Nonlocality of Majorana bound states revealed by electron waiting times in a topological Andreev interferometer

The analysis of waiting times of electron transfers has recently become experimentally accessible owing to advances in noninvasive probes working in the short-time regime. We study electron waiting times in a topological Andreev interferometer: a superconducting loop with controllable phase difference connected to a quantum spin Hall edge, where the edge state helicity enables the transfer of electrons and holes into separate leads, with transmission controlled by the loop's phase difference $\phi$. This setup features gapless Majorana bound states at $\phi=\pi$. The waiting times for electron transfers across the junction are sensitive to the presence of the gapless states, but are uncorrelated for all $\phi$. By contrast, at $\phi=\pi$ the waiting times of Andreev-scattered holes show a strong correlation and the crossed (hole-electron) distributions feature a unique behavior. Both effects exclusively result from the nonlocal properties of Majorana bound states. Consequently, electron waiting times and their correlations could circumvent some of the challenges for detecting topological superconductivity and Majorana states beyond conductance signatures.

Fluctuations in electron transport can greatly impact the performance of electronic circuits but, at the same time, provide us with invaluable information about the quantum-coherent behavior of conductors [1].Charge fluctuations are often analyzed by full counting statistics [2][3][4], which usually concerns the zero-frequency or long-time limit.However, experimental advances with noninvasive probes [5][6][7] have now enabled access to the short-time regime with almost single-event resolution.
In this Letter, we analyze the potential of electron WTDs and their correlations for identifying topological superconductors hosting Majorana bound states (MBSs) [23][24][25][26].There is currently an intense research activity focused on obtaining reliable signatures of MBSs, since the simplest one, a robust and quantized zero-bias conductance peak [27][28][29], has proven insufficient [30,31].Electron waiting times in superconducting hybrid junctions have already been proposed to characterize the entanglement between the electrons in Cooper pairs [32][33][34] and to detect the presence of MBSs [35][36][37][38][39].These theo-retical proposals extended the concept of waiting times to both spin and electron-hole degrees of freedom, but they have still primarily only focused on the local properties of MBSs [35][36][37][38][39]. Instead, we here suggest a Majorana platform without magnetic materials that is both conceptually simple and presents important advantages for measuring waiting times of electrons and holes and their nonlocal properties: an Andreev interferometer built on the edge of a quantum spin Hall insulator (QSHI) [40][41][42][43][44][45][46] [Fig.1(a)].
Due to the lack of a gap, the topological MBSs dominate the local and nonlocal transport across the interferometer.We find that the waiting times for electron transfers across our junction are sensitive to the MBSs, but are uncorrelated with each other.By contrast, the waiting times of Andreev reflected holes are less sensitive to the MBSs, but instead present a strong correlation at ϕ ∼ π.Importantly, the crossed (hole-electron) distributions and their correlations feature a unique behavior characteristic of a gapless nonlocal MBS.Consequently, electron waiting times and their correlations constitute an alternative signature of MBSs, sensitive to their nonlocal nature, thus circumventing the problems arising from trivial resonant levels that naturally form in many Majorana platforms [30].
Topological Andreev interferometer.-Weconsider an Andreev interferometer at the edge of a QSHI (Fig. 1), which comprises of a superconducting loop with a short SL-SR junction that is attached to the normal metal leads NL and NR.For simplicity, we fix the length of each superconductor segment to be equal, L S , and only analyze the situation where SL and SR share an interface [Fig.1(b)] [69].Low-energy excitations are described in the basis Ψ(x) = (ψ ↑ , ψ ↓ , ψ † ↓ , −ψ † ↑ ) T , with ψ † σ (x) the creation operator for electrons with spin σ ∈ {↑, ↓} at position x, by the Bogoliubov-de Gennes Hamiltonian [62] Here, v F is the Fermi velocity, µ the chemical potential, and the Pauli matrices ηj and σj act in Nambu and spin spaces, respectively.We set the pair potential ∆(x) = ∆ for SL, ∆(x) = ∆e iϕ for SR (ϕ is the superconducting phase difference), and zero otherwise.Henceforth, we set v F = ℏ = ∆ = 1 so that the superconducting coherence length is ξ = ℏv F /∆ = 1 [70], and set µ = 0 and eV = ∆/2 [71], see Supplementary Material (SM) [72].
The QSHI Andreev interferometer forms a topological Josephson-like junction that hosts gapless MBSs at the SL-SR interface at ϕ = π [25,65].For any other trivial junction, the bound states develop a gap around ϕ ∼ π.
To distinguish between topological (gapless) and trivial (gapful) bound states, we compare below the prototypical cases ϕ = 0 and ϕ = π.Although our junction is always topological, results at ϕ = 0 are qualitatively equivalent to those of a trivial bound state (for any ϕ), as long as its gap is comparable or larger than the bias eV .Electron waiting times.-WTDsfor phase-coherent transport of noninteracting electrons are evaluated from the scattering matrix [9,10,36].Generally an Andreev interferometer has four effective transport channels [electrons or holes (e, h), incoming or outgoing (i, o), from the left or right leads (L, R)], represented by the spinor hR ) T .For a given energy E, the scattering matrix connects outgoing and incoming solutions of Eq. (1) as Ψ (o) = SΨ (i) [72].Owing to the spin-momentum locking at the QSHI edge, here only the normal transmissions S αL,αR and S αR,αL , with α = e, h, and the Andreev reflections S eX,hX and S hX,eX , with X = L, R, are nonzero.From the scattering matrix we define the idle-time probability Π({τ γ }) that no particles of type γ = αX are detected during the time interval τ γ (for the stationary processes considered here only time intervals are relevant).Following Refs.[10,36], we have where I is the identity matrix and K is a diagonal matrix, The linear dispersion relation of the QSHI helical edge states allows us to naturally divide the transport window [µ, µ+eV ] in intervals of width κ = eV /N , where N is the total number of intervals and eV the applied bias.Due to the inversion symmetry of our setup, we only consider voltages applied to the left lead, V L ≡ V , V R = 0. We work in the limit N → ∞ where Eq. ( 3) correctly applies to stationary transport [10].
We can now define W αβ (τ ) = −⟨τ α ⟩∂ 2 τ Π(τ ) as the probability density of detecting a particle of type β at a time τ after having measured a particle of type α [72].Here, the mean waiting time ⟨τ α ⟩ is related to the average current for α particles, I α = 1/⟨τ α ⟩.Analogously, we define the joint waiting time τ2 Π, which generalizes the waiting time distribution between particles of type α and β to include the extra detection of a particle of type γ at an intermediate time τ 1 , such that 0 ≤ τ 1 ≤ τ 2 [72].The joint WTD describes correlations between consecutive waiting times.When the waiting times are uncorrelated, the joint distribution factorizes as the product of two waiting time distributions [14], . We can further quantify the correlations between consecutive waiting times using the correlation function A main feature of the QSHI topological Andreev interferometer is that for an electron (say, spin-up) injected in NL, only (spin-down) holes and (spin-up) electrons can scatter into NL and NR, respectively.Thus, with electrons and holes always scattering into different leads, all local or same detector WTDs are necessarily given by W ee and W hh , while all nonlocal WTDs are given by W eh and W he , where measurements take place at different detectors.Similarly, W eee and W hhh are local joint WTDs, while joint distributions combining electron (NR) and hole (NL) measurements, like W ehe , are nonlocal.
Local waiting times.-Westart with the local WTDs W αα , representing two consecutive detections at either NL (α = h) or NR (α = e).It was established in Ref. [10] that the WTD of a quantum-coherent channel with energy-independent transmission is determined by its scattering probability: highly-transmitting channels result in a Wigner-Dyson distribution [orange area in Fig. 2 low-energy electron transmission probability becomes strongly energy-dependent for long junctions due to the resonant-tunneling through the MBS [65,72].Consequently, we find that W ee converges to the WTD of a resonant level in the tunnel limit [10], [solid magenta line in Fig. 2 Even though W hh is not very sensitive to the presence of the MBS, the correlations between consecutive waiting times for hole transfers, δW hhh , contain very relevant information.We focus on long junctions, L S > ξ, which are dominated by Andreev reflection processes and feature a more pronounced dependence on the phase ϕ.For gapful states (ϕ = 0), the Andreev interferometer behaves like an electron-hole beam splitter, featuring the same correlations as a standard quantum point contact for electrons [14] [Fig.3(a)]: When the time between two hole transfers is small, τ 1 < ⟨τ h ⟩ (or long, τ 1 > ⟨τ h ⟩), the next hole detection at τ 2 will require a long (short) waiting time (red color signals positive correlations).By contrast, gapless MBSs around ϕ ∼ π exhibit correlations that seemingly explode at short waiting times, with values increasing an order of magnitude compared to the gapful case [Fig.3(b)].This means that short time intervals between detections are the most likely.The waiting times between electron transfers, on the other hand, are completely uncorrelated, i.e., δW eee (τ 1 , τ 2 ) ≃ W ee (τ 1 )W ee (τ 2 ) [72].
Nonlocal waiting times.-Wenow fully exploit the multi-terminal advantage of the topological Andreev interferometer by exploring the nonlocal WTDs.By definition, the distribution W αβ , with β ̸ = α, assumes that the first particle α has been detected; no matter how unlikely that event is.Therefore, the nonlocal waiting times are determined by the probability of the second detection.Consequently, W eh (W he ) is determined by the Andreev reflection (electron transmission) probability, following a behavior similar to W hh (W ee ), which we verify in Fig. 4.
The one marked difference between local and nonlocal distributions is that, as particles transfer into different detectors, nonlocal WTDs can be finite at zero waiting time and also always fulfill W eh (0) = W he (0) [36].
The nonlocal WTDs at zero waiting time have already been established to increase in the presence of MBSs, independently of the scattering probabilities [36].Here, we interestingly also find that W he (τ ), which for ϕ = π is determined by the Majorana-assisted electron tunneling, is further strongly altered.Specifically, W he (τ ) presents a dip at short but finite waiting times.We explain this behavior as being due to the transition between a regime dominated by the Andreev reflection probability at τ → 0 into a regime where electron transmissions dominate at long waiting times.The former initially reduces the probability, while the latter imposes a behavior similar to the local distribution, W ee .The dip, or local minimum at short waiting times, reflects this transition and is particularly visible in the presence of Majorana-induced resonant tunneling when the corresponding local WTD becomes anomalous, see Fig. 2(a).With W eh being primarily determined by W hh , we find no such dip in W eh .This behavior of W he is unique to the topological Andreev interferometer, which we confirmed by checking both local and nonlocal WTDs for an ordinary interferometer, in the absence of any topology.
We further find that the correlations between nonlocal waiting times also show a unique dependence on the phase ϕ.We focus on alternate electron-hole-electron transfers in the long junction regime, where the phase dependence is stronger, and study the behavior of W ehe containing the correlations between W he and W eh .Very short waiting times, τ 1 < ⟨τ h ⟩, show only a weak correlation with long waiting times (τ 2 > ⟨τ e ⟩) for gapful states (ϕ = 0), but this behavior is enhanced two orders of magnitude for gapless MBS around ϕ ∼ π [Fig.5].As mentioned above, the sequential tunneling of three electrons, W eee , is uncorrelated.However, including one hole detection between the electron transfers drastically changes the statistics in the presence of MBSs.These results indicate that to completely characterize gapless states in Andreev interferometers, we need to compare transport processes in both arms of the circuit.We note that W heh , which is dominated by Andreev reflections and thus less sensitive to the presence of the SL-SR junction, shows negative correlations and a weak phase dependence [72].The behavior of all studied WTDs and correlations is summarized in Table I.
Concluding remarks.-Wehave analyzed the distribution of waiting times and their correlations for electrons and holes emitted from a topological Andreev interferometer: a NL-SL-SR-NR junction on the quantum spin Hall edge.Two special features of this setup are (i) the emergence of gapless MBSs and (ii) it acting as an electron-hole beam splitter, sending holes and electrons to different leads.This topological Andreev interferometer is thus one of the simplest multi-terminal setups without magnetic elements that features MBSs and allows us to test their nonlocal behavior.We find that the gapless property of the MBSs makes the waiting times involving Majorana-assisted electron transfers, W ee and W he , very special around ϕ ∼ π.Most importantly, the nonlocal property of MBSs is captured in the correlations between waiting times, see Table I.For example, the waiting times for consecutive hole reflections and for alternate electron-hole detections are strongly correlated for gapless Majorana states [Fig.3(b), Fig. 5(b)], even if their distributions, W hh and W eh , are almost insensitive to the phase ϕ.
The search for signatures of Majorana states currently faces several challenges [30,31,[73][74][75].In particular, the presence of trivial low-energy modes that mimic the local properties of MBSs, obscuring their detection in many platforms.To circumvent the latter, the quantum spin Hall effect is a promising platform where signatures of gapless MBSs have already been identified experimentally, even in the presence of extra trivial modes [51][52][53].Thus, Andreev reflection and normal transmissions, which determine all our results, are the dominant scattering processes.Furthermore, our analysis of nonlocal properties of Majorana states addresses the need to go beyond local signatures.In fact, previous works have already analyzed the WTD of electrons tunneling into a Majorana state [35,36,38,39] focusing on its local properties and showing that the resonant transport through a MBS yields WTDs very similar to that of a single (trivial) level resonance [10,36].Therefore, our results complement earlier work by showing that taking into account independent and simultaneous transfers of electrons and holes into separate detectors (i.e., both local and nonlocal WTDs), and the correlations between them, yields distinctive signatures of Majorana modes.

SUPPLEMENTARY MATERIAL FOR "NONLOCALITY OF MAJORANA BOUND STATES REVEALED BY ELECTRON WAITING TIMES IN A TOPOLOGICAL ANDREEV INTERFEROMETER"
In this supplemental material, we provide details of the theoretical formalism and some additional results to support the discussions of the main text about the scattering probabilities, the bias dependence of the waiting time distributions, and the nonlocal joint waiting time correlations.

THEORETICAL METHODS
In this section, we summarize the most important steps in order to arrive at the scattering matrices used in the main text and describe the theory for WTDs and joint WTDs.

Scattering matrix formalism
We consider a NL-SL-SR-NR junction, with NL and NR normal leads and SL and SR superconducting ones, along the x direction on one edge of a QSHI, where the NL-SL interface is placed at x = 0, the SL-SR interface at x = L S , and the SR-NR one at x = 2L S .The scattering states at the different regions of the NL-SL-SR-NR junction can be written as [65] ) where The a i , b i , c i , and d i are the scattering parameters to be determined.The wave vectors in normal regions NL and NR read as and in the superconducting regions SL and SR take the form, We solve the above equations by matching the wave functions at the interfaces x = 0, x = L S , and x = 2L S , to obtain The local normal reflection and nonlocal crossed Andreev transmission are forbidden by the conservation of helicity of the edge states of the QSHI [57,80] allowing only two processes: (i) local Andreev reflections, where an incident electron is reflected as a hole at NL, and (ii) nonlocal electron transmission at NR with the amplitudes as follows [65].
• Andreev reflection in NL for an incident electron from NL: • Transmission of an electron into NR for an incident electron from NL:

Waiting time distributions
The waiting time is a fluctuating quantity, which must be described by a probability distribution.The waiting time distribution (WTD) is the conditional probability density of detecting a particle of type β at time t e β , given that the last detection of a particle of type α occurred at the earlier time t s α .Here, the types α and β may refer to the out-going channel, the spin of the particle, and the particle being an electron or a hole.The WTD is denoted as W α→β (t s α , t e β ).For the systems considered here, with no explicit time dependence, the WTD is a function only of the time difference, such that W α→β (t s α , t e β ) = W α→β (τ ) with τ = t e β − t s α .To evaluate the WTD, we proceed as in Ref. [14] and express the WTD as time-derivatives of the idle time probability.The idle time probability Π(t s α , t e α ) is the probability that no particles of type α are detected in the time interval [t s α , t e α ] by a detector at position x α .The idle time probability can be a function of several different particle types and associated time intervals.
The idle time probability can be evaluated using scattering theory, leading to the determinant formula [14] where the set {t s γ , t e γ } corresponds to all relevant particles γ and associated time intervals.The Hermitian operator Q({t s γ , t e γ }) is a matrix in a combined energy and particle type representation.It has the block form having omitted the time arguments.The scattering matrix S(E), for particles with excitation energy E, and the kernel K(E) are matrices in the space of particle types.The kernel is the block diagonal matrix given by the direct sum of kernels corresponding to each particle of type γ detected at position x γ .Equation (S 12) corresponds to Eq. ( 3) in the main text.We work close to the Fermi level, where the dispersion relation E = ℏv F k is linear and all quasi-particles propagate with the Fermi velocity v F .To implement the matrix in Eq. (S 10), we discretize the transport window [E F , E F + eV ] in N intervals of width κ = eV /N .The width κ explicitly enters in Eq. (S 12), and we always consider the limit N → ∞, for which the transport is stationary.
The WTD can be related to the idle time probability by realizing that time derivatives correspond to detection events [10].When taking derivatives of operators and determinants, we use Jacobi's formula Applied to Eq. (S 9), the derivative takes the form where we have defined and Qα = ∂ tα Q.After recasting the kernel definition, Eq. [3] of the maintext and Eq.(S 12), as it is straightforward to see that the derivative with respect to time t α of the operator Q({t s γ , t e γ }) depends only on time t α , that is, Qα = Qα (t α ).Consequently, we have The first-passage time distributions are defined as given that a particle of type α was detected at the initial time t = 0. Similarly, Eq. (S 19) concerns the time τ we have to wait until a particle of type α is detected, given that we start the clock at time t = 0.For the single-channel stationary case, where all quantities depend only on the time difference τ = t e − t s and we have only one particle type, we find that these two probability distributions are the same, The distribution of waiting times between particles of type α and particles of type β can be expressed as [14] where I α is the average particle current of type α particles, and the minus sign comes together with the derivative with respect to the starting time t s α .In addition, after having performed the derivatives, we set the starting times to zero, i.e., t s α = t s β = 0, while for the end times we set t e α = 0 and t e β = τ .The waiting time is then measured from the time when a particle of type α is detected until the later time when a particle of type β is detected.During this waiting time, additional particles of type α may be detected, but not of type β.
By combining Eqs.(S 9) and (S 20), we find having made repeatedly use of Jacobi's formula for derivatives of determinants.Finally, for evaluating Eq. (S 21) we note that the average particle current of type α particles can be expressed as I α = F α (0).In combination, Eqs.(S 9-S 21) allow us to evaluate the distributions of waiting times for the superconducting systems that we consider in the main text.We have plotted them as a function of τ i normalized by the mean time ⟨τ i ⟩.

Joint waiting time distributions
We can now introduce the joint waiting time distribution W α→γ→β (t s α , t m γ , t e β ).This probability distribution generalizes the waiting time distribution between particles of type α and β, Eqs.(S 20) and (S 21), to include the extra detection of a particle of type γ at an intermediate time t m γ , such that t s α < t m γ < t e β .To find the joint waiting time distribution, we introduce an auxiliary, virtual particle channel of type γ.
where we have extended the idle time probability to include the auxiliary channel.From Eq. (S 17), second partial derivatives over the auxiliary channel are zero.In fact, all second derivatives are zero, since t s α and t e β are independent variables even when α = β.Consequently, the joint waiting time distribution has the following structure: where ∂ j , with j = 1, 2, 3, are time derivatives over some generic and independent variables.follows the opposite behavior with relatively more energy intervals with high probability than those with zero.

BIAS DEPENDENCE OF THE WAITING TIME DISTRIBUTIONS
In the main text, we have shown results for eV = ∆/2 only.To better understand the effect of the voltage bias, particularly for the parameter regime where the WTDs show anomalous behaviors, we now plot local W ee and nonlocal W he at ϕ = π for several transport windows.In Fig. S 2(a), W ee shows a plateau region for eV = ∆/2, which is exactly the same result shown in the main text for the same parameter values (just scaled with respect to the axis labels).Increasing the bias, small oscillations appear in the WTD which give rise to the plateau feature.The frequency and amplitude of the oscillations increase with eV , particularly for eV ≥ ∆, since, in addition to the zero energy anomaly from the MBSs, the transmission also features small peaks around E ∼ ∆, cf.Fig. S 1(b).The period of these oscillations is related to the energy difference between peaks, that is, it is inversely proportional to ∆ [10,36,39].
We have already described the anomalous behavior of W he at ϕ = π in the main text.We now plot the nonlocal WTD with the same parameters for other bias values and concentrate on short waiting times τ ; longer waiting times are mostly unaffected by the bias voltage [36].With increasing the bias, W he (0) increases, see Ref. [36], and so does the value of the dip or minimum in the nonlocal WTD.We also note an oscillatory behavior for larger biases.Therefore, the behavior of the WTDs for voltages larger than the one used in the main text, but that are still comparable to the superconducting gap, does not affect our main conclusions.

NONLOCAL JOINT WAITING TIME CORRELATIONS
Finally, we show some results for the correlations between transmitted electrons and/or reflected holes to complete Table I.
First, the sequential tunneling of three electrons into the right lead is almost uncorrelated, i.e., δW eee (τ 1 , τ 2 ) ≃ W ee (τ 1 )W ee (τ 2 ), as seen in Next, we show that the correlation between waiting times for the hole-electron-hole sequence of detections, i.e., δW heh , is not informative for the Majorana states.In the absence of any phase, there is a very small correlation between two nonlocal waiting times when both of them are at very short waiting times, i.e., τ 1(2) < ⟨τ e(h) ⟩, see t e x i t s h a 1 _ b a s e 6 4 = " R 3 a Wx T z k Z d u w t Q L M O V U t g F Q s K O E = " > A A A E 4 n i c d Z T d b t M w F M e 9 L c A o X x t c c h N R T R o S T M 3 4 F F e T G B J c I A q i 7 a S m q p z k J L U a 2 5 H t r O 2 s v A B 3 E 3 f A 6 / A Q v A 1 2 W 0 V K k / r C P v L / d 3 L + s n M c Z C m R q t P5 t 7 O 7 5 9 y 4 e W v / d u v O 3 X v 3 H x w c P u x L n o s Q e i F P u b g I s I S U M O g p o l K 4 y A R g G q Q w C K b v r T 6 4 B C E J Z 9 / V I o M R x Q k j M Q m x M l u D + P j D c + g / H R + 0 O y e d 5 X D r g b c O 2 m g 9 u u P D v b 9 + x M O c A l N h i q U c e p 1 M j T Q W i o Q p F C 0 / l 5 D h c I o T 0 J h K i t X k m W u C m D M l l 5 F c 0 K D K W U h x n s r q d i J w N i H h v L I 7 j C 5 J J h m m I E d 6 v j y H x q K

FIG. 1 .
FIG. 1.(a) Andreev interferometer at the QSHI edge with magnetic flux ϕ applied through a superconductor loop.(b) Incident electrons (blue balls) with energy within the gap ∆ scatter off the NL-SL-SR-NR junction only as electron transmissions to NR or Andreev-reflected holes (red balls) into NL.Detectors D1 and D2, respectively placed at NL and NR, detect electrons or holes either individually or simultaneously.NL is biased by a voltage V and NR is at equilibrium.
(a)], instead of evolving into a Poisson distribution like for ϕ = 0.At high transmission (short junctions), the variation with the phase of W ee is less noticeable [green lines in Fig. 2(a)].This is also the case for the distribution of reflected holes W hh at any transparency [Fig.2(b)], since the probability of Andreev reflection is |S hL,eL | 2 ∼ 1 for all the energies in the transport window (|E| ≤ eV ).

Fig. S 3
(c).However, when ϕ = π, the correlation dies out.Mostly negative values of the correlation exist indicating dominance of the uncorrelated part W he (τ 1 )W eh (τ 2 ), as observed in Fig. S 3(d).
This new channel allows us to perform a third derivative, representing the intermediate detection event, by adding an extra kernel K(t s γ , t e γ ) to Eqs. (S 9) and (S 11).Eventually, we set t s γ = t e γ = t m , effectively closing this channel's contribution to the idle time probability.
It is important to stress that both Π and G in Eq. (S 23) depend only on the time coordinates of the original channels, since both t s,e γ → t m .The only dependence on t m is thus on the intermediate derivative, Q2 .Based on the general structure of Eq. (S 23), we denote as γ the auxiliary channel where the intermediate detection event takes place, and define the same channel joint waiting time distributionI α W α→γ→α (τ 1 , τ 2 ) = ∂ t eHere, we have assumed a stationary case and set the initial time t s α = 0.The intermediate detection time interval is then t m − t s α = τ 1 , and the total time interval is t e α,β − t s α = τ 1 + τ 2 .