# Dynamical leakage of Majorana mode into side-attached quantum dot

J. Barański<EMAIL_ADDRESS>Military University of Aviation, ul.
Dywizjonu 303, 08-521 Dęblin, Poland  M. Barańska  Military University of
Aviation, ul. Dywizjonu 303, 08-521 Dęblin, Poland  T. Zienkiewicz  Military
University of Aviation, ul. Dywizjonu 303, 08-521 Dęblin, Poland  R. Taranko
Institute of Physics, M. Curie-Skłodowska University, 20-031 Lublin, Poland
T. Domański<EMAIL_ADDRESS>Institute of Physics, M. Curie-
Skłodowska University, 20-031 Lublin, Poland

(August 27, 2024)

###### Abstract

We study a hybrid structure, comprising the single-level quantum dot attached
to the topological superconducting nanowire, inspecting dynamical transfer of
the Majorana quasiparticle onto normal region. Motivated by the recent
experimental realization of such heterostructure and its investigation under
the stationary conditions [L. Schneider et al., Nature Communications 11, 4707
(2020)] where the quantum dot energy level can be tuned by gate potential we
examine how much time is needed for the Majorana mode to leak into the normal
region. We estimate, that for typical hybrid structures this dynamical process
would take about 20 nanoseconds. We propose a feasible empirical protocol for
its detection by means of the time-resolved Andreev tunneling spectroscopy.

##  I Motivation

Topological superconductors, hosting the Majorana boundary modes, are
currently of great interests both for basic science Aguado (2017); Lutchyn
_et al._ (2018); Prada  _et al._ and potential applications Aguado (2020); Liu
_et al._ (2016). Topological protection and nonabelian character make them
promising candidates for realization of stable qubits Schrade and Fu (2018)
and quantum computations Karzig  _et al._ (2017). Signatures of their
fractional statistics can tested, for instance, by a sequence of the charge-
transfer operations using the quantum dots attached to topological
superconductor Flensberg (2011); Steiner and von Oppen (2020). Such dynamical
transfer of the charge between the quantum dot and topological superconductors
might enable nonabelian operations on the Majorana bound states Seoane Souto
_et al._ (2020); Posske  _et al._ (2020).

Development of time-resolved spectroscopies with their resolution down to sub-
picosecond regime allows to probe the physical structures in response to
abrupt change of the model parameter(s) or other nonequilibrium conditions
Tuovinen  _et al._ (2019). Such dynamics has been recently investigated in
topological phases by a number of groups, considering the fermionic Mazza  _et
al._ (2015); Wang  _et al._ (2017); Yang  _et al._ (2018) and bosonic systems
Maffei  _et al._ (2018). In particular, there has explored dynamical
teleportation due to nonlocality of the zero-energy boundary modes Li and Xu
(2020) and it has been shown, that dynamical techniques based on the noise
measurements could unambiguously identify the true Majorana quasiparticles
Jonckheere  _et al._ (2019); Manousakis  _et al._ (2020); Radgohar and
Kargarian (2020); Perrin  _et al._ (2020). Time-dependent measurements have
been also proposed to detect topological invariants of the higher order
topological superconductor, harboring the corner states Mizoguchi  _et al._
(2020).

Hybrid structures consisting of the topological superconducting nanowires
side-attached to the quantum dots allow for a tunable control of the zero-
energy bound states Deng  _et al._ (2016, 2018); Schneider  _et al._ (2020).
Their variations with respect to the gate potentials or magnetic field could
discriminate the true Majorana quasiparticles from the trivial Andreev bound
states, appearing accidentally at zero energy Pan and Das Sarma (2020). More
complex magnetic-superconducting heterostructures enable coexistence of the
localized and chiral modes Crawford  _et al._ (2020).

Here we analyze a dynamical transfer of the Majorana mode from the
superconducting nanowire to the quantum dot (see Fig. 1). We consider such
evolution driven by their sudden coupling or abrupt change of the QD energy
level imposed by applying external gate potential. We examine how much time is
needed for the Majorana mode to leak into the quantum dot region. We also
show, that time-resolved process provide valuable insight into the effective
quasiparticles of topological superconductors.

Figure 1: Schematic representation of considered system comprising quantum dot
(QD) deposited between metallic (N) and superconducting (S) electrode and side
coupled to one end of monoatomic chain hosting Majorana edge state represented
by $\eta_{1}$. Additional electrode stands for gate voltage ($V_{g}$) tuning
of quantum dot’s energy level.

The paper is organized as follows. In Sec. II we introduce the microscopic
model and outline the procedure for studying the time-dependent evolution.
Next, in Sec. III we present the time-dependent signatures of the Majorana
quasiparticle leaking on the side-attached quantum dot. In Sec. IV we provide
realistic estimations of the characteristic time- and energy-scales which
could be practically verified experimentally. Finally, Sec. V summarizes the
main results.

##  II Formulation of the problem

We study dynamical properties of the heterostructure comprising the quantum
dot deposited on s-wave superconductor and weakly coupled to another metallic
lead that can be thought as STM tip. The quantum dot is additionally coupled
to the topological superconducting nanowire, hosting the Majorana end modes
(Fig. 1). Our considerations aim to determine the characteristic time needed
for development of the Majorana features transmitted onto the QD region. In
particular, we shall investigate the quantum evolution (i) driven by abrupt
formation of the hybrid structure, inspecting the zero-energy features
appearing for various positions of the quantum dot energy level, and b) study
how long it takes to qualitatively transform the existing Majorana signatures
after a sudden change of the QD level.

It has been predicted Vernek  _et al._ (2014) that in QD attached to the
topological superconductor the zero-energy quasiparticle would be transmitted
to the normal region. The first empirical evidence for such Majorana leakage
has been reported by M.T. Deng et al Deng  _et al._ (2016) and recently in STM
method by L. Schneider et al Schneider  _et al._ (2020). The quantum dot -
topological superconductor hybrid structures have studied by a number of
groups (see Ref. Pan  _et al._ (2020) for survey). In N-QD-S configurations
(Fig. 1) such leakage process qualitatively depends on the QD level Barański
_et al._ (2016); Zienkiewicz  _et al._ (2019). Namely, when the energy level
is away from zero, the signatures of Majorana mode are manifested by the peak
appearing in the density of states (DOS), thereby enhancing the zero-bias
conductance. Contrary to such picture, when the QD level happens to be near
zero energy, the quantum interference induces the fractional Fano-type
depletion in the QD spectrum Calle  _et al._ (2020). Dynamical changeover from
one regime to another has not been analyzed yet, and this is the main purpose
of our present study.

###  II.1 Microscopic scenario

Let us start, by formulating the microscopic model and presenting the
computational method used for determination of time-dependent quantities. We
describe our hybrid structure (Fig. 1) by the following Hamiltonian

$\displaystyle\hat{H}=\hat{H}_{QD}+\sum_{\beta}\left(\hat{H}_{\beta}+\hat{T}_{QD-\beta}\right)+\hat{H}_{MQD},$
(1)

where $\hat{H}_{QD}$ refers to the quantum dot (QD), $\hat{H}_{\beta}$ denote
the metallic ($\beta=N$) and superconducting ($\beta=S$) leads, whereas
$\hat{H}_{MQD}$ stands for the zero-energy Majorana quasiparticles and
coupling of one of them to QD. The normal lead can be treated as a Fermi sea
$\hat{H}_{N}=\sum_{k,\sigma}\xi_{Nk}\hat{c}^{\dagger}_{Nk\sigma}\hat{c}_{Nk\sigma}$,
where $\hat{c}^{\dagger}_{Nk\sigma}$ $(\hat{c}_{Nk\sigma})$ is the creation
(annihilation) of $\sigma$ spin electron whose energy
$\xi_{Nk}=\varepsilon_{Nk}-\mu_{N}$ is measured from the chemical potential
$\mu_{N}$. The superconducting electrode is taken in BCS-type form
$\hat{H}_{S}=\sum_{k,\sigma}\xi_{Sk}\hat{c}^{\dagger}_{Sk\sigma}\hat{c}_{Sk\sigma}-\sum_{k}\left(\Delta_{S}\hat{c}^{\dagger}_{Sk\uparrow}\hat{c}^{\dagger}_{Sk\downarrow}+h.c.\right)$,
where $\Delta_{S}$ is the isotropic pairing gap. Itinerant electrons are
hybridized with the quantum dot through the term
$\hat{T}_{QD-\beta}=\sum_{k,\sigma}\left(V_{\beta
k}\hat{d}^{\dagger}_{\sigma}\hat{c}_{\beta k\sigma}+h.c.\right)$. We introduce
the coupling functions $\Gamma_{\beta}=2\pi\sum_{k}\left|V_{\beta
k}\right|^{2}\delta\left(\omega-\xi_{\beta k}\right)$ assuming them to be
constant in the low-energy regime.

As regards the quantum dot we represent it by the single level impurity term
$\hat{H}_{QD}=\sum_{\sigma}\epsilon_{d}\hat{d}_{\sigma}^{\dagger}\hat{d}_{\sigma}$,
where $\hat{d}^{\dagger}_{\sigma}$ ($\hat{d}_{\sigma}$) is the creation
(annihilation) operator of spin $\sigma=\uparrow$, $\downarrow$ electrons. We
focus the low energy regime $|\omega|\ll\Delta_{S}$, when the fermion degrees
of freedom outside the pairing gap can be integrated out. Their influence
simplifies to the induced pairing
$\hat{H}_{S}+\hat{T}_{QD-S}\approx\Delta_{d}\left(\hat{d}_{\downarrow}\hat{d}_{\uparrow}+\mbox{\rm
h.c.}\right)$ with the on-dot potential $\Delta_{d}=\Gamma_{S}/2$ Bauer  _et
al._ (2007); Barański and Domański (2013); Yamada  _et al._ (2011).

The ‘proximitized’ QD is thus effectively modeled by

$\displaystyle\hat{H}_{QD}\\!+\\!\hat{H}_{S}\\!+\\!\hat{T}_{QD-S}\\!=\\!\sum_{\sigma}\\!\epsilon_{d}\hat{d}_{\sigma}^{\dagger}\hat{d}_{\sigma}\\!+\\!\Delta_{d}\left(\hat{d}_{\downarrow}\hat{d}_{\uparrow}\\!+\\!\mbox{\rm
h.c.}\right)\\!.$ (2)

Restricting our considerations to the low-energy regime (within
milielectronvolts around the chemical potential) we describe the topological
superconductor by

$\displaystyle\hat{H}_{MQD}=i\epsilon_{m}\hat{\eta}_{1}\hat{\eta}_{2}+\lambda(d_{\uparrow}\hat{\eta}_{1}+\hat{\eta}_{1}\hat{d}^{\dagger}_{\uparrow}),$
(3)

where the selfhermitian operators $\hat{\eta}_{1}$, $\hat{\eta}_{2}$ represent
the Majorana end-modes and $\epsilon_{m}$ stands for their overlap. The last
part appearing in Eq. (3) accounts for hybridization of the left h.s. Majorana
quasiparticle with the quantum dot, where $\lambda$ is the coupling strength.
For convenience we recast the Majorana operators by the standard fermion
operators defined via
$\hat{\eta}_{1}=\frac{1}{\sqrt{2}}(\hat{f}+\hat{f}^{\dagger})$,
$\eta_{2}=\frac{-i}{\sqrt{2}}(f-f^{\dagger})$. This transformation implies,
that (3) can be rewritten as

$\displaystyle\hat{H}_{MQD}$ $\displaystyle=$
$\displaystyle\epsilon_{m}\hat{f}^{\dagger}\hat{f}+t_{m}(\hat{d}_{\uparrow}^{\dagger}-\hat{d}_{\uparrow})(\hat{f}+\hat{f}^{\dagger})$
(4)

with a shorthand notation $t_{m}=\lambda/\sqrt{2}$.

###  II.2 Outline of computational method

Fingerprints of the Majorana mode would be practically observed in the quantum
dot region, examining the charge current $I_{\sigma}(V,t)$ induced via N-QD-S
junction by the bias voltage $V$. Its differential conductance
$G_{\sigma}(V,t)=\frac{d}{dV}I_{\sigma}(V,t)$ can be obtained numerically,
using the following expression for the current

$\displaystyle
I_{\sigma}(V,t)=-e\left<\frac{d\hat{N}_{N\sigma}}{dt}\right>=-\frac{ie}{\hbar}\langle[\hat{H},\hat{N}_{N\sigma}]\rangle.$
(5)

Since all parts of the Hamiltonian (1), except $\hat{T}_{QD-N}$, commute with
the number operator $\hat{N}_{N\sigma}$ therefore the tunneling current reads

$\displaystyle I_{\sigma}(V,t)$ $\displaystyle=$
$\displaystyle\frac{-ie}{\hbar}\sum_{k}\left(V_{Nk}\langle\hat{d}^{\dagger}_{\sigma}(t)\hat{c}_{Nk\sigma}(t)\rangle-\mbox{\rm
h.c.}\right)$ (6) $\displaystyle=$
$\displaystyle\frac{2e}{\hbar}\;\Im\left[\sum_{k}V_{Nk}\langle\hat{d}^{\dagger}_{\sigma}(t)\hat{c}_{Nk\sigma}(t)\rangle\right].$

To simplify the notation from now onwards we set $\hbar=e=\Gamma_{N}=1$,
expressing the energies, currents and time in units of $\Gamma_{N}$,
$e\Gamma_{N}/\hbar$ and $\hbar/\Gamma_{N}$, respectively. Additionally we
choose the chemical potential of superconductor as a reference level
($\mu_{S}=0$), when $\mu_{N}=eV$. Since the itinerant electrons degrees of
freedom refer solely to the metallic lead, for brevity we also skip the
subindex $N$ appearing in $\hat{c}^{(\dagger)}_{Nk\sigma}$, $V_{Nk}$, and
$\xi_{Nk}$.

Using the standard expression for $\hat{c}_{k\sigma}(t)$ Taranko and Domański
(2018)

$\displaystyle\hat{c}_{k\sigma}(t)=\hat{c}_{k\sigma}(0)e^{-i\xi_{k}t}-iV_{k}\int_{0}^{t}d\tau
e^{-i\xi_{k}(t-\tau)}\hat{d}_{\sigma}(\tau)$ (7)

and imposing the wide band limit approximation we get

$\displaystyle
I_{\sigma}(t)=2\Im\left[\sum_{k}V_{k}\langle\hat{d}^{\dagger}_{\sigma}(t)\hat{c}_{k\sigma}(0)\rangle-i\frac{\Gamma_{N}}{2}\langle\hat{n}_{\sigma}(t)\rangle\right].$
(8)

Here $\langle...\rangle$ denotes the statistically averaged value. In order to
determine the current (6) one needs the correlation function
$\langle\hat{d}_{\sigma}^{\dagger}(t)\hat{c}_{k\sigma}(0)\rangle$ and the
time-dependent QD occupancy
$\langle\hat{n}_{\sigma}(t)\rangle=\langle\hat{d}_{\sigma}^{\dagger}\hat{d}_{\sigma}\rangle$.

To find these quantities we employ the equation of motion approach (EOM)
Taranko and Domański (2018) and make use of the 4-th order Runge-Kutta method
to solve numerically the appropriate set of differential equations.
Traditionally such EOM applied to any correlation function generates a
sequence of the additional functions, for which the next equations of motion
must be constructed, until they are finally closed (or terminated). For the
uncorrelated QD embedded in our heterostructure this procedure mixes the
operators $\hat{d}^{(\dagger)}_{\sigma}$ and $\hat{f}^{(\dagger)}$ with the
operators of the normal lead electrons. They have a general form
$\sum_{k}V_{k}\langle\hat{O}(t)\hat{c}_{k\sigma}(0)\rangle$, where $\hat{O}$
stands for one of the following six operators $\hat{d}_{\downarrow}$,
$\hat{d}^{\dagger}_{\downarrow}$, $\hat{d}_{\uparrow}$,
$\hat{d}^{\dagger}_{\uparrow}$, $\hat{f}$, and $\hat{f}^{\dagger}$. We thus
have to determine 12 momentum-dependent correlation functions
$f_{ki}(t)=\langle\hat{O}(t)\hat{c}_{k\sigma}(0)\rangle$ and 9 momentum-
independent expectation values
$f_{i}(t)=\langle\hat{O}(t)\hat{\tilde{O}}(t)\rangle$ listed explicitly in
table 1. Their differential equations are presented in the appendix [see Eqs
(10,11)].

$f_{i}$ | $f_{ki}$
---|---
$f_{1}=\langle\hat{d}_{\uparrow}^{\dagger}(t)\hat{d}_{\uparrow}(t)\rangle$ | $f_{k1}=\langle\hat{d}_{\uparrow}^{\dagger}(t)\hat{c}_{k\uparrow}(0)\rangle$
$f_{2}=\langle\hat{d}_{\downarrow}^{\dagger}(t)\hat{d}_{\downarrow}(t)\rangle$ | $f_{k2}=\langle\hat{d}_{\uparrow}(t)\hat{c}_{k\downarrow}(0)\rangle$
$f_{3}=\langle\hat{f}^{\dagger}(t)\hat{f}(t)\rangle$ | $f_{k3}=\langle d_{\uparrow}(t)\hat{c}_{k\uparrow}(0)\rangle$
$f_{4}=\langle\hat{d}_{\downarrow}(t)\hat{d}_{\uparrow}(t)\rangle$ | $f_{k4}=\langle d_{\downarrow}^{\dagger}(t)\hat{c}_{k\downarrow}(0)\rangle$
$f_{5}=\langle\hat{d}_{\uparrow}^{\dagger}(t)\hat{f}(t)\rangle$ | $f_{k5}=\langle\hat{f}^{\dagger}(t)\hat{c}_{k\uparrow}(0)\rangle$
$f_{6}=\langle\hat{d}_{\uparrow}^{\dagger}(t)\hat{f}^{\dagger}(t)\rangle$ | $f_{k6}=\langle\hat{f}(t)\hat{c}_{k\uparrow}(0)\rangle$
$f_{7}=\langle\hat{d}_{\downarrow}(t)\hat{f}(t)\rangle$ | $f_{k7}=\langle\hat{f}(t)\hat{c}_{k\downarrow}(0)\rangle$
$f_{8}=\langle\hat{d}_{\downarrow}(t)\hat{f}^{\dagger}(t)\rangle$ | $f_{k8}=\langle\hat{f}^{\dagger}(t)\hat{c}_{k\downarrow}(0)\rangle$
$f_{9}=\langle\hat{d}_{\uparrow}^{\dagger}(t)\hat{d}_{\downarrow}(t)\rangle$ | $f_{k9}=\langle\hat{d}_{\downarrow}(t)\hat{c}_{k\downarrow}(0)\rangle$
|
$f_{k10}=\langle\hat{d}_{\uparrow}^{\dagger}(t)\hat{c}_{k\downarrow}(0)\rangle$
| $f_{k11}=\langle\hat{d}_{\downarrow}(t)\hat{c}_{k\uparrow}(0)\rangle$
|
$f_{k12}=\langle\hat{d}_{\downarrow}^{\dagger}(t)\hat{c}_{k\uparrow}(0)\rangle$
Table 1: The list of 12 momentum-dependent functions $f_{ki}$ and 9 momentum-
independent functions $f_{i}$ constituting a closed set of the equations of
motion.

Let us note, that in such treatment the parameters (such as the coupling
$t_{m}$ or the energy level $\epsilon_{\sigma}$) can be either static or may
depend on time in arbitrary way.

##  III Time-resolved features

The most profound consequence of bringing the quantum dot in contact with
topological superconductor is emergence of the zero-energy quasiparticle in
the spectrum of QD. This feature (measurable by charge transport through
N-QD-S circuit) is a signature of the Majorana mode. Dynamical leakage process
provides information about the time required to perform logical operations
with use of the Majorana quasiparticles.

###  III.1 Dynamical Majorana leakage

Let us estimate the time required for transferring the zero-energy mode onto
QD region, after abruptly coupling the quantum dot to topological chain. Until
$t=0$ all parts of our setup (Fig. 1) are assumed to be completely
disconnected and the quantum dot unoccupied. Thus our initial conditions are
$\langle\hat{d}^{\dagger}_{\sigma}(0)\hat{d}_{\sigma}(0)\rangle=0$,
$\langle\hat{d}_{\downarrow}(0)\hat{d}_{\uparrow}(0)\rangle=0$,
$\langle\hat{c}^{\dagger}_{k\sigma}(0)\hat{c}_{k\sigma}(0)\rangle=f_{FD}(\varepsilon_{k}-eV)$
and the mixed terms $\langle\hat{O}(0)\hat{c}_{k\sigma}(0)\rangle=0$. At
$t=0^{+}$ the QD is abruptly connected to external reservoirs by the couplings
$\Gamma_{\beta}$. To ensure the subgap quasiparticle states to be well
separated we impose the asymmetric couplings ($\Gamma_{S}=3\Gamma_{N}$). The
superonducting proximity leads to gradual buildup of the in-gap Andreev bound
states. We noticed that these states reach their equilibrium positions and
amplitudes after time $t\simeq\hbar/\Gamma_{N}$ Taranko and Domański (2018).

Figure 2: The time-dependent differential conductance as a function of bias
$V$ (horizontal axis) and time $t$ (vertical axis) obtained for
$\epsilon_{\uparrow}=\epsilon_{\downarrow}=\Gamma_{N}$. At $t=0$ the quantum
dot is abruptly connected to the external leads (imposing
$\Gamma_{S}=3\Gamma_{N}$) when the Andreev bound states (ABS) begin to emerge.
Once the position and amplitude of ABS are established at
$t=20\hbar/\Gamma_{N}$ the quantum dot is additionally connected to the
topological superconductor, assuming $t_{m}=0.5\Gamma_{N}$. From this moment
onward the zero-energy peak gradually emerges, signalizing the Majorana
leakage into the quantum dot region. For the chosen set of model parameters
this zero-energy feature stabilizes its shape after approximately $15-20$
units of time, i.e. at $t\simeq 40\hbar/\Gamma_{N}$.

Safely later after the Andreev states are stabilized we abruptly connect the
quantum dot to topological superconductor. For computations we assume
$t_{m}=0.5\Gamma_{N}$, but more detailed discussion concerning influence of
$t_{m}$ on the time required for the MZM to leak onto QD region is given in
Sec. IV.2. Starting from $t=20\hbar/\Gamma_{N}$ the Majorana mode gradually
leaks to the QD region, as manifested by enhancement of the zero bias
conductance (Fig. 2). We can notice that its amplitude establishes within the
time interval $\Delta t\sim 15-20\hbar/\Gamma_{N}$. This result brings us the
needed information on the characteristic time of the Majorana leakage. We need
to keep in mind, however, that the coupling of QD to metallic reservoir
$\Gamma_{N}$ in different experimental realizations can take various values.
To estimate the order of magnitude of the leakage time (in nanoseconds), in
Sec. IV.1 we compare the qualitative results with typical energy scales used
in experiments on in-gap states and hybrids comprising quantum dots and
topological superconducting chains.

###  III.2 Quench-driven dynamics

The transient evolution discussed in Sec. III.1 allowed us to estimate the
time required for the MZM leakage. Abrupt coupling of the QD to topological
superconductor would be however hardly feasible in practice. More realistic
scenario can rely on employing the gate voltage potential to vary the QD’s
energy level in controlled manner. Depending on the specific value of
$\epsilon_{\sigma}$, the Majorana mode should be evidenced either by the
interferometric depletion (when $\epsilon_{\sigma}\simeq 0$) or constructive
enhancement (for $|\epsilon_{\sigma}|\gg t_{m}$ ) of the zero-bias tunneling
conductance Barański  _et al._ (2016); Zienkiewicz  _et al._ (2019). To
provide some experimentally verifiable result we propose to test how long it
takes to transform the ditch into the peak feature. Time needed for such
changeover can be subsequently confronted with the timescale of the MZM
leakage driven by abrupt coupling of the QD to the topological chain.

Figure 3: Time-resolved differential conductance obtained for
$\Gamma_{S}=3\Gamma_{N}$,
$t_{m}=0.5\Gamma_{N}\theta(t-10\frac{\hbar}{\Gamma_{N}})$ and the QD energy
level $\epsilon_{\sigma}=1.5\Gamma_{N}\theta(t-25\frac{\hbar}{\Gamma_{N}})$,
where $\theta$ is step function. Similarly as in Fig. 2 we initially couple
the QD to the external N/S leads, imposing the energy level
$\epsilon_{\sigma}=0$. After $t=10\hbar/\Gamma_{N}$ the quantum dot is
connected to topological superconductor, inducing the destructive interference
ditch in the zero-bias conductance. At $t=25\hbar/\Gamma_{N}$) we subsequently
lift the QD energy level to
$\epsilon_{\uparrow}=\epsilon_{\downarrow}=\Gamma_{N}$ and from this moment
onward the ditch transforms into the zero-bias peak. This feature establishes
after approximately $15\hbar/\Gamma_{N}$, i.e. at $t\simeq
40\hbar/\Gamma_{N}$.

For this purpose we consider the following three step procedure. (i) As
previously, we start by forming the N-QD-S circuit with the initial energy
level $\epsilon_{\sigma}=0$ and let the Andreev bound states to establish.
(ii) Once the differential conductance saturates at its static value (at
$t\simeq 10\hbar/\Gamma_{N}$ fluctuations become almost negligible) we couple
the quantum dot to the topological chain. After certain amount of time there
appears the interference ditch in the zero-bias conductance. (iii) When ‘the
dust is settled’ we abruptly lift the QD energy level to
$\epsilon_{\sigma}=1.5\Gamma_{N}$ by applying the gate voltage. In this way
the destructive interference would be gradually replaced by the conventional
MZM leakage regime.

We have estimated that this transition takes approximately $\Delta
t=15[\frac{\hbar}{\Gamma_{N}}]$. Such timescale to transform one Majorana
feature to another is comparable to the time interval needed for emergence of
the zero-energy peak after abrupt coupling of the QD to topological
superconductor (see Fig. 2). Observation of such dynamical changeover could
thus indirectly probe the MZM leakage time itself. In Sec. IV we provide
quantitative evaluation of this characteristic time, having in mind the
typical energy scales in experiments using with various quantum dots coupled
to superconductors and/or topological superconductors.

##  IV Quantitative evaluations

To deliver reliable information on the timescale in some tangible units one
has to take into account the specific energy scales for the experimentally
achievable setups in analysis of the Andreev/Majorana bound states. Let us
consider a few realistic examples.

###  IV.1 Typical energy and time scales

The energy gap of conventional ($s$-wave) superconductors, which are often
used in experiments with quantum dots varies from a few tens to hundreds of
microelectronvolts. For instance, vanadium electrode used in Ref. Lee  _et
al._ (2013) was characterized by $\Delta_{S}\simeq 0.55$ meV. The energy gap
of titanium electrode used by Deacon et al. Deacon  _et al._ (2010) was about
$152$ $\mu$eV. In the present context more useful would be the proximity
induced on-dot pairing gap, which is roughly equal to $\Gamma_{S}$ Barański
and Domański (2013). Its value in experiment performed by Schönenberger et al.
Jünger  _et al._ (2019) varied from $10$ $\mu$eV to $165$ $\mu$eV, whereas in
the setup of Deng et al. Deng  _et al._ (2016) using Ti/Al its magnitude was
$250$ $\mu$eV.

To observe well pronounced subgap states the hybridization with metallic
electrode $\Gamma_{N}$ (which controls the inverse life time) should be
considerably smaller than both $\Gamma_{S}$ and $\Delta_{S}$. For this reason
we have enforced $\Gamma_{N}\ll\Gamma_{S}$, otherwise the in-gap states would
overlap with each other. In numerous experiments devoted to investigations of
the in-gap bound states, such coupling to metallic electrode $\Gamma_{N}$ was
kept about ten or hundred times smaller than $\Gamma_{S}$. For example Ref.
Lee  _et al._ (2013) reported the experimental value $\Gamma_{N}\simeq 50$
$\mu$eV. In our present approach (where $\Gamma_{N}$ is used as energy unit)
we thus assume the following realistic value $\Gamma_{N}\simeq 5-50$ $\mu$eV,
implying the time unit $\frac{\hbar}{\Gamma_{N}}\simeq 0.125-1.25$
nanoseconds. Taking such quantities into account, the time of MZM leakage into
the quantum dot region $\simeq 15[\frac{\hbar}{\Gamma_{N}}]$ (estimated for
$t_{m}=0.5\Gamma_{N}$) would be approximately $2-20$ ns. Transport
measurements have temporal resolution in sub-picosecond regime Tuovinen  _et
al._ (2019), so this dynamical process should be observable in real time.

One should note that the coupling $t_{m}$ between QD and topological
superconducting chain is expressed in terms of $\Gamma_{N}$. To reconcile the
specific influence of $t_{m}$ on the Majorana leakage time, we shall briefly
analyze in Sec. IV.2 a few representative values. As concerns a quantitative
effect of the hybridization $\Gamma_{S}$ we have checked, that its influence
on the Majorana mode leakage time onto the QD region is rather negligible.

###  IV.2 Influence of $t_{m}$

Figure 4: Evolution of the differential conductance obtained for setup shown
in Fig. 1, assuming the couplings $t_{m}=0.25\Gamma_{N}$ (upper left),
$0.5\Gamma_{N}$ (upper right), $1\Gamma_{N}$ (lower left), $1.5\Gamma_{N}$
(lower right). The quantum dot is abruptly hybridized with the Majona end-mode
at $t=20\hbar/\Gamma_{N}$. We used the model parameters
$\Gamma_{S}=3\Gamma_{N}$, $\epsilon_{\sigma}=\Gamma_{N}$. Figure 5: Blue dots
represent time evolution of zero-bias differential conductivity obtained for
$\Gamma_{S}=3\Gamma_{N}$, energy of the quantum dot
$\epsilon_{\sigma}=\Gamma_{N}$ and quantum dot - chain couplings
$t_{m}=0.25\Gamma_{N}$ (upper left), $0.5\Gamma_{N}$ (upper right),
$1\Gamma_{N}$ (lower left), $1.5\Gamma_{N}$ (lower right). Quantum dot is
abruptly connected to Majorana mode at time $t=20\hbar/\Gamma_{N}$. Red dashed
lines represent exponential fitting function with the characteristic time
scale $\tau$.

Fig. 4 shows the time-resolved differential conductance obtained for several
values of the dot-chain hybridization $t_{m}$. We clearly notice, that the
zero-energy feature develops more rapidly for stronger couplings $t_{m}$.
Besides this zero-energy mode one also observes that the Andreev quasipartice
states split into two branches. One of them (the low-energy Andreev branch) is
located at $\pm\sqrt{\Gamma_{S}^{2}+\epsilon^{2}}$. The other (high-energy
branch) arising from hybridization with the Majorana mode is formed at
$\pm\sqrt{\Gamma_{S}^{2}+\epsilon^{2}+2t_{m}^{2}}$, in agreement with the
static solution predictions Barański  _et al._ (2016); Zienkiewicz  _et al._
(2019).

To quantify the time interval needed for formation of the zero-energy Majorana
feature we have fitted (Fig. 5) a difference between the initial and final
($t=\infty$) zero-bias conductance by an exponential function

$G(0,t)=G(0,\infty)-\left[G(0,\infty)-G(0,t_{1})\right]e^{-(t-t_{1})/\tau},$
(9)

where the initial moment $t_{1}=20$. The phenomenological parameter $\tau$
characterizes the temporal interval, in which a mismatch between the initial
conductivity and the equilibrium conductance diminishes $e$-times. Values of
such numerically evaluated parameter $\tau$ indicate, that development of the
MZM features occurs the faster the stronger $t_{m}$ is. A few examples are
listed in Tab. 2.

$t_{m}[\Gamma_{N}]$ | $\tau[\hbar/\Gamma_{N}]$
---|---
0.25 | 8.2
0.5 | 6.7
0.75 | 5.4
1 | 4.6
1.5 | 3.8
Table 2: Leakage time $\tau$ obtained for several couplings $t_{m}$.

##  V Summary

We have investigated time-resolved development of the Majorana features
transmitted onto the quantum dot due to its coupling to the topological
superconductor. For its feasible detection we have considered the tunneling of
charge through a circuit, in which the quantum dot is strongly hybridized with
the bulk superconductor and weakly coupled to the normal metallic lead. Our
hybrid structure can be realized in practice, depositing the topological
nanowire (for instance a chain of magnetic $Fe$ atoms) with the side-attached
quantum dot (e.g. nonmagnetic atom) on a surface of conventional
superconductor Schneider  _et al._ (2020); Kim  _et al._ (2018). Approaching
the conducting STM tip to QD its low-energy quasiparticles could be observable
in the differential conductance, originating from the Andreev (particle-to-
hole) scattering that is efficient in the low-bias regime, smaller or
comparable to the superconducting gap.

We have evaluated the characteristic time, needed for inducing the Majorana
features in the zero-bias conductance. When the QD energy level is distant
from the chemical potential, the Majorana leakage is manifested by enhancement
of the differential conductance. In the opposite limit, when the quantum dot
level is nearby the chemical potential, the Majorana mode has detrimental
influence on the subgap spectrum of QD, producing the interferometric dip
structure. We have evaluated the time interval during which these features
emerge after: (i) abrupt coupling of the quantum dot to the topological
superconducting nanowire, and (ii) sudden change of the quantum dot energy
level by external gate potential. For empirically realistic parameters we have
found, that emergence of the Majorana features would take in both cases about
$2-20$ nanoseconds, in perfect agreement with estimations obtained by a full
counting statistics analysis Seoane Souto  _et al._ (2020). This dynamical
process should be detectable with use of the currently available state-of-the-
art tunneling spectroscopies. Such leakage seems to be fast enough to
guarantee the practical realizations of braiding protocols designed for the
Majorana quasiparticles.

In future studies it would be worthwhile to consider the correlated quantum
dot, where the superconducting proximity effect competes with the Coulomb
repulsion qualitatively affecting the in-gap bound states. In particular, they
may cross one another at, so called, zero-pi transition. Their dynamical
interplay with the Majorana mode would require more sophisticated many-body
techniques (e.g. time-dependent numerical renormalization approach) what is
beyond a scope of our study.

###### Acknowledgements.

This research has been conducted within a framework of the project Analysis of
nanoscopic systems coupled with superconductors in the context of quantum
information processing No. GB/5/2018/209/2018/DA funded in the years 2018-2021
by the Ministry of National Defence Republic of Poland (JB, MB, TZ). The work
is also supported by the National Science Centre (Poland) under the grant
2017/27/B/ST3/01911 (RT, TD).

*

##  Appendix A Equations of motion

In this appendix we present the explicit equations of motion for the momentum-
dependent $f_{ki}$ functions

$\displaystyle\frac{df_{k1}}{dt}$ $\displaystyle=$
$\displaystyle\left(i\epsilon_{\uparrow}-\frac{\Gamma_{N}}{2}\right)f_{k1}+i\frac{\Gamma_{S}}{2}f_{k11}+it_{m}(f_{k5}+f_{k6})+iV_{k}e^{i\xi_{k}t}f_{FD}[(\xi_{k}-eV),T]$
$\displaystyle\frac{df_{k2}}{dt}$ $\displaystyle=$
$\displaystyle\left(-i\epsilon_{\uparrow}-\frac{\Gamma_{N}}{2}\right)f_{k2}-i\frac{\Gamma_{S}}{2}f_{k4}-it_{m}(f_{k7}+f_{k8})$
$\displaystyle\frac{df_{k3}}{dt}$ $\displaystyle=$
$\displaystyle\left(-i\epsilon_{\uparrow}-\frac{\Gamma_{N}}{2}\right)f_{k3}-i\frac{\Gamma_{S}}{2}f_{k12}-it_{m}(f_{k5}+f_{k6})$
$\displaystyle\frac{df_{k4}}{dt}$ $\displaystyle=$
$\displaystyle\left(i\epsilon_{\downarrow}-\frac{\Gamma_{N}}{2}\right)f_{k4}-i\frac{\Gamma_{S}}{2}f_{k2}+iV_{k}e^{i\xi_{k}t}f_{FD}[(\xi_{k}-eV),T]$
$\displaystyle\frac{df_{k5}}{dt}$ $\displaystyle=$ $\displaystyle
i\epsilon_{m}f_{k5}+it_{m}(f_{k1}-f_{k3})$ $\displaystyle\frac{df_{k6}}{dt}$
$\displaystyle=$ $\displaystyle-i\epsilon_{m}f_{k6}+it_{m}(f_{k1}-f_{k3})$
$\displaystyle\frac{df_{k7}}{dt}$ $\displaystyle=$
$\displaystyle-i\epsilon_{m}f_{k7}+it_{m}(f_{k10}-f_{k2})$
$\displaystyle\frac{df_{k8}}{dt}$ $\displaystyle=$ $\displaystyle
i\epsilon_{m}f_{k8}+it_{m}(f_{k10}-f_{k2})$ $\displaystyle\frac{df_{k9}}{dt}$
$\displaystyle=$
$\displaystyle\left(-i\epsilon_{\downarrow}-\frac{\Gamma_{N}}{2}\right)f_{k9}+i\frac{\Gamma_{S}}{2}f_{k10}$
$\displaystyle\frac{df_{k10}}{dt}$ $\displaystyle=$
$\displaystyle\left(i\epsilon_{\uparrow}-\frac{\Gamma_{N}}{2}\right)f_{k10}+i\frac{\Gamma_{S}}{2}f_{k9}+it_{m}(f_{k7}+f_{k8})$
$\displaystyle\frac{df_{k11}}{dt}$ $\displaystyle=$
$\displaystyle\left(-i\epsilon_{\downarrow}-\frac{\Gamma_{N}}{2}\right)f_{k11}+i\frac{\Gamma_{S}}{2}f_{k1}$
$\displaystyle\frac{df_{k12}}{dt}$ $\displaystyle=$
$\displaystyle\left(i\epsilon_{\downarrow}-\frac{\Gamma_{N}}{2}\right)f_{k12}-i\frac{\Gamma_{S}}{2}f_{k3}$
(10)

and for the momentum-independent $f_{i}$ functions

$\displaystyle\frac{df_{1}}{dt}$ $\displaystyle=$ $\displaystyle
2\Im\left[-\frac{\Gamma_{S}}{2}f_{4}+t_{m}(f_{5}+f_{6})+S_{k1}-\frac{i\Gamma_{N}}{2}f_{1}\right]$
$\displaystyle\frac{df_{2}}{dt}$ $\displaystyle=$ $\displaystyle
2\Im\left[-\frac{\Gamma_{S}}{2}f_{4}+S_{k4}-\frac{i\Gamma_{N}}{2}f_{2}\right]$
$\displaystyle\frac{df_{3}}{dt}$ $\displaystyle=$ $\displaystyle
2\Im\left[t_{m}(f_{6}-f_{5})\right]$ $\displaystyle\frac{df_{4}}{dt}$
$\displaystyle=$
$\displaystyle[-i(\epsilon_{\uparrow}+\epsilon_{\downarrow})-\Gamma_{N}]f_{4}-i\frac{\Gamma_{S}}{2}(1-f_{1}-f_{2})-it_{m}(f_{7}+f_{8})+i(S_{k2}-S_{k11})$
$\displaystyle\frac{df_{5}}{dt}$ $\displaystyle=$
$\displaystyle\left[i(\epsilon_{\uparrow}-\epsilon_{m})-\frac{\Gamma_{N}}{2}\right]f_{5}+i\frac{\Gamma_{S}}{2}f_{7}+it_{m}(f_{3}-f_{1})+iS_{k5}^{*}$
$\displaystyle\frac{df_{6}}{dt}$ $\displaystyle=$
$\displaystyle\left[i(\epsilon_{\uparrow}+\epsilon_{m})-\frac{\Gamma_{N}}{2}\right]f_{6}+i\frac{\Gamma_{S}}{2}f_{8}+it_{m}(1-f_{1}-f_{3})+iS_{k6}^{*}$
$\displaystyle\frac{df_{7}}{dt}$ $\displaystyle=$
$\displaystyle\left[-i(\epsilon_{\downarrow}+\epsilon_{m})-\frac{\Gamma_{N}}{2}\right]f_{7}+i\frac{\Gamma_{S}}{2}f_{5}-it_{m}(f_{4}+f_{9})+iS_{k7}$
$\displaystyle\frac{df_{8}}{dt}$ $\displaystyle=$
$\displaystyle\left[-i(\epsilon_{\downarrow}-\epsilon_{m})-\frac{\Gamma_{N}}{2}\right]f_{8}+i\frac{\Gamma_{S}}{2}f_{6}-it_{m}(f_{4}+f_{9})+iS_{k8}$
$\displaystyle\frac{df_{9}}{dt}$ $\displaystyle=$
$\displaystyle\left[-i(\epsilon_{\downarrow}-\epsilon_{\uparrow})-{\Gamma_{N}}\right]f_{9}-it_{m}(f_{7}+f_{8})+i(S_{k12}^{*}-S_{k10}),$
(11)

where $S_{ki}=\sum_{k}V_{k}e^{-i\xi_{k}t}\;f_{ki}$.

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