Time-resolved energy dynamics after single electron injection into an ...

Time-resolved energy dynamics after single electron injection into an interacting helical liquid Alessio Calzona1,2 , Matteo Acciai1 , Matteo Carrega2 , Fabio Cavaliere1,2 , and Maura Sassetti1,2

arXiv:1604.03323v1 [cond-mat.mes-hall] 12 Apr 2016

1

Dipartimento di Fisica, Universit` a di Genova,Via Dodecaneso 33, 16146, Genova, Italy. 2 SPIN-CNR, Via Dodecaneso 33, 16146, Genova, Italy. (Dated: April 13, 2016)

The possibility to inject a single electron into ballistic conductors is at the basis of the new field of electron quantum optics. Here, we consider a single electron injection into the helical edge channels of a topological insulator. Their counterpropagating nature and the unavoidable presence of electron-electron interactions dramatically affect the time evolution of the single wavepacket. Modeling the injection process from a mesoscopic capacitor in presence of non-local tunneling, we focus on the time resolved charge and energy packet dynamics. Both quantities split up into counterpropagating contributions whose profiles are strongly affected by the interactions strength. In addition, stronger signatures are found for the injected energy, which is also affected by the finite width of the tunneling region. Indeed, the energy flow can be controlled by tuning the injection parameters and we demonstrate that, in presence of non-local tunneling, it is possible to achieve situation in which charge and energy flow in opposite directions. PACS numbers: 73.23.-b, 71.10.Pm, 42.50.-p

I.

INTRODUCTION

Electron-electron (e-e) interactions in one dimensional systems play a prominent role1,2 . The celebrated Fermi liquid theory dramatically fails and intriguing phenomena appear, such as the fractionalization of the charge3–7 and the spin7–11 degrees of freedom. Here, an electron injected into an interacting system splits up originating two collective excitations which carry a fraction of the electron charge and spin. In this context, many other theoretical predictions have been put forward5,6,8,12 and, recently, some of them have also been experimentally tested13–15 . Among all, it is worth to mention the direct observation of charge fractionalization in chiral conductors by means of time-resolved charge current measurements reported by Kamata et al., see Ref.14 and 16. Despite the great interest on fractionalization phenomena, up to now little attention has been devoted to the study of the energy associated to electrons injected into an interacting system. In Ref. 17 it has been shown that the DC energy current along a quantum wire is partitioned between left- and right- moving excitations, but in a distinct way with respect to that of the injected charge. In particular, it has been shown that, differently from the charge, the energy partitioning depends on the injection process and its evidence can be already tested in a DC configuration. Nevertheless, an accurate description and understanding of energy dynamics for time dependent single electron injection in an interacting system is still lacking, despite it will play an important role for the fast developing field of electron quantum optics18,19 . This very promising field relies on the possibility of injecting single electrons and holes into 1D systems. On-demand single electron sources can be experimentally realized by means of driven mesoscopic capacitors20–23 or properly designed Lorentzian voltage pulses24–27 .

Injected wave packets propagate ballistically along 1D systems such as integer quantum Hall edge states, allowing for optics-like experiments where e.g. quantum point contacts act as the analog of beam splitters. In this regard it is worth mentioning two seminal experiments, based on the chiral edge state of a ν = 2 quantum Hall system, dealing with the so-called HanburyBrown-Twiss28 and the Hong-Ou-Mandel29 effects. Different theoretical works have investigated single electron injection in chiral conductors18,22,23 and the role of e-e interactions in copropagating edge channels30–34 , aiming to the explanation of recent experimental observations. In addition, the heat and energy transport has been also considered35,36 in presence of external drive but only in absence of e-e interactions. Recently there have been suggestions that counterpropagating helical edge states of two-dimensional topological insulators (2DTI)37,38 can also be used as electronic wave guides. They can be realized in CdTe/HgTe39–41 and InAs/GaSb42–44 quantum wells. Importantly, they are topologically protected from backscattering and characterized by the so called spinmomentum locking. These features allow for a richer phenomenology, in comparison with quantum Hall-based setups45,46 , and for the study of effects related to spinentanglement47–49 , relevant for quantum computation implementations. In this context e-e interactions between counterpropagating edge channels can lead to remarkable effects, also in comparison to the case of interacting copropagating edge states of chiral conductors30–34 . A deep understanding of the role of interactions is thus of great importance in view of all these realizations. In this work we consider the on-demand injection of a single polarized electron from a quantum dot (QD) mesoscopic capacitor into a couple of interacting helical edge states, modeled as helical Luttinger liquid (HLL)45,50,51 . Our goal is to study how the presence of e-e interactions

2

B

Spin ↑ Spin ↓

σ 0

x

FIG. 1. (color online) Sketch of the setup. A mesoscopic capacitor, quantum dot, is tunnel coupled to the helical edge states of a 2DTI trough an extended tunneling region of width σ. Solid red lines refer to 1D non-interacting electron states with spin-up. Dashed blue lines are associated to spin-down electrons. By means of a top gate (in light gray) it is possible to shift the quantum dot energy levels. Their spin degeneracy can be broken with a magnetic field B present in the QD region and perpendicular to its plane.

affects the dynamics after an injection of a single electron into the edge channels of a 2DTI. In particular, we will focus on the time evolution of both charge and energy densities in the ballistic helical conductor. We will demonstrate that both quantities display fractionalization phenomena, due to the presence of e-e interactions, resulting in left- and right- moving charge and energy profiles. Interestingly, the time-evolution of the injected energy presents different features that strongly depend on the nature of the tunneling process, in sharp contrast to what happens for the charge degree of freedom. We will investigate these features, considering that the quantum dot has finite dimension and thus allowing for nonlocal tunneling process. In this case it is also possible to achieve situations in which the charge and energy packets flow in opposite directions, simply by tuning external parameters (such as gate voltages). The manuscript is organized as follows. In Sec. II we describe the setup, presenting the time dependent density matrix approach to single electron tunneling injection. The charge density is investigated in Sec. III, where charge fractionalization factors and time evolution of the single wave packet are derived. Here, the case of local injection is discussed in detail. Sec. IV is devoted to the study of the energy dynamics. Here, we focus on energy density profiles and on energy partitioning, highlighting the role of e-e interactions and finite width of the tunneling region.

screens e-e interactions in the QD region47,48 with energy levels dominated by confinement rather than Coulomb charging energy. Due to its finite dimension l, the QD has discrete single particle energy levels that come in Kramers pairs and are spaced by ∆ ∼ vF /l, where vF is the Fermi velocity (throughout this paper we set ~ = 1). The energy spectrum of the QD can be tuned and shifted (with respect to the Fermi energy EF of the whole system) by properly acting on a top gate. Moreover, spin degeneracy can be lifted47 by means of a perpendicular magnetic field B in the island region (see Fig. 1). The injection of a single polarized electron into the EC can be thus achieved with an abrupt change of the top gate potential at time t = 0. Consequently, the most energetic electron in the QD, chosen here with spin up, is suddenly brought above EF and leaves it19,47 tunneling into the helical edges. In this paper, we will assume that the spin-preserving tunneling is the dominant mechanism and we thus restrict the discussion to this case. The Hamiltonian of the whole system reads ˆ =H ˆ EC + H ˆ QD + H ˆt H

ˆ EC = H ˆ0 +H ˆ e−e . Here where the edge channels term is H the free Hamiltonian reads Z h i ˆ 0 = vF dx ψˆ† (x) i∂x ψˆL (x) − ψˆ† (x) i∂x ψˆR (x) (2) H L

R

where ψˆr (x) is the fermionic field annihilating electrons in the right- (r = R) or left- (r = L) branches. As shown in Fig. 1, we consider R-electrons (L-electrons) having spin up (down). The presence of short-range e-e interactions can be taken into account by the additional contribution Z XZ 2 ˆ e−e = g4 H dx (ˆ nr (x)) + g2 dx n ˆ R (x)ˆ nL (x), (3) 2 r with n ˆ r (x) =: ψˆr† (x)ψˆr (x) :

SETUP AND GENERAL MODEL

We consider helical edge channels (EC) of a 2DTI tunnel coupled with a quantum dot acting as a mesoscopic capacitor, as schematically shown in Fig. 1. The QD can be realized by means of metallic gates separating the island region from the 2DTI or by means of mechanical etching, in close analogy with what was done in quantum Hall based devices20 . The presence of electrostatic gates

(4)

the electron density on the r channel and gi coupling constants referring to inter- (i = 4) and intra-channel (i = 2) interactions2,52 . The QD is represented in terms of a spin-up electron level 0 , measured with respect to the Fermi energy, ˆ QD = 0 dˆ† dˆ. H

II.

(1)

(5)

This situation can be achieved by a sudden shift of the uppermost occupied electron above EF , with 0 > 0 bounded by the level spacing ∆47 . The tunneling of the spin-up electron between the dot ˆt = H ˆ t+ + H ˆ t− and the helical EC is represented by H where Z +∞ † ˆ t+ = λ ˆ t− = (H ˆ t+ )† H dy w(y) ψˆR (y)dˆ, H (6) −∞

3 respectively adds or removes a spin-up electron to the edge right channel with constant tunneling amplitude λ. The tunneling region is characterized by the envelope function w(y), whose precise shape will be specified later53,55? . We assume w(y) centered around y = 0 (the injection point) with a spatial extension given by σ. For geometrical reasons (see Fig. 1), σ is bounded by the QD dimension σ < l and thus, in view of the constraint 0 < ∆, also by σ . vF /0 . Concerning the momentum involved in the tunneling process, the bottom edge of the QD (described as a system of spin up and left moving electrons) has momentum kQD , while the right electrons in the helical EC have momentum kF . One can then define the total variation as k0 = kQD − kF . This quantity can be tuned by means of gate voltages applied to the QD and/or to the edge channels56–58 . It will be incorporated into the envelope function w(y) as a complex phase factor w(y) = ξ(y)eik0 y , with ξ(y) real. This phase factor plays a relevant role in the case of non-local tunneling, as we will show. Note that in the case of local tunneling, with σ → 0, one has w(y) = ξ(y) = δ(y).

A.

Single electron injection

In this section we model the single electron injection process. Assume that at time t = 0 the edge channels of the 2DTI are at thermal equilibrium (at temperature T ) with fixed particle number N and with an equilibrium density matrix ρˆEC (0). On the contrary, the single QD level is initially occupied and described by the density matrix ρˆQD (0) = |1ih1|. ˆ Let O(x) be a generic hermitian and numberconserving operator that acts on the EC, such as, for example, the particle density or the Hamiltonian density of the edge channels. In the interaction picture, with ˆ t , the time evorespect to the tunneling Hamiltonian H ˆ t) reads lution average of O(x, n o ˆ t)i = Tr O(x, ˆ t)ˆ hO(x, ρ(t) ,

(7)

with the time dependent density matrix ˆ (t, 0)ˆ ˆ † (t, 0) ρˆ(t) = U ρ(0)U

neling one has δO(x, t) = Z t Z = 2< dt2 0

n h io ˆ t− (t1 ) O(x, ˆ t), H ˆ t+ (t2 ) dt1 Tr ρˆ(0)H

1,N

0

(11) where the symbol Tr {. . . }1,N denotes the trace over system’s excitations with fixed particle numbers: one electron in the QD and N in the EC. In the following we will consider the low temperature limit (temperature smaller than the energy level splitting of the QD and of the energy excitations of the helical edge), setting T → 0 and thus ρˆEC (0) = |ΩN ihΩN |, with |ΩN i the N-particles EC ground state. Moreover we explicitly take into account the finite lifetime 1/2γ of the QD level19,59 by assuming the time evolution of QD ˆ = β(t)d(0) ˆ with operator d(t) β(t) = e−i0 t e−γt .

(12)

Note that, at this level, 2γ is a phenomenological parameter which describes the broadening of the QD level or the inverse lifetime of electron in the QD. Its precise value will be self-consistently evaluated in the next section and it will strongly depend on e-e interactions. In any case, the single electron injection implies that the QD level is sufficiently well-defined, with γ smaller than both the level position 0 and the spin level splitting. All these assumptions allow to express δO(x, t) as t

Z

2

δO(x, t) = |λ| 2
0) =

k0

2 2π e−a/u  a 2A− A2 ( + uk) − A2− Γ2 (A2− ) 2u (39)

A2− −1

( − uk) k

k

FIG. 2. (Color online) Sketch of the overlap between the 2 ˜ spectral function AR (k, ) (in gray) and |ξ(k)| (in red). The latter is represented with a horizontal line at energy 0 since we are considering injection of an electron with well defined energy. Panel (a): local injection (σ → 0). Panel (b): nonlocal injection (σ ∼ 2u−1 0 ) with a finite extension in k region 2 ˜ for |ξ(k)| centered around k0 .

A.

Charge fractionalization and inverse lifetime

The total amount of injected charge that travels in a given direction (η = ±) is Z +∞ Qη = dx δnη (x, t → ∞). (33) −∞

This integral can be easily performed from Eq. (32) for δnη (zη ). One finds Qη = qη Q where Z ∞ Z t2 Z Z +∞ Q = |λ|2 dt2 dt1 dy1 dy2 [Ξ G + h.c.] (34) 0

0

is the spectral function of the right edge channel1 . Recall that k and  are defined as momentum and energy with respect to kF and EF respectively. Equation (37) has a clear physical interpretation: γ represents a tunneling rate and is proportional to the overlap between the spectral function AR (k, 0 ) and the k “spectrum” of the ˜ 0 − k)|2 . In Fig. 2 one injected electron, described by |ξ(k can see this overlap in the energy and momentum space. The region where AR (k, ) 6= 0 is filled in gray, showing that in the presence of e-e interactions the spectral function broadens and does not vanish away from the mass shell ( = uk). The injected electron has a well defined ˜ 0 − k)|2 is repenergy 0 > 0 and thus the function |ξ(k resented as a red horizontal line at 0 , centered around k = k0 with an extension of the order of σ −1 . Panel (a) ˜ 0 − k)|2 ∼ 1. refers to local injection: σ → 0, with |ξ(k Here, the momentum k0 is not relevant and the overlap is along the darker red line over the gray cone. In this limit the integral in Eq. (37) can be solved analytically, giving the local rate 2A2

γ

−∞

where Q is the total amount of charge injected in the system. We thus recover the well-known5,6 expression for charge fractionalization factors Qη 1 + ηK = qη = Q+ + Q− 2

(35)

that depend only on the interaction strength K. As discussed in Appendix A all contributions to Qη are due to (b) (a) In and not to the polarization term In . For t  1/(2γ) the QD level is empty and the total amount of injected charge Q = Q+ +Q− must satisfy Q = 1. This constraint allows to self-consistently determine the inverse lifetime 2γ. To this end, we observe that Eq. (34) can be rewritten as Z Z +∞ Z Z +∞ Q = |λ|2 dt1 dt2 dy1 dy2 Ξ G . (36)

θ( − u|k|),

loc

(K¯ a) − = Kγ0 e−K¯a , Γ(1 + 2A2− )

(40)

with a ¯ = a0 /vF the dimensionless cut-off. Note that γ0 in Eq. (38) represents the asymptotic value of γ loc < γ0 in the non interacting limit K → 1. ˜ 0 −k)|2 Fig. 2(b) shows a non-local injection. Here, |ξ(k is centered around k0 , chosen in the figure to be negative, with a width ∼ σ −1 . The overlap between the two functions is significantly smaller with respect to (a) and it further reduces as long as k0 is pushed away from the gray cone. In addition, for a given interaction strength K, and momentum k0 , the overlap decreases as σ increases with the result γ < γ loc < γ0 . In order to discuss quantitative results, we consider a gaussian envelope function with extension σ 2 2 1 ξ(y) = √ e−y /σ πσ

(41)

−∞

0

Imposing now Q = 1 with the assumption γ  0 one can indeed determine (see Appendix A) Z 2 Ku γ = γ0 dk AR (k, 0 ) ξ˜ (k0 − k) , (37) 2π with γ0 =

|λ|2 . 2vF

(38)

Note that for σ → 0 we recover the point-like injection ξ(y) = δ(y), while increasing σ the injection extension increases with a decreasing amplitude. For convenience we introduce the dimensionless parameters σ ¯=

σ0 , vF

k0 vF k¯0 = . 0

(42)

The dependence of the ratio γ/γ0 on different parameters is reported in Fig. 3, where the relation γ < γ0

6

FIG. 4. (Color online) Charge current j+ (τ ) (in units of 0 ) flowing in the right direction through the detection point xD > 0 as a function of time (in units of −1 0 ). Different interaction strengths are considered: solid red line K = 1 (non-interacting case), dashed blue K = 0.8, and green dotted K = 0.6. The inset shows the function C1 (τ ) with the same color coding. Parameters: γ0 = 0.05 0 and a ¯ = 1/40.

mirrored shape δn− (−x, t) δn+ (x, t) = . q+ q−

FIG. 3. (Color online) Panel (a): ratio γ/γ0 as a function of ¯0 = −1.2. Panel interaction strength K with σ ¯ = 0.9 and k ¯0 (b): density plot of γ/γ0 as a function of σ ¯ (x-axis) and k (y-axis) with K = 0.6. In both panels a ¯ = 1/40.

(44)

As a consequence, we can focus only on the right-moving packet (η = +). We analyze the corresponding charge current j+ (τ ) = uδn+ (τ ) with τ = t − xD /u, flowing through a “detection” point xD > 0 away from the injection region. The integral over t2 in Eq. (43) can be easily performed yielding j+ (τ ) = 2q+ γ0 θ(τ ) exp [−2τ γ] < [C1 (τ )] ,

clearly emerges. Panel (a) shows the suppression of the tunneling rate as the interaction strength increases, a well-known feature of HLL. Parameter k¯0 , considered in panel (b), does not affect γ as long as local-tunneling is concerned but becomes more and more relevant as σ ¯ increases. In particular, γ significantly diminishes when k0 is pushed away from the momentum range where the spectral function AR (k, 0 ) has finite values (see also Fig. 2). B.

Charge density profile after local injection

We now focus on the local-injection limit ξ(y) = δ(y), in order to study interactions effects on the charge density profile. Integrating Eq. (32) one has Z t Z t2 qη |λ|2 δnη (x, t) = 2< dt2 dt1 β ∗ (t1 )β(t2 ) 2πau 0 0  1+2A2− (43) ηx a ) . δ(t2 − t − u a + iu(t1 − t2 ) We observe that, apart from the fractionalization factors qη , the two chiral charge density packets share the same

(45)

where (m ∈ N) 0 Cm (τ ) = π¯ am

Z

0 −sγ is0

ds e −τ

e



a ¯ a ¯ + is0 K −1

m+2A2−

.

(46) First of all we note that, because of causality, j+ (τ ) 6= 0 only for τ > 0, since an excitation created in x = 0 takes exactly a time xD /u to reach the detection point. Another clear feature is the exponential decrease e−2γτ due to the QD single level inverse lifetime (2γ). The presence of the interacting helical Fermi sea is taken into account by the function C1 (τ )61 . Fig. 4 shows all these features. The decreasing exponential behavior is clearly visible as well as the increase of the QD level lifetime (2γ)−1 as interactions strength increases. Function C1 (τ ), plotted in the inset, is characterized by a global decrease, while increasing interaction strength. It also presents oscillations with a period given by 2π−1 0 and an amplitude damped by interactions. This fact is due to the smearing of the Fermi function, which weakens the effects of the Fermi sea. Similar qualitative features are expected in the case of non-local injection, where however the pulse will be less

7 localized. Bigger effects related to the nature of the injection process manifest at the level of energy partitioning and therefore will be discussed more in detail later.

IV.

ENERGY DENSITY

The injected electron transfers into the helical edge not only charge but also energy. We then start focusing on the evaluation of the energy density (see Eq. (24)) variation, proceeding along the lines discussed in the previous Section. Considering IO=H in Eq. (15) and the commutator relation in Eq. (A5) one can derive the following expression IH =

Xu η

2

  2 † ˆ ˆ ˆ ψR (y1 , t1 ) : ∂x φη (zη ) : , ψR (y2 , t2 ) Ω

   X uηAη √π  1 a (a) (b) √ ∂ M +M =− x η η η π a2 + (zη − z2 )2 2 η (47) with D E ˆR (y1 , t1 ) φˆη (zη ) ψˆ† (y2 , t2 ) M(a) = ψ η R D EΩ † (b) ˆ ˆ ˆ Mη = ψR (y1 , t1 ) ψR (y2 , t2 ) φη (zη ) .

(48) (49)

Ω

These average functions are evaluated in Appendix C with the final result

FIG. 5. (Color online) Instant energy power P+ (τ ) (in units of 20 ) flowing through the detection point xD > 0 as a function of time τ (in units of −1 0 ). Different interaction strengths are considered: solid red line (K = 1), dashed blue K = 0.8, and green dotted K = 0.6. Inset: function CH (τ ) with the same color code for interactions. Parameters: γ0 = 0.05 0 and a ¯ = 1/40.

Eq. (51) over space with ξ(y) = δ(y) one obtains Z t Z t2 A2η |λ|2 < δHη (zη ) = dt2 dt1 β ∗ (t1 )β(t2 ) πa2 0 0 "  2A2− +1 a ηx a ∂t2 δ(t2 − t + ) 2ui a + iu(t1 − t2 ) u #  2A2− +2 a ηx + δ(t2 − t + ) . a + iu(t1 − t2 ) u (52)

   η 1 a η =u G i ∂z2 2 π a2 + (zη − z2η )2 η   (50)  a 1 1 . + π a2 + (zη − z2η )2 a + iη(zη − z1η )

Similarly to charge, the two chiral energy density packets share the same mirrored shape as long as local-injection is concerned

This formula allows to express the total energy density profile δH(x, t) in Eq. (13) as a sum P of the left and right moving contributions δH(x, t) = η δHη (zη ), with

We then focus on the right moving energy packet (η = +), by analyzing the instantaneous energy power P+ (τ ) = u δH+ (τ ) that flows through the “detection” point xD . Integration of (52) over t2 leads to (τ = t − xD /u)

IH

X

A2η

uA2η |λ|2 δHη (zη ) = < πa2 

Z

t

Z

dt2

t2

ZZ

+∞

dy1 dy2 Ξ G  a ηa + i ∂z2η δ(zη − z2η ). a + iη(zη − z1η ) 2 (51) 0

δH+ (x, t) δH− (−x, t) = . A2+ A2−

dt1

0

−∞

P+ (τ ) = A2+ γ0 0 θ(τ ) exp [−2τ γ] < [CH (τ )] ,

Energy density profile after local-injection

In order to highlight the effects of e-e interactions we start by discussing the local-injection limit. Integrating

(54)

with CH (τ ) =

A.

(53)


0 − iγ 1 C1 (τ ) + 1 − 2A2− C2 (τ ) 0 K

(55)

and Cm (τ ) (m = 1, 2) given in Eq. (46). In Fig. 5 the instantaneous energy power P+ (τ ) is plotted as a function of time for different interaction strength. As for charge current, it reflects causality, ensured by θ(τ ), and the exponential decay related to the QD level inverse lifetime

8 2γ, with analogous behaviors. The function CH (τ ) (plotted in the inset) features also a spike at τ = 0, even in the non-interacting case, reflecting the sudden turning on of the injection process and the consequent excitation of energy modes.

B.

Energy partitioning

To analyze energy partitioning phenomena, we now focus on the total amount of energy that travels in a given direction once the injection is concluded Z +∞ Eη = dx δHη (x, t → ∞) . (56) −∞

Using the expression (51) for δHη (x, t) one has Eη =

Z Z t2 Z Z +∞ uA2η |λ|2 +∞ dt dt1 dy1 dy2 2 2πa2 0 0 −∞ i (57) h 2πG(z2+ −z1+ )gη+ 2πG(z1− −z2− )gη− + h.c. , e Ξ e

where gη± = A2± + (1 ± η)/2. In passing we note that (b)

the term IH , present in Eq. (15), does not contribute to this integrated quantity62 . The above expression can be conveniently represented in Fourier space (similarly to what done in Appendix B) as Eη =

2A2−

  Z +∞ KA2η γ0 K¯ a 1 d+ 2π 20 Γ(gη− )Γ(gη+ ) 0 Z ++ 2 + ˜ −K¯a +0 d− (+ + − )gη −1 β( ) e +

(58)

−+

(+ − − )

gη− −1

2 ˜ ξ (k0 − − /u) .

The key quantities to discuss are the energy partitioning factors defined as pη =

Eη . E+ + E−

(59)

They indeed represent the fraction of the total energy E = E+ + E− that propagates in the direction η = ±. Concerning the total contribution E = E+ + E− , we demonstrate in Appendix D that E = 0 as long as γ  0 . ˜ In the local injection limit ξ(k) = 1 one has (see Appendix D) ploc η =

A2−

FIG. 6. (Color online) Energy partitioning factor p+ as a ¯0 = 0, function of the interaction strength K. In panel (a) k with σ ¯ → 0 (solid red), σ ¯ = 2 (dashed blue), σ ¯ = 3 (dotted green) and σ ¯ = 3.75 (dot dashed orange). Panel (b) shows ¯0 = −1.2 with σ → 0 (solid red), σ k ¯ = 0.9 (dashed blue), σ ¯ = 1.5 (dotted green) and σ ¯ = 1.95 (dot dashed orange). Parameters: γ0 = 0.05 0 and a ¯ = 1/40.

A2η (1 + ηK)2 = . 2 + A+ 2(K 2 + 1)

(60)

Namely, energy partitioning has a “universal” character, i.e. ploc does not depend on injection parameters but η only on interaction strength, in agreement with the partitioning of DC energy transport found in Ref. 17. On the other hand, it can be shown that such universality

breaks down as the tunneling region increases. In order to quantitatively highlight this deviation we present below results for the right moving energy fraction p+ in Eq. (59), using the gaussian envelope ξ(y) (see Eq. (41)). Fig. 6 shows two representative cases of energy partitioning as a function of interaction strength. The “universal” limit ploc + (60) is drawn with a solid red line. Panel (a), has k¯0 = 0, and shows deviations from the universal limit as σ ¯ increases, with 0.5 < p+ (K) < ploc + (K). These deviations are even more striking for negative values of k0 as shown in panel (b) with k¯0 = −1.2. Here, it is even possible to achieve p+ (K) < 0.5 for a wide range of interaction strength (dot-dashed curve). This means that, due to interactions and non local tunneling, an electron injected into the right branch, creates an energy packet that mostly travels to the left, while the charge still continues to move mainly to the right (q+ > q− ). Fig. 7 represents the cartoon of this opposite charge and energy propagation. To clarify the physical interpretation of this effect, we consider in Fig. 8 the energy partitioning factor p+ as a function of σ ¯ for dif-

9 Energy

u

Charge

u

Injection region

x

FIG. 7. (Color online) Cartoon showing the strong direction separation of energy (solid green) and charge (dashed orange) ¯0 = −1.2. The majority of charge for K = 0.8, σ ¯ = 1.95 and k (80%) travels to the right while most of the energy (about 65%) moves to the left.

ferent interaction strength. In panel (a) k¯0 = 0 while in panel (b) k¯0 = −1.2. For σ ¯ → 0 one recovers the “universal” behavior, while deviations from it become relevant as σ ¯ increases and reaches σ ¯ & 1. Comparing the two panels, note that these deviations emerge at smaller σ ¯ when k¯0 is significantly different from k¯0 = 0. This fact can be understood considering again the overlap between the spectral function AR (, k) and the injected electron ˜ 0 − k)|2 represented as insets momentum “spectrum” |ξ(k of the two main panels in Fig. 8. Here, we sketched two typical situations with the same interaction and momentum k¯0 as given in the main panel. Non-universal effects appear only when the red line does not cover the whole gray region, whose extension at  = 0 is given by 20 K/vF (see Eq. (39)). Therefore, if one considers k¯0 = 0 (panel (a)) it is necessary σ ¯ & K −1 in order to brake the energy partitioning universality. By contrast, for a negative k¯0 = −1.2 (panel (b)), a smaller σ ¯ will be required since the overlap is already smaller. Note that all these deviations are much less pronounced (and then not shown) for k¯0 > 0 since even with extended tunneling, the transferred momentum lies near the right electron branch, leading to p+ (K) > ploc + . As a last comment, the non-interacting limit K → 1 shows always p+ = 1, regardless of all the other parameters. Energy partitioning is indeed a manifestation of e-e interactions and so, if they’re absent, all the energy added to the system after an R-electron injection goes to the right. In the end, we want to stress that the condition σ ¯ ∼ 1, although challenging, is consistent with the boundary imposed by the setup. Non-universal features of energy partitioning can thus play an important role when a nonlocal injection is concerned. In particular, it is possible to directly control the energy flow after a single electron injection, being able even to invert its direction with respect to the charge flow.

V.

CONCLUSIONS

In this work we have investigated the injection process of a single electron from a mesoscopic capacitor into the counterpropagating edge states of a topological insula-

FIG. 8. (Color online) Energy partitioning factor p+ as a function of the tunneling region width σ ¯ . Each line refers to different interaction parameter: K = 0.8 (solid red), K = 0.6 ¯0 = 0 (dashed blue) and K = 0.5 (dotted green). In panel (a) k ¯0 = −1.2. The insets show the overlap, at the while panel (b) k same interaction strength, between the edge spectral function (in gray) and the momentum “spectrum” of the injected elec¯0 is tron (in red), along the lines of Fig. 2. The momentum k the same of the hosting panel. Parameters: γ0 = 0.05 0 and a ¯ = 1/40.

tor. Particular attention has been devoted to the role played by e-e interactions and how their presence affects the dynamics of both charge and energy density. We have presented a time-dependent density matrix formalism to evaluate their time evolution after a single electron injection. The charge and energy profiles have been analyzed in presence of local and non-local tunneling. Fractionalization phenomena, due to interactions, have been discussed, elucidating the differences between charge and energy. We have found that the latter is strongly affected not only by interactions but also by the nature of the tunneling process itself. Indeed, we have shown that in presence of non local tunneling from a mesoscopic capacitor, it is possible to have situations in which charge and energy profiles flow in opposite directions and are completely decoupled. These results shed new lights on the single electron injection into an interacting system, with relevant implications for the field of electron quantum optics.

10 ACKNOWLEDGMENTS

We acknowledge the support of the MIUR-FIRB2012 Project HybridNanoDev (Grant No.RBFR1236VV), EU FP7/2007-2013 under REA grant agreement no 630925 – COHEAT, MIUR-FIRB2013 – Project Coca (Grant No. RBFR1379UX) and the COST Action MP1209.

This Appendix is devoted to the evaluation of the average function D h iE † In = In(a) − In(b) = ψˆR (y1 , t1 ) n ˆ (x, t), ψˆR (y2 , t2 ) , Ω

(A1) defined in Eq. (15) and necessary in order to compute the density variation δn(x, t) in Eq. (13). Let us start with the commutator in (A1), which can be written in terms of chiral fields as r i h i K X h ˆ † † ˆ η ∂x φη (zη ), ψˆR (y2 , t2 ) , n ˆ (x, t), ψR (y2 , t2 ) = − 2π η (A2) with zη = x − ηut. Using the bosonized expression (19) with (20) one has h i † ∂x φˆη (zη ), ψˆR (y2 , t2 ) = i √ − + 1 h ˆ ˆ ˆ =√ ∂x φη (zη ), ei 2π(A+ φ+ (z2 )+A− φ− (z2 )) 2πa (A3) with the boson fields satisfying c-number commutation relations2,52 h i 1 a ∂x φˆη (x), φˆη0 (y) = iη δη,η0 . (A4) 2 π a + (x − y)2 This allows to use the Baker Hausdorff relation2 among ˆ ˆ two h operators i h Aiand B (with a c-number commutator) ˆ ˆ B B ˆ ˆ ˆ A, e = A, B e , arriving to i † ∂x φˆη (zη ), ψˆR (y2 , t2 ) =   √ 1 a † = −ηAη 2π ψˆR (y2 , t2 ) . π a2 + (zη − z2η )2 (A5)

Then, using Eq. (25), we arrive at h i † n ˆ (x, t), ψˆR (y2 , t2 ) =  X 1 a † = qη ψˆR (y2 , t2 ) . 2 + (z − z η )2 π a η 2 η

Ω

with G the bosonic Green function defined in Eq. (30). In writing Eq. (32), as long as a is the smallest length scale, it is possible to approximate 1 a → δ(zη − z2η ). π a2 + (zη − z2η )2

Appendix A: Calculation of In

h

As a final step the fermionic Green function G = † hψˆR (y1 , t1 )ψˆR (y2 , t2 )iΩ , is expressed using the identity2 E D   ˆ ˆ e−iαφη (x) eiαφη (y) = exp α2 G(−η(x − y)) , (A7)

(A6)

The average function In in Eq. (A1) is then given by   X 1 a † In = qη hψˆR (y1 , t1 )ψˆR (y2 , t2 )iΩ 2 + (z − z η )2 π a η 2 η=±1

(A8)

Finally, we comment on the role played by the term D E † In(b) = − ψˆR (y1 , t1 )ψˆR (y2 , t2 )ˆ n(x, t) , (A9) Ω

present in Eq. (15) for the evaluation of the total amount of charge Qη which travels in the direction η after the injection. As shown in Eq. (33), it is obtained integrating the chiral charge density δnη (zη ) over the whole system. Since one has Z +∞ dx n ˆ (x, t) |Ωi = 0, (A10) −∞ (a)

it turns out that all contributions to Qη are due to In only. Appendix B: Determination of the tunneling rate

In this Appendix we derive the integral representation (37) for the tunneling rate γ, starting from the expression of the total charge given in Eq. (36). Let us consider the Fourier representation63 Z Ez Ea 1  a g +∞ 2πgG(z) e = dE E g−1 e−i u e− u . (B1) Γ(g) u 0 This allows to express the fermionic function G in Eq. (31) in terms of its Fourier transform  a 1+2A2− Z Z +∞ 1 1 G= dE1 dE2 2πa Γ(A2− )Γ(A2+ ) u 0 E1 +E2 (B2) A2 A2 −1 E1 − E2 − e−a u e−it1 (E1 +E2 ) eit2 (E1 +E2 ) e−iy1

E2 −E1 u

eiy2

E2 −E1 u

.

We also introduce the Fourier transform of functions β(t) and w(y) Z +∞ 1 ˜ β(E) = dtβ(t)eiEt = , (B3) i(E − 0 ) + γ 0 Z +∞ w ˜ (k) = dy w(y) eiky = −∞ (B4) Z +∞ ˜ + k0 ) . = dy ξ(y) eiy(k+k0 ) = ξ(k −∞

11 Now we can rewrite Q in terms of a double integral over energies  a 1+2A2− 1 |λ|2 Q= 2πa Γ(A2− )Γ(A2+ ) u   2 Z Z +∞ E2 − E1 A2− A2− −1 dE1 dE2 E1 E2 ˜ w u 0 2 E1 +E2 ˜ β(E1 + E2 ) e−a u . (B5) Moreover, since the energy level of the dot should be well defined, with a long lifetime, we can assume γ  0 , verifying “a posteriori” the validity of this assumption. In this limit one can perform the following approximation on the Fourier transform of the time dependent dot function 2 1 π ˜ → δ(E − 0 ) . (B6) β(E) = 2 2 γ + (E − 0 ) γ Inserting this δ-function in Eq. (B5) we are left with a single integral  a 2A2− 1 |λ|2 −a 0 1 0 e u 2 2 2 Q= γ 2u A− Γ (A− ) 2u Z +1
2 2 2 dχ (1 + χ)A− (1 − χ)A− −1 w ˜ −0 χu−1 , −1

(B7) ˜ thus proving Eq. (37). Note that if ξ(k) = 1 (local tunneling) the above integral can be done yielding Z +1 2 2 dχ (1 + χ)A− (1 − χ)A− −1 = −1 2

= 22A−

Z

2

1

dx 0

xA− 1−A2−

(1 − x)

2

= 22A−

A2− Γ2 (A2− ) Γ(1 + 2A2− )

where we have used Eq. (19) and the identity ˆ φˆη (x, t) = −i∂ν eiν φη (x,t) ν=0 .

(C3)

By means of the Baker-Hausdorff identity, one can rewrite √ ± ± ± ± M(a) η (zη , z1 , z2 ) = −iAη 2π G(z1 , z2 ) (G (ηzη − ηz1η ) − G (ηz2η − ηzη )) (C4) where the bosonic Green function G and the fermionic correlation function G have been defined in Eq. (30) and (b) in Eq. (31) respectively. It is easy to show that Mη has the same expression apart from a different sign in the argument of the second bosonic Green function. As a result one has √ η ± ± (b) M(a) η + Mη = −iAη 2π G(z1 , z2 ) [2G (ηzη − ηz1 ) −G (ηzη − ηz2η ) − G (ηz2η − ηzη )] . (C5) Eq. (50) can now be readily obtained simply taking the derivative of functions G. Appendix D: Behavior of Eη

In this Appendix we present details on the energy Eη that travels in the η direction. Such quantity, defined in Eq. (56) is expressed as in Eq. (58). Let us start to discuss the total energy E = E+ + ˜ + )|2 → E− . In the limit γ  0 we can approximate |β( δ(+ − 0 ) π/γ, see Eq. (B6), writing  2A2− Z +0 A2η Kγ0 K¯ a −K¯ a Eη = e d− 2γ 20 Γ(gη− )Γ(gη+ ) −0 +

−

2

(0 + − )gη −1 (0 − − )gη −1 |w ˜ (−− /u)| .

that allows to obtain Eq. (40) valid in the case of local injection with σ → 0.

(D1) Recalling that gη± = A2± + (1 ± η)/2, one then has

Appendix C: Calculation of IH

Here we evaluate the average function D h iE ˆ t), ψˆ† (y2 , t2 ) IH = ψˆR (y1 , t1 ) H(x, R

(C1)

Ω

demonstrating the validity of Eq. (50), necessary in order to evaluate the energy density fluctuations δH(x, t). (a/b) In particular we have to compute functions Mη , in(a) troduced in Eq. (47). Focusing on Mη we get D E ˆ ˆ ˆ† M(a) η = ψR (y1 , t1 ) φη (zη ) ψR (y2 , t2 ) Ω i D −i√2πA−η φˆ−η (z1−η ) i√2πA−η φˆ−η (z2−η ) E = − e e D2πa√ E Ω √ ˆη (z η ) iν φ ˆη (zη ) i 2πAη φ ˆη (z η ) −i 2πAη φ 1 2 ∂ν e e e Ω ν=0

(C2)

Kγ0 γ



K¯ a 2

2A2−

1 e−K¯a A2− Γ2 (A2− ) Z +1
2 2 2 dχ (1 + χ)A− (1 − χ)A− −1 w ˜ −0 χu−1 .

E = 0

−1

(D2) By comparing this result with the behavior of the total charge Q in Eq. (B7) we can conclude that E = 0 Q = 0 , since Q = 1. The “universal” limit present for local ˜ injection (ξ(k) → 1) and given in Eq. (60) is recovered using in Eq. (58) the relation + − Z ++ (+ + − )gη −1 (+ − − )gη −1 = d− Γ(gη− )Γ(gη+ ) −+ (D3) 2 (2+ )1+2A− = . Γ(2 + 2A2− )

12 We therefore have that Eη = A2η E with 

E=

independent of η. Such an expression immediately allows to recover the energy partitioning factors ploc η given in Eq. (60).

2A2−

Kγ0 K¯ a 1 π 0 Γ(2 + 2A2− ) Z ∞ 2 + 1+2A2 ˜ + ) e−K¯a 0 d+ + − β(

(D4)

0

1 2 3

4

5 6 7

8 9 10 11 12 13

14 15

16 17

18 19

20

21

22

23 24

25 26

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Note that all physical observables are obtained in the limit a → 0, where one can set ~vF /a as the highest energy scale in the problem, of the order of the energy gap for a 2DTI (20 − 30 meV). It is possible to show that, for K = 1 and in the limit 0 → ∞ (where one can actually forget about Fermi sea) function C1 (τ ) does not contribute to j+ (τ ) since one has < [C1 (τ )] ∼ 1. R +∞ ˆ ˆ This can be easily shown since one has −∞ dx H(x, t) = H R (b) ˆ and H|Ωi = 0. As a result dx IH = 0. L. Vannucci, F. Ronetti, G. Dolcetto, M. Carrega, and M. Sassetti, Phys. Rev. B 92, 075446 (2015).