Auto- versus Cross-Correlation Noise in Periodically Driven Quantum Coherent Conductors

Abstract

Expressing currents and their fluctuations at the terminals of a multi-probe conductor in terms of the wave functions of carriers injected into the Fermi sea provides new insight into the physics of electric currents. This approach helps us to identify two physically different contributions to shot noise. In the quantum coherent regime, when current is carried by non-overlapping wave packets, the product of current fluctuations in different leads, the cross-correlation noise, is determined solely by the duration of the wave packet. In contrast, the square of the current fluctuations in one lead, the autocorrelation noise, is additionally determined by the coherence of the wave packet, which is associated with the spread of the wave packet in energy. The two contributions can be addressed separately in the weak back-scattering regime, when the autocorrelation noise depends only on the coherence. Analysis of shot noise in terms of these contributions allows us, in particular, to predict that no individual traveling particles with a real wave function, such as Majorana fermions, can be created in the Fermi sea in a clean manner, that is, without accompanying electron–hole pairs.

1. Introduction

Recently, the quantum tomography of a single-electron wave function was demonstrated experimentally [1,2]. In both experiments, a periodic stream of single-electron wave packets was mixed with a low-amplitude electrical probe signal at the electron wave splitter, a quantum point contact, and the resulting electrical noise averaged over long time was measured. However, if the fluctuations of an electrical current within one output lead were measured in one experiment, the correlations of currents flowing within both output leads were measured in another. In a sense, these works are the culmination of a number of recent works where the cross-correlation noise was measured to count electrons emitted per period [3,4] to demonstrate a tunable fermionic anti-bunching [4,5,6,7], and the auto-correlation noise at high [8,9] and low [10,11] frequencies was measured to identify a single-electron emission regime. These experimental advances motivate us to take a closer look at how exactly the quantum properties of wave packets manifest themselves in the measured electrical noise [12,13,14].

Note that tomography of the density profile of solitary electrons was successfully realized in [15] using the measurement of the electrical current rather than noise.

Here, in contrast to previous works [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]—for a review, see in [33]—I will focus on comparing auto- and cross-correlation noise. I will show that, in the case of a periodic train of non-overlapping single-electron wave packets scattered off the wave splitter with reflection probability R, there are two contributions to noise. These contributions can be addressed separately by measuring both auto- and cross-correlation noise in the weak back-scattering regime, $R\ll 1$. As I will show in the subsequent sections, for a single-electron wave packet $\mathsf{\Psi }\left(t,x\right)=e^{-\frac{i}{\hslash }\mu t}\psi \left(t,x\right)$ injected during one period into a one-dimensional chiral Fermi sea and reflected into a detector at a wave splitter, see Figure 1, the auto-correlation, ${\mathcal{P}}_{auto}^{ex}$, and cross-correlation, ${\mathcal{P}}_{cross}^{}$, noise power, see Equation (10), at frequency $\omega$ and at zero temperature are calculated as follows:

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{auto}^{ex}\left(\omega \right)& =& R\frac{e^2}{{\mathcal{T}}_0}\underset{-\infty }{\overset{\infty }{\int }}d\tau e^{i\omega \tau }\frac{\mathrm{Im}\mathcal{C}\left(\tau \right)}{\pi \tau },\hfill \\ \hfill {\mathcal{P}}_{cross}^{}\left(\omega \right)& =& -R\frac{e^2}{{\mathcal{T}}_0}{\left|\mathcal{N}\left(\omega \right)\right|}^2,\hfill \end{array}$$

(1)

where the superscript $ex$ (unnecessary for cross-correlation noise) indicates the excess over the equilibrium value, e is an electron charge, ${\mathcal{T}}_0$ is a period, $\mathcal{C}\left(\tau \right)$ is the integrated over time coherence of the wave function envelope, [34,35]

$$\begin{array}{ccc}\hfill \mathcal{C}\left(\tau \right)& =& \underset{-\infty }{\overset{\infty }{\int }}dt{\psi }^{*}\left(t+\tau ,x_D\right)\psi \left(t,x_D\right),\hfill \end{array}$$

(2)

and $\mathcal{N}\left(\omega \right)$ is the Fourier transform of the wave packet density,

$$\begin{array}{ccc}\hfill \mathcal{N}\left(\omega \right)& =& \underset{-\infty }{\overset{\infty }{\int }}d\tau e^{i\omega \tau }{\left|{\psi }^{}\left(\tau ,x_D\right)\right|}^2.\hfill \end{array}$$

(3)

In Equations (2) and (3), $x_D$ is the coordinate of the detector used to measure the current. For simplicity, I assume that the coordinates of both detectors in the outgoing channels $\alpha =3$ and $\alpha =4$ are the same. Below, I discard $x_D$ and use the following notation: $\mathsf{\Psi }\left(t\right)\equiv \mathsf{\Psi }(t,x_D)$.

Equation (1) is the main result of this work, which shows us that in the weak backscattering regime, the auto-correlation noise and cross-correlation noise are determined by essentially different quantities. One can say that they provide somehow complementary information. The cross-correlation noise is sensitive merely to the shape of a wave packet, that is, to its duration in time, while the auto-correlation noise is rather sensitive to how different parts of the wave packet correlate with each other. Such correlations are related to the spread of the wave packet in energy, which is supported by the reasoning in [9] that only those particles whose energy exceeds the Fermi energy by more than $\hslash \omega$ contribute to the auto-correlation noise at the frequency $\omega$.

Below, I use three known wave functions of particles injected into the Fermi sea by various electronic sources and calculate the corresponding auto- and cross-correlation noise in accordance with Equation (1). These results provide additional evidence of the duality of the information provided by both types of noise.

A Lorentzian voltage pulse, one per period, $eV\left(t\right)=2\hslash {\mathsf{\Gamma }}_{\tau }{\left(t^2+{\mathsf{\Gamma }}_{\tau }^2\right)}^{-1}$, applied to the Fermi sea with energy $\mu$ and at zero temperature creates a single-electron wave packet [36,37], named a leviton [4]. This excitation should be understood in such a way that the voltage pulse shakes the Fermi sea and excites just a single electron on its surface [38]. The wave function of a leviton, ${\mathsf{\Psi }}^L\left(t\right)=e^{-\frac{i}{\hslash }\mu t}{\psi }^L\left(t\right)$, has the following envelope function [39],

$$\begin{array}{c}\hfill {\psi }^L\left(t\right)=\frac{1}{\sqrt{\pi {\mathsf{\Gamma }}_{\tau }}}\frac{{\mathsf{\Gamma }}_{\tau }}{t-i{\mathsf{\Gamma }}_{\tau }}.\end{array}$$

(4)

Here, the subscript $\tau$ indicates that ${\mathsf{\Gamma }}_{\tau }$ is the half-width of the wave packet in time. I assume ${\mathsf{\Gamma }}_{\tau }\ll {\mathcal{T}}_0$ to avoid overlap between the successive wave packets.

Using the above equation in Equation (1), I calculate

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{auto}^{ex,L}\left(\omega \right)=-{\mathcal{P}}_{cross}^L\left(\omega \right)& =& R\frac{e^2}{{\mathcal{T}}_0}{e}^{-\left|\omega \right|2{\mathsf{\Gamma }}_{\tau }}.\hfill \end{array}$$

(5)

The fact that the auto- and cross-correlation noise show the same frequency dependence is because there is only one time parameter in the problem, ${\mathsf{\Gamma }}_{\tau }$, which defines both the characteristic energy of the wave packet, $\hslash /\left(2{\mathsf{\Gamma }}_{\tau }\right)$ [39], and the characteristic width in time of the wave packet, $2{\mathsf{\Gamma }}_{\tau }$.

In the next example, the shape and the energy distribution are not related so tight and ${\mathcal{P}}_{auto}^{ex}\left(\omega \right)$ and $-{\mathcal{P}}_{cross}^{}\left(\omega \right)$ become different.

Let us consider a quantum level of half-width $\delta$ filled with one electron and tunnel-coupled to a one-dimensional Fermi sea at zero temperature. The energy of a level raises at a constant rapidity c, and crosses the Fermi level at $t=0$ when an electron is injected into the Fermi sea. Such regime of injection can be realized using the quantum capacitor [40,41,42]. The wave function of the injected electron was calculated in [43], ${\mathsf{\Psi }}^c\left(t\right)=e^{-\frac{i}{\hslash }\mu t}{\psi }^c\left(t\right)$, with

$$\begin{array}{ccc}\hfill {\psi }^{\left(c\right)}\left(t\right)& =& \frac{1}{\sqrt{\pi {\mathsf{\Gamma }}_{\tau }}}\underset{0}{\overset{\infty }{\int }}dx{e}^{-x}{e}^{-ix\frac{t}{{\mathsf{\Gamma }}_{\tau }}}{e}^{i{x}^2\frac{{\tau }_D}{{\mathsf{\Gamma }}_{\tau }}},\hfill \end{array}$$

(6)

where ${\mathsf{\Gamma }}_{\tau }=\delta /c$ is the crossing time, the time it takes for a raising widened quantum level to cross the Fermi level, ${\tau }_D=\hslash /\left(2\delta \right)$ is the dwell time, an average time spent an electron on a quantum level before escaping to the Fermi sea provided that such an escape is possible, that is, after the quantum level has risen above the Fermi level. Note that if ${\tau }_D\ll {\mathsf{\Gamma }}_{\tau }$, then ${\psi }^c$, Equation (6), is essentially ${\psi }^L$, Equation (4). Notice, to get a stream of electrons we need a set of levels. Subsequent crossings occur with a delay of ${\mathcal{T}}_0\gg {\mathsf{\Gamma }}_{\tau },{\tau }_D$. Now Equations (1) and (6) give us

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{auto}^{ex,c}\left(\omega \right)& =& R\frac{e^2}{{\mathcal{T}}_0}{e}^{-\left|\omega \right|2{\mathsf{\Gamma }}_{\tau }},\hfill \\ \hfill {\mathcal{P}}_{cross}^c\left(\omega \right)& =& -R\frac{e^2}{{\mathcal{T}}_0}\frac{e^{-\left|\omega \right|2{\mathsf{\Gamma }}_{\tau }}}{1+{\left(\omega {\tau }_D\right)}^2}.\hfill \end{array}$$

(7)

For this model, the crossing time $2{\mathsf{\Gamma }}_{\tau }$ is the only parameter that determines the energy distribution, the same as for the source of levitons [44]. This is why the auto-correlation noise is the same as in the first example.

However, the shape of the wave packet is different from that of a leviton. It is determined by both time parameters ${\mathsf{\Gamma }}_{\tau }$ and ${\tau }_D$. Namely, if the dwell time is comparable with the crossing time, ${\tau }_D\gtrsim {\mathsf{\Gamma }}_{\tau }$, the density profile becomes larger than $2{\mathsf{\Gamma }}_{\tau }$, asymmetric, and with some wavy structure developing at later times. All this leads to additional suppression of $\mathcal{N}\left(\omega \right)$ and ${\mathcal{P}}_{cross}^c\left(\omega \right)$ with increasing frequency compared to the first example. In the case when ${\tau }_D\gg {\mathsf{\Gamma }}_{\tau }$, the dwell time, not the crossing time, determines how the cross-correlation noise decreases with frequency.

The connection between the auto-correlation noise and energy becomes even more transparent in the final example, where the dwell time is the only characteristic time.

The final (third) example is injection from a quantum dot with the equidistant ladder of levels, which is suddenly raised by one level spacing $\Delta$ at $t=0$ [42,45]. The Fermi level is exactly between the two successive levels. The probability of tunneling between the dot and the Fermi sea is small. The wave function, ${\mathsf{\Psi }}^{\Delta }\left(t\right)=e^{-\frac{i}{\hslash }\mu t}{\psi }^{\Delta }\left(t\right)$, has an envelope [35,46]

$$\begin{array}{ccc}\hfill {\psi }^{\Delta }\left(t\right)& =& \theta \left(t\right)\frac{e^{-i{\omega }_{0}t}}{\sqrt{{\tau }_D}}{e}^{-\frac{t}{2{\tau }_D}}.\hfill \end{array}$$

(8)

Here, $\theta \left(t\right)$ is the Heaviside step function and $\hslash {\omega }_0=\Delta /2$. Notice, in this case, the wave packet width is determined by the dwell time, while the energy of an injected electron is $\hslash {\omega }_0$, which is unrelated to ${\tau }_D$.

For not too large frequencies, $\omega \ll \Delta /\hslash$, the straightforward calculations based on Equation (1) lead to the following result (see Appendix A for details):

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{auto}^{ex,\Delta }\left(\omega \right)& =& R\frac{e^2}{{\mathcal{T}}_0},\hfill \\ \hfill {\mathcal{P}}_{cross}^{\Delta }\left(\omega \right)& =& -R\frac{e^2}{{\mathcal{T}}_0}\frac{1}{1+{\left(\omega {\tau }_D\right)}^2}.\hfill \end{array}$$

(9)

As the particles are injected far above the Fermi sea, $\Delta /2\gg \hslash \omega$, all of them contribute to noise. As a result, there is no energy related suppression (for $\omega \ll \Delta /\hslash$ and $R\ll 1$). Therefore, the auto-correlation noise is independent on frequency. On the other hand, the density profile has a finite width, ${\tau }_D$. Therefore, the cross-correlation noise gets suppressed at $\omega \ge {\tau }_D^{-1}$.

One more important conclusion can be drawn from Equation (1). In the case when a single-particle envelope wave function, $\psi$, is real-valued, for example, as for Majorana fermions [47,48], the corresponding contribution to the auto-correlation noise is identically zero in the weak backscattering regime. I emphasize that this conclusion applies to traveling single particles in the Fermi sea, and not to localized states.

For some injection protocols the auto-correlation noise is also zero for $R=1$, see in [49].

Note that the charge conservation implies ${\mathcal{P}}_{auto}^{ex}\left(0\right)+{\mathcal{P}}_{cross}^{}\left(0\right)=0$, see Appendix B.3 [14]. This fact imposes some indirect constraint on the wave function of a single-electron wave packet that can be injected into a one-dimensional Fermi sea. In particular, no single particle with a real wave function can be injected in a clean manner, that is, without accompanying electron–hole pairs. Indeed, as Equation (1) predicts, the cross-correlation noise at zero frequency is not zero, ${\mathcal{P}}_{cross}^{}\left(0\right)=-R{e}^2/{\mathcal{T}}_0\ne 0$. While in the case of a real-valued wave function, the excess auto-correlation noise vanishes for any frequency, ${\mathcal{P}}_{auto}^{ex}\left(\omega \right)=0$. To eliminate the apparent violation of charge conservation and ensure ${\mathcal{P}}_{auto}^{ex}\left(0\right)=-{\mathcal{P}}_{cross}^{}\left(0\right)\ne 0$, we must assume that if such a particle is injected, then the additional excitations are unavoidable created. The example is a half-leviton [50], a particle with a real wave function whose creation is accompanied by the creation of an electron–hole cloud.

The rest of the paper is structured as follows. In Section 2, within the framework of the Floquet scattering matrix approach, a connection is established between the correlation functions of the electrical current, the auto- and cross-correlation noise power, and the first-order correlation function of a periodic stream of electrons injected into a chiral Fermi sea. This relationship allows a detailed analysis of the similarities and differences between auto- and cross-correlation noise, which is illustrated in Section 3 using some examples. The conclusion is given in Section 4. Some technical details of calculations are presented in Appendix A, Appendix B, Appendix C and Appendix D.

2. Electrical Noise and Electron Correlation Function

In general, an external source is required to pass current through a conductor. The role of the source can be played, for example, by a constant or time-dependent voltage applied across a conductor; a time-dependent gate voltage, which changes the position of the quantum levels of electrons in a part of the conductor; etc. If the characteristics of the source are known, the current can be calculated. In the quantum coherent regime, when the current is carried by individual electrons, the characteristics of carriers, for example, their wave function, also can be calculated using the characteristics of the source. The measurements of electrical current and its fluctuations were already used to acquire information on quantum state of carriers [1,2,15]. Therefore, it is desirable to have a direct relation between the electrical and electron characteristics without explicit recursion to the characteristics of the source. Some efforts in this direction have already been made [44,50,51,52]. Below the fluctuations of an electrical current are expressed in terms of the wave functions, more precisely, in terms of the excess first-order correlation function [17,18,35,46] of electrons responsible for those fluctuations.

To be specific, here I am interested in a quantum coherent conductor connected via one-channel (chiral) leads [53] to several electron reservoirs in equilibrium. Some (or all) incoming leads are fed by external sources working periodically with period ${\mathcal{T}}_0$.

Below, I use the theory of non-interacting electrons. This is justified by the fact that such a theory has proved useful in describing the injection of single electrons [4,42], the results of quantum tomography of single electrons [1,2], and, in particular, the frequency-dependent noise [8,9] of interest here.

2.1. Frequency-Dependent Noise

The correlation function of currents, $I_{\alpha },I_{{\alpha }^{\prime }}$, flowing in leads $\alpha$ and ${\alpha }^{\prime }$ of a multi-terminal conductor are defined as follows [14]:

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{\alpha {\alpha }^{\prime }}\left(\omega \right)& =& \underset{0}{\overset{{\mathcal{T}}_0}{\int }}\frac{dt}{{\mathcal{T}}_0}\underset{-\infty }{\overset{\infty }{\int }}d\tau e^{i\omega \tau }\left\{\frac{〈{\widehat{I}}_{\alpha }(t+\tau ){\widehat{I}}_{{\alpha }^{\prime }}\left(t\right)+{\widehat{I}}_{{\alpha }^{\prime }}\left(t\right){\widehat{I}}_{\alpha }(t+\tau )〉}{2}-I_{\alpha }(t+\tau )I_{{\alpha }^{\prime }}\left(t\right)\right\},\hfill \end{array}$$

(10)

where ${\widehat{I}}_{\alpha }$ and $I_{\alpha }=〈{\widehat{I}}_{\alpha }〉$ are an operator in second quantization and a corresponding measurable for a current in the lead $\alpha$; the angular brackets $〈\cdots 〉$ denote the quantum statistical average; for a periodic drive, ${\mathcal{T}}_0$ is a period, for a non-periodic drive ${\mathcal{T}}_0\to \infty$. Note the difference in the factor of 2 compared to the definition used in [14].

The current operator ${\widehat{I}}_{\alpha }$ is expressed in terms of creation and annihilation operators ${\widehat{a}}_{\alpha }^{†}\left(E\right),{\widehat{a}}_{\alpha }\left(E\right)$ of electrons with energy E incoming from the reservoir $\alpha$ and operators ${\widehat{b}}_{\alpha }^{†}\left(E\right),{\widehat{b}}_{\alpha }\left(E\right)$ of electrons with energy E scattered into the reservoir $\alpha$ [54]. In the wide band limit, that is, when the relevant energy scales, such as a voltage applied, a temperature, the energy quantum $\hslash \mathsf{\Omega }$ with $\mathsf{\Omega }=2\pi /{\mathcal{T}}_0$, etc., are all small compared to the Fermi energy ${\mu }_{\alpha }$, the current operator reads [55],

$$\begin{array}{ccc}\hfill {\widehat{I}}_{\alpha }\left(t\right)& =& \frac{e}{h}\underset{}{\overset{}{\int \int }}dEd{E}^{\prime }{e}^{i\frac{E-E^{\prime }}{\hslash }t}\left\{{\widehat{b}}_{\alpha }^{†}\left(E\right){\widehat{b}}_{\alpha }^{}\left(E^{\prime }\right)-{\widehat{a}}_{\alpha }^{†}\left(E\right){\widehat{a}}_{\alpha }^{}\left(E^{\prime }\right)\right\}.\hfill \end{array}$$

(11)

In the case of a periodically driven conductor, the operators ${\widehat{b}}_{\alpha }$ are related to various operators ${\widehat{a}}_{\alpha }$ via the elements of the unitary Floquet scattering matrix $S_F$ [56],

$$\begin{array}{ccc}\hfill {\widehat{b}}_{\alpha }\left(E\right)& =& \sum _{\beta }^{}\sum _{n=-\infty }^{\infty }{S}_{F,\alpha \beta }\left(E,E_n\right){\widehat{a}}_{\beta }\left(E_n\right),\hfill \end{array}$$

(12)

where the short notation $E_n=E+n\hslash \mathsf{\Omega }$ is introduced. Charge conservation requires the scattering matrix to be unitary, which means

$$\begin{array}{c}\hfill \sum _{\gamma }^{}\sum _{n=-\infty }^{\infty }{S}_{F,\gamma \alpha }^{*}\left(E_n,E_m\right)S_{F,\gamma \beta }^{}\left(E_n,E_{}\right)={\delta }_{\alpha \beta }{\delta }_{m,0},\\ \hfill \\ \hfill \sum _{\gamma }^{}\sum _{n=-\infty }^{\infty }{S}_{F,\alpha \gamma }^{*}\left(E_m,E_n\right)S_{F,\beta \gamma }^{}\left(E_{},E_n\right)={\delta }_{\alpha \beta }{\delta }_{m,0},\end{array}$$

(13)

where ${\delta }_{n,0}$ is the Kronecker delta.

Equation (12) allows to express the quantum-statistical average of the product of b-operators in terms of that of a-operators. Because the reservoirs are in equilibrium, the latter average is known. In the case of reservoirs of non-interacting electrons forming the Fermi sea, we have $〈{\widehat{a}}_{\beta }^{†}\left(E\right){\widehat{a}}_{\beta }^{}\left(E^{\prime }\right)〉=f_{\beta }\left(E\right)\delta \left(E-E^{\prime }\right)$, where $f_{\beta }\left(E\right)$ is the Fermi distribution function with temperature ${\theta }_{\beta }$ and chemical potential ${\mu }_{\beta }$, and $\delta \left(E-E^{\prime }\right)$ is the Dirac delta.

2 × 2 Circuit

Our aim is to compare auto- and cross-correlation noise. The minimal circuit that allows cross-correlation noise is an electronic wave splitter, a quantum point contact (QPC) with two incoming, $\beta =1,2$, and two outgoing, $\alpha =3,4$, channels, see Figure 1.

Below, I am interested in current fluctuations in outgoing channels, that is, $\alpha ,{\alpha }^{\prime }=3,4$ in Equation (10). For this case, the general equation for noise within the Floquet scattering matrix approach [57,58] gives us

$$\begin{array}{c}\hfill {\mathcal{P}}_{33}^{}\left(\omega \right)=\frac{e^2}{h}\underset{}{\overset{}{\int }}dE\{F_{33}(E,E+\hslash \omega )+\sum _{n,m,q=-\infty }^{\infty }\sum _{\delta =1}^2\sum _{\gamma =1}^2{F}_{\gamma \delta }(E_q+\hslash \omega ,E_{})\\ \hfill \times S_{F,3\delta }^{*}(E_n,E_{})S_{F,3\delta }(E_m,E_{})S_{F,3\gamma }^{*}(E_m+\hslash \omega ,E_q+\hslash \omega )S_{F,3\gamma }(E_n+\hslash \omega ,E_q+\hslash \omega )\},\end{array}$$

(14a)

and

$$\begin{array}{c}\hfill {\mathcal{P}}_{34}\left(\omega \right)=\frac{e^2}{h}\underset{}{\overset{}{\int }}dE\sum _{n,m,q=-\infty }^{\infty }\sum _{\delta =1}^2\sum _{\gamma =1}^2{F}_{\gamma \delta }(E_q+\hslash \omega ,E_{})S_{F,3\delta }^{*}(E_n,E_{})S_{F,4\delta }(E_m,E_{})\\ \hfill \times S_{F,4\gamma }^{*}(E_m+\hslash \omega ,E_q+\hslash \omega )S_{F,3\gamma }(E_n+\hslash \omega ,E_q+\hslash \omega ),\end{array}$$

(14b)

with

$$\begin{array}{c}\hfill F_{\gamma \delta }(E_1,E_{})=\frac{f_{\gamma }\left(E_1\right)+f_{\delta }\left(E\right)}{2}-f_{\gamma }\left(E_1\right)f_{\delta }\left(E\right).\end{array}$$

(15)

Let us also introduce excess noise, that is, an increase in noise due to the source, which is defined as the following difference:

$$\begin{array}{c}\hfill {\mathcal{P}}_{\alpha {\alpha }^{\prime }}^{ex}\left(\omega \right)={\mathcal{P}}_{\alpha {\alpha }^{\prime }}^{}\left(\omega \right)-{\mathcal{P}}_{\alpha {\alpha }^{\prime }}^{off}\left(\omega \right),\end{array}$$

(16)

where the upper index $off$ indicates that the source is switched off.

I suppose a unitary $2\times 2$ scattering matrix of the QPC to be energy-independent,

$$\begin{array}{c}\hfill S^{QPC}=\left(\begin{array}{cc}\sqrt{R}& i\sqrt{T}\\ i\sqrt{T}& \sqrt{R}\end{array}\right),\end{array}$$

(17)

a real number $0\le R\le 1$ is the reflection probability, the transmission probability $T=1-R$. We need an energy independent $S^{QPC}$ in order to use noise to get information on injected wave packets only. If the properties of the electronic circuit that connects the incoming and outgoing channels do depend on energy, the outgoing signal also carries nontrivial information about the circuit [59].

In addition, for the sake of simplicity, I suppose that the periodic source is present only in the incoming channel $\beta =1$. This source characterized by the Floquet scattering amplitude, which is a matrix in an energy space with elements $S_F\left(E,E_n\right)$. The results presented below can be directly generalized to the case when another source is added in the second incoming channel, see Appendix B.4.

For the circuit with single source and single QPC, the elements of the total Floquet scattering matrix are represented as follows:

$$\begin{array}{ccc}\hfill S_{F,31}\left(E,E_n\right)& =& \sqrt{R}{S}_F\left(E,E_n\right),\hfill \\ \hfill S_{F,41}\left(E,E_n\right)& =& i\sqrt{T}{S}_F\left(E,E_n\right),\hfill \\ \hfill \\ \hfill S_{F,32}\left(E,E_n\right)& =& i\sqrt{T}{\delta }_{n,0},\hfill \\ \hfill S_{F,42}\left(E,E_n\right)& =& \sqrt{R}{\delta }_{n,0}.\hfill \end{array}$$

(18)

All other elements are zero.

2.2. First-Order Correlation Function

To characterize a quantum state injected by the source into a ballistic one-dimensional electronic waveguide, I use the first-order correlation function, ${\mathcal{G}}_{}^{\left(1\right)}$. This function is defined as a quantum statistical average of the product of two field operators for electrons calculated in the electronic wave-guide $\beta$ just after the source, ${\mathcal{G}}_{\beta }^{\left(1\right)}\left(t_1;t_2\right)=〈{\widehat{\mathsf{\Psi }}}_{\beta }^{†}\left(t_1\right){\widehat{\mathsf{\Psi }}}_{\beta }\left(t_2\right)〉$ [17]. Strictly speaking, this object is a $2\times 2$ matrix in the spin space. However, here I consider the spin-polarized case and suppress the spin index.

When the source is placed in lead $\beta =1$ and is characterized by the Floquet scattering amplitude, $S_F\left(E_n,E\right)$, the corresponding correlation function is calculated as follows [51]:

$$\begin{array}{c}\hfill v_{\mu }{\mathcal{G}}_1^{\left(1\right)}(t_{}+\tau ;t_{})=\frac{1}{h}\int dE{f}_1\left(E\right)e^{\frac{i}{\hslash }E\tau }\sum _{n,m=-\infty }^{\infty }{e}^{i\mathsf{\Omega }n\tau }{e}^{i\mathsf{\Omega }\left(n-m\right)t}{S}_F^{*}\left(E_n,E\right)S_F\left(E_m,E\right).\end{array}$$

(19)

Here, $v_{\mu }$ is a velocity of electrons at the Fermi level, which is originated from the density of states being energy independent in the wide band limit used here.

When the source is switched off, $S_F\left(E_n,E\right)={\delta }_{n,0}$, the above equation is reduced to the correlation function of the Fermi sea in equilibrium, which depends on the difference of times rather than on two times separately. I denote the equilibrium correlation function by the complementary subscript 0 and obtain for lead $\beta$, $\left[{\mathcal{G}}_{0,\beta }^{\left(1\right)}\left(\tau \right)\equiv {\mathcal{G}}_{0,\beta }^{\left(1\right)}(t_{}+\tau ;t_{})\right]$

$$\begin{array}{ccc}\hfill v_{\mu }{\mathcal{G}}_{0,\beta }^{\left(1\right)}\left(\tau \right)& =& \frac{1}{h}\int dE{f}_{\beta }\left(E\right)e^{i\frac{E}{\hslash }\tau }=\frac{e^{i\tau \frac{{\mu }_{\beta }}{\hslash }}}{2\pi i}\frac{1/{\tau }_{{\theta }_{\beta }}}{sinh\left(\tau /{\tau }_{{\theta }_{\beta }}\right)}.\hfill \end{array}$$

(20)

Here, ${\tau }_{{\theta }_{\beta }}=\hslash /\left(\pi k_B{\theta }_{\beta }\right)$ is the thermal coherence time for the reservoir, where the lead $\beta$ is attached to.

The difference of correlation functions with the source being on and off is the excess correlation function, which characterizes what is injected by the source into an electron waveguide $\beta =1$ [18],

$$\begin{array}{ccc}\hfill G_1^{\left(1\right)}\left(t_1;t_2\right)& =& {\mathcal{G}}_1^{\left(1\right)}\left(t_1;t_2\right)-{\mathcal{G}}_{0,1}^{\left(1\right)}\left(t_1-t_2\right).\hfill \end{array}$$

(21)

In the case when the source injects a single electron with wave function $\mathsf{\Psi }\left(t\right)$ per period, the excess correlation function during that period takes on a very simple form, $G_1^{\left(1\right)}\left(t_1;t_2\right)={\mathsf{\Psi }}^{*}\left(t_1\right)\mathsf{\Psi }\left(t_2\right)$. In the case of injection of N electrons, $G_1^{\left(1\right)}\left(t_1;t_2\right)={\sum }_{j=1}^N{\mathsf{\Psi }}_j^{*}\left(t_1\right){\mathsf{\Psi }}_j\left(t_2\right)$ [63].

2.3. Noise Power in Terms of ${\mathcal{G}}_{}^{\left(1\right)}$

For non-interacting electrons, the correlation function ${\mathcal{G}}_{}^{\left(1\right)}$ contains complete information about the system of electrons. In particular, all measurables can be expressed in terms of correlation function, see, e.g., in [51] for some examples. Such expressions are notably useful when transport is due to only a few electrons per period.

Here, I express the excess noise in terms of $G_{}^{\left(1\right)}$ in the case when all incoming channels have the same temperature ${\theta }_{\beta }=\theta$ and Fermi energy ${\mu }_{\beta }=\mu$. Therefore, the equilibrium electronic correlation functions are the same, ${\mathcal{G}}_{0,\beta }^{\left(1\right)}={\mathcal{G}}_0^{\left(1\right)}$. For more general case and for details of calculations, see Appendix B.

First, let us substitute Equations (15) and (18) into Equations (14a) and (14b) and calculate the excess noise, Equation (16). As a result, I find for the auto-correlation noise (see Equation (A20)),

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{33}^{ex}\left(\omega \right)& =& e^2{v}_{\mu }^2\underset{0}{\overset{{\mathcal{T}}_0}{\int }}\frac{dt}{{\mathcal{T}}_0}\underset{-\infty }{\overset{\infty }{\int }}d\tau e^{i\omega \tau }\left\{-R^2{\left|G_1^{\left(1\right)}\left(t+\tau ;t\right)\right|}^2-2R\mathrm{Re}{G}_1^{\left(1\right)}\left(t+\tau ;t\right){\mathcal{G}}_0^{\left(1\right)*}\left(\tau \right)\right\},\hfill \end{array}$$

(22a)

and for the cross-correlation noise (see Equation (A18)),

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{34}^{}\left(\omega \right)& =& -RT{e}^2{v}_{\mu }^2\underset{0}{\overset{{\mathcal{T}}_0}{\int }}\frac{dt}{{\mathcal{T}}_0}\underset{-\infty }{\overset{\infty }{\int }}d\tau e^{i\omega \tau }{\left|G_1^{\left(1\right)}\left(t+\tau ;t\right)\right|}^2.\hfill \end{array}$$

(22b)

The above equations are the central result of this work.

Notice that when all reservoirs are in the same conditions, cross-correlation noise disappears when the source is turned off. This is why the superscript $ex$ is omitted.

The important difference between auto- and cross-correlation noise (Equations (22a) and (22b)) is that the latter is determined solely by what is injected by the source, while the former in addition depends explicitly on the properties of the Fermi sea.

The part of the noise that is determined by ${\left|G_1^{\left(1\right)}\right|}^2$ depends on the possible quantum exchange [14] between the injected electrons. At zero temperature and when electrons are injected one at a time without overlapping, this part of the noise is reduced to the product of currents in Equation (10). In the wide band approximation used here, the electric current is proportional to the density profile of the wave packet, hence Equation (1), the second line.

In contrast, the part of the noise that is determined by the product of $G_1^{\left(1\right)}$ and ${\mathcal{G}}_0^{\left(1\right)}$ takes into account the quantum exchange of an injected electron and electrons of the Fermi sea. Such an exchange does not contribute to cross-correlation noise, unless the two incoming Fermi seas are different, see Equation (A23).

The formal difference between the auto- and cross-correlation noise becomes especially pronounced in the weak back-scattering regime, $R\ll 1$, when we can discard the terms $\sim R^2$ in Equations (22a) and (22b) and get the following,

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{33}^{ex}\left(\omega \right)& =& -R{e}^2{v}_{\mu }^2\underset{0}{\overset{{\mathcal{T}}_0}{\int }}\frac{dt}{{\mathcal{T}}_0}\underset{-\infty }{\overset{\infty }{\int }}d\tau e^{i\omega \tau }2\mathrm{Re}{G}_1^{\left(1\right)}\left(t+\tau ;t\right){\mathcal{G}}_0^{\left(1\right)*}\left(\tau \right),\hfill \\ \hfill \\ \hfill {\mathcal{P}}_{34}^{}\left(\omega \right)& =& -R{e}^2{v}_{\mu }^2\underset{0}{\overset{{\mathcal{T}}_0}{\int }}\frac{dt}{{\mathcal{T}}_0}\underset{-\infty }{\overset{\infty }{\int }}d\tau e^{i\omega \tau }{\left|G_1^{\left(1\right)}\left(t+\tau ;t\right)\right|}^2.\hfill \end{array}$$

(23)

Notice that using Equation (20) for ${\mathcal{G}}_0^{\left(1\right)}$ at zero temperature, $\theta =0\Rightarrow {\tau }_{\theta }\to \infty$, and for a single-particle injection, $G_{}^{\left(1\right)}\left(t+\tau ;t\right)=e^{i\frac{\mu }{\hslash }\tau }{\psi }^{*}\left(t+\tau \right)\psi \left(t\right)$, we arrive at Equation (1) with ${\mathcal{P}}_{auto}^{ex}={\mathcal{P}}_{33}^{ex}$ and ${\mathcal{P}}_{cross}^{}={\mathcal{P}}_{34}^{}$, where integration over t is extended to infinity, as the duration of the wave packet is much less than the period ${\mathcal{T}}_0$.

3. Examples

In this section, I will consider two examples: one when auto- and cross-correlation noise are perfectly anti-correlated at any frequency, and the other when they can be different. The corresponding equations are valid for arbitrary R, and not only for $R\ll 1$, as in the examples given in the introductory section.

3.1. Energy-Independent Source

In the case when the properties of the source do not change on the scale of the energy of the injected particles, the corresponding Floquet scattering amplitude can be represented as a Fourier coefficient of a certain energy-independent scattering amplitude, $S_F\left(E_n,E\right)={\int }_0^{{\mathcal{T}}_0}\frac{dt}{{\mathcal{T}}_0}{e}^{2\pi in\frac{t}{{\mathcal{T}}_0}}S\left(t\right)$ [46,56]. In a one-dimensional case, unitarity implies that $S\left(t\right)$ is a pure phase factor, that is, ${\left|S\left(t\right)\right|}^2=1$. For example, if a voltage $V\left(t\right)$ plays the role of a source, then this phase factor reads $S\left(t\right)=exp\left(i\frac{e}{\hslash }{\int }_{}^{t}d{t}^{\prime }V\left(t^{\prime }\right)\right)$.

In such a case, Equation (19) gives us ${\mathcal{G}}_1^{\left(1\right)}(t_1;t_2)=S^{*}\left(t_1\right)S\left(t_2\right){\mathcal{G}}_{0,1}^{\left(1\right)}(t_1-t_2)$. Note that ${\mathcal{G}}_{0,1}^{\left(1\right)}$ describes the Fermi sea in equilibrium at temperature ${\theta }_1$. Using this result in Equations (22a) and (22b), and taking into account that

$$\begin{array}{c}\hfill G_1^{\left(1\right)}(t_1;t_2)=\left\{S^{*}\left(t_1\right)S\left(t_2\right)-1\right\}{\mathcal{G}}_{0,1}^{\left(1\right)}(t_1-t_2),\end{array}$$

(24)

we find that the excess auto- and cross-correlation noise are perfectly anti-correlated at any frequency,

$$\begin{array}{c}\hfill {\mathcal{P}}_{33}^{ex}\left(\omega \right)+{\mathcal{P}}_{34}^{}\left(\omega \right)=0,\end{array}$$

(25)

not only at zero frequency, $\omega =0$, as the charge conservation requires [14]. An example was shown in Equation (5).

For $R=1$, when cross-correlation noise does not exist, that is, formally ${\mathcal{P}}_{34}^{}=0$, the above equation tells us that whatever emitted by the source under consideration is silent on any frequency, the excess auto-correlation noise is zero, ${\mathcal{P}}_{33}^{ex}\left(\omega \right)=0$. In particular, any voltage applied to a ballistic channel produces no excess noise at any frequency. Note that for a generic source injecting electrons into a ballistic waveguide, a similar conclusion can be drawn for noise only at zero frequency, see Equation (A25). Note that in the case of $R=1$, zero excess auto-correlation noise tells us nothing about the number of excited particle–hole pairs. In order to count the number of excitations, we need a wave-splitter with $R<1$.

Some general conclusions can be made regarding the effect of temperature on noise. Indeed, Equations (20) and (24) allow us to relate the excess correlation function at zero (the extra subscript 0) and non-zero (the extra subscript $\theta$) temperatures as follows:

$$\begin{array}{c}\hfill G_{1,\theta }^{\left(1\right)}\left(t+\tau ;t\right)=\frac{\tau /{\tau }_{{\theta }_{}}}{sinh\left(\tau /{\tau }_{{\theta }_{}}\right)}{G}_{1,0}^{\left(1\right)}\left(t+\tau ;t\right).\end{array}$$

(26)

Then, I use the above equation in Equations (22a) and (22b), utilize the inverse Fourier transformation with respect to $\omega$, and express the noise at temperature $\theta$, ${\mathcal{P}}_{\theta }^{ex}$, in terms of the noise at zero temperature, ${\mathcal{P}}_0^{ex}$, as follows,

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{\theta }^{ex}\left(\omega \right)& =& \underset{-\infty }{\overset{\infty }{\int }}d\tau e^{i\omega \tau }{\left(\frac{\tau /{\tau }_{{\theta }_{}}}{sinh\left(\tau /{\tau }_{{\theta }_{}}\right)}\right)}^2\underset{-\infty }{\overset{\infty }{\int }}\frac{d{\omega }^{\prime }}{2\pi }{e}^{-i{\omega }^{\prime }\tau }{\mathcal{P}}_0^{ex}\left({\omega }^{\prime }\right).\hfill \end{array}$$

(27)

Here, I introduce ${\mathcal{P}}^{ex}\equiv {\mathcal{P}}_{33}^{ex}=-{\mathcal{P}}_{34}^{}$ according to Equation (25).

3.2. Injection from a Quantum Level Raising at a Constant Rapidity

Now, let us consider a single-electron injection from a source, whose properties depend on energy. The corresponding scattering amplitude and the wave function of the injected electron were discussed in [43] at zero temperature and in [60] at nonzero temperatures. In this case, the auto- and cross-correlation noises do not stick together unless at zero frequency.

At zero temperature, the excess correlation function is $v_{\mu }{G}_1^{\left(1\right)}\left(t+\tau ;t\right)=e^{i\frac{\mu }{\hslash }\tau }{\psi }^{\left(c\right)*}$$\left(t+\tau \right){\psi }^{\left(c\right)}\left(t\right)$, where ${\psi }^{\left(c\right)}$ is shown in Equation (6). Using this equation in Equations (22a) and (22a) and assuming that the width of the wave packet is small compared to the period, $max\left({\mathsf{\Gamma }}_{\tau },{\tau }_D\right)\ll {\mathcal{T}}_0$, I find (see Appendix C for details),

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{33}^{ex}\left(\omega \right)& =& R\frac{e^2}{{\mathcal{T}}_0}\frac{T+{\left(\omega {\tau }_D\right)}^2}{1+{\left(\omega {\tau }_D\right)}^2}{e}^{-\left|\omega \right|2{\mathsf{\Gamma }}_{\tau }},\hfill \\ \hfill {\mathcal{P}}_{34}^{}\left(\omega \right)& =& -R\frac{e^2}{{\mathcal{T}}_0}\frac{T}{1+{\left(\omega {\tau }_D\right)}^2}{e}^{-\left|\omega \right|2{\mathsf{\Gamma }}_{\tau }}.\hfill \end{array}$$

(28)

At $R\ll 1$ ($T\approx 1$), Equation (7) is reproduced.

The above equations are illustrated in Figure 2 in the case of $2{\mathsf{\Gamma }}_{\tau }\ll {\tau }_D$, when the difference between them is most pronounced. As I have already discussed in the introduction after Equation (7), the excess auto-correlation noise and the cross-correlation noise demonstrate significantly different dependencies on frequency. The cross-correlation noise (its absolute value) decreases monotonically with frequency, see Figure 2, a red dashed line. In contrast, the excess auto-correlation noise is non-monotonically dependent on frequency, which is a manifestation of the existence of two contributions. The first contribution, which is responsible for the quadratic increase at low frequencies, is similar to the cross-correlation noise, compare the first term in Equations (22a) and (22b), while the second contribution is different. This contribution dominates in the limit of $T\to 1$ and at high frequencies, see a black solid line in Figure 2.

At non-zero temperature, $\theta >0$, auto- and cross-correlation noise is modified by the same factor, ${\mathcal{P}}_{33,\theta }^{ex}\left(\omega \right)={\eta }_{}\left(\omega ,\theta \right){\mathcal{P}}_{33}^{ex}\left(\omega \right)$ and ${\mathcal{P}}_{34,\theta }^{}\left(\omega \right)={\eta }_{}\left(\omega ,\theta \right){\mathcal{P}}_{34}^{}\left(\omega \right)$ (see Appendix D for details), where

$$\begin{array}{c}\hfill {\eta }_{}\left(\omega ,\theta \right)=e^{\left|\omega \right|2{\mathsf{\Gamma }}_{\tau }}\underset{-\infty }{\overset{\infty }{\int }}\frac{dx}{\pi }\frac{e^{ix\omega 2{\mathsf{\Gamma }}_{\tau }}}{x^2+1}{\left(\frac{x2{\mathsf{\Gamma }}_{\tau }/{\tau }_{{\theta }_{}}}{sinh\left(x2{\mathsf{\Gamma }}_{\tau }/{\tau }_{{\theta }_{}}\right)}\right)}^2.\end{array}$$

(29)

The thermal coherence time ${\tau }_{\theta }$ is defined after Equation (20).

Interestingly, the above equation is independent of the dwell time ${\tau }_D=\hslash /\left(2{\mathsf{\Gamma }}_{\tau }c\right)$, where c is a rapidity, see Equation (6). Therefore, the same factor ${\eta }_{}\left(\omega ,\theta \right)$ applies for the case of ${\tau }_D=0$, which is the case for the source of levitons of half-width ${\mathsf{\Gamma }}_{\tau }$. However, this analogy is not complete.

Namely, for the case of levitons, the noise at non-zero temperatures can be expressed directly in terms of the noise at zero temperature, see Equation (27). On the contrary, for the case of electrons emitted from the quantum level, this is generally not the case due to factors that depend on $\omega {\tau }_D$. Yet, in the weak backscattering regime, $R\ll 1$, the auto-correlation noise obeys Equation (27), because it has no such factors, see Equation (7).

The temperature-dependent factor $\eta \left(\omega ,\theta \right)$, Equation (29), is shown in Figure 3 for several frequencies $\omega$. Remarkably, the maximum occurs at $\omega {\tau }_{\theta }\sim 1$, which is independent of properties of the source. The non-monotonic temperature behavior at non-zero frequencies is due to two counteracting effects, both due to the fact that the quantum state of electrons injected at non-zero temperatures is a mixed quantum state [51].

The first effect, which leads to noise suppression, comes from the fact that each component of a mixed state is scattered independently at the wave splitter. As a result, the effect of charge quantization becomes less pronounced, and shot noise decreases with increasing temperature. At $\theta \gg {\theta }_0$, where the characteristic temperature, ${\theta }^{*}$, is determined by the energy of an injected electron, $k_B{\theta }_0=\hslash /\left(2\pi {\mathsf{\Gamma }}_{\tau }\right)$, the shot noise decays as ${\theta }_0/\theta$ [61].

Suppression of a zero-frequency shot noise with temperature has been reported in [3,7,62,63]. For the source of levitons, this effect was predicted in [19].

The second effect, leading to an increase in noise, is associated with an effective broadening of the energy distribution of injected particles due to the broadening of the probability density of the components of the mixed state with increasing temperature, $p\left(ϵ\right)=-\partial f_1\left(ϵ\right)/\partial ϵ$, see Equation (A40). As a result, the injected particle is more likely to be able to emit energy $\hslash \omega$ in order to contribute to noise at frequency $\omega$ [9]. This increase reaches saturation at $4k_B\theta \sim \hslash \omega$, which leads to the maxima in Figure 3.

4. Conclusions

The finite-frequency fluctuations of an electric current in multi-terminal conductors were analyzed at zero as well as non-zero temperatures. The focus was on the quantum coherent regime, when the current is carried by non-overlapping single-particle wave packets periodically injected into a unidirectional, chiral wave guide.

To highlight similarities and differences between auto- and cross-correlation noise, the fluctuations of an electric current were expressed in terms of the wave functions of injected electrons, bypassing the use of explicit source characteristics. Two contributions to shot noise have been identified. The first, which depends on the possible quantum exchange between the injected electrons, determines the cross-correlation noise and part of the auto-correlation noise. In the case of single-particle injection at zero temperature, this part is determined by the density profile of the injected wave packets. The second contribution, which depends on the quantum exchange of an injected electron and electrons of the Fermi sea, contributes only to the auto-correlation noise. This part is determined by the coherence of the injected wave packets multiplied by the coherence of the Fermi sea.

At zero frequency, the charge conservation tightly links both contributions to shot noise. Such a connection allows us to make some general conclusions related to the properties of excitations that can be injected/created in the Fermi sea. In particular, no excitations with a real wave function can be created in the Fermi sea without accompanying electron–hole pairs, which follows from the fact that otherwise Equation (1) would be incompatible with the conservation of charge [14]. At non-zero frequencies, the two contributions in question are generally different. They can be addressed separately by measuring both auto- and cross-correlation noise in the weak backscattering regime, $R\ll 1$.

For several experimentally available single-electron sources for which the wave function was calculated, I calculated and compared auto- and cross-correlation noise. For the family of so-called energy-independent sources, the source of leviton [4] is an example, the contribution related to the density profile and the contribution related to the coherence of the wave packet turn out to be the same, see Equation (5). Therefore, the auto- and cross-correlation noises are the same (up to the minus sign) at any frequency, not only at zero frequency as the charge conservation requires, see Equation (25). For another source, which relies on tunneling [42], the two contributions are manifestly different, see Equations (7) and (9) for different working regimes of the source.

I analyzed the effect of temperature on shot noise in the case when electrons are injected on top of the Fermi. It turns out that temperature affects both contributions equally, see Equation (29) for the temperature-dependent factor. As one of the contributions to shot noise depends on the quantum state of electrons in the Fermi sea, I conclude that temperature affects the quantum state of both the electrons in the Fermi sea and the injected electrons in the same way. Namely, a pure state at zero temperature becomes a mixed state at non-zero temperatures [51]. At zero frequency, changing the quantum state from pure to mixed leads to noise suppression, while at non-zero frequencies, the temperature dependence of noise is non-monotonic, see Figure 3. The temperature-dependent factor peaks when the thermal coherence time becomes of the order of the inverse of the frequency at which the noise is measured. Importantly, the position of this maximum does not depend on the properties of the sources of electrons under consideration.