A Derivation of the Hamiltonian

Time scales for Majorana manipulation using Coulomb blockade in gate-controlled superconducting nanowires

Abstract

We numerically compute the low-energy spectrum of a gate-controlled superconducting topological nanowire segmented into two islands, each Josephson-coupled to a bulk superconductor. This device may host two pairs of Majorana bound states and could provide a platform for testing Majorana fusion rules. We analyze the crossover between (i) a charge-dominated regime utilizable for initialization and readout of Majorana bound states, (ii) a single-island regime for dominating inter-island Majorana coupling, (iii) a Josephson-plasmon regime for large coupling to the bulk superconductors, and (iv) a regime of four Majorana bound states allowing for topologically protected Majorana manipulations. From the energy spectrum, we derive conservative estimates for the time scales of a fusion-rule testing protocol proposed recently [arXiv:1511.05153]. We also analyze the steps needed for basic Majorana braiding operations in branched nanowire structures.

pacs:
71.10.Pm, 74.50.+r, 68.65.La

I Introduction

Systems with topologically nontrivial phases have become a focal point of condensed-matter research over the past decade Bernevig and Hughes (2013) and especially systems hosting Majorana bound states (MBS) Kitaev (2001) have been heavily pursued Alicea (2012); Leijnse and Flensberg (2012); Beenakker (2013); Stanescu and Tewari (2013); Elliott and Franz (unpublished); Nayak et al. (2015). Two MBS can form a fermionic mode, which can be occupied at the cost of zero energy, that is, the ground state has a fermion-parity degeneracy. This degeneracy is topologically protected against perturbations, which implies that MBS obey non-Abelian exchange statistics Rowell and Wang (2015). Hence, exchanging MBS (braiding) changes the ground-state in a nontrivial way, a key ingredient for topological quantum computation Nayak et al. (2008a); Kitaev (2003); Hassler (2014); Ivanov (2001); Nayak et al. (2008b).

Identifying a suitable platform for realizing and manipulating MBS, however, remains challenging. MBS exist only in superconductors with triplet pairing Read and Green (2000), which appears intrinsically in SrRuO Das Sarma et al. (2006) or can be induced extrinsically as a proximity effect Fu and Kane (2008). The first candidate systems for MBS were vortices of 2D triplet superconductors, in which the MBS might be manipulated through gate-voltage controlled point contacts Liang et al. (2012) or supercurrents Lee et al. (in preparation). As an arguably more feasible alternative, 1D systems have been considered Fu and Kane (2009); Sau et al. (2010); Choy et al. (2011) and among those magnetic atom chains Nadj-Perge et al. (2013) and semiconductor nanowires Lutchyn et al. (2010); Oreg et al. (2010) have been suggested and seem experimentally promising. Here, the combined effect of strong spin-orbit coupling, (proximity-induced) superconductivity, and exchange interactions or Zeeman splitting Alicea (2010) induces MBS located at the opposite ends of a topological phase region. Experiments have so far focused on probing transport signatures of MBS Rokhinson et al. (2012); Nadj-Perge et al. (2014); Finck et al. (2013), such as a zero-bias conductance peak Mourik et al. (2012); Das et al. (2012); Deng et al. (2012); Churchill et al. (2013), but they could not conclusively rule out other topologically trivial origins.

Figure 1: Segmented nanowire setup for gate-controlled fusion of MBS. The device consists of two tunnel-coupled superconducting islands (orange), formed by a nanowire in proximity to a superconductor (not shown). In addition to that, both islands are connected to nontopological bulk superconductors (blue) with a tunable coupling. In (a), when the outer valves are maximally open (largest coupling) and the central valve is closed, the device hosts four MBS at zero energy (crosses). In (b), when all junctions are closed (minimal coupling), the pairs of MBS on each island are fused (connected circles). In this situation, the device may still possess a subgap state provided (), rendering the sketched setup distinct from a conventional nontopological Cooper pair box.

Clear evidence in favor of the topological nature of MBS would instead be provided by verifying their distinctive exchange characteristics. Various approaches to braiding have been suggested, which fall into two categories: One way is to move topological phase boundaries, for example, in nanowire-based proposals through keyboard gates Alicea et al. (2011); Bonderson (2013); Clarke et al. (2011); Halperin et al. (2012) or supercurrents Romito et al. (2012). Another way is to slowly change the couplings between the MBS to adiabatically manipulate the ground state Burrello et al. (2013). This can be achieved, for example, by magnetic-field control for magnetic atom rings Li et al. (2014) or by tuning electric gates Sau et al. (2011) as well as magnetic fluxes Hyart et al. (2013) in nanowire devices. With the ongoing progress in nanowire fabrication Gül et al. (2015); Krogstrup et al. (2015); Chang et al. (2015), especially the fabrication of branched structures Dick et al. (2004); Plissard et al. (2013) essential for braiding, experiments have begun to move forward in this direction.

The successful implementation of braiding necessitates, however, also initialization and readout of MBS and this requires lifting the ground-state degeneracy — a process called fusion of MBS Nayak et al. (2008b). MBS obey nontrivial fusion rules that, in fact, imply braiding Rowell and Wang (2015) and are therefore an interesting subject in itself. Avoiding errors by tuning between the fused and degenerate regime will practically limit the operation speed of topological devices in experiments. The limitations arising from these steps may even be more restrictive than those for manipulating MBS during braiding as we specifically show for nanowire setups. Estimating such time scales, inferred from studying the behavior of the energy spectrum, is thus of the utmost importance to devise future experiments.

A viable strategy to fuse MBS controllably in nanowire setups is to form mesoscopic superconducting islands (see Fig. 1), which form the basis of various nontopological qubits Makhlin et al. (2001); Devoret et al. (2004), including Cooper pair boxes and transmon qubits Cottet (2002); Koch et al. (2007). In the topological case, the charging energy of the islands introduces an energy splitting between states of different fermion parity. This fuses the MBS as indicated in Fig. 1(b) by connected circles. By coupling the island through a junction to a bulk superconductor, the parity splitting can be made exponentially small by tuning the Josephson energy of this junction  Hassler et al. (2010). This introduces MBS at zero energy, at least up to exponential accuracy as indicated in Fig. 1(a) by crosses. The first proposals of this kind envisaged the Josephson couplings to be controlled by magnetic fluxes van Heck et al. (2012); Hyart et al. (2013), and parity readout to be accomplished by a cavity, analogous to transmon qubits Hassler et al. (2011).

A complementary, all-electrical proposal was put forward in Ref. Aasen et al. (2015). In contrast to superconducting qubits typically implemented in metallic systems Makhlin et al. (2001), this approach is based on gateable semiconductor nanowire Josephson junctions. Such junctions have been demonstrated experimentally recently for nontopological devices de Lange et al. (2015); Larsen et al. (2015) with prospects also for nontopological quantum computation Shim and Tahan (2015). This approach allows the application of experimental tools from quantum-dot experiments, including parity readout by charge sensing and charge pumping. Based on this, a sequence of stepping stones interpolating between MBS detection and quantum computing was suggested, among these the detection of MBS fusion rules and braiding Aasen et al. (2015). While braiding requires branched nanowire structures, a fusion-rule test could already be realized with a single nanowire: A prototypical device would consist of a superconducting nanowire hosting two islands in series, which are each coupled to a bulk superconductor as sketched in Fig. 1.

In this paper, we numerically compute the low-energy spectrum for this setup, extending prior studies on coupled Cooper-pair boxes Shim and Tahan (2015) by accounting for both the parity degree of freedom and the charge state of the islands. This goes beyond several other studies of MBS devices for braiding that either exclude charging energy Alicea et al. (2011); Clarke et al. (2011); Sau et al. (2011); Halperin et al. (2012); Bonderson (2013), or treat charging only effectively Hyart et al. (2013); van Heck et al. (2012).

Accounting for charging effects in the entire range between and is necessary to study the crossover between the degenerate and the fused regime both used in the proposal of Ref. Aasen et al. (2015). In the latter case, the charge stored in the superconducting Cooper pair condensate cannot be disregarded any more.

In our study, we will identify four different operating regimes for the coupled topological superconducting islands: (i) a charging-dominated regime [see Fig. 1(b)], (ii) a single-island regime, (iii) a double-island regime, and (iv) a regime of four MBS [see Fig. 1(a)]. These regimes can be divided into several subregimes depending on the “fine structure” of the energy spectrum. Mapping out these regimes is useful for the experimental characterization and needed to study the time scales for operating this device.

We further derive conservative time-scale conditions for the fusion-rule testing protocol suggested in Ref. Aasen et al. (2015). In this protocol, one changes the Majorana couplings () and the Josephson couplings (), which are indicated in Fig. 1(a), in order to tune the system through the above-mentioned regimes (i) (iv). The manipulations have to be made on a time scale that is slow in the sense that

(1)

but at the same time fast in the sense that

(2)

(We set .) The above conditions depend on the minimal and maximal values of the couplings, which have to satisfy the condition

(3)

Moreover, Eqs. (2) and (3) incorporate the superconducting gap in the nanowires and the bulk superconductors. The first criterion (1) guarantees that the evolution proceeds adiabatically, that is, transitions from the ground-state manifold into any of the excited state are suppressed. However, there is also a diabaticity condition because the ground-state degeneracy is changed during the protocol. This degeneracy is not perfect in practice (as for any realistic topological device) but a small energy splitting remains. Proceeding adiabatically with respect to this (unwanted) remaining splitting could take the system to the lowest of all energy eigenstates. This has to be avoided by proceeding fast enough so that the system state has no time to evolve within the ground-state manifold, which leads to the criterion in Eq. (2). The second condition in Eq. (2) is needed to avoid relaxation processes (tunneling of single electrons between the islands) when they are unwanted. By contrast, resetting the system by going back from (iv) to its initial state at (i) involves a charge-relaxation processes, which contributes to the period of a full cycle [not accounted for by Eq. (2)].

With these insights from the single-wire geometry, we additionally analyze basic operations on MBS in nanowire networks essential for braiding. We show that similar time-scale conditions as Eqs. (1) and (2) have to be satisfied. In the context of braiding, diabatic corrections due to a finite operation time have been discussed in the literature Cheng et al. (2011), mostly when the MBS are braided through moving domain walls Scheurer and Shnirman (2013); Karzig et al. (2013, 2015a), but also through changing their couplings Karzig et al. (2015b); Knapp et al. (2016). The latter works also suggest error correction procedures based on introducing counterdiabatic correction terms in the Hamiltonian Karzig et al. (2015b), or smoother parameter changes as well as intermediate measurements Knapp et al. (2016). The focus of our work is on the time scales that are necessary to suppress transition rates into excited states from the start.

Our study does not include the effect of quasiparticles which can lead to parity flips detrimental to the operation of the devices. Thus, should also be much smaller than quasiparticle poising times and at least for closed islands this seems to satisfied in view of recent experimental results Higginbotham et al. (2015). Our results thus apply when the low-energy spectrum lies below the superconducting gap , which is ensured by the last inequality in Eq. (3). In particular, the charging energies has to be smaller than the superconducting gap . Moreover, has to be much larger than temperature since thermal fluctuations of the charge states have to be suppressed for initialization and readout.

The paper is structured as follows: In Sec. II, we introduce our model for the double-island setup shown in Fig. 1. After briefly discussing our numerical approach in Sec. III, we analyze the numerically computed energy spectrum in Sec. IV. We find different operating regimes, which can be understood in simple limits from analytic approximations, closely resembling the behavior of coupled oscillators. Based on the energy spectrum, we discuss the time-scale conditions for the fusion-rule testing protocol in Sec. V and for basic manipulations in branched nanowire structures in Sec. VI. Finally, Sec. VII concludes and summarizes our findings. Our analysis is complemented by several appendices, in which we discuss among other things our parameter choices (App. B), extend our analysis of the parameter regimes mentioned above, e.g., for asymmetric setups (App. D), and analyze the time scales for the readout (App. E).

Ii Model: Gate-tunable coupled superconducting Majorana islands

The setup we investigate in this work is sketched in Fig. 1: It consists of a segmented superconducting nanowire, which is at both ends coupled to a bulk superconductor. The superconductivity in the nanowire is proximity induced, for example, by metal deposition Mourik et al. (2012) or an epitaxially grown shell coating the nanowire Krogstrup et al. (2015); Chang et al. (2015). The combined proximal superconductor and the nanowire form what we refer to as superconducting islands (red in Fig. 1). The coupling of these islands to the bulk superconductors and the coupling between the two islands can be controlled electrically by nearby gates. The junctions can thus be operated as valves, which are open for maximal coupling and closed for minimal coupling. When the nanowire is driven into a topologically nontrivial regime, as already mentioned in Sec. I, the islands possess an additional ground-state degeneracy associated with the MBS (see Sec. II.2).

We note that the analysis in this paper is not necessarily tied to the specific physical realization mentioned above. One might, for example, also control the Josephson energies by magnetic fluxes Hyart et al. (2013). However, it is important that all the couplings can be tuned over a wide range from to . To achieve large ratios with gate control but without perturbing the MBS, it may be experimentally useful to deviate from the geometry shown in Fig. 1 and instead form a contact from the middle of the nanowire to the ground via a second nanowire allowing gate control of the Josephson energy.

Our goal is to investigate the energy spectrum of this device at energies below , the minimum of the superconducting gaps on the island and in the bulk. We model the device by the following Hamiltonian,

(4)

which consists of three parts: The first two parts, and , describe the two individual superconducting islands and their coupling to the bulk superconductors, and the third part, , accounts for the tunnel coupling between the islands when opening the central junction. We neglect here capacitive coupling between the islands since there are indications that they are much smaller than the local charging energies Aasen et al. (2015). We have verified that small cross-capacitive couplings (much smaller than the local charging energies) do not impair our results for the operating regimes and the time scales discussed below. The reason is that capacitive couplings are relevant only when the charging energies are the dominating energy scale; and otherwise they introduce only minor corrections for the energy spectrum.

ii.1 Hamiltonian for superconducting islands

The superconducting islands are modeled in the standard way for a Cooper pair box Koch et al. (2007):

(5)

The first term incorporates the classical Coulomb interaction between the electrons on the islands,

(6)

while the second term accounts for the Josephson couplings to the bulk superconductors:

(7)

Here, the operator counts the number of excess electrons on island relative to an arbitrary offset. The electron number with the minimal energy can be tuned by nearby gates changing . The operator is conjugate to the operator of the phase difference between island and bulk:

(8)

This means that the phase operator generates changes of the charge Fu (2010),

(9)

where denotes the orthonormal number basis consisting of states with electrons on island .

One may question at this point whether Eq. (7) is an appropriate model for the Josephson energy for semiconductor nanowire junctions. Recent experiments on gatemons Larsen et al. (2015); de Lange et al. (2015) indicate that such junctions connect the island with the bulk through a few, say , channels with large transmission probability when the valve is opened. Then a different expression for the Josephson Hamiltonian involving higher harmonics in should be used as discussed in Refs. Aasen et al. (2015); de Lange et al. (2015). Moreover, the charging energy might be renormalized when the transmission amplitudes are not small, similar to Refs. Molenkamp et al. (1995); Schoeller and Schön (1994). Our model is thus strictly valid only if the transmission probabilities of all channels are small. This implies that we can study the regime only under the assumption because the Josephson couplings scale as Nazarov and Blanter (2009). However, we emphasize that all the above-mentioned effects will serve to enlarge the window for the time scales compared with our derivation, which therefore remains a useful conservative estimate.

ii.2 Majorana bound states and basis

At first sight, Eq. (5) does not seem to differ from a standard Cooper pair box for a topologically trivial superconducting island. What is different here, however, is that the fermion number can be both even and odd, i.e., we have to account for both fermion parity sectors. Introducing the fermion-parity operator,

(10)

one can project Eq. (5) onto the two subspaces according to its eigenvalues (even fermion parity) and (odd fermion parity), respectively, which yields

(11)

There are no off-diagonal blocks between even and odd parity because the Hamiltonians () conserve the fermion parity for each island 1:

(12)

In a topologically trivial superconductor one would account only for the even parity part, , and omit the odd parity part, . The reason is that an odd parity state requires an additional quasiparticle mode to be occupied, which is associated with an energy at least as large as the superconducting gap , which is outside the energy regime we are interested in here.

Majorana bound states. The topologically nontrivial superconducting islands each possess an additional fermionic mode, associated with the field operators for the left island and for the right island, respectively. When the outer valves in our device are opened and the central valve is closed, these additional modes derive from pairs of Majorana bound states (MBS) at zero energy, (), localized at the opposite ends of the wire segments Alicea (2012); Leijnse and Flensberg (2012) as sketched in Fig. 1(a). The associated self-conjugate Majorana operators, , are denoted with hats and satisfy anticommutation relations . We assume that the two Majorana wave functions on each island do not overlap in space. As a consequence, occupying the fermionic modes and is associated with zero “orbital” energy. Hence, both fermion parity sectors are accessible at low energies , in contrast to a topologically trivial superconductor. The degeneracy between the even and odd parity sectors can be lifted either by the charging energy [see Fig. 1(b)] Fu (2010) or by the tunnel coupling of the central valve, which in both cases fuses MBS.

Phase basis and Majorana operators. We give an intuitive definition of the Majorana operators first in the phase basis , which is the Fourier transform of the number basis. First considering only the left island, we define (see App. A) Fu (2010)

(13)
(14)

where the variable is continuous and denotes the occupation of fermionic mode and thus the fermion parity of the island 2. When used as a label, we add a subscript (here ) to the numbers and to denote the fermionic mode that is meant; however, in mathematical expressions should be evaluated as the numbers and . The wave functions can be represented in phase space as , which obey the periodicity condition

(15)

The action of the Majorana operators and on the wave functions can then be defined through

(16)
(17)

where . We added the phase factors here because the Majorana operators appear only in combination with them in the tunneling Hamiltonian (20) discussed below (which is all we need). Moreover, because of the phase factors , the phase-space wave functions on the right-hand side of Eqs. (16) and (17) obey automatically the boundary conditions (15). We give a derivation of the above relations in App. A.1 starting from a standard BCS description of the island.

Since we express the Hamiltonian in the number basis for our numerics, we further note the following useful relations:

(18)
(19)

Corresponding relations hold when replacing and for the right island and a full basis can be formed by tensor-product states.

ii.3 Majorana-Josephson coupling

Without tunnel coupling across the center junction, the even and odd parity sectors for each of the islands decouple. This changes with a tunnel coupling, which we model with the following Hamiltonian:

(20)

The first term is the “conventional” Cooper pair tunneling associated with a Josephson energy , which conserves the fermion parities of both islands. The second term, also known as the fractional Josephson effect Fu and Kane (2009), involves parity flips. We derive the Majorana-Josephson term in App. A.2, which also shows that the combination appears naturally since the tunnel coupling is local. The Majorana-Josephson term can be interpreted most clearly by comparing with its representation in the number basis [using Eq. (19)]:

(21)

Equations (20) and (21) together show that the transfer of single electrons across the central junction [described by Eq. (21)] leads to a transfer of charge between the islands [through the phase-dependent terms in Eq. (20)] as well as a flip of their fermion parities [through the Majorana operators in Eq. (20)].

We note that in combining Eq. (20) with the island Hamiltonians (5) in the full Hamiltonian (4), we assume that there is no phase difference across the two bulk superconductors. This is further discussed in App. B.1 and motivated mainly by the fact that any phase difference would increase the ground-state energy (and for the protocols we discuss the system should stay mostly in the ground state, at least when phase differences could be relevant).

Even though and appear as independent parameters in Eq. (4), they cannot be controlled individually in an experiment with a gate at the central junction. We estimate in App. B.2 that they are related by for typical parameters. The Josephson energy and the Majorana coupling may therefore become of comparable size only when approaches the superconducting gap . Since we assume during MBS manipulations, we will thus set in some parts of our analysis, which simplifies the considerations. However, we point out that a nonzero central Josephson coupling is not detrimental to the gate-controlled approach for manipulating MBS.

Total parity conservation. The Hamiltonian (4) conserves the total fermion parity:

(22)

This is seen from the representation (21) of the Hamiltonian in the number basis. Only the fermion parity of the individual islands may be changed by the tunneling process. In contrast to the total charge, the total parity is thus always a good quantum number 3. The coupling to the environment can break fermion parity conservation (so-called quasiparticle poisoning), which is an experimentally relevant issue. Including such processes is beyond the scope of this paper; a brief discussion of this issue can be found in Ref. Aasen et al. (2015). Basically, we expect that such processes happen on time scales long compared to those on which MBS will be operated in such devices.

ii.4 Hamiltonian in sum and difference variables

The above terms of the Hamiltonian are expressed in the phase and number operators referring to the individual islands. For our physical discussion of the energy spectra, it will be useful to express the Hamiltonian instead in the sums and differences of the phase and number operators,

(23)
(24)

which is merely a canonical transformation (up to normalization constants). These operators form again a set of canonically conjugate operator pairs:

(25)
(26)

In terms of these operators, the Hamiltonian reads

(27)

This contains the averages,

(28)

and asymmetry parameters,

(29)

as well as the sum and difference of the gatings:

(30)
(31)

This completes our discussion of the model Hamiltonian and we explain next how we diagonalize it numerically.

Iii Numerical diagonalization

We compute the spectrum of the model Hamiltonian (4) by expressing it as a matrix in the number basis based on Eqs. (6), (7), and (20) together with Eqs. (9) and (21). Since the Hamiltonian (4) conserves the total parity, both the total parity sectors can be diagonalized individually for all parameters. We restrict our calculations to the subspace of even total parity unless stated otherwise.

For the numerical diagonalization, we introduce a cutoff for the maximal electron number that we include: . This is sufficient because the charging energy acts like a parabolic potential for a “particle” that is confined in number space. In this analogy, the kinetic energy of the particle is given by the energy scale . If the potential energy exceeds the kinetic-energy scale , the contributions from the corresponding number states become exponentially small for the lowest eigenstates. Thus, one may neglect states for . Since we keep in our simulations, choosing a cutoff already yields the low-energy spectrum with high accuracy.

Diagonalizing the Hamiltonian in the number basis turns out to be much more convenient in the presence of charging energies as compared to diagonalizing it in the phase basis as we further substantiate in App. C.

Iv Low-energy spectrum

We next discuss the low-energy spectrum of the coupled topological superconducting islands. In addition to providing an overview of the different operating regimes, this is also of experimental interest: A thorough characterization of such devices will be needed as a preparation step before the protocols presented in Ref. Aasen et al. (2015) can be implemented experimentally.

To keep the analysis simple in this Section, we neglect the Josephson energy of the central junction () and assume symmetric islands ( and ). We dedicate App. D to investigating effects of deviating from these assumptions. Analyzing the spectrum under the simplifying assumptions has the advantage that it provides a conservative estimate of the time scales for operating MBS in these devices (see Sec. V).

The characteristics of the spectrum can be divided into four different parameter regimes as illustrated in Fig. 2: (i) If the charging energy dominates all other energy scales, , the system behaves as two closed, uncoupled islands (green in Fig. 2, see Sec. IV.1). The islands host a definite number of electrons except for charge-degeneracy points, depending on the gatings and . (ii) If the Majorana tunneling dominates, , the system behaves as a single, larger island at low energies (orange in Fig. 2, see Sec. IV.2). The superconducting phases of both islands are then locked to each other. At larger energies , the dynamics of their phase difference has to be taken into account.

(iii) If instead the Josephson coupling dominates, , the system has to be treated rather as two separate open islands (blue in Fig. 2, see Sec. IV.3). Here, the phases of the islands have to be treated as individual degrees of freedom even at low energies. (iv) Finally, there is the four-MBS regime (yellow in Fig. 2, see Sec. IV.4), which appears for two open uncoupled islands. Here, both the tunnel coupling of the MBS across the center junction and the charging-mediated couplings of the MBS on each island are strongly suppressed.

Figure 2: Sketch of the parameter regimes for the energy spectra characteristics of the coupled topological superconducting island Hamiltonian (4). In the boxes, we sketch the energy level spectrum and indicate the different energy scales dominating the level structure. The lower part of the boxes shows the couplings (lines) between the MBS, which are fused when denoted as circles and at close to zero energy when denoted as crosses. We assume symmetric islands, , , and have set the central Josephson coupling to zero, . The white arrows indicate the paths taken in the plane for the plots in Fig. 3

. Representative plots of the energy spectra along paths in the plane shown in Fig. 2 are shown in Fig. 3. The color of the horizontal axes in Fig. 3 corresponds to the regimes shown in Fig. 2. From Fig. 3  it is clear that the dashed lines in Fig. 2, marking the boundaries of the different regimes, should not be understood as lines of a “phase transition”. The transition from one regime to the other is gradual and may even be shifted for higher-lying excited states. Figure 2 should thus be understood rather as a rough guide for the characteristics of the spectra. We next discuss the different regimes in detail.

iv.1 Two closed uncoupled islands:

When the charging energy dominates, the eigenstates are close to the number states except at degeneracy points where . This results in a charge-stability diagram similar to nonsuperconducting double-dot devices van der Wiel et al. (2003) but with the constraint that the total parity is conserved (parity-switching processes are not considered here). The eigenenergies are roughly given by , which approximately reproduces the low-energy spectra shown in the left (green) parts of Figs. 3(a) and (d) for .

The MBS and are fused in this regime, which is a viable way to initialize and readout MBS (see Sec. V). Away from degeneracy points, the eigenstates of different fermion parity possess a different charge – parity and charge are not independent.

Figure 3: Low-energy spectrum of coupled topological superconducting islands. Solid lines are the numerically computed energy splittings (blue) between succeeding excited states (energy ) and the ground state (energy ) in the even total parity sector. Dashed lines indicate analytic approximations to the energy gaps (see below). The parameters are changed along the paths in the plane as denoted by the white arrows in Fig. 2 and the colors on the horizontal axis correspond to the regimes in Fig. 2. In (a), we show the spectrum as a function of a the Majorana coupling for . We indicate in darker color those states with zero excess charge, , while the bright color corresponds to other excess excess charge, . The red-dashed lines indicate the analytic approximations from Eq. (34). In (b)–(d), we show the spectra as a function of the bulk Josephson coupling for different values of as indicated. Here, the darker color highlights the 12 lowest eigenstates and the bright color corresponds to higher-lying states. The dashed lines indicate the transition energies related to Josephson plasma oscillations (red), the parity splitting , Eq. (37) (yellow), and the Majorana splitting (green). We assume , , , , and use a number-state cutoff for the numerical calculations.

iv.2 Single island:

When the Majorana coupling between the islands is the largest energy scale, the physics can be understood most clearly from the representation (27) of the Hamiltonian in the sum and difference variables. Equation (27) can here be interpreted analogous to a pair of strongly coupled oscillators, identifying the phases with positions and the number of electrons with momentum. For , the dynamics of the “relative coordinate” is fast since it is subject to a strong confining potential, while the dynamics of the “center coordinate” is slow since it is much more weakly confined.

To understand the energy spectrum, we first decompose the Hamiltonian into with

(32)

On a rough energy scale, the spectrum is determined by , while can be treated as a perturbation for . First setting , the Hamiltonian is formally the same as that of a Cooper pair box, with the physical difference that the Majorana term couples consecutive electron number states on the islands instead of consecutive Cooper pair number states 4. The eigenenergies are approximately given by

(34)

with . We obtained this result by expanding in and including anharmonic corrections perturbatively along the lines of Ref. Koch et al. (2007). The resulting lowest-order energy gaps of to the ground state are indicated as red-dashed lines in Fig. 3(a). We note that the energy levels are altered when the central Josephson energy is included, which we discuss further in App. D.1.

We next discuss the effect of the “fine-structure” term , which gives rise to smaller energy splittings than those induced by . Depending on the bulk Josephson coupling , we find two subregimes, which are marked in Fig. 2 in two shades of orange.

(i) Closed islands: . Let us first consider the simplest case of , the situation shown in the orange part of Fig. 3(a): Here, the total number of electrons on the islands is conserved, , and can thus be treated as a number . In Fig. 3(a), we highlight the energy gaps of all states with zero excess charge, , labeled by in darker color (corresponding to ). The full energy spectrum including all other states with (even) is then obtained as replicas of the spectrum for by adding the charging energies . The corresponding energy differences to the total ground state are shown in Fig. 3(a) in brighter color. Among those, we see a sequence of constant energy gaps. These correspond to the energetically lowest states () for the different values of . Correspondingly shifted “replicas” of the states with excited oscillation quanta can also be identified.

(ii) Open islands: . Treating the “fine-structure” term (LABEL:eq:finestructure) as a perturbation, we replace in leading order by its average:

(35)

and the eigenenergies are with

(36)

Here, and . Thus, each is associated with a ladder of states and each of these ladder states is specified by . The different ladders are spaced by the large energy and the states within each ladder are spaced by an energy . Note that for , anharmonic corrections to can be neglected as compared to the latter splitting.

The crossover from regime (i) to (ii) is indicated by the transition from the dark orange to the light orange part in Fig. 3(b). If we focus on low energies , all states correspond to , i.e., there are no excitations corresponding to oscillations in the difference phase involved. This physically means that both phases are rigidly coupled at low energies and the two islands behave as one.

In view of the Majorana physics, this means that the “nonlocal” parity , indicating that the inter-island fermionic mode, associated with the annihilator , is empty. The MBS pair is thus fused – the occupation of requires a finite energy , which is much larger than all other energy scales. For low energies, the fermion parity degree of freedom is gapped out (recall that the spectra are shown here only for even total parity).

iv.3 Two open islands:

We next turn to the more intricate case when the Josephson energies of the junctions with the bulk superconductors dominate. Here, and are not rigidly coupled to each other at low energies (as in the foregoing section) and their individual dynamics becomes important. The low-energy spectrum depends crucially on how the Majorana coupling energy compares with (i) the Josephson plasma frequency and (ii) the charging-induced energy splitting of even and odd fermion parity states of the individual islands. The corresponding energy scale van Heck et al. (2012) is

(37)

which is much smaller than . This yields three possibilities, which are shown in Fig. 2 as three shades of blue and which we discuss next.

(i) Weak inter-island coupling: : In this regime, the system behaves as two weakly coupled topological superconducting islands. To discuss the physics, let us first set so that simply decomposes into the two island parts. The eigenstates of the system are trivially the product states

(38)
(39)

Here, are the excitations of the Josephson plasma oscillations of each island with excited quanta and parity of the fermionic mode . The total energies are approximated by

(40)

On a rough energy scale, the spectrum can be grouped into pairs of -fold degenerate states, split by the Josephson plasma frequency (see Fig. 2). The corresponding splittings are denoted by red-dashed lines in the right blue parts of Figs. 3(c) and (d) with . These are valid not only for the weak inter-island coupling, , but also for the other subregimes discussed in this section.

Within each of the “parity pairs”, the odd-odd parity combination is split from the lower even-even parity combination 5 by the smaller energy (we focus now on the case ). For the two lowest lying states with , the excited state is therefore split from the ground state by as given by Eq. (37), which is indicated by the yellow-dashed line in the dark blue part of Fig. 3(d). We have verified that a nonzero capacitive coupling between the islands, modeled by a term (, influences this parity energy splitting only slightly and does not affect the exponential suppression. We also neglected anharmonic corrections of to the Josephson-plasma frequency in Eq. (40) because these only shift levels of the same by the same amount but do not contribute their splitting. Compared to the rough energy scale , these anharmonic corrections are negligible.

When , the Majorana tunneling has little effect on the spectrum. For the lowest parity pair, a nonzero results in a small shift of the energy levels. This is different for the higher-lying excited states, which exhibit degeneracies that may be lifted for nonzero (not resolved on the scale shown in Fig. 3).

(ii) Intermediate inter-island coupling: . When the Majorana coupling exceeds the charging-induced parity splitting, the inter-island tunneling strongly mixes the even-even and odd-odd parity sectors. The energy eigenstates are therefore “bonding” and “antibonding” combinations of the “local” parity states:

(41)
(42)

split by an energy [green dashed in Figs. 3(c) and (d)]. The crossover from regime (i) to (ii) can be clearly seen for the lowest-lying excited state in Fig. 3(d) from the dark blue to the lighter blue part. As our notation in Eqs. (41) and (42) suggests, increasing the tunnel coupling fuses the MBS in a complementary way: The MBS at the central junction become more strongly fused than the pairs on each island. This is a key ingredient to test the Majorana fusion rules as discussed in Sec. V.

(iii) Strong inter-island coupling: . In this regime, the tunnel coupling between the two islands is not a “fine-structure” effect: Roughly speaking, the lower end of the spectrum is given by bonding states of the two islands, while the upper end of the spectrum is given by antibonding states (see Fig. 2). This effectively removes the parity degree of freedom from the low-energy spectrum, similar to the single-island regime. Accordingly, the Josephson plasmon excitations do not appear in parity pairs.

This can be seen in Fig. 3(c) in the light-blue part: Here, the lowest state of flipped nonlocal parity, labeled by , is at a much higher energy than other states with the same nonlocal parity as the ground state. This contrasts with the situation for the darker blue parts shown in the Fig. 3(c) and in Fig. 3(d), where the state is closest to the ground state.

Finally, we emphasize that there is still a difference between the regime of strongly coupled individual islands () and the regime of a single island (). In the former case and are not locked to each other and one may still use , as independent quantum numbers to give a rough construction of the low-energy spectrum. In contrast to the single-island regime, the oscillator levels are degenerate here [compare Eq. (36) and Eq. (40)]. It is only the parity degree of freedom that is “gapped out” in both regimes.

iv.4 Regime of four zero-energy MBS

The system hosts four zero-energy MBS ,.., as sketched Fig. 1 when both and become negligibly small. Equation (37) shows that the charging-mediated energy splitting can become exponentially small in . The lowest-energy parity states and () are then degenerate up to exponential accuracy. Including also states of odd total parity, the ground state becomes four-fold degenerate and is spanned by ().

V Time scales for gate-controlled fusion-rule testing protocol

The segmented nanowire structure shown in Fig. 1 has recently been proposed as en experimental testbed for Majorana physics that could be realized in the near future Aasen et al. (2015). MBS may be manipulated by opening and closing junctions through gate control. In this Section, we derive the time-scale conditions (1) and (2) stated in Sec. I, which are required to perform the fusion-rule test suggested in Ref. Aasen et al. (2015).

The fusion rules of MBS can lead to nontrivial parity correlations by fusing four MBS in complementary pairs. This is rooted in the nonlocal character of the MBS and not possible for local fermions. To prepare and probe such parity correlations, one goes through the steps sketched in Fig. 4(a), changing the parameters along the path in the plane as shown in Fig. 4(b). The starting point (A) is to initialize the system in the ground state when the MBS are fused in pairs (,) and (,). This corresponds to a superposition of even-even and odd-odd fermion parities in the complementary pairs (,) and (,). To detect these parity correlations, one first forms all four zero-energy MBS (C) and and then converts them into charge states for the islands (D). Subsequent charge detection then probes the prepared parity correlations. To repeat the experiment by going back from point D to A, a resetting step is needed, in which the system has to relax into the ground state again.

We now go step by step through the protocol and verify the time-scale criteria (1) and (2). Our considerations here concern the cycle sketched in Fig. 4; the time-scale conditions on the readout are discussed in App. E.

Figure 4: Protocol for testing the Majorana fusion rules. The steps of the protocol are sketched in (a) and the corresponding path taken in the parameter space in (b) (compare with Fig. 2). White arrows indicate that these processes have to be adiabatic, while the orange arrow indicates a diabatic step, in which the system should not follow the ground state evolution. The yellow arrow indicated the resetting step, which should be done on the time scale of the charge relaxation to the ground state. The color scale gives the ground-state expectation value of [see Eq. (46)], indicating the preferred parity combination of the ground state. In the charge-dominated regime (), the Majorana pairs () and () are fused [], while in the tunneling-dominated regime (), the Majoranas () are fused []. To bring all the parity states of the two islands as close to zero energy as possible, the path should cover the Majorana regime ( and should be, in principle, as large as possible). We use , , , , and .

v.1 Time-scale conditions

To derive the time-scale conditions for each step of the fusion-rule protocol, one has to demand the following adiabaticity condition Schiff (1968):

(43)

with

(44)

Here, labels all excited states at energy  and 0 labels the ground state at energy [except for step 2, in which the first excited state has to be excluded from condition (43)].

Our first goal is to turn Eq. (43) into a condition for the entire time interval when changing the parameters, labeled in the following by , from at time to at time . In our case, the parameters are given by . The optimal way to change the parameters is to keep the function constant during the parameter sweep because that minimizes the sweeping time for a given value of . We can then turn Eq. (43) into the following condition by integrating over time:

(45)

Here, is the path in the parameter space connecting and . We emphasize that Eq. (43) is sufficient only if is constant (in practice remaining of the same order of magnitude), which might correspond to a rather complicated time dependence for . This means that one has to know the properties of the system quite well to design the best gate pulse (in that respect the estimate (45) is optimistic).

Our estimates for the energy gaps are based on the numerically computed energy spectrum of Hamiltonian (4) and the analytic approximations worked out in Sec. IV. The gaps to the lowest and further selected excited states along the path shown in Fig. 4 are shown in Fig. 5. We motivate our parameter choices for Figs. 4 and 5 in App. B.3. In the following, we work out Eq. (45) only for the lowest accessible excited state and show in App. F that the time scale derived from this is not modified if the effect of transitions into the entire spectrum of higher-lying excited states is included.

Provided optimal pulsing shapes can be achieved, our time-scale estimates are conservative in four respects: (i) We implement the cosine approximation for the Josephson energies in Eq. (4). As we explained in the Appendix of Ref. Aasen et al. (2015), we expect corrections due to higher harmonics to enhance energy gaps between the ground state(s) and the excited states and to reduce splittings within the ideally degenerate ground-state manifold. The time-scale window we estimate here is therefore narrower than what one could expect including these corrections. (ii) While our numerical results in Figs. 4 and 5 include a nonzero Josephson energy for the central junction, we use for our analytic estimates below. Since nonzero also enhances the relevant gaps, this again tends to underestimate the actual time-scale window. (iii) For the estimates, we take the islands to be symmetric, , and . In practice, they will be asymmetric and therefore the energy gaps associated with one of the islands will be less constraining for the time scales than those associated with the other island. (iv) We estimate all matrix elements in Eq. (45) by maximal values (if nonzero) even though they can be smaller in practice. We next go through each of the steps of the protocol sketched in Fig. 4 in detail.

Figure 5: Energy gap between the ground state and first excited state for even total parity (in some intervals the gap to other excited states is shown as well). The parameters (and accordingly ), as well as on the horizontal axis are changed logarithmically along the path marked in Fig. 4(b). The solid lines correspond to the numerically computed splittings while the dashed lines are the approximation formulas for the splittings (see text). Gray arrows indicate allowed transitions and we mark the lowest accessible excited state from the ground state in blue. In several steps transitions into the lowest excited states (red, green) are prohibited by parity or charge conservation as explained next: In step 2, the lowest excited state is decoupled from the ground state for a large energy range because the tunneling Hamiltonian conserves the nonlocal parities , . However, when becomes dominant, the parity character of the ground state changes to . Thus, when approaches , transitions to the lowest excited state become possible. By tuning fast the system stays in the desired state , i.e., the system can be both in the ground and first excited state after step 2. Since the islands are decoupled then, the local parities are conserved in step 3 and transitions from even-even to odd-odd parity or vice versa are prohibited. From the ground state, one can therefore only reach the third-lowest excited state [blue, parity ]. In addition, transitions from the first to the second excited state are possible [red, parity ], which determines the adiabaticity condition. Finally, in step 4, the islands host a different total charge in the lowest excited state, while transitions are only possible into the lowest state ( is the total number of electrons). The parameters are , , (see text), and the number-state cutoff is . Note that we chose a rather large value for to be consistent with the assumption for our numerical approach. We expect that the protocol should also work for smaller values of as we further explain in App. B.

v.2 Initial point

We start at point A in the parameter space in Fig. 4(b). Here, the central valve is open , while the valves to the bulk superconductors are closed