Lasing and antibunching of optical phonons in semiconductor DQDs

# Lasing and antibunching of optical phonons in semiconductor double quantum dots

## Abstract

We theoretically propose optical phonon lasing in a double quantum dot (DQD) fabricated on a semiconductor substrate. No additional cavity or resonator is required. An electron in the DQD is found to be coupled to only two longitudinal optical phonon modes that act as a natural cavity. When the energy level spacing in the DQD is tuned to the phonon energy, the electron transfer is accompanied by the emission of the phonon modes. The resulting non-equilibrium motion of electrons and phonons is analyzed by the rate equation approach based on the Born-Markov-Secular approximation. We show that the lasing occurs for pumping the DQD via electron tunneling at rate much larger than the phonon decay rate, whereas a phonon antibunching is observed in the opposite regime of slow tunneling. Both effects disappear by an effective thermalization induced by the Franck-Condon effect in a DQD fabricated in a suspended carbon nanotube with strong electron-phonon coupling.

## 1 Introduction

In conventional lasers, two-level systems couple to a single mode of photon in a cavity. The pumping of electrons to the upper level results in a light amplification through the stimulated emission of radiation. Recently, the lasing was reported for a single atom in a cavity, which is called microlaser [1]. Such a system is being intensively studied in the context of cavity quantum electrodynamics (QED) [2], which also works as a single photon source to produce antibunched photons [3].

Quantum dots are electrically tunable two-level systems. The cavity QED using a quantum dot has a potential for wider application to the quantum information processing [4] as well as the single photon source [5]. When the quantum dot is connected to an external circuit, the electronic state in the quantum dot can be controlled by the electric current. The microlaser was realized in the so-called circuit QED, in which a superconducting quantum dot in a circuit is coupled to a microwave resonator [6]. In this case, the pumping is realized using the superconducting circuit. The electric current drives the lasing when the level spacing is tuned to the microwave energy [7, 8, 9, 10, 11].

In the present work, we theoretically examine the transport through a semiconductor double quantum dot (DQD) in the presence of electron-optical-phonon coupling and propose a phonon lasing without a cavity or resonator. The electron-phonon interaction in quantum dots reveals itself in the transport phenomena, which was investigated in various contexts until now. For DQDs fabricated in InAs nanowire and graphene, an interference pattern of electric current was observed as a function of level spacing in the DQDs, which is ascribable to the emission of acoustic phonons [12]. It is the Dicke-type interference between two transport processes in which a longitudinal acoustic (LA) phonon is emitted in one dot or another [13]. In a single quantum dot fabricated in a suspended carbon nanotube (CNT), the Franck-Condon blockade was reported [14, 15]. Due to the strong electron-phonon interaction in the CNT, the electric transport is accompanied by the lattice distortion, which results in the current suppression under a small bias voltage [16]. This is the manifestation of the Franck-Condon effect in the electric transport, which was originally known in the optical absorption of molecules [17]. Regarding the study of optical phonons, Amaha and Ono observed the phonon-assisted transport through a DQD. The current is markedly enhanced when the level spacing in the DQD is tuned to an integer multiple of the energy of longitudinal optical (LO) phonons in the semiconductor substrate [18].

In this paper, we show the LO-phonon lasing in the phonon-assisted transport through a DQD. First, we show that a DQD effectively couples to only two LO phonon modes. The phonon modes do not diffuse and act as a natural cavity since the optical phonons have a flat dispersion relation. Thus our laser does not require a cavity or resonator. The pumping to the upper level is realized by an electric current through the DQD under a finite bias voltage, in a similar manner to the microlaser in the circuit QED [6]. Thus the pumping rate is determined by the tunneling rate between the DQD and leads, . As we discuss in section 6, the amplified LO phonons occasionally escape from the “cavity” by decaying into so-called daughter phonons [19] that can be observed externally. When the pumping rate is much larger than the phonon decay rate , the stimulated emission of phonons, i.e., phonon lasing, takes place. We proposed a basic idea of the optical phonon lasing in our previous letter [20]. In this paper, we present further comprehensive discussion on the phonon lasing and address the possible experimental realization.

We also find the phonon antibunching in the same system if the pumping rate is smaller than . In this situation, the phonon emission is regularized by the single electron transport through the DQD. We emphasize that the phonon statistics can be changed by electrically tuning the tunnel coupling between DQD and leads. Note that LO-phonon-assisted transport through a DQD was theoretically studied by Gnodtke et al[21]. We also note that lasing for acoustic phonons was demonstrated using phonon-assisted transport through quantum wells fabricated in a semiconductor superlattice, which works as a cavity for acoustic modes [22]. Acoustic phonon lasers by optical pumping [23] and transport in the spin blockade regime [24] were proposed in a single quantum dot embedded in a superlattice.

The electron-phonon coupling in DQDs fabricated in CNTs is much stronger than the electron-optical-phonon coupling in DQDs made in GaAs substrate, as we discuss in section 2. Both the phonon lasing and antibunching are spoilt by phonon thermalization via the Franck-Condon effect in the former case. In the electric transport, the number of electrons in the DQD fluctuates, which is accompanied by the lattice distortion and thus the creation of bunched phonons. We show that this effect is negligible in weak coupling case of semiconductor-based DQDs but surpasses the lasing and antibunching in strong coupling case of CNTs1. We also show that the strong electron-phonon coupling brings about the Franck-Condon blockade in a DQD with finite bias voltages, as in the case of single quantum dots [14, 15, 16].

This paper is organized as follows. In section 2, we explain our model and calculation method. Starting from the microscopic electron-optical-phonon interaction, we show that only two phonon modes, - and -phonons, are coupled to an electron in the DQD. The effective Hamiltonian is then derived in terms of the phonon modes. Based on the Born-Markov-Secular approximation, we obtain the rate equation for the non-equilibrium dynamics of electrons and phonons in the DQD under a finite bias. In section 3, we take into account -phonons and disregard -phonons. We examine the electron transport accompanied by the phonon emission. This results in the phonon lasing or antibunching in the weak coupling case, whereas it brings about the phonon thermalization in the strong coupling case. These different situations are elucidated by the analytical solution of the rate equation as well as the numerical studies. In section 4, -phonons are examined without -phonons. -phonons do not contribute to the phonon-assisted tunneling in the DQD, in contrast to -phonons, and hence they are irrelevant to the phonon lasing and antibunching. We examine the Franck-Condon blockade under a finite bias by the coupling to -phonons as well as -phonons. Section 5 is devoted to the investigation of general situations in the presence of both - and -phonons. We show that -phonons do not disturb the lasing or antibunching of -phonons. In section 6, we discuss the validity of our theory and address possible experimental realizations to observe the phonon lasing and antibunching. Finally, we present our conclusions in section 7.

## 2 Model and calculation method

### 2.1 Phonon modes coupled to DQD and effective Hamiltonian

Figure 1(a) depicts our model of a DQD embedded in a semiconductor substrate, in which two single-level quantum dots, and , are connected by tunnel coupling . The energy levels, and , are electrically tunable. We choose and denote the level spacing by . We assume that the total number of electrons in the DQD is restricted to one or zero due to the Coulomb blockade. The electron couples to LO phonons of energy in the substrate by the Fröhlich interaction. Our system Hamiltonian is ,

 He =Δ2(nL−nR)+VC(d†LdR+d†RdL), (1) Hph =ℏωph∑\biqN\biq, (2) Hep =∑α=L,R∑\biqMα,\biq(a\biq+a†−\biq)nα, (3)

using creation (annihilation) operators () for an electron in dot and () for a phonon with wavevector . and are the number operators. The spin index is omitted for electrons. The coupling constant is given by

 Mα,\biq= ⎷ℏωphe22V[1ϵ(∞)−1ϵ(0)]1q∫\rmd\bir|ψα(\bir)|2\rme\rmi\biq⋅\bir, (4)

where [] is the dielectric constant at high [low] frequency, is the volume of substrate, and is the electron wavefunction in dot of radius . The LO phonons only around the point, such as , are coupled to the DQD because of an oscillating factor in the integral over . This fact justifies the dispersionless phonons in . We assume equivalent quantum dots and , whence with being a vector joining their centers.

In , an electron in dot couples to a single mode of phonon described by

 aα=∑\biqMα,\biqa\biq(∑\biq|Mα,\biq|2)1/2. (5)

We perform a unitary transformation for phonons from to - and -phonon modes,

 aS=aL+aR√2+(S+S∗),aA=aL−aR√2−(S+S∗), (6)

and others orthogonal to and , where is the overlap integral between and phonons in equation (5)2. Disregarding the modes decoupled from the DQD, we obtain the effective Hamiltonian

 H=He +ℏωph[NS+λS(aS+a†S)(nL+nR)] (7) +ℏωph[NA+λA(aA+a†A)(nL−nR)],

where and , with dimensionless coupling constants

 λS/A=12ℏωph⎛⎝∑\biq|ML,\biq±MR,\biq|2⎞⎠1/2. (8)

The mode functions for - and -phonons are shown in figure 1(b) along a line through the centers of the quantum dots. The definition and calculation of the mode functions are given in A. Since the phonons are dispersionless, they do not diffuse and act as a cavity including the DQD3. -phonons play a crucial role in the phonon-assisted tunneling between the quantum dots and thus in the phonon lasing, as discussed below, whereas -phonons do not since it couples to the total number of electrons in the DQD, . Both phonons are relevant to the Franck-Condon effect.

Our Hamiltonian in equation (7) is applicable to DQDs fabricated in a semiconductor substrate, where and for in GaAs. It also describes a DQD in a suspended CNT when an electron couples to a vibron, longitudinal stretching mode with , , and in experimental situations [14, 15], as shown in B.

### 2.2 Rate equation in energy eigenbasis

The DQD is connected to external leads and in series, which enables the electronic pumping by the electric current under a finite bias. The tunnel coupling between lead and dot is denoted by and that between lead and dot is by . We also introduce the phonon decay rate to take into account a natural decay of LO phonons into so-called daughter phonons due to the lattice anharmonicity [19]. We describe the dynamics of the DQD-phonon density matrix using a Markovian master equation

 ˙ρ =−\rmiℏ[H,ρ]+Leρ+Lphρ, (9)

where and describe the electron tunneling between the DQD and leads and the phonon decay, respectively. is written as

 Leρ=∑α=L,R;i,jΓα2 [fα(ϵi−ϵj)(|i⟩⟨i|d†α|j⟩⟨j|ρdα+d†αρ|j⟩⟨j|dα|i⟩⟨i| (10) −ρ|j⟩⟨j|dα|i⟩⟨i|d†α−dα|i⟩⟨i|d†α|j⟩⟨j|ρ) +¯fα(ϵi−ϵj)(|j⟩⟨j|dα|i⟩⟨i|ρd†α+dαρ|i⟩⟨i|d†α|j⟩⟨j| −ρ|i⟩⟨i|d†α|j⟩⟨j|dα−d†α|j⟩⟨j|dα|i⟩⟨i|ρ)],

with and being an eigenstate of and the corresponding energy eigenvalue, respectively, and being the Fermi distribution function for electrons [holes] in lead [25]. The Fermi levels in leads and are given by and , respectively, with bias voltage between the leads.

In the limit of large bias voltage, is reduced to

 Leρ=(ΓLD[d†L]+ΓRD[dR])ρ, (11)

where is a Lindblad dissipator. In this case, an electron tunnels into dot from lead with tunneling rate and tunnels out from dot to lead with in one direction. We examine this situation in the main part of this paper. With finite bias voltages, equation (10) is evaluated in sections 3.3 and 4.2, where the electron tunneling takes place in both directions unless is far beyond the temperature .

The phonon dissipator is given by

 Lphρ=Γph(D[aS]+D[aA])ρ, (12)

on the assumption that the temperature in the substrate is much smaller than and daughter phonons immediately escape from the surroundings of the DQD.

In the following, we adopt the Born-Markov-Secular (BMS) approximation [26] to equation (9). We diagonalize the Hamiltonian in equation (7) and set up the rate equation in the energy eigenbasis,

 ˙Pi =∑jLijPj (13)

for the probability to find the system in eigenstate . Here, The solution of equation (13) with determines the steady state properties. The condition to justify the BMS approximation will be given in section 6.

## 3 Lasing and antibunching of A-phonons

In this section, we examine -phonons, disregarding -phonons by fixing at . The results in this section are not modified by a finite coupling to -phonons, as seen in section 5.

### 3.1 Phonon-assisted transport and phonon lasing

First, we present our numerical results in the case of . We consider the limit of large bias voltage. Figure 2(a) shows the current through the DQD as a function of level spacing , with (solid line) and (dotted line). Beside the main peak at , we observe subpeaks at () due to the phonon-assisted tunneling4. At the th subpeak, electron transport through the DQD is accompanied by the emission of phonons. As a result, the phonon number is markedly enhanced at the subpeaks, as shown in figure 2(b), in both cases of and . However, the physics is very different for the two cases, as we will show below.

For and , the electronic state with phonons is coherently coupled to with phonons [27], similarly to cavity QED systems, if the lattice distortion is neglected. To examine the amplification of -phonons, we calculate the phonon autocorrelation function

 g(2)A(τ)=⟨:NA(0)NA(τ):⟩⟨NA⟩2. (14)

The numerator includes the normal product, . is proportional to the probability of phonon emission at time on the condition that a phonon is emitted at time [28, 29]. A value of indicates a Poisson distribution of phonons which is a criterion of phonon lasing, whereas [] represents the phonon antibunching [bunching]. We thus find phonon lasing at the current subpeaks in figure 2(c) in the case of (solid line).

When , the strength of the electron-phonon interaction is comparable to the phonon energy. In this case, the lattice distortion by the Franck-Condon effect seriously disturbs the above-mentioned coherent coupling between an electron and phonons in the DQD and, as a result, suppresses the phonon lasing. Indeed, at the current subpeaks, indicating the phonon bunching.

To compare the two situations in detail, we present the number distribution of -phonons in figures 3(a) and (b) at the current main peak and subpeaks. In the case of , a Poisson-like distribution emerges at the subpeaks, whereas a Bose distribution with effective temperature is seen at the main peak. is determined from the number of phonons in the stationary state as . When , on the other hand, the distribution shows an intermediate shape between Poisson and Bose distributions at the subpeaks and the Bose distribution at the main peak.

In figures 4(a) and (b), we plot the autocorrelation function as a function of . In the case of , , regardless of the time delay , which supports the phonon lasing at the current subpeaks. At the main peak, . This is a character of thermal phonons with temperature . When , we find an intermediate behavior, (), at the th subpeak. This indicates that the phonons are partly thermalized by the Franck-Condon effect. For larger , the distribution is closer to the Poissonian with smaller .

### 3.2 Competition between phonon lasing and Franck-Condon thermalization

To elucidate the competition between the phonon lasing and thermalization by the Franck-Condon effect, we analyze the rate equation in equation (13), focusing on the current peaks in the large bias-voltage limit. We introduce polaron states for an electron in dot and phonons with lattice distortion:

 |L,n⟩eA=|L⟩e⊗TA|n⟩A,|R,n⟩eA=|R⟩e⊗T†A|n⟩A, (15)

where

 TA=\rme−λA(a†A−aA) (16)

and its Hermitian conjugate describe the shift of equilibrium position of the lattice when an electron stays in dot and , respectively. Note that the lattice distortion produces extra phonons: . When (), the eigenstates of Hamiltonian are given by the zero-electron states , bonding and anti-bonding states between the polarons,

 |±,n⟩eA=1√2(|L,n⟩eA±|R,n+ν⟩eA) (17)

, and polarons localized in dot , (). This is a good approximation provided that . The rate equations for these states are

 ˙P0,n=−ΓLP0,n+∞∑m=0ΓR2|A⟨n|T†A|m+ν⟩A|2Pmol,m+ν−1∑m=0ΓR|A⟨n|T†A|m⟩A|2PR,m +Γph[(n+1)P0,n+1−nP0,n], (18) ˙Pmol,n=−ΓR2Pmol,n+∞∑m=0ΓL|A⟨n|TA|m⟩A|2P0,m +Γph[(n+1+ν2)Pmol,n+1−(n+ν2)Pmol,n], (19)

where (), and

 ˙PR,n =−ΓRPR,n+Γph[(n+1)PR,n+1−nPR,n], (20)

with (). As shown in C, these equations yield the current and electron number in the DQD, , in terms of the number of polarons localized in dot , , as

 I=eΓR1+⟨~nR⟩2+γ,⟨ne⟩=2−γ⟨~nR⟩2+γ, (21)

with . The number of -phonons is given by

 ⟨NA⟩=(ν+2λ2A)IeΓph+λ2A⟨ne⟩. (22)

The first two terms in equation (22) indicate the emission of phonons by the phonon-assisted tunneling (from dot to dot ) and creation of phonons by the lattice distortion (with two tunnelings between the DQD and leads) per transfer of a single electron through the DQD. The last term describes the average number of polarons in the stationary state.

When , we obtain

 I=I0+O(Γph/ΓL,R), (23)

where is the current at the main peak in the absence of electron-phonon interaction, and

 g(2)A(0)=ν+4λ2Aν+2λ2A+O(Γph/ΓL,R). (24)

These explain the numerical results in figure 2 at the current subpeaks. The formula in equation (24) indicates (phonon lasing) for and (thermalized phonons by the lattice distortion) for . In the latter case, the phonons follow the Bose distribution with to deduce in equation (22).

We comment on the peak width of the electric current in figure 2(a). The electron transfer around the th current peak is dominated by the tunneling between polaron states and with . Thus the peak width is determined by the effective tunnel coupling

 Wν =|eA⟨R,n+ν|He|L,n⟩eA|n≃⟨NA⟩ (25) =∣∣∣n!(n+ν)!(−2λA)ν\rme−2λ2ALνn(4λ2A)VC∣∣∣n≃⟨NA⟩

, where is the Laguerre polynomial5. The factor of in equation (25) stems from the electron localization by dressing the phonons in forming the polarons. This explains the narrower subpeaks in the case of than that of .

When , equation (25) yields

 Wν≃√⟨NA⟩+1λνAVC. (26)

This is in quantitative accordance with the peak widths in the case of [solid line in figure 2(a)].

So far we have considered the large bias-voltage limit. In this subsection, we examine the case of finite bias voltages to elucidate the Franck-Condon blockade [14] in our system. Figures 5(a) and (b) show the electric current as a function of bias voltage when the level spacing is tuned to the main and subpeaks in figure 2(a). The electron-phonon coupling is (a) and (b) 1. At the main peak () in case (a), the current is almost identical to in the large bias-voltage limit when exceeds the interdot tunnel coupling . (The current vanishes when , reflecting the formation of bonding and antibonding orbitals at energy level , from two orbitals at in the DQD.) The influence of electron-phonon coupling is hardly observable. At the subpeaks (, ) in case (a), on the other hand, the current is suppressed at small and it increases stepwise to the value in the large limit. This is due to the electron-phonon coupling, as explained below. The current suppression is much more prominent in case (b) with larger . We observe the suppression even at the main peak in this case.

The reason for the current suppression is as follows. When an electron tunnels between the DQD and leads, the equilibrium position of the lattice is suddenly changed to form the polaron, or , in equation (15). While all the phonon states participate in the polaron formation in the large bias-voltage limit, the phonon states are limited under finite bias voltages due to the energy conservation. This weakens the tunnel coupling between the DQD and leads and also between the quantum dots, which is known as the Franck-Condon blockade. In figures 5(a) and (b), the current increases stepwise as increases by because higher-energy states become accessible (Franck-Condon steps) and converges to in the large bias-voltage limit. The larger voltage is required to lift off the Franck-Condon blockade for larger [16].

Figures 5(c)–(f) show the phonon number and autocorrelation function as a function of . The phonon number shows the Franck-Condon steps in both cases of (c) and (d) . The autocorrelation function, on the other hand, is qualitatively different for the two cases. In figure 5(e) with , even at the first Franck-Condon step except for anomalous behavior around the beginning of the step. This indicates that the phonon lasing is robust against the current suppression by the Franck-Condon blockade and hence it is observable under finite bias. In figure 5(f) with , changes slowly with , reflecting -dependence of the thermalization due to the Franck-Condon effect.

### 3.4 Phonon antibunching

In subsections 3.1 to 3.3, we have restricted ourselves to the case of to examine the phonon lasing. If the tunnel coupling is tuned to be , we observe another phenomenon, antibunching of -phonons [30]. Figure 6(a) presents a color-scale plot of in the plane when is tuned to be at the first current subpeak (). We assume that , , and large limit of bias voltage. At and , for example, , representing a strong antibunching of phonons. This is because the phonon emission is regularized by the electron transport through the DQD. In figure 6(b), we plot the autocorrelation function of the electric current

 g(2)current(τ)=⟨:nR(0)nR(τ):⟩⟨nR⟩2, (27)

where is the electron number in dot . It fulfills , indicating the antibunching of electron transport, since dot is empty just after the electron tunnels out [29]. Remarkably, almost coincides with . When , the emitted phonon escapes from the natural cavity soon after the electron tunneling between the quantum dots. Thus the stimulated emission for the lasing does not take place.

At strong couplings of , neither phonon antibunching nor phonon lasing can be observed because of an effective phonon thermalization due to the Franck-Condon effect. More than one phonon is created by the polaron formation, which spoils the regularized phonon emission by single electron tunneling and results in the phonon bunching.

Even with small , bunched phonons are emitted if is too small. Then the number of phonons created by the tunneling is exceeded by that accompanied by the polaron staying in dot [the first two terms are much smaller than the last term in equation (22)], as discussed in C.4. The analytical expression of is also given for in the appendix.

Figure 6(c) shows a color-scale plot of when is tuned to be at the second current subpeak (). The antibunching does not occur even when because two phonons are emitted simultaneously by the electron tunneling, which are bunched to each other.

## 4 Franck-Condon effect of S-phonons

In this section, we examine -phonons and disregard -phonons with .

### 4.1 Franck-Condon thermalization

We begin with the large bias-voltage limit. The electric current has a single-peaked structure as a function of [Lorentzian with center at and width of , as will be seen in equation (30)]. We do not observe subpeaks at since -phonons are not relevant to the phonon-assisted tunneling between the quantum dots because they couple to the total number of electron, in the DQD. The polaron states involving -phonons are given by

 |L,n⟩eS=|L⟩e⊗TS|n⟩S,|R,n⟩eS=|R⟩e⊗TS|n⟩S, (28)

for an electron in dot or , with phonons, where the lattice distortion

 TS=\rme−λS(a†S−aS) (29)

is common for and . -phonons do not show the phonon lasing nor antibunching.

We derive the rate equation for arbitrary level spacing in D. By tracing out -phonon degrees of freedom, we obtain the reduced rate equation for electrons, which is the same as that in the absence of electron-phonon coupling. We obtain the electric current and electron number in the DQD,

 I =eΓR(Δ/VC)2+2+γ,⟨ne⟩ =(Δ/VC)2+2(Δ/VC)2+2+γ. (30)

The number of -phonons is given by

 ⟨NS⟩=2λ2SIeΓph+λ2S⟨ne⟩. (31)

The first term in equation (31) indicates the creation of phonons by the lattice distortion with two tunnelings between the DQD and leads per a single electron transfer through the DQD. The second term describes the average number of polarons. In contrast to equation (22) for -phonons, -phonons are not created by the interdot tunneling.

We also examine the autocorrelation function

 g(2)S(τ)=⟨:NS(0)NS(τ):⟩⟨NS⟩2. (32)

is independent of , for arbitrary . When , we find

 g(2)S(0)=2+O(Γph/ΓL,R), (33)

which indicates the thermalization induced by the Franck-Condon effect.

Next, we examine the Franck-Condon blockade under finite bias voltages. Figures 7(a) and (b) show the current as a function of when . The dimensionless coupling constant is (a) and (b) 1. While the Franck-Condon blockade suppresses the current under small bias voltages in the case of , the current suppression is negligible in the case of . In the former case, a larger voltage is needed to lift off the Franck-Condon blockade for larger .

When , the dependence of the current is almost the same as in figure 5 for -phonons with . The phonon number and its autocorrelation function also change with the bias voltage in a similar manner to those at the current main peak for -phonons.

## 5 Coupling with both phonon modes

Now we consider both - and -phonons. Here, we examine a DQD fabricated in the semiconductor substrate where an electron is weakly coupled to both phonons; , .

First, we analytically derive that the coupling to -phonons does not influence the calculated results in section 3 for the electron--phonon system in the large bias-voltage limit. Consider the current main peak () and subpeaks (), assuming that . The eigenstates of are given by the zero-electron states , bonding and anti-bonding states between the polarons, , and polarons localized in dot , . The rate equations for these states yield

 ˙P0,n;n′=−ΓLP0,n;n′+∞∑m,m′=0ΓR2|A⟨n|T†A|m+ν⟩A|2|S⟨n′|T†S|m′⟩S|2Pmol,m;m′ +ν−1∑m=0∞∑m′=0ΓR|A⟨n|T†A|m⟩A|2|S⟨n′|T†S|m′⟩S|2PR,m;m′ +Γph[(n+1)P0,n+1;n′+(n′+1)P0,n;n′+1−(n+n′)P0,n;n′], (34) ˙Pmol,n;n′=−ΓR2Pmol,n;n′+∞∑m,m′=0ΓL|A⟨n|TA|m⟩A|2|S⟨n′|TS|m′⟩S|2P0,m;m′ +Γph[(n+1+ν2)Pmol,n+1;n′+(n′+1)Pmol,n;n′+1−(n+n′+ν2)Pmol,n;n′], (35)

where (), and

 ˙PR,n;n′ =−ΓRPR,n;n′+Γph[(n+1)PR,n+1;n′+(n′+1)PR,n;n′+1−(n+n′)PR,n;n′], (36)

with . We trace out -phonon degrees of freedom by summing up both sides of equations (34)–(36) over . We then obtain the reduced rate equations for the electron--phonon system, which are just the same as equations (18)–(20) with , , and . This fact indicates that -phonons do not affect the dynamics of the electron--phonon system if the bias voltage is sufficiently large.

In figure 8, we plot (a) the electric current, (b) - and -phonon numbers, and (c) their autocorrelation function, as a function of , in the case of and . The current, phonon number, and autocorrelation function for -phonons are the same as in figure 2 with (solid line) where -phonons are disregarded, in accordance with the above-mentioned consideration. An increase in -phonon number is induced by the current via the Franck-Condon effect. It is explained by equation (31) using the current and electron number . at the current peaks, indicating the thermalization of -phonons.

When the bias voltage is finite, -phonon degrees of freedom cannot be traced out in the rate equation. Therefore -phonons can influence the current and distribution of -phonons. However, the influence is very small, provided that , because the current suppression by the Franck-Condon blockade with -phonons is negligible, as shown in figure 7(a).