically reset leading to the 2 π -periodicity ofthe Josephson current. However, in the ”cigars” regime,sin( β/ > √ D , depicted on Fig. 3, the levels are iso-lated from the continuum, and the level population mayremain unchanged during the time greatly exceeding theJosephson period. This will lead to the the 4 π -periodicityof the ac Josephson current, and could be experimentallydetected by observing anomalous Shapiro effect with onlyeven Shapiro steps present. The effect would be the mostpronounced for the π -junctions, β = π . Similar effect hasbeen discussed in a different context of unconventionalsuperconductor junctions. Manipulation with the Andreev level spin affects thespin polarization of the Andreev levels, and thus, at firstglance, a relevant relaxation mechanism would requiresome spin active scattering. Since magnetic interactionsin superconductors are usually rather small compared tonon-magnetic interactions, e.g. with electromagnetic en-vironment, one would expect a long lifetime of Andreevspin excitations. However, one should take into accountthe relation between the spin polarization and populationof the Andreev level pair Eq. (B9): The non-equilibriumspin polarization is associated with non-equilibrium pop-ulation of the Andreev level pair, which can be relaxed byany non-magnetic interaction. Consider, for example, theprocess of approaching the equilibrium state in magneticcontact after the phase bias has been suddenly changed.This will first create a non-equilibrium state in the An-dreev level system, both in terms of individual level popu-lations, and total population of the level pair, which thenwill rapidly relax to local equilibrium via the interlevel,and level-continuum quasiparticle transitions induced by(presumably) strong non-magnetic interaction. Such in-teraction does not change the total spin polarization ofcontact electrons, since it preserves the spin rotationsymmetry, however it is able to transfer the polarizationfrom the Andreev levels to the continuum states. Thetotal polarization is maintained in the local equilibriumby shifting the energy argument in the Fermi distribu-tion function by an energy independent constant. Thisconstant will slowly relax to the zero value in a secondrelaxation stage, due to spin-flip processes.Thus we conclude that decoherence of the states gen-erated by the magnetic manipulations basically resultsfrom the same physical interactions that destroy excitedstates produced by the phase manipulations, and there-fore one should not expect significant differences of therespective lifetimes.
VII. CONCLUDING REMARKS
In conclusion, we studied the properties of Andreevbound level system, and various ways of manipulationwith them in Josephson quantum point contacts contain-ing magnetic scatterers.In practice, such contacts can be realized by attach-ing magnetic nanoparticles or molecules to the contactbridge; another possibility is to insert magnetic macro-molecules, e.g. doped metallofullerene in the contact. Coulomb blockade regime in molecular dots offers ad-ditional possibility due to uncompensated spin of oddelectronic configurations on the dot.
In the studied cases of resonant driving the super-conducting phase difference and the direction of mag-netic scatterer, the contact response consists of a time-oscillation of the Josephson current, and for the magneticdrive, also oscillation of the Andreev level spin polariza-tion. We identified the regions of external parameterswhere these oscillations can be excited. The correspond-0ing selection rule results from specific symmetry proper-ties of the bound Andreev states in magnetic contacts:4 π -periodicity of the level spectrum and strong sponta-neous spin polarization.In all the studied cases, the non-stationary contact re-sponse results from resonant dynamics of two physicallyidentical two-level systems (for one conducting mode),whose evolutions mirror each other. This is the man-ifestation in a non-stationary regime of the redundancy(double counting) of the four-component Nambu descrip-tion of the Josephson effect in magnetic contacts. Dueto the fundamental constraint, Eq. (7), imposed by thesinglet pairing the four-component Nambu field possessesthe algebraic structure of Majorana fermion, thus de-scribing only two physical degrees of freedom rather thanfour. In contacts with magnetic impurities under discus-sion, these two physical degrees of freedom relevant forthe stationary Josephson effect correspond to the twoAndreev bound states per conducting mode, similar tothe case of non-magnetic contacts. Acknowledgments
The work was supported by the Swedish ResearchCouncil, and the SSF-OXIDE Consortium. We arethankful to M. Fogelstr¨om and T. L¨ofwander for usefuldiscussions.
APPENDIX A: BOUND STATE WAVEFUNCTIONS
To explicitly construct the wave functions of the An-dreev states, we consider a quasiclassical approximationfor φ ( x ) by separating rapidly oscillating factors e ± ik F x ,and slowly varying envelopes φ ± ( x ), φ ( x ) = φ + ( x ) e ik F x + φ − ( x ) e − ik F x . (A1)The envelopes, φ ± ν , satisfy a quasiclassical Bogoliubov-deGennes (BdG) equation, i ~ ∂ t φ ± = (cid:18) ± v F ˆ p σ z + ~ ∂ t ϕσ z + ∆ σ x e − iσ z ϕ sgn x/ (cid:19) φ ± , (A2)Furthermore the superconducting phase can be elimi-nated from this equation and moved to the boundarycondition by means of the gauge transformation, φ → exp( iσ z ϕ sgn x/ φ .The boundary condition at the contact for the qua-siclassical envelopes, φ ± ν (0), follows from the electronictransfer matrix in Eq. (3). We assume for simplicity theshort contact limit, L ≪ ξ , where ξ is the supercon-ducting coherence length, thus neglecting the energy dis-persion of the scattering amplitudes. Then it is easy toestablish that the transfer matrix for holes has the sameform as for the electrons. Thus the boundary condition connecting the left (L) and right (R) electrode wave func-tions can be written on the form, (cid:18) φ + φ − (cid:19) L = e iσ z ( ϕ/ T e (cid:18) φ + φ − (cid:19) R . (A3)Elementary solutions to a stationary BdG equation,( ± v F ˆ pσ z + ∆ σ x ) φ ± = Eφ ± , have the form for given en-ergy | E | < ∆, φ ± α ( x ) = 1 √ (cid:18) e ± iαγ/ e ∓ iαγ/ (cid:19) e − α ( ζ/ ~ v F ) x , α = ± , (A4)wherecos γ = E ∆ , sin γ = ζ ∆ , ζ = p ∆ − E . (A5)Index α is defined by the zero boundary condition at theinfinity. The matching condition, Eq. (A3) then reads, (cid:18) A + φ + A − φ − (cid:19) α = − = e iσ z ( ϕ/ T (cid:18) B + φ + B − φ − (cid:19) α =+ , (A6)where the coefficients A ± , B ± are to be determined bythis equation and the normalization condition. The solv-ability of this matching requires,cos(2 γ + β ) = R + D cos ϕ, (A7)where R = r ↑ r ↓ and D = d ↑ d ↓ play the role of effective,spin-symmetric reflection and transmission coefficients (cf. Ref. 27 where a more general form of this equationhas been derived). Introducing a phase η through therelation, cos 2 η = R + D cos ϕ , we obtain a solution forthe quantity γ , γ s = sη − β πn s , s = ± , (A8)from which the energies of the Andreev bound states arefound, E s = θ s ∆ cos( sη − β/ , θ s = sgn[sin( sη − β/ . (A9)The factor θ s = ± γ >
0, which guarantees the exponential decay ofthe bound state wave functions into the superconductingleads. To simplify the further discussion, we assume theabsence of spin selection, d ↑ = d ↓ , r ↑ = r ↓ . In this case,the relation D + R = 1 holds, and the parameter η canbe chosen as follows,sin η = √ D sin ϕ , (A10)To write down an explicit form of the Andreev levelwave functions, it is convenient to combine the envelopes,Eq. (A1), in a four-vector, ˆ φ s = ( φ + s , φ − s ), thenˆ φ s ( x >
0) = (cid:18) v s v ∗ s (cid:19) (cid:18) F s iθ s F − s (cid:19) G s ( x ) , (A11)1ˆ φ s ( x <
0) = sθ s (cid:18) − σ x σ x (cid:19) ˆ φ s ( x > , (A12)where v s = 1 √ (cid:18) e iγ s / e − iγ s / (cid:19) , (A13)and F s = r ε − s √ D cos ϕ ,G s ( x ) = r ζ s ~ v F ε e − ζ s ~ v F | x | ,ε = q − D sin ( ϕ/ . (A14) APPENDIX B: SYMMETRY RELATIONS
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