Noise spectrum of a quantum dot–resonator lasing circuit

We consider the electron transport setup schematically shown in figure 1, where electrons tunnel through a semiconducting double quantum dot coupled to a high-Q electromagnetic resonator such as a superconducting transmission line. The Hamiltonian includes the interacting dot–resonator system, H_S = H_d + H_r + H_I, which is responsible for the coherent dynamics. The double dot is described by

$$\begin{equation} H_{\rm d}=\sum_{j}\varepsilon_{ j}d^\dagger_{j} d_{j} +Ud^\dagger_{\mathrm{l}} d_{\mathrm{l}}d^\dagger_{\mathrm{r}} d_{\mathrm{r}}+\frac{t_{\mathrm{c}}}{2}\big(d^\dagger_{\mathrm{l}} d_{\mathrm{r}}+d^\dagger_{\mathrm{r}} d_{\mathrm{l}}\big) \end{equation} \tag{ 1 }$$

with d^†_j being the electron creation operators for the two levels in the dots j (j = l,r) with energies ε_j, separated by ε = ε_l − ε_r, which are coupled coherently with strength t_c. Both ε_j and t_c can be tuned by gate voltages [10, 11, 26–28]. The interdot Coulomb interaction is denoted by U. The transmission line can be modeled as a harmonic oscillator, H_r = ω_ra^†a, with frequency ω_r and a^† denoting the creation operator of photons in the resonator. The dipole moment induces an interaction between the resonator and the double dot, H_I, which will be specified below.

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We further account for electron tunneling between the dots and electrodes, $H_{\mathrm { t}}=\sum _{ k} ( V_{\mathrm {L} k } c^{\dagger }_{L k} d_{\mathrm {l}}+ V_{\mathrm {R} k } c^{\dagger }_{R k} d_{\mathrm {r}}+\mbox {H.c.})$, with tunneling amplitudes V_αk (with α = L,R). The electrodes with $H_{\mathrm { b}} =\sum _{\alpha k}\varepsilon _{\alpha k} c^{\dagger }_{\alpha k}c_{\alpha k}$ act as baths. Here c^†_αk is the electron creation operator for an electron state in the electrode α. Below, the tunneling between the electrodes and the dots is assumed to be an incoherent process.

The double dot can be biased such that at most one electron occupies each dot. The two charge states |L〉 and |R〉 serve as the basis of a charge qubit [29, 30]. In this work, we consider two limits, (i) strong U and (ii) weak U, respectively. In case (i), transport through the double dots involves only one extra third state, namely the empty dot |0〉, whereas in case (ii), two extra states, |0〉 and the double occupation state |2〉 ≡ |LR〉, are involved in the transport. In both limits the dipole interaction between the resonator and the double dot is H_I = ℏg₀(a^† + a)τ_z, with Pauli matrices acting in the space of the two charge states, τ_z = |L〉〈L| − |R〉〈R|.

In the eigenbasis of the double dot and within rotating wave approximation, the Hamiltonian of the coupled dot–resonator system, for strong interdot Coulomb interaction, can be reduced to

$$\begin{eqnarray} &&H_{\mathrm{S}} = \frac{\hbar\omega_0}{2} \sigma_z + \hbar \omega^{}_{\mathrm{r}} a^\dagger a + \hbar g (a^\dagger \sigma_- +a^{} \sigma_+), \end{eqnarray} \tag{ 2 }$$

while for weak interdot interaction an extra term U|2〉〈2| is to be included. In the restricted space of states we have d_l = |0〉〈L| + |R〉〈2| and d_r = |0〉〈R| − |L〉〈2|, and the Pauli matrix operates in the eigenbasis, i.e. σ_z = |e〉〈e| − |g〉〈g| with

$$\begin{eqnarray} &&\begin{array}{ll} |e\rangle = \cos\left(\theta/2\right)|L\rangle + \sin\left(\theta/2\right)|R\rangle, \\ |g\rangle = \sin\left(\theta/2\right)|L\rangle - \cos\left(\theta/2\right)|R\rangle. \end{array} \end{eqnarray} \tag{ 3 }$$

Here, we fix the zero energy level by ε_l + ε_r = 0. The angle $\theta = \arctan (t_{\mathrm {c}}/\varepsilon )$ characterizes the mixture of the pure charge states, the coupling strength is g = g₀ sin θ and $\omega _0 = \sqrt {\varepsilon ^2+t^2_c}/\hbar$ denotes the level spacing of the two eigenstates. It can be tuned via gate voltages, which allows control of the detuning Δ = ω₀ − ω_r from the resonator frequency.

The dynamics of the coupled dot–resonator system, which is assumed to be weakly coupled to the electron reservoirs with smooth spectral density, can be described by a master equation for the reduced density matrix ρ in the Born–Markov approximation [31, 32]. Throughout this paper, we consider low temperatures, T = 0, with vanishing thermal photon number and excitation rates. Consequently, the master equation is

$$\begin{equation} \dot \rho = -\frac{\mathrm{i}}{\hbar}\left[H_{\mathrm{S}}, \rho\right] + \mathcal L_{\rm L}\, \rho+\mathcal L_{\mathrm{R}}\, \rho +\mathcal L_{\mathrm{r}}\, \rho \equiv \mathcal L_{\rm tot} \, \rho, \end{equation} \tag{ 4a }$$

where the dissipative dynamics is described by Lindblad operators of the form

$$\begin{equation} {\cal L}_{i}\rho=\frac{\Gamma_i}{2}(2L_i\rho L_i^{\dagger}-L_i^{\dagger}L_i\rho-\rho L_i^{\dagger}L_i). \end{equation} \tag{ 4b }$$

The first two terms $\mathcal L_{\mathrm { L/R}}$ account for the incoherent sequential tunneling between the electrodes and the dots with $\Gamma _\alpha (\omega )=2\pi \sum _k |V_{\alpha k}|^2\delta (\omega -\varepsilon _{\alpha k})\equiv \Gamma _\alpha$. For the assumed high voltage and low temperature, i.e. in the absence of reverse tunneling processes, we have L_L = d^†_l and L_R = d_r with tunneling rates Γ_L and Γ_R, respectively. For the oscillator we take the standard decay term L_r = a with rate Γ_r = κ. Here, we ignore other dissipative effects, such as relaxation and dephasing of the two charge states, which were studied in [8], since such effects only weakly affect the main points we wish to study.

From the definition $I_\alpha (t)\equiv -e\frac {\mathrm {d}\langle n_\alpha (t)\rangle }{\mathrm {d}t}$ with $n_\alpha =\sum _k c^\dagger _{\alpha k}c_{\alpha k}$, it is straightforward to obtain the transport current from the electrodes to the dots [33, 34], $I_\alpha (t)={\mathrm { Tr}}[ \skew3\hat{I}_{\alpha }\rho (t)]$, with current operators given by

$$\begin{equation} \skew3\hat{I}_{\rm L}\rho(t)= \frac{e}{\hbar}\Gamma_{\rm L} d^\dagger_{\mathrm{l}}\rho(t) d_{\mathrm{l}}, \end{equation} \tag{ 5a }$$

$$\begin{equation} \skew3\hat{I}_{\mathrm{R}}\rho(t)= -\frac{e}{\hbar}\Gamma_{\mathrm{R}} d_{\mathrm{r}}\rho(t) d^\dagger_{\mathrm{r}}. \end{equation} \tag{ 5b }$$

$I=\frac {1}{2}(I_{\mathrm {L}}-I_{\mathrm {R}})=I_{\mathrm {L}}=-I_{\mathrm {R}}$

We consider the symmetrized current noise spectrum

$$\begin{eqnarray} S(\omega)&=&{\cal F}\langle \{\delta \skew3\hat{I}(t),\delta \skew3\hat{I}(0)\}\rangle \nonumber \\ &\equiv&\int^\infty_{-\infty}\mathrm{d}t\, \mathrm{e}^{\mathrm{i}\omega t}\langle \{\delta \skew3\hat{I}(t),\delta \skew3\hat{I}(0)\}\rangle \nonumber \\ &=&2 \,{\rm Re}\big\{ \widetilde{G}_I(\omega)+\widetilde{G}_I(-\omega)\big\}, \end{eqnarray} \tag{ 6 }$$

where $\delta \skew3\hat{I}(t)=\skew3\hat{I}(t)- I$ and $\widetilde {G}_I(\pm \omega )=\int ^\infty _{0}\mathrm {d}t\, \mathrm {e}^{\pm \mathrm {i}\omega t}G_I(t)$ with $G_I(t)=\langle \delta \skew3\hat{I}(t) \delta \skew3\hat{I}(0)\rangle$. In the Born–Markov approximation, the current noise spectrum can be calculated via the widely used MacDonald's formula [35] or the quantum regression theorem [31]. Since we already know the current operators, as expressed in equation (5), it is more convenient to calculate the current correlation function via the quantum regression theorem

$$\begin{equation} G_I(t)={\rm Tr} \big[\skew3\hat{I}\,\mathrm{e}^{{\cal L}_{\rm tot} t} \skew3\hat{I} \rho^{\rm st}\big]-I^2, \end{equation} \tag{ 7 }$$

where ρ^st denotes the steady-state density matrix.

According to the Ramo–Shockley theorem, the measured quantity in most experiments [19] is the total circuit current I(t) = aI_L(t) − bI_R(t), with coefficients, a + b = 1, depending on the symmetry of the transport setup (e.g. the junction capacitances). The circuit noise spectrum is thus composed of three components: S(ω) = a²S_L(ω) + b²S_R(ω) − 2abS_LR(ω) [19, 36], where $S_{\mathrm { \alpha }}(\omega )={\cal F}\langle \{\delta \skew3\hat{I}_\alpha (t),\delta \skew3\hat{I}_\alpha (0)\}\rangle$ are the auto-correlation noise spectra of the current from lead-α, and $S_{\mathrm { LR}}(\omega )= ({\cal F}\langle \{\delta \skew3\hat{I}_{\mathrm {L}}(t),\delta \skew3\hat{I}_{\mathrm {R}}(0)\}\rangle +{\cal F}\langle \{\delta \skew3\hat{I}_{\mathrm {L}}(t),\delta \skew3\hat{I}_{\mathrm {R}}(0)\}\rangle$)/2 is the current cross-correlation noise spectrum between different leads. Alternatively, in view of the charge conservation, i.e. I_L = I_R + dQ/dt, where Q is the charge on the central dots, the circuit noise spectrum can be expressed as [37–39] S(ω) = aS_L(ω) + bS_R(ω) − ab S_C(ω) with $S_{\mathrm { C}}(\omega )\equiv{\cal F}\langle \{\delta \dot {Q}(t),\delta \dot {Q}(0)\}\rangle =2S_{\rm L R}(\omega )+S_{\rm L}(\omega )+S_{\mathrm {R}}(\omega )$ [40]. Thus, from the behavior of the auto-correlation and cross-correlation noise spectra, which will be studied in the following, we can fully understand the circuit noise spectrum even including the charge fluctuation spectrum in the central dots, S_C(ω). At zero frequency, we have S(0) = S_L(0) = S_R(0) = −S_LR(0) and S_C(0) = 0 due to current conservation in the steady state.

Let us first consider the low-frequency regime around ω ∼ 0 displayed in figure 4. We find a zero-frequency peak and dip in the auto- and cross-correlation noise spectra, respectively. Both decrease and finally disappear when one approaches the lasing state. The height of the zero-frequency peak as a function of a detuning is shown in figure 3(d). Since in the absence of a resonator we have S_α(ω ≈ 0)/2I = −S_LR(ω ≈ 0)/2I ≃ 1, the peak/dip feature at zero frequency in the noise spectra must be the effect of the resonator.

The noise spectra in figure 4 have a Lorentzian shape with linewidth γ₀ ∼ κ, determined by the emission spectrum of the photons [32]. In the regime around zero frequency, corresponding to the long-time limit, the noise spectra are determined by the single minimum eigenvalue λ_min with the real part dominated by the weakest decay rate, i.e. κ. For weak interdot Coulomb interaction, where we have to account for one more incoherent tunneling channel (see figures 2(b) and (d)) the low-frequency noise spectra display a similar behavior as in figure 4(d), except for the enhancement of the zero-frequency peak as shown in figure 3. It is worth noting that in this low-frequency regime, the relation S_L(ω ∼ 0) = S_R(ω ∼ 0) = −S_LR(ω ∼ 0) is still satisfied. However, as we will show below, the cross-correlation noise changes sign beyond the low-frequency regime.

At higher frequencies but still within the range |ω| < ω_r, the spectra are no longer Lorentzian due to the contributions from several λ_k in equation (12). We find characteristic features showing a step and peak in the auto- and cross-correlation noise spectra, respectively, as shown in figure 5. The position of the step/peak is nearly independent of the detuning, while the magnitude is sensitive to it. With increasing dot–resonator interaction, both the step and peak are shifted as shown in figure 6. These characteristics are a consequence of the coherent dynamics of the coupled dot–resonator system. The step/peak occurs at ω = δE, where $\delta E=|E_{+,\langle n\rangle }-E_{-,\langle n\rangle }|=\sqrt {4g^2(\langle n\rangle +1)+\Delta ^2}\approx 2g\big (\langle n\rangle +1\big )$ is the Rabi frequency corresponding to the photon number 〈n〉. As expected, this coherent signal of the step/peak becomes weak and even disappears with increasing incoherent tunneling rate Γ (not shown in the figure). Interestingly, as shown in figure 6(b), we find that with increasing dot–resonator interaction, the coherent signal for weak interdot Coulomb interaction is not only shifted, but the step also turns into a dip. This is consistent with the coherent signal of the Rabi frequency in the double dot in the absence of the resonator showing a dip and peak in the auto- and cross-correlation noise spectra, respectively [39, 46]. It arises from the recovered symmetrical transition tunneling channels (figures 2(b) and (d)). In the low-frequency regime, ω < |ω_r|, the auto-correlation noise spectra of left and right leads satisfy S_L(ω) = S_R(ω). This is no longer true in the high-frequency regime ω ≳ |ω_r| for strong interdot Coulomb interaction, as will be shown in the following subsection.

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Before addressing the noise spectrum in the high-frequency regime, let us briefly discuss its properties in the absence of the resonator. It has been demonstrated [38, 39] that the signal of the intrinsic Rabi frequency ω₀ of the double dots can be extracted from the noise spectra. For instance, the auto-correlation noise spectrum shows a dip-peak structure and a dip at ω = ω₀ for strong and weak interdot Coulomb interactions, respectively [38, 39]. Considering the present parameter regime, where lasing is induced for ω₀ ≈ ω_r with very weak incoherent tunneling, Γ = 10⁻³ω₀, we find in the strong Coulomb interaction case nearly Poissonian noise in the full-frequency regime, with a small correction due to a weak coherent Rabi signal, i.e. S_α(ω₀)/2I ∼ 1 ± 5 × 10⁻⁵. The correction can be neglected compared to the signal induced by the coupled resonator as shown in figure 7.

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The signals in the high-frequency noise spectrum arise because of transitions with the energy E_±,n − E_±,n−1 ≈ ω_r. They depend on the detuning in the same way as the spectrum of the oscillator [47]. Namely, for positive detuning we find a signal at frequencies somewhat higher than ω_r and for negative detuning at frequencies below ω_r.

In contrast to the low-frequency case, for high frequencies the spectra of the current in the left and right junction, S_L(ω) and S_R(ω), do not have to be identical due to the overall symmetry of the circuit broken by the resonator. This feature has been demonstrated by the previous studies in [48, 49] for investigating the spectral properties of a resonator coupled to a single-electron transistor (SET) and a superconducting single-electron transistor (SSET), respectively, in the nonlasing regime. For the present studied setup in the lasing regime, in this case we find significant differences between the cases (i) and (ii) of strong and weak Coulomb interactions, as illustrated in the left and right columns of figure 7, respectively. For strong Coulomb interaction the correlators are

$$\begin{eqnarray} &&\begin{array}{ll} \displaystyle \langle I_{\mathrm{L}}(t) I_{\mathrm{L}}(0)\rangle = \sum_n\langle n| \langle 0| \rho_{I_{\mathrm{L}}}(t) |0\rangle |n\rangle,\\ \displaystyle \langle I_{\mathrm{R}}(t) I_{\mathrm{R}}(0)\rangle = \sum_n\langle n| \langle R| \rho_{I_{\mathrm{R}}}(t) |R\rangle |n\rangle , \end{array} \end{eqnarray} \tag{ 16 }$$

while for weak Coulomb interaction we have

$$\begin{eqnarray} &&\begin{array}{ll} \displaystyle \langle I_{\mathrm{L}}(t) I_{\mathrm{L}}(0)\rangle =\sum_n\langle n|\left[ \langle 0| \rho_{I_{\mathrm{L}}}(t) |0\rangle+\langle R| \rho_{I_{\mathrm{L}}}(t) |R\rangle \right] |n\rangle, \\ \displaystyle \langle I_{\mathrm{R}}(t) I_{\mathrm{R}}(0)\rangle = \sum_n\langle n|\left[ \langle R| \rho_{I_{\mathrm{R}}}(t) |R\rangle+\langle 2| \rho_{I_{\mathrm{R}}}(t) |2\rangle \right] |n\rangle. \end{array} \end{eqnarray} \tag{ 17 }$$

Here we introduced the density matrix ρ_Ii(t) which satisfies the master equation (4) with the initial condition $\rho _{I_i}(0)=\skew3\hat{I}_i \rho ^{\mathrm { st}}$ (i = L,R).

For strong Coulomb interaction only S_R(ω) couples directly to the state |R〉, which in turn couples resonantly to the oscillator. As a result we observe the signal at ω ≈ ω_r only in S_R(ω), while S_L(ω) ≈ 1 is unaffected by the oscillator. In contrast, in case (ii), where we allow the state |2〉 to participate, we again find a symmetry between the currents through the right and left junctions and S_L(ω) = S_R(ω), as well as the anti-symmetry between the auto- and cross-correlation noise spectrums, i.e. roughly S_α(ω)/2I ≈ 1 + ΔS(ω) and S_LR(ω)/2I ≈ −ΔS(ω) with the signal ΔS(ω) changing sign leading to a peak and a dip as function of frequency. Furthermore, in contrast to the low-frequency regime, the noise spectrum at high frequency shows a Fano-resonance profile. It displays the same mechanism as observed by Rodrigues [50] that the detector (here the double quantum dot) feels the force in two ways, namely the original one from the voltage-driven tunneling and the one from the resonator. It arises from a destructive inference between the two transition paths between |g〉 and |e〉, i.e. a direct tunneling channel through the leads and a transition assisted by the absorption and emission at the detection frequency. We would like to mention that the present result differs from the observation mode in [50]. These authors found a Fano-resonance in the charge noise spectrum of a single-electron transistor coupled to a resonator and conjectured that it should also show up in other experimentally more accessible variables. We find the Fano-resonance in the current noise spectrum, which is directly observable.