Quantum-to-classical crossover in single-electron emitter Y . Yin1 1Department of Physics Sichuan University Chengdu Sichuan 610065 China

2025-04-29 0 0 1.37MB 17 页 10玖币

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Quantum-to-classical crossover in single-electron emitter

Y. Yin1, ∗

1Department of Physics, Sichuan University, Chengdu, Sichuan, 610065, China

(Dated: November 17, 2022)

We investigate the temperature-driven quantum-to-classical crossover in a single-electron emitter. The emitter

is composed of a quantum conductor and an electrode, which is coupled via an Ohmic contact. At zero temper-

ature, it has been shown that a single electron can be injected coherently by applying an unit-charge Lorentzian

pulse on the electrode. As the electrode temperature increases, we show that the electron emission approaches

a time-dependence Poisson process at long times. The Poissonian character is demonstrated from the time-

resolved full counting statistics. In the meantime, we show that the emission events remain correlated, which is

due to the Pauli exclusion principle. The correlation is revealed from the emission rates of individual electrons,

from which a characteristic correlation time can be extracted. The correlation time drops rapidly as the elec-

trode temperature increases, indicating that correlation can only play a non-negligible role at short times in the

high-temperature limit. By using the same procedure, we further show that the quantum-to-classical crossover

exhibits similar features when the emission is driven by a Lorentzian pulse carrying two electron charge. Our

results show how the electron emission process is affected by thermal ﬂuctuations in a single-electron emitter.

I. INTRODUCTION

The on-demand control of coherent electron emission in

mesoscopic conductors has attracted much interest in recent

decades [1, 2]. In a simple setup, the electrons are emitted

from an electrode into a conductor though an Ohmic contact,

which are driven by voltage pulses V(t). For a single-channel

quantum conductor, the current I(t)follows instantly to V(t)

as I(t) = e2V(t)/h, with ebeing the electron charge and h

being the Planck constant. However, the details of the emis-

sion process can be quite different in quantum and classical

limits.

In the quantum limit, the current is due to the coherent

emission of electrons or holes, which are usually accompa-

nied by neutral electron-hole pairs. Their wave functions are

well-deﬁned, which can be extracted by quantum tomogra-

phy methods [3–7]. In particular, by using an unit-charge

Lorentzian pulse V(t)with time width W,i.e., V(t) =

hW/[πe(W2+t2)], a single electron can be emitted on top of

the Fermi sea without accompanied electron-hole pairs [8, 9].

It has been called a leviton, whose wave function can be given

as ψL(t) = pW/(2π)/(t+iW/2). The emission proba-

bility density of the leviton follows instantly to the voltage

pulse as |ψL(t)|2=I(t)/e =eV (t)/h, indicating an excel-

lent synchronization between the electron emission and driv-

ing voltage. In contrast, the current corresponds to the in-

coherent emission of electrons and/or holes in the classical

limit. This typically occurs at high temperatures, when ther-

mal ﬂuctuations in the electrode can play an important role.

The emission events can be treated as random and indepen-

dent, which follow Poisson statistics. In this limit, the voltage

pulse does not control the emission probability density of in-

dividual electrons. Instead, it decides the overall emission rate

of the emission process.

One expects that the quantum to classical crossover occurs

as the temperature of the electrode increases. Indeed, the

∗Author to whom correspondence should be addressed; yin80@scu.edu.cn.

many-body quantum state of the emitted electrons can evolve

from a pure state into a mixed one due to thermal ﬂuctuations

[10, 11]. This can lead to a reduction of the dc shot noise

[12–14]. But the electron anti-bunching is robust against tem-

perature, which has been revealed from the Hong-Ou-Mandel

interference [12]. This indicates that the quantum coher-

ence between different electrons is still preserved and hence

the temperature-induced quantum-to-classical crossover is in-

complete. In the case of dc driving when V(t) = V0, this

has been clearly demonstrated from the waiting time distribu-

tion W(τ)(WTD), which gives the distribution of time delays

τbetween successive electrons [15–19]. At zero tempera-

ture, W(τ)follows the Wigner-Dyson distribution, which has

a zero dip around τ= 0 and a Gaussian tail as τ→+∞[16].

As the electrode temperature Tincreases, the tail approaches

an exponential distribution when Tis comparable to eV0/kB,

with kBbeing the Boltzmann constant. In contrast, the es-

sential shape of W(τ)remains unchanged, especially around

the zero dip [17]. This implies that the emission approaches a

Poisson process at long times, but the emission events are still

correlated at short times.

However, the crossover can have a different nature when the

emission is driving by the Lorentzian pulse. First, the pulse

width Wprovides an important time scale in this case. In

fact, it decides the crossover temperature, which separates the

high- and low-temperature regions for the dc shot noise [10].

Secondly, the full counting statistics (FCS) changes drasti-

cally as the temperature increases: In the classical limit, the

emission tends to follow the Poisson statistics. While in the

quantum limit, the voltage pulse injects exactly one electron

into the quantum conductor. This make the WTD less suitable

for the study of emission in this limit. Finally, as the emis-

sion is triggered by a time-dependent voltage, it is helpful to

elucidate the relation between the electron emission and the

driving voltage, which is also absent from the WTD.

To solve this problem, in this paper we discuss the

quantum-to-classical crossover by combining the time-

resolved full counting statistics and electron emission rates.

The time-resolved FCS Pn(t)gives the probability of emitting

nelectrons up to a given time t, which can be extracted from

arXiv:2210.14497v2 [cond-mat.mes-hall] 16 Nov 2022

}

n=1

n=2

n=3

...

−4−2 0 2 4

0.0

0.5

1.0

𝑃𝑛(𝑡)

(b)

𝑇/𝑇𝑊=0.01

−4−2 0 2 4

𝑡/𝑊

(c)

𝑇/𝑇𝑊=0.1

−4−2 0 2 4

(d)

𝑇/𝑇𝑊=1.0

−4−2 0 2 4

𝑡/𝑊

0.0

0.5

1.0

1.5

𝜆1(𝑡)𝑊

(e)

𝑇/𝑇𝑊=0.01

𝑇/𝑇𝑊=0.1

𝑇/𝑇𝑊=1.0

𝜆𝑞(𝑡)

𝜆𝑐(𝑡)

−4−2 0 2 4

−4

−2

𝑡1/𝑊

(f)

−4−2 0 2 4

(g)

−4−2 0 2 4

(h)

−4−2 0 2 4

0.0

0.5

1.0

𝜆2(𝑡|𝑡1)𝑊

(i)

−4−2 0 2 4

𝑡/𝑊

(j)

−4−2 0 2 4

(k)

𝜆𝑐(𝑡)

𝜆𝑞(𝑡)

FIG. 1. (a) Illustration for the real-time electron counting mea-

surements. A single measurement is represented by a sequence of

random points in the time trace. The k-th point in each time trace

represents the emission time of the k-th electron. The emission of

the k-th electron is governed by the corresponding emission rate

λk(t|t1,...,tk−1). In the ﬁgure, we mark the emission of the ﬁrst,

second and third electrons by the red, green and blue dots, which

correspond to the emission rate λ1(t),λ2(t|t1)and λ3(t|t1, t2), re-

spectively. The time-resolved FCS Pn(t)is obtained from the statis-

tics gathered from a large number of measurements. (b-d) The time-

resolved FCS Pn(t)at different temperatures T/TW= 0.01 (b),

T/TW= 0.1(c) and T/TW= 1.0(d). The red solid, green dashed

and blue dotted curves correspond to n= 0,1and 2, respectively.

The thin gray solid curves correspond to the quantum limit from

Eq. (1). The thin black dotted curves represent the classical limit

from Eq. (2). (e) The emission rate λ1(t)of the ﬁrst electron at

different temperatures. The gray solid and black dotted curves rep-

resent the quantum and classical limit for the emission rate. See text

for details. (f-h) The emission rate λ2(t|t1)of the second electron

at different temperatures T/TW= 0.01 (f), T/TW= 0.1(g) and

T/TW= 1.0(h). They are shown as contour plots in the t-t1plane.

(i-j) The emission rate λ2(t|t1)as a function of tfor three typical t1

at different temperatures. The red solid, green dash-dotted and blue

dashed curves correspond to t1/W =−2.0,−1.0and 0.0, while the

red dots, green squares and blue triangles represent the point t=t1,

respectively. The classical and quantum limit of the emission rates

λq(t)and λc(t)are also plotted by the gray solid and black dotted

curves for comparison.

the statistics gathered on a large number of measurements

[37]. Each single measurement is represented by a random

time sequence {t1, t2, . . . , tk,...tn}, where each tkstands

for the emission time of the k-th electron [See Fig. 1(a) for

illustration]. The emission of the k-th electron is governed by

the corresponding emission rate λk(t|t1, t2, . . . , tk−1), which

describes the expected number of electrons emitted in a given

inﬁnitesimal time interval [t, t +dt]. As the emission events

are generally correlated in quantum conductors [20], the emis-

sion rate depends not only on the time t, but also on the emis-

sion times {t1, t2, . . . , tk−1}of all previously emitted elec-

trons.

The typical temperature-dependence of Pn(t)is illustrated

in Fig. 1(b-d). In the quantum limit when the temperature T

is very low, Pn(t)can be well-approximated from the wave

function of leviton as

Pn(t) = 









1−Rt

−∞ dτ |ψL(t)|2, n = 0

−∞ dτ |ψL(t)|2, n = 1

0, n ≥2

.(1)

The quantum limit of Pn(t)is shown by the gray solid curves

in the ﬁgure. At high temperatures, Pn(t)tends to follow the

classical limit, which corresponds to the time-dependent Pois-

son distribution

Pn(t) = hRt

−∞ dτλc(τ)in

n!exp −Zt

−∞

dτλc(τ),(2)

where the emission rate λc(t)is decided by the driving pulse

as λc(t) = eV (t)/h. The classical limit of Pn(t)is plotted by

the black dotted curves. The ﬁgure shows that Pn(t)evolves

from the quantum limit to the classical limit as the tempera-

tures Tapproaches TW=~/(kBW), which providing a clear

signature of the quantum-to-classical crossover.

The behavior of Pn(t)suggests that the emission rates of all

the electrons should approach the classical limit λc(t)at high

temperatures. We ﬁnd that this only holds for the emission

rate of the ﬁrst electron λ1(t), which is illustrated in Fig. 1(e).

The ﬁgure shows that λ1(t)evolves from the quantum limit

λq(t) = V(t)/[1 −Rt

−∞ dτV (τ)] (gray solid curve) to the

classical limit λc(t)(black dotted curve) as the temperature T

approaches TW. However, other emission rates do not fully

follow this behavior. This can be seen from the emission rate

of the second electron λ2(t|t1), which are displayed as con-

tour plots in the t-t1plane [Fig. 1(f-h)]. As the emission of the

second electron is the coeffect of the driving pulse and ther-

mal ﬂuctuations, it remains rather small at low temperatures

[Fig. 1(f)]. At moderate temperatures [Fig. 1(g)], it depends

signiﬁcantly on both tand t1, indicating strong correlations

between the emission of the ﬁrst and second electrons. In par-

ticular, λ2(t|t1)always drops to zero as tapproaches t1. This

is better demonstrated in Fig. 1(j), where the points t=t1for

three typical t1are marked by the red dots, blue triangles and

green squares. This indicates that the emission of the second

electron is coupled to the ﬁrst one due to the Pauli exclusion

principle. At high temperatures [Fig. 1(h)], the correlations re-

main pronounced, but can only be seen at short times. We ﬁnd

that λ2(t|t1)tends to follow Θ(t−t1)λc(t)in the high temper-

ature limit. This can be better seen from Fig. 1(k), where λc(t)

is plotted by the black dotted curve for comparison. These be-

haviors show that, as the electrode temperature increases, the

electron emission approaches a time-dependent Poisson pro-

cess at long times, but the correlations between the emission

events are always preserved at short times. By using a simi-

lar procedure, we ﬁnd that similar behaviors can also be seen

when the emission is driven by a Lorentzian pulse carrying

two electron charge.

This paper is organized as follows. In Sec. II, we introduce

a procedure to calculate the time-resolved statistics and emis-

sion rates for the electron emission in quantum conductors. In

Sec. III, we demonstrate the procedure by studying the sin-

gle and two electron emissions at zero temperature. The ﬁnite

temperature effect is discussed in Sec. IV. We summarize our

results in Sec. V.

II. TIME-RESOLVED STATISTICS OF A QUANTUM

EMITTER

The statistics of the electron emission process can be de-

scribed by two equivalently ways. On one hand, it can be

characterized by n-fold delayed coincidences, which gives the

joint probability that one electron is emitted in each inﬁnites-

imal time interval [ti, ti+dt](with i= 1,2, . . . , n). It can be

expressed as

gn(t1, . . . , tn) = ha†(t1). . . a†(tn)a(tn). . . a(t1)i

− ha†(t1). . . a†(tn)a(tn). . . a(t1)i0,(3)

where a†(t)and a(t)represent the creation and annihilation

operations of electrons, respectively. Here the two angular

brackets h. . . iand h. . . i0denote the quantum-statistical aver-

age over the many-body electron states with and without the

emitted electrons, respectively. In doing so, one excludes the

contribution from the undisturbed Fermi sea [4, 21]. The coin-

cidences functions are just the diagonal part of the Glauber’s

correlation functions [22–27], which can be extracted from

the current correlation measurements [3, 4, 28].

In contrast, the emission process can also be investigated by

real-time electron counting techniques [29–33]. This can be

used to extract the information of electron process in single-

shot experiments. In this case, the emission process can be de-

scribed by recording each individual emission event in a time

trace [See Fig.1(a) for illustration]. This allows us to repre-

sent emission events by random points in a line. One usually

further assumes that two emission events cannot occur exactly

at the same time, i.e., there can only exist at most one emis-

sion event in an arbitrary inﬁnitesimal time interval [t, t +dt].

This assumption has been proved to be valid for typical emis-

sion processes, such as photon emission in quantum optics

and neuronal spike emission in neuroscience [24, 34–36].

The emission process can be characterized by the time-

resolved statistics of these events. For example, it can be char-

acterized by the time-dependent counting statistics Pn(ts, te),

which gives the probability to emit nelectrons in a given

time interval [ts, te]. It can also be described by statistics of

times, such as idle time distribution Π0(ts, te), which gives

the probability that no electron is emitted in the time inter-

val [ts, te]. The WTD can be obtained from Π0(ts, te)by its

second derivative [16]. Note that all these quantities usually

depend on two times tsand te, which can be treated as the

starting and ending times of the electron counting measure-

ments.

It is inconvenient to describe the time-resolved statis-

tics directly in terms of the n-fold delayed coincidences

gn(t1, . . . , tn). In contrast, the exclusive joint probability

density fn(t1, . . . , tn;ts, te)(with n≥1) has been intro-

duced [24, 38]. It gives the joint probability that one electron

is emitted in each inﬁnitesimal time interval [ti, ti+dt](with

i= 1,2, . . . , n and all ti∈[ts, te]), while no other electron

is emitted in the time interval [ts, te]. It has been shown that

fn(t1, . . . , tk;ts, te)can be related to gn(t1, . . . , tn)as

fn(t1, . . . , tn;ts, te) =

+∞

k=n

(−1)k−n

(k−n)! Zte

dtn+1... Zte

dtkgk(t1, . . . , tn, tn+1, . . . , tk).(4)

This relation is derived from the deﬁnition of

fn(t1, . . . , tn;ts, te)and gn(t1, . . . , tn), which is inde-

pendent on the nature of the emitted particles [24]. So it can

be applied to both Bosons and Fermions.

All the time-resolved statistics can be obtained from

fn(t1, . . . , tn;ts, te)in a direct way. In particular, the FCS

Pn(ts, te)for n≥1can be given as

Pn(ts, te) = 1

n!Zte

dt1· · · Zte

dtnfn(t1, t2, . . . , tn;ts, te).

(5)

For n= 0, the corresponding FCS P0(ts, te), or equivalently

the idle time distribution Π0(ts, te), can be obtained via the

normalization relation

Π0(ts, te) = P0(ts, te)=1−

+∞

n=1

Pn(ts, te).(6)

Comparing to the FCS, the exclusive joint probability den-

sities fn(t1, . . . , tn;ts, te)contain much detailed informa-

tion on the emission process. In particular, one can ex-

tract the emission rate of each individual electron from them

[36, 38, 39]. This can be better understood by starting from

a stationary Poisson process with emission rate λ0. In this

case, the emission events are random and independent. The

corresponding coincidence function can be simply given as

gk(t1, . . . , tn) = λn

0. From Eq. (4), one ﬁnds that

fn(t1, . . . , tn;ts, te) = λn

0exp[−λ0(te−ts)],(7)

whose FCS follows the well-known Poisson statistics

Pn(ts, te) = [λ0(te−ts)]n

n!exp[−λ0(te−ts)].(8)

The corresponding idle time distribution can be expressed as

Π0(ts, te) = exp[−λ0(te−ts)].

The physical meaning of Eq. (7) can be better seen by

rewriting it in the form

fn(t1, . . . , tn;ts, te) = exp[−λ0(t1−ts)]

n−1

i=1

{λ0exp[−λ0(ti+1 −ti)]}

×λ0exp[−λ0(te−tn)].(9)

Here each λ0gives the emission probability of an electron

in each inﬁnitesimal time interval [ti, ti+dt]. The exponen-

tial factors are just idle time distributions, they ensure that no

other electron can be emitted between these inﬁnitesimal time

intervals. From the above expression, one ﬁnds that the emis-

sion rate can be obtained from fn(t1, . . . , tn;ts, te)as

λ0=fn+1(t1, . . . , tn, tn+1;ts, tn+1)

fn(t1, . . . , tn;ts, tn).(10)

Alternatively, it can also be related to the idle time distribution

λ0=f1(t1;ts, t1)

Π0(ts, t1).(11)

In general cases, the emission rate of an electron has a

more complicated form. On one hand, it depends explic-

itly on the time t, which is due to the time-dependence of

the driving voltage. On the other hand, it also depends on

the history of the emission process, which can be induced by

the quantum coherence between electrons [20]. So the emis-

sion rates are different for different electrons. For the n-th

electron, the corresponding emission rate can be written as

λn(t|ts, t1, . . . , tn−1). It is essentially a conditional intensity,

which gives the expected number of electrons emitted in the

inﬁnitesimal time interval [t, t +dt], under the condition that

there are n−1electrons emitted previously in each inﬁnitesi-

mal time interval [ti, ti+dt](with i= 1,2, . . . , n −1). Here

the history of the emission process is represented by the or-

dered time sequence t1, . . . , tn−1, which satisﬁes

ts< t1<· · · < tn−1< t. (12)

Equation (9) can be generalized to incorporate the history-

dependence of the emission rate. For example, f1(t1;ts, te)

can be expressed in terms of λ1(t)and λ2(t|t1), which has

the form

f1(t1;ts, te) = exp −Zt1

dτλ1(τ|ts)

×λ1(t1|ts) exp −Zte

dτλ2(τ|ts, t1).(13)

The expression looks complicated at the ﬁrst sight, but

it can be understood in the similar way as Eq. (9). It

is composed by three factors: 1) The exponential fac-

tor exp h−Rt1

tsdτλ1(τ|ts)iis just the idle time distribution

Π0(ts, t1), it ensures that no electron can be emitted in the

time interval [ts, t1]; 2) λ(t1|ts)gives the emission probability

of the ﬁrst electron in the inﬁnitesimal time interval [t1, t1+

dt]; 3) The exponential factor exp h−Rte

t1dτλ2(τ|ts, t1)ican

be treated as a conditional idle time distribution. It guarantees

that if the ﬁrst electron has been emitted at the time t1, the

second electron cannot be emitted before the time te. From

the deﬁnition of f1(t1;ts, te), one ﬁnds that the emission rate

λ1(t|ts)of the ﬁrst electron can be given as

λ1(t|ts) = f1(t;ts, t)

Π0(ts, t).(14)

The exclusive joint probability density for arbitrary ncan

be constructed in a similar way, which depends on the emis-

sion rates up to the (n+ 1)-th electron. To write it in a more

compact form, one reminds that the emission times tifollows

the relation given in Eq. (12). Due to this restriction, all the

emission rates can be composed into a piece-wise function

λ∗(t), which has the form

λ∗(t) = 









λ1(t|ts), ts< t ≤t1

λ2(t|ts, t1), t1< t ≤t2

λn+1(t|ts, t1, . . . , tn), tn< t ≤te

(15)

In doing so, fn(t1, . . . , tn;ts, te)can be written as [38]

fn(t1, t2, . . . , tn;ts, te) =

i=1

λ∗(ti) exp −Zte

dτλ∗(τ).

(16)

From this expression, one ﬁnds that the emission rate for the

n-th electron (n≥2) can be given as

λn(t|ts, t1, . . . , tn−1) = fn(t1, . . . , tn−1, t;ts, t)

fn−1(t1, . . . , tn−2, t;ts, t).(17)

Generally speaking, one can calculate the FCS and emis-

sion rates from the coincidences gn(t1, . . . , tn)by using

Eqs. (4), (5), (6), (14) and (17) . However, the calculation

is rather involved in general cases. For non-interacting sys-

tems, this can be greatly simpliﬁed, as the system can be fully

decided by the corresponding ﬁrst-order Glauber correlation

functions G(t, t0) = ha†(t)a(t0)i − ha†(t)a(t0)i0[40–44]. In

a previous work, Macchi has shown that fn(t1, . . . , tn;ts, te)

can be calculated from G(t, t0)by solving the eigenvalue

equation [35]:

Zte

dt0G(τ, τ0)ϕα(τ0) = να(ts, te)ϕα(τ),(18)

with α= 1,2, . . . being the index of the eigenvalues and

eigenfunctions. The eigenvalue να(ts, te)satisﬁes 0≤να≤

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Quantum-to-classicalcrossoverinsingle-electronemitterY.Yin1,1DepartmentofPhysics,SichuanUniversity,Chengdu,Sichuan,610065,China(Dated:November17,2022)Weinvestigatethetemperature-drivenquantum-to-classicalcrossoverinasingle-electronemitter.Theemitteriscomposedofaquantumconductorandanelectrode,whichi...

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Quantum-to-classical crossover in single-electron emitter Y . Yin1 1Department of Physics Sichuan University Chengdu Sichuan 610065 China

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