Noise spectrum of quantum transport through double quantum dots: Renormalization and non-Markovian effects

Quantum transport in serial double quantum dots has attracted tremendous attention for both the interest of fundamental quantum physics and possible applications in nano-devices. Due to the intrinsic quantum coherence, double quantum dots have become ideal systems to investigate various quantum mechanical effects.^1–3 Furthermore, they have been widely considered as promising candidates for solid-state qubits (charge or spin)^4–8 and quantum states detector^9,10 for the realization of quantum information processing and quantum computation.¹¹ 

Recently, as a consequence of the quantum dots tunneling coupled with the electron reservoirs, the energy level renormalization effect has drawn attentions.^12–17 For the serial coupled double dots, the renormalization effect are obvious in the average current (or conductance) even in the limit of weak dot-reservoir coupling region.¹⁴ It has also been demonstrated that the renormalization gave rise to a negative differential conductance^14,15 and super-Poissonian shot noise.¹⁷ However, lots of studies so far focus on the stationary current and the zero-frequency shot noise,^12–17 the investigation of this effect on the finite-frequency noise spectrum of the current fluctuations is still unavailable.

It is well-known that the shot noise of nonequilibrium current fluctuations carries rich information beyond the average current and thus it has been studied extensively.^18–20 The finite-frequency noise spectrum would even access the energetic and dynamical information on the system, and the system-electrode composite systems.^21–32 The advances in nano-technology allow now to control and fabricate the double-dot structures. Recent experiments have realized for the measurements of quantum level repulsion,³³ the demonstration of coherent dynamics between the dot states,³ and the observation of the zero-frequency shot noise.³⁴ Furthermore, experimental progress has also been made toward measuring the finite-frequency noise spectrum of electron transport.³⁵ 

Motived by the interested renormalization effect and the experimental advances in nano-technology of the double dot, in this paper, we study the full frequency-dependent noise spectrum of the electron sequential tunneling through serial double dots. The calculations are based on the time-nonlocal particle-number resolved master equation, and the MacDonal’s formula. Our discussion focus on the renormalization effect on the average current, zero-frequency shot noise, and the finite-frequency noise spectrum with the non-Markovian effect.

The paper is organized as follows. In Sec. II, we introduce the model of Hamiltonian and the time-nonlocal particle-number resolved master equation approach for the construction of the current expression and its symmetric noise spectrum. Sec. III is devoted to the results, including the average current as well as zero-frequency shot noise, and finite-frequency noise spectrum with pay attention to the renormalization effect and non-Markovian feature. We finally give summary in Sec. IV.

Considering the transport through two coupled quantum dots (CQD) in series contacted by the two electron reservoirs (left α = L and right α = R), as sketched in Fig. 1. It is described by the standard entire Hamiltonian H_T = H_B + H + H′. The first part describes the contacted electron reservoirs $H B = ∑ k , α ϵ α k c α k † c α k$⁠, with $c α k †$ (c_αk) denoting the creation (annihilation) operator for electrons in the reservoir α ∈ L, R. The second and the third terms describe the double coupled dots and the tunneling between the dots and the reservoirs, respectively. They are given by

where $a u †$ (a_u) denotes the creation (annihilation) operator for electrons in the dot-u (u = l, r). U is the inter-dot Coulomb interaction, Ω describes the inter-dot electron coherent tunneling, and t_αk is the tunneling coefficient between the reservoir and the dot. Here we consider each dot at most one electron occupied for the assumption of the infinite intra Coulomb interaction and large Zeeman split in each dot. The involved states thus are |0〉, |l〉, |r〉 and |d〉 ≡ |lr〉 representing the two dots empty, the left dot occupied, the right dot occupied, and double dots occupied, with the energy 0, ϵ_l, ϵ_r and ϵ_d = ϵ_l + ϵ_r + U, respectively.

As well-known, the states of |l〉 and |r〉 are not the eigenstates of the double-dot Hamiltonian H (see Eq. (1a)) which contains the intrinsic coherent Rabi oscillation as illustrated by the coherent coupling strength Ω. The corresponding Rabi frequency denoted by Δ is the energy difference between the eigenstates, i.e., $Δ= ε + − ε − = ε 2 + 4 Ω 2$ with the eigen-energies $ε + = ε 0 + Δ 2$ and $ε − = ε 0 − Δ 2$⁠. Here, we denoted ε ≡ ϵ_l − ϵ_r and ε₀ = (ϵ_l + ϵ_r)/2.

For late use to the description of the effect of the reservoirs on the transient dynamics of the system (CQD), we introduce the correlation function of the bath,

where σ = + , − and $σ ̄ =−σ$⁠, 〈⋯〉_B stands for the statistical average over the bath (electron reservoirs) in thermal equilibrium, $F α + ( t ) = ∑ k c α k † e i ϵ α k t$ and $F α − ( t ) = ∑ k c α k e − i ϵ α k t$ are determined by the tunneling Hamiltonian in Eq. (1b). The bath correlation function is related to the reservoir spectral density $Γ α ( ω ) =2π ∑ k t α k t α k ∗ δ ( ω − ϵ α k )$ via fluctuation dissipation theorem,^36–38 

where

with $f α + ( ω )$ the Fermi-function of α-lead and $f α − ( ω ) =1− f α + ( ω )$⁠. Without loss of generality, we assume the reservoir spectral density as a Lorentzian-type form,^36,39,40

where μ_α is the chemical potential of the leads, Γ_α describes the dot-lead coupling strength and W_α is the band width of the source (drain) reservoir with α = L(R). Obviously the wideband limit (WBL), Γ_α(ω) = Γ_α, is achieved by simply letting W_α → ∞. Throughout this work, we set e = ħ = 1, and assume the anti-symmetric time-independent bias voltage as μ_L = − μ_R = V/2.

In the frequency-domain via Laplace transformation, i.e., $C α ( ± ) ( ω ) = ∫ 0 ∞ dt e i ω t C α ( ± ) ( t )$⁠, we have⁴¹ 

It can be further calculated as

where the real and the imaginary parts are the so-called bath interaction spectrum and dispersion, respectively.⁴² They are related via the Kramers-Kronig relations:

where $P$ denotes the principle value of the integral, and Ψ(x) is the digamma function.

In this part, we briefly review the Kernel master equation (ME) for the calculation of the current and noise spectrum (for the details see Ref. 27). The ME having the memory kernel Σ(t − τ) is described by

where the first term describes the intrinsic coherent dynamics determined by the system’s Hamiltonian in Eq. (1a), i.e., $L•= [ H , • ]$⁠, the second term arises from the effect of the coupled reservoir containing the memory effect. Considering the weak dot-reservoir coupling limit, i.e., Γ ≪ |μ_α − ϵ_u| and/or Γ ≪ k_BT with Γ = Γ_L + Γ_R, where sequential transport dominates, the self-energy for the expansion up to second-order of the coupling Hamiltonian is given by $Σ ( t − τ ) ≡〈 L ′ ( t ) e − i L ( t − τ ) L ′ ( τ ) 〉 B$ in the H_B-interaction picture. The explicit formalism for the self-energy in Eq. (9) thus reads,

where σ = + , −, and $C α u ( σ ) ( L , t ) = e − i L t C α ( σ ) ( t ) a u σ$⁠, with $a u + ≡ a u †$ and $a u − ≡ a u$⁠. Note that in the summation, only two sets of (αu) are considered, i.e., (α = L, u = l) and (α = R, u = r) due to the tunneling Hamiltonian of Eq. (1b) considered.

For the calculation of the transport output, i.e., the current and noise spectrum, we recur to the particle-number resolved master equation, implemented with memory kernel prescription. The key quantity is ρ⁽ⁿ⁾, the reduced system of interest (double dots) conditioned on n = n_α electrons passed through the tunnel junction to α–lead. It is given by²⁷ 

where α ≠ α′, with a specifying the junction lead of electrons counting performed. The unconditional ME of Eq. (9) is related to ρ⁽ⁿ⁾ via $ρ ( t ) = ∑ n ρ ( n ) ( t )$⁠.

With the knowledge of P(n, t) = Tr[ρ⁽ⁿ⁾], all transport properties can be obtained. For the current to the α–lead, it reads

where we introduced

The symmetric noise spectrum of the circuit current is defined as $S ( ω ) = ∫ − ∞ ∞ dt e i ω t { 〈 δ I ( t ) , δ I ( 0 ) 〉 }$⁠, with $δI ( t ) =I ( t ) − I ̄$ and $I ̄$ being the steady current. The circuit current is a combination of the left and right junction currents, I(t) = aI_L(t) − bI_R(t), which is the quantity accessible by experiment, and the coefficients a and b depend on the junction capacitances and satisfy a + b = 1.¹⁸ This leads to the noise spectrum contains three parts, i.e.,

which can be obtained via Via MacDonald’s formula,⁴³ e.g., $S α ( ω ) =2ω ∫ 0 ∞ dtsin ( ω t ) d d t [ 〈 n 2 〉 − ( I ̄ t ) 2 ]$ with $〈 n 2 〉= ∑ n n 2 P ( n , t ) = ∑ n n 2 Tr [ ρ ( n ) ( t ) ]$⁠.²⁷ Here, for simplicity, we denote S_αα(ω) ≡ S_α(ω). The specific current noise spectrum is given by²⁷ 

where all the quantity is in the frequency-domain via Laplace transformation, e.g., $f ( ω ) = ∫ 0 ∞ dt, e i ω t f ( t )$⁠, including the current operator of Eq. (12) and the self-energy of Eq. (10). Explicitly, they are given by

and

Here, we introduced the notation of $a → u σ •= a u σ •$ and $a ← u σ •=• a u σ$⁠.

For auto-correlation noise spectrum, S_α(ω) ≡ S_αα(ω), Eq. (16) is reduced to

The stationary state and current read, $ρ ̄ = [ − i ω ρ ( ω ) ] ω = 0$ and $I ̄ α =−Tr [ J α ( − ) ( ω = 0 ) ρ ̄ ]$⁠, respectively.

In the present work, for the demonstration of the renormalization effect as well as non-Markovian effect, we consider the strong inter-dot charging energy regime, U + ε₀ > μ_L > ε₀ > μ_R. It can be realized by, for instance, setting the positions of the two dots in a relatively short distance.

Let us first focus on the stationary properties. The interested quantities are the average current, $I ̄ =− I ̄ L = I ̄ R$⁠, and the zero-frequency noise (so-called shot noise), S(0) = S_L(0) = S_R(0) = − S_LR(0).^28,31 They correspond to the quantities in the long time limit where non-Markovian effect vanishes, however, the renormalization plays significant roles.^14,17

The numerically results are shown in Fig. 2 for wide band limit (WBL) suggesting the flat spectral density of the bath. It is worth pointing out that the renormalization is completely induced by the dispersion function Λ_α(ω) given in Eq. (8). This would be further demonstrated by the following analytical results. Consequently, the characteristics in Fig. 2 with and without renormalization correspond to the consideration with and without the dispersion function, respectively. We find that the renormalization effect plays prominent roles in the weak coherent inter-dot coupling regime with Ω ≲ Γ, while it has minor effect on strong inter-dot coupling regime with Ω > Γ, as shown in the left panel [(a) and (b)] and right panel [(c) and (d)] in Fig. 2, respectively. This is due to the fact that the central coupled dots is dominated by the intrinsic coherent dynamics for the latter case. While in the former case with Ω ≲ Γ, the transport tunneling processes dominates the dissipative dynamics that induces the renormalization effect.

For the weak inter-dot coupling regime (Ω ≲ Γ), under the approximation of f_L(ε₀) = 1 and f_L(ε₀ + U) = f_R(ε₀) = f_R(ε₀ + U) = 0 for WBL, we can get analytical results for stationary current $I ̄$ and the zero frequency shot noise $S ( 0 ) =2 I ̄ F ( 0 )$ as follows,

where γ = Γ_R/Γ_L, δΓ = Γ_L − Γ_R and the renormalized energy difference

The energy shift δε_α of the energy level in dot-u caused by the coupled reservoir-α, is given by

which is consistent with the result of Ref. 14. It has been demonstrated¹⁴ that the renormalized level difference $ε ̃$ reaches a extremum when the Fermi energy of the lead aligns with the energy needed for either single or double occupation, e.g., μ_α = ϵ₀ = 10Γ and μ_α = ϵ₀ + U = 30Γ in Fig. 2. This renormalization induces the asymmetric behavior of the current and shot noise as compared to those without renormalization, see Fig. 2. For Eq. (17b), it indicates that the super-Poissonian can be expected for asymmetrical coupling strength with Γ_L ≪ Γ_R.¹⁷ Similar result for shot noise (fano factor $F ̄ ( 0 ) =S ( 0 ) /2 I ̄$⁠) as expressed in Eq. (17b) has been obtained in Ref. 17 which considered the electron-spin in each dot based on full counting statistics technique.

For strong inter-dot coupling regime with Ω > Γ, we approximately have

which obvious has no renormalization effect and does not affected by the dispersion function as indicated in Fig. 2(c). Although the renormalization has minor effect on the shot noise, it still can be neglected [see Fig. 2(d)].

With the consideration of finite bandwidth as Lorentz-structured described by Eq. (5), the renormalization is strongly affected by the bandwidth W_α. We find that the energy shift in Eq. (18) for finite bandwidth can be approximated as

where $Λ α ( ω ) = Λ α ( + ) ( ω ) + Λ α ( − ) ( ω )$ with $Λ α ( ± ) ( ω )$ the dispersion function being given by Eq. (8). For WBL, Eq. (20) reduces to Eq. (19). Thus the presence and absence of the renormalization are corresponding to the dispersion Λ_α ≠ 0 and Λ_α = 0, respectively.

The corresponding numerical result of the renormalized level difference $ε ̃$ as a function of the bias voltage with different bandwidth is shown in Fig. 3. It displays the dips and the peaks when the Fermi energy of a reservoir becomes resonant with the energy level of the dot, single occupation (μ_α = ϵ₀) and double occupation (μ_α = ϵ₀ + U), respectively, as demonstrated in Ref. 14. Moreover, we find that the renormalized level separation is increasing with widening bandwidth. The reason is that the wide bandwidth enlarges the range of the electrons in the reservoirs involving in the tunneling processes and thus enhances the resonant tunneling. Consequently the magnitude of the peak-dip at μ_α = ε₀ + U = 30Γ (μ_α = ± V/2) is monotone increasing with the bandwidth and finally saturates at wideband limit. However and counter-intuitively, the magnitude of the peak-dip at μ_α = ε₀ = 10Γ reaches maximum at W = 20Γ and then decreases at the saturated value of WBL [see Fig. 3]. This is because the electrons of the reservoirs are not only resonant with the energy level of the dot ε₀, but also can tunneling through Coulomb-blockade channel for W > ε₀ + U = 20Γ.

The effect of the finite bandwidth on the average current and shot noise is displayed in Fig. 4. Due to the reduction of the range of the electrons involved in the tunneling processes, the magnitude of the current decreases with narrowing the bandwidth. Simultaneously, the asymmetrical behavior of shot noise as well as the average current becomes significant. Note that for finite bandwidth, the renormalization is not the only reason for negative differential conductance, which is different from WBL as studied in Ref. 14 and Ref. 17. The difference of the characteristic between bandwidth of W = 50Γ and WBL is minor on the renormalized level separation $ε ̃$⁠, however it is distinct on the average current and shot noise. Moreover, the shot noise is more sensitive to the renormalization than the average current as expected. These features are reasonable since the average current and the shot noise are related to $ε ̃ 2$ and $ε ̃ 4$ [see Eq. (17)], respectively.

Consider now the finite-frequency current noise spectrum. It describes the transient dynamics of the double-dot system under the electron-tunneling process accompanied by the energy (ħ ω) absorption/emission of detection. The full frequency regime of noise spectrum covers the short time scale (high-frequency regime) as well as long time scale (low-frequency regime) of the system dynamics. Consequently, the dominated behavior shall be the non-Markovian feature, especially in the high-frequency regime.

The numerical result for circuit noise spectrum expressed in Eq. (13) under wide-band limit is displayed in Fig. 5. Regardless of the dispersion function Λ(ω), the noise spectrum always shows the typical non-Markovian quasisteps as consistent with the demonstration in the electron transport through single quantum dot.^26–31 These non-Markovian quasisteps appear in the so-called high frequency regime around the energy-resonance of ω ≈ ω_α0 = |ε_± − μ_α| and ω ≈ ω_αU = |ε_± + U − μ_α|. On the formula, these non-Markovian quasisteps come from the bath interaction spectrum describing energy-dependent tunneling rate $Γ α ( ± ) ( ω )$ expressed in Eq. (4) (the real part of bath correlator Eq. (7)) as illustrated in Ref. 26. However, the magnitude of the noise spectrum is modified by the dispersion function both for weak and strong inter-dot coupling strength, which is different from zero-frequency shot noise. This means that the dispersion function not only induces the renormalization effect but also non-Markovian corrections.

Interestingly, the effect of the dispersion on the non-Markovian feature is vanished in the high-frequency limit where we find the noise spectrum at α-lead S_α(ω) → Γ_α and that of cross-lead S_LR(ω) → 0. This feature can be seen in Fig. 5 for circuit noise spectrum and further understood as follows. In the high frequency region, where the relevant time scale is short, the electron correlation between different leads is yet to be established and thus we have S_LR(ω) → 0. Consequently, the fluctuation arising now completely from the reservoir background is proportional to the system-reservoir coupling strength, i.e., S_α(ω) → Γ_α and the dispersion function induced the energy shift of the central dot is yet to be worked. This is the common non-Markovian feature of the transport through quantum dots.^27,30

In this frequency-dependent noise spectrum, the renormalization effect is still observable in the position shift of the Rabi coherence signal showing a dip at $ω≈Δ= ε 2 + 4 Ω 2$⁠. In the absence of dispersion function, say, Λ_α(ω) = 0, the Rabi coherence signal exactly appears at ω = Δ. In the presence of dispersion function, the position of this signal is slightly moved away from Δ for weak inter-dot coupling strength and still exact at Δ for strong inter-dot coupling strength, which can be seen in the inset of Fig. 5(a) for Ω = 0.5Γ and Fig. 5(b) for Ω = 3Γ, respectively. Reasonably, this renormalization effect is consistent with that of zero-frequency shot noise as discussed above. It is worth noting that Rabi coherence signal can be extracted even under Markovian approximation as shown in Fig. 5 (gray dash-line), due to the intrinsic dynamics of the central double-dot. For weak coherent coupling strength Ω ≲ Γ [see Fig. 5(a)], the Rabi signal appears at low-frequency region with the renormalization effect. In the strong inter-dot coupling regime of Ω > Γ, the Rabi signal appears at relatively high-frequency region that strongly modifies non-Markovian quasisteps as displayed in Fig. 5(b).

Furthermore, we would like to mention the effect of finite bandwidth on the noise spectrum. The non-Markovian flat quasisteps become tilted steps and S(ω → ∞) = 0, due to the restriction of the channels for electron transfer between the dots and leads. The behavior is quite similar to that of single quantum dot studied in Ref. 27 and thus is not shown here.

Based on the time-nonlocal particle number-resolved master equation approach, we have studied the sequential transport through serial two double dots in the strong charging energy regime. We mainly consider the renormalization effects on the stationary current as well as its zero-frequency shot noise and the non-Markovian feature in the finite-frequency noise spectrum.

We find that the energy renormalization effect of the central dots completely induced by the dispersion of the bath correlator. The renormalization effect plays the significant role on the stationary properties for weak inter-dot coherent coupling strength. It induces the asymmetrical behavior on the average current and zero-frequency shot noise as function of bias voltage for the emergence of the dip at the Coulomb-induced channel (i.e., μ_α ≈ ε₀ + U). The asymmetrical feature becomes more and more obvious with narrowing the bandwidth. However, the renormalization effect can be negligible in strong inter-dot coupling strength due to the dominated intrinsic coherence dynamics in the coupled dots.

For the finite-frequency noise spectrum, it displays typical non-Markovian feature with the appearance of the quaisteps at the energy-resonance. We find that the dispersion not only induces the renormalization effect but also brings non-Markovian corrections. Both for the weak and strong inter-dot coupling strength, the magnitude of the noise spectrum is modified by the dispersion function, which however does not affect the noise spectrum in the high-frequency limit. This is because the electron coherent (intrinsic) dynamics inside the double dot is not yet to be established at the relevant short time scale and the fluctuations arises completely from the reservoir background. The renormalization effect can be further observed in the position shift of the Rabi coherence signal showing a dip in the noise spectrum.

We acknowledge the support from the national NSFC (Grant No. 11274085), NSF of Zhejiang Province (Grant No. LZ13A040002), and Hangzhou Innovation Funds for Quantum Information and Quantum Optics.