Majorana oscillations and parity crossings in semiconductor-nanowire-based transmon qubits

We show that the microwave (MW) spectra in semiconductor-nanowire-based transmon qubits provide a strong signature of the presence of Majorana bound states in the junction. This occurs as an external magnetic field tunes the wire into the topological regime and the energy splitting of the emergent Majorana modes oscillates around zero energy owing to spatial overlap in finite-length wires. In particular, we discuss how the zero-energy fermion parity crossings arising from Majorana oscillations result in distinct spectroscopic features. In split-junction geometries, the plasma mode couples to the phase-dispersing subgap levels resulting from Majorana hybridization via a Jaynes-Cummings-like interaction. As a consequence of this interaction, higher order plasma excitations in the junction inherit Majorana properties, including the $4\pi$ effect. Our results, based on a fully microscopic description of the junction, suggest that MW spectroscopy of nanowire-based transmon qubits provides an interesting alternative to Majorana detection by transport spectroscopy.

We here focus on a specific proposal where the JJ is based on a semiconducting nanowire (NW) that can be driven into a topological superconductor phase by means of an external Zeeman field B [27 -30], see Fig. 1 (a).In this topological regime, the JJ contains MBSs which coherently interact with the superconducting island degrees of freedom.While previous theoretical studies of this problem have focused on either effective low energy toy-models [9,10] or partial microscopic descriptions [16], our work contains a fully microscopic description, which allows to unveil new physics from the E J /E C 1 Cooper pair box (CPB) to the E J E C transmon regimes.This includes the magnetic field dependence of NW junction parameters and the emergence of MBSs in the topological phase.Importantly, the corresponding MW spectroscopy presents signatures that fully map parity crossings in the NW spectrum owing to the oscillatory energy splitting of overlapping Majoranas, hence providing a powerful tool for Majorana detection.In split junctions, the resulting physics generalizes that of the Jaynes-Cummings model, with multiple replicas of the plasma mode reflecting the 4π effect and Majorana oscillations.
Model -The Josephson potential is defined, on a microscopic level, as the operator are Nambu spinors and H BdG is the Bogoliubov-de Gennes (BdG) Hamiltonian and where H NW is the normal NW junction Hamiltonian.It consists of two segments (left/right) with normal Hamiltonians H L/R , coupled across a short weak link of transparency T N ∈ [0, 1].Each segment contains all the microscopic NW details (Rashba coupling α, Zeeman field B and chemical potential µ) and is described by a single-band model momentum operator and σ i Pauli matrices in spin space).These NWs undergo a topological phase transition at B c ≡ ∆ 2 + µ 2 with the appearance of MBSs at their edges, see blowup in Fig. 1 (a).∆(x, ϕ) = iσ y ∆e ±iϕ/2 (where the ± corresponds to x ∈ L/R, respectively) is the induced pairing term [31,32].While V J (ϕ) ∼ −E J cos(ϕ) is a good approximation at B = 0 in the T N → 0 tunneling limit, it can strongly deviate from this form under relevant conditions (finite T N , finite B, etc).This deviation is reflected in important qubit parameters such as the Josephson inductance and the anharmonicity [1,19,32].
Our goal is to derive a quantitatively precise but simple low-energy approximation for V J , starting from its Sketch and spectrum of a NW-based superconducting qubit.(a) Sketch of the simplified transmon/CPB qubit circuit, with a split-junction variation on the right.EJ is implicit in the Josephson potential VJ (ϕ) of the nanowire junction (in red), while the combination of a shunting capacitor CJ and the gate capacitance Cg define the charging energy EC = e 2 /2(CJ +Cg).γi represent Majorana bound states.(b-i) Blue/orange colours denote fermionic even/odd parities, while intermediate gradient signals parity mixing (see text).Panel (b): spectrum versus ng = Vg/(2eCg) in the CPB limit (EJ /EC = 0.5, EM /EC ∼ 0.12) for B = 0 (dashed lines) and B = 1.25Bc corresponding to δ = 0 (coloured).Panel (c): same as (b) but in a transmon regime with EJ /EC = 25, EM /EC ∼ 6.5.(d) and (e) show the same CPB and transmon cases but for δ = 0 (B = 1.4Bc).Panels (f,g): Magnetic field dependence for the CPB in (d), at ng = 0.25 and ng = 0.5, respectively.Panels (h,i): same as (f,g) but for the transmon (arrows mark parity crossings).Rest of parameters: LS = 2.2µm, τ = 0.8, µ = 0.5meV, ∆ = 0.25meV.microscopic form, that retains both standard Josephson events due to Cooper pair tunneling, as well as anomalous Majorana-mediated events where a single electron is transferred across the junction.While the total fermion parity n mod 2 is conserved (where n = n L +n R and n L,R are the fermion occupations in the left/right segment of the junction), an anomalous tunneling term changes the parity n L/R mod 2 on each superconducting left/right segment.For instance, assuming even global parity, Majorana-mediated tunnelling corresponds to the process (In what follows, even/odd parity will always refer to the partial fermion parity n L mod 2 = n R mod 2).Physically, this suggests that to compute an effective lowenergy BdG .The first one takes into account the bulk contribution of BdG levels above the gap, whose occupation is assumed in thermal equilibrium, V bulk J (ϕ) = − p >∆ p (ϕ).The second contribution corresponds to the subgap sector.By projecting H BdG onto this subspace, one gets This effective form describes four subgap states (two electron-hole copies of two spin-resolved subgap states) in terms of four Majorana operators, γ 1,2 ∈ L and γ 3,4 ∈ R, interacting pairwise through the λ ij terms [33,34].
Projection method-To derive the different λ ij from the microscopic model itself we first calculate the Nambu spinors ψ0 is of the empty/full (i = −, +) lowest subgap eigenstate of each decoupled s = L, R segments.These states span a fermion basis {c L , c † L , c R , c † R } of our 4-dimensional projection space, in terms of which we can write fermion numbers as . We now perform a low-energy projection of the form: where G(ω) = (ω + iε − H BdG ) −1 is the resolvent of the full BdG Hamiltonian.Using this projection, Eq. ( 2) can be written as H sub BdG = 1 2 ψ0 † H ψ0 , which yields the different λ ij .After projecting onto the parity basis {|p } = {|0 , |1 }, the final effective Hamiltonian reads It describes two different parity copies of a superconducting island, which are mixed through a non-diagonal term The parity content of the eigenstates of Eq. ( 4) can be calculated by a projection onto the parity axis defined by τz ≡ |0 0| − |1 1|.
We emphasize that, despite the superficial similarity with the effective low-energy model in Refs.[9,10], Eq. ( 4) is derived by projecting the fully microscopic H BdG .Thus, H is able to describe, in particular, the evolution of the junction from trivial to topological.This physics is captured by the three B-field-dependent microscopic energy scales [32] relevant for this problem: the Josephson coupling J 00 (ϕ) cos(ϕ), the energy splitting between different fermionic parities δ = V sub J 11 (ϕ)−V sub J 00 (ϕ)| ϕ=0 and the Majorana contribution to the Josephson coupling E M = 2π 0 dϕ π V sub J 01 (ϕ) cos(ϕ).Results-In what follows, the ratio E J /E C and the plasma frequency ω pl ≡ √ 8E J E C / are defined respect to the zero-field junction, for which a microscopic calculation of I c [31,35,36] gives E J ≡ I c (B = 0)/2e at fixed E C .The fermionic parity content is represented by different colours (even/odd=blue/orange for τz = ±1).In the CPB limit (E J /E C = 0.5) each even (odd) parabola in the spectrum, see Fig. 1 (b), has a minimum on n g = m + n 0 g , where m ∈ Z and n 0 g = 0 (0.5) (only the m = 0, 1 cases are shown).For B = 0 (dashed lines), odd parabolae are energy shifted from even ones by exactly δ = 2∆ (outside plot range in Fig. 1 (b)), since in the state |n L = 1, n R = 1 each fermion must overcome an energy gap ∆ in the left/right segment of the wire.As expected, the spectrum is 2e-periodic.Increasing B, the gap gets reduced until an odd ground state (GS) around n g = 0.5 is possible when δ < E C .In the δ → 0 limit, both parity sectors have minima at zero energy and the periodicity becomes e [37,38], Fig. 1 (b).Furthermore, both sectors are coherently mixed around n g = 0.25 and n g = 0.75 (lighter regions) due to MBSs [9,10,16].In the transmon limit, Fig. 1 (c), the presence of MBSs manifests as splitting of the transmon lines which show strong parity mixing for all n g .
The above phenomenology depends on the ratio ξ M /L S , with ξ M the Majorana coherence length (which depends on B [39]) and L S the NW length, which in turn governs the energy splitting δ [40].Figures 1 (d,e) illustrate this effect where we plot the same spectra but at a different B corresponding to δ = 0. Interestingly, the spectrum is now 2e-periodic for both the CPB and transmon limits.In this latter case, Fig. 1 (e), the overall n g dependence differs considerably from a standard transmon [compare with Fig. 1 (c)].Note also that the (odd) parity of the GS is well defined for all n g .We now focus on the magnetic field evolution of the island spectrum.In the CPB regime, this evolution strongly depends on gate (owing to the large charge dispersion).Since parity mixing occurs near n g = 0.25, the B-field dependence at this gate considerably differs from the one at n g = 0.5, where parity mixing is negligible, compare Figs. 1 (f) and (g).In the transmon limit, the spectrum mimics Majorana oscillations and shows alternating GS parities (change of colour from blue to orange and back) after each parity crossing [see arrows in Fig. 1  (h,i)].This behaviour occurs for all n g .These parity transitions in the GS have important consequences for MW spectroscopy, as we shall discuss next.Before that, we just mention that these switches of GS parity are possible since, for the single band case considered here, E M is non-negligible as compared to E J [31,32,35,36,41].Consequently, in a transmon regime with E J /E C 1, the ratio E M /E C is not small.Considering typical crit-ical current values I c ∼ 0.2I 0 , with I 0 = e∆/ the maximum supercurrent of a single open channel, this gives E J ∼ 0.1∆ ∼ 25µeV∼ 6GHz, assuming an induced gap of the order ∆ ∼ 250µeV.A conservative estimate E M ∼ 0.5E J ∼ 0.05∆ yields E M ∼ 12.5µeV∼ 3GHz, which compared to typical charging energies in the transmon regime E C ∼ 300MHz [17][18][19] indeed gives E M /E C ∼ 10.This relevant regime has hitherto remained unexplored, even at the level of the effective models in Refs.[9,10,16] which focus on the opposite E M /E C 1 regime.This would require much larger charging energies (which is detrimental for the transmon since they induce charge dispersion) or Majorana couplings well below the above estimation.The latter can be, however, somewhat difficult to reach in a few-channel topological wire: while more than one channel can contribute to E J [19,42,43] , the value of E M is in turn governed by the topological minigap from a single-channel Majorana Josephson effect.In this few channel situation, the above estimation of E M /E C would be reduced by a factor ∼ 1/N , with N the number of trivial channels contributing to E J [32].
MW spectroscopy-By considering a capacitive coupling to a small periodic perturbation in the gate voltage, the resulting linear-response MW absorption spectrum reads , where Ψ n are the eigenstates of Eq. ( 4) and In what follows we focus on the transmon limit with E M /E C 1, realized by a single band nanowire in the topological regime.A full discussion about other relevant parameter regimes, including the CPB and the transmon regime with E M /E C 1, can be found in Ref. [32].In Fig. 2 (a) we plot the magnetic field dependence of the MW spectrum in the transmon regime for n g = 0.5.Majorana oscillations and parity switches in Fig. 1 (i) are found to result in abrupt spectroscopic changes at B fields where Majorana oscillations have nodes.They include spectral holes in the first transition (corresponding to a low-energy transition owing to Majorana-mediated parity mixing) accompanied by higher transition lines suddenly disappearing/appearing.The spectral holes can be understood as exact cancellations of spectral weight owing to parity degeneracy at crossings.This is illustrated in Fig. 2 (b) where we plot the transition frequencies weighted by their respective matrix element (represented as the width of the line).Together with the absence of spectral weight of the ω 01 transition at crossings, there is a complete spectral weight transfer between higher energy transitions, where one thick line becomes thin or viceversa (see arrows).Figs. 2 (c-e) show the n g dependence for three fixed B fields [coloured bars in (a)] across a parity crossing.Interestingly, all the spectroscopic features before and after the crossing [(c) and (e), respectively] are shifted exactly by one e unit, which reflects the change of parity of the GS.See also the transitions weighted by their matrix elements in Figs. 2 (f-h).This results in distinct spectroscopic features like spectral holes that shift from half-integer to integer values of n g and viceversa, and changes of curvatures of the involved transitions.Right at the δ = 0 parity crossing, (d,g), we recover a standard transmon spectrum.Such unique behaviour of transmon spectra across a Majorana oscillation should provide a strong signature of the presence of MBSs and their associated parity crossings.
Split junction geometry-We next consider a splitjunction where a standard ancillary JJ in parallel with the NW JJ forms a loop which is threaded by an external flux Φ, see Fig. 1 (a) right panel.The Josephson potential is then split into two terms −E L cos( φ) + V J (φ − φ), where φ ≡ 2πΦ/Φ 0 with Φ 0 = h/2e the flux quantum.When E L E C , fluctuations of φ are small and centered around zero, while the external phase mostly drops over the NW JJ.In this limit, the dependence on n g is irrelevant and one is left with an effective LC harmonic oscillator with the phase-dispersing levels of the NW JJ through an inductive term [16,44].In practice, we calculate the current operator in the eigenbasis that diagonalizes the NW JJ effective Hamiltonian, Î = ∂V J (ϕ) ∂ϕ = ∂ ∂ϕ ( k E k |k k|).In this configuration, the visibility of the allowed MW transitions is just given by the matrix elements n| Î|0 .The above model generalizes the Jaynes-Cummings model that results from considering only one Andreev level in the junction [45][46][47][48].Indeed, the topologically trivial B < B c regime captures this Jaynes-Cummings physics where an Andreev level strongly dispersing with phase E A (φ) anticrosses with the plasma mode [Figs.3 (a,b)], in good agreement with previous experiments [47][48][49].When B > B c , the levels show a characteristic phase dispersion with a zero-energy crossing at φ = π (the socalled 4π effect), see Figs. 3 (c,d).Apart from the fundamental transition, the MW response shows higher order processes where m plasma modes are excited, which results in transitions occurring at mω pl and E A (φ) + mω pl [49], see Figs. 3 (c,d).Interestingly, the residual splittings at φ = π owing to Majorana overlaps [33,34] are imprinted on each of these replicas, which, near φ = π, mimic the oscillatory Majorana behavior as a function of B [Figs. 3 (e,f)].As before, minima of the Majorana oscillations result in spectral holes.
In summary, our results demonstrate that MW spectroscopy of NW-based transmon qubits is a powerful tool to detect the presence of MBSs in the JJ junction, including Majorana oscillations, parity crossings and the 4π effect.This provides a detection scheme alternative to tunneling spectroscopy [50].Our projection method can be readily extended to other relevant NW regimes not discussed here, like multiband NWs [36] or other qubit regimes [32].Other geometries currently under intensive experimental study, including junctions with quantum dots [51,52] and superconducting islands in the fluxonium regime [53], should be the subject of future studies.

FIG. 2 .
FIG. 2. MW spectroscopy of a NW-based transmon.Panel (a): contour plot of MW absorption spectrum SN (ω) versus ω and B/Bc at ng = 0.5.Bright lines signal allowed transitions in the MW response.Spectral holes in the n = 1 transition line and abrupt jumps in the n > 1 ones, where lines suddenly disappear/appear, coincide with parity crossings in the GS owing to Majorana oscillations.(b): Transition frequencies and spectral weights (shadowed widths).At minima of the oscillations (arrows), the spectral weight of different transitions gets exchanged.Panels (c-h): ng dependence.Same parameters as the transmon in Fig. 1.

FIG. 3 .
FIG. 3. MW spectroscopy in a split-junction geometry with EL/EJ = 60.NW parameters as in Fig. 2. Panel (a): Phase dispersion of the transition frequencies and their spectral weights (widths of the lines) in the trivial regime B = 0.33Bc.(b) Blowup of the corresponding MW spectrum showing the avoided crossing between the plasma mode and the Andreev level in the NW junction.(c) and (d): Same in the topological regime B = 1.25Bc showing plasma replicas of the underlying 4π effect in the NW junction.(e) and (f): Magnetic field dependence near φ = π.Panels (g) and (h) show a blowup near the first minimum at B ≈ 1.25Bc.