Is Telegraph Noise A Good Model for the Environment of Mesoscopic Systems?

Abstract

Some papers represent the environment of a mesosopic system (e.g., a qubit in a quantum computer or a quantum junction) by a neighboring fluctuator, which generates a fluctuating electric field—a telegraph noise (TN)—on the electrons in the system. An example is a two-level system, that randomly fluctuates between two states with Boltzmann weights determined by an effective temperature. To consider whether this description is physically reasonable, we study it in the simplest example of a quantum dot which is coupled to two electronic reservoirs and to a single fluctuator. Averaging over the histories of the TN yields an inflow of energy flux from the fluctuator into the electronic reservoirs, which persists even when the fluctuator’s effective temperature is equal to (or smaller than) the common reservoirs temperature. Therefore, the fuluctuator’s temperature cannot represent a real environment. Since our formalism allows for any time dependent energy on the dot, we also apply it to the case of a non-random electric field which oscillates periodically in time. Averaging over a period of these oscillations yields results which are very similar to those of the TN model, including the energy flow into the electronic reservoirs. We conclude that both models may not give good representations of the true environment.

Appendix: Dyson Equations for the Green’s Functions

The Green’s functions in Eq. (7) are derived from the Dyson equations,

$$\begin{aligned} G^{}_{\mathbf{k}d}(t,t')=\int dt^{}_{1}g^{}_\mathbf{k}(t,t^{}_{1})V^{}_{L}G^{}_{dd}(t^{}_{1},t'),\ \ G^{}_{d\mathbf{k}}(t,t')=\int dt^{}_{1}G^{}_{dd}(t,t^{}_{1})V^{*}_{L}g^{}_\mathbf{k}(t^{}_{1},t'), \nonumber \\ \end{aligned}$$

(A.1)

where $g^{}_\mathbf{k}(t,t^{}_1)$ is the decoupled Green’s function in the lead. Here and below, all the unspecified time integrals begin at $t^{}_0=-\infty$ and end at t. Substituting Eq. (A.1) into Eq. (7), we have

$$\begin{aligned} I^{}_{L}(t) =\int dt^{}_{1}[\varSigma ^{}_{L}(t,t^{}_{1})G^{}_{dd}(t^{}_{1},t)-G^{}_{dd}(t,t^{}_{1})\varSigma ^{}_{L}(t^{}_{1},t)]^{<}_{}, \end{aligned}$$

(A.2)

where $\varSigma ^{}_{L}(t,t')$ is the self energy due to the tunnel coupling with the left lead,

$$\begin{aligned} \varSigma ^{}_{L}(t,t')=|V^{}_{L}|^{2}g^{}_{L}(t,t')\equiv |V^{}_{L}|^{2}\sum _\mathbf{k} g^{}_\mathbf{k}(t,t^{}_1). \end{aligned}$$

(A.3)

Noting that

$$\begin{aligned} g^{r(a)}_\mathbf{k}(t-t')=\mp i\varTheta (\pm t\mp t')\langle \{ c^{}_\mathbf{k}(t),c^{\dagger }_\mathbf{k}(t')\}\rangle =\mp i\varTheta (\pm t \mp t')e^{-i\epsilon ^{}_{k}(t-t')}, \end{aligned}$$

(A.4)

and

$$\begin{aligned} g^{<}_\mathbf{k}(t-t')=if(\epsilon ^{}_{k})e^{-i\epsilon ^{}_{k}(t-t')},\ \ \ f^{}_L(\epsilon ^{}_{k})=\big \langle c^{\dagger }_\mathbf{k}c^{}_\mathbf{k}\big \rangle , \end{aligned}$$

(A.5)

The superscript r(a) indicates the retarded (advanced) Green’s function and corresponds to the upper (lower) sign on the right hand-side, while

$$\begin{aligned} f^{}_{L(R)}(\omega )=[e^{(\omega -\mu ^{}_{L(R)})/(k^{}_\mathrm{B}T^{}_{L(R)})}+1]^{-1} \end{aligned}$$

(A.6)

is the Fermi distribution in the left (right) lead], and using the wide-band limit, in which the densities of states in the reservoirs are assumed to be independent of the energy [22], we have

$$\begin{aligned} \varSigma ^{r(a)}_{L}(t,t')=\mp i\varGamma ^{}_{L}\delta (t-t'),\ \ \ \ \varSigma ^<_{L}(t,t')=2i\varGamma ^{}_{L}\int \frac{d\omega }{2\pi }e^{-i\omega (t-t')}f^{}_{L}(\omega ). \end{aligned}$$

(A.7)

Applying the Langreth rule, $[AB]^<=A^rB^<+A^<B^a$ [22], to Eq. (A.2), and substituting Eqs. (A.7), we find

$$\begin{aligned} I^{}_L(t)=2i\varGamma ^{}_L\left( \int \frac{d\omega }{2\pi }f^{}_L(\omega )\big [e^{-i\omega (t-t^{}_1)}G^a_{dd}(t^{}_1,t)-e^{-i\omega (t^{}_1-t)}G^r_{dd}(t,t^{}_1)\big ] -G^<_{dd}(t,t)\right) .\nonumber \\ \end{aligned}$$

(A.8)

The time-dependent particle flux into the right lead is derived from these equations by interchanging $L\Leftrightarrow R$ and $\mathbf{k}\Leftrightarrow \mathbf{p}$.

The Dyson equation for the Green’s functions on the dot reads

$$\begin{aligned} G^{}_{dd}(t,t') =g^{}_{d}(t,t')+\int \int dt^{}_{1}dt^{}_{2}g^{}_{d}(t,t^{}_{1})\big [V^*_L G^{}_{Ld}(t^{}_1,t')+V^*_R G^{}_{Rd}(t^{}_1,t')\big ], \end{aligned}$$

(A.9)

where

$$\begin{aligned} G^{}_{Ld}(t,t')=\sum _\mathbf{k}G^{}_{\mathbf{k}d}(t,t'),\ \ \ G^{}_{Rd}(t,t')=\sum _\mathbf{p}G^{}_{\mathbf{p}d}(t,t'), \end{aligned}$$

(A.10)

while $g^{}_d$ is the Green’s function on the isolated dot, with

$$\begin{aligned} g^{r(a)}_{d}(t,t')=\mp i\varTheta (\pm t\mp t')e^{ -i\int _{t'}^{t}dt^{}_{1}\epsilon ^{}_{d}(t^{}_{1})}, \end{aligned}$$

(A.11)

and $g^{<}_{d}=0$, since it is assumed that the dot is empty in the decoupled junction. Inserting Eq. (A.1) into Eq. (A.9) yields

$$\begin{aligned} G^{}_{dd}(t,t') =g^{}_{d}(t,t')+\int \int dt^{}_{1}dt^{}_{2}g^{}_{d}(t,t^{}_{1})\varSigma ^{}_{}(t^{}_{1},t^{}_{2})G^{}_{dd}(t^{}_{2},t'), \end{aligned}$$

(A.12)

where

$$\begin{aligned} \varSigma (t,t')=\varSigma ^{}_{L}(t,t')+\varSigma ^{}_{R}(t,t'). \end{aligned}$$

(A.13)

Using Eqs. (A.7), Eq. (A.9) becomes

$$\begin{aligned} G^{r(a)}_{dd}(t,t')=g^{r(a)}_{d}(t,t')\mp i \varGamma \int dt^{}_1g^{r(a)}_d(t,t^{}_1)G^{r(a)}_{dd}(t^{}_1,t'),\ \ \ \ \varGamma =\varGamma ^{}_L+\varGamma ^{}_R,\qquad \quad \end{aligned}$$

(A.14)

with the solution given in Eq. (17).

Using the Langreth rules, the lesser Green’s function on the dot obeys

$$\begin{aligned} G^<_{dd}(t,t')=\int dt^{}_1\int dt^{}_2 g^r_d(t,t^{}_1)\big [\varSigma ^r(t^{}_1,t^{}_2)G^<_{dd}(t^{}_2,t')+\varSigma ^<(t^{}_1,t^{}_2)G^a_{dd}(t^{}_2,t')\big ]\ .\qquad \quad \end{aligned}$$

(A.15)

Inserting Eqs. (A.7), this becomes

$$\begin{aligned} G^<_{dd}(t,t')= & {} -i\varGamma \int dt^{}_1 g^r_d(t,t^{}_1)G^<_{dd}(t^{}_1,t')\nonumber \\&+\int dt^{}_1\int dt^{}_2 g^r_d(t,t^{}_1)\varSigma ^<(t^{}_1,t^{}_2)G^a_{dd}(t^{}_2,t')\big ]. \end{aligned}$$

(A.16)

Differentiating this equation with respect to t, one finds

$$\begin{aligned} \frac{\partial G^<_{dd}(t,t')}{\partial t}=-i[\epsilon ^{}_d(t)-i\varGamma ]G^<_{dd}(t,t')-i\int dt^{}_1\varSigma ^<(t,t^{}_1)G^a_{dd}(t^{}_1,t'), \end{aligned}$$

(A.17)

with the solution

$$\begin{aligned} G^<_{dd}(t,t')=\int dt^{}_1\int dt^{}_2 G^r_{dd}(t,t^{}_1)\varSigma ^<(t^{}_1,t^{}_2)G^a_{dd}(t^{}_2,t'). \end{aligned}$$

(A.18)

With Eqs. (17), and setting $t'=t$, this becomes

$$\begin{aligned} G^{<}_{dd}(t,t)=\int \frac{d\omega }{2\pi }\varSigma ^{<}_{}(\omega )\int ^{t}dt^{}_{1}\int ^{t}dt^{}_{2}e^{\varGamma (t^{}_{1}+t^{}_{2}-2t)}\ e^{i\omega (t^{}_{2}-t^{}_{1})}X(t^{}_1,t^{}_2), \end{aligned}$$

(A.19)

where [see Eq. (A.7)]

$$\begin{aligned} \varSigma ^<(\omega )=\varSigma ^{<}_L(\omega )+\varSigma ^<_R(\omega )=2i[\varGamma ^{}_Lf^{}_L(\omega )+\varGamma ^{}_Rf^{}_R(\omega )]\ . \end{aligned}$$

(A.20)

Changing the double integration,

$$\begin{aligned} \int ^{t} dt^{}_{1}\int ^{t}dt^{}_{2}F(t^{}_{1},t^{}_{2})= \int ^{t} dt^{}_{1}\int ^{t^{}_{1}}dt^{}_{2}[F(t^{}_{1},t^{}_{2} )+F(t^{}_{2},t^{}_{1})], \end{aligned}$$

(A.21)

and changing variables, $t^{}_2\rightarrow \tau =t^{}_1-t^{}_2$, yield Eq. (24).

We now turn to the energy fluxes. The energy flux into the left lead is given in Eq. (15) In terms of the Green’s functions on the dot, this energy flux reads

$$\begin{aligned} I^{E}_{L}(t) =\int dt^{}_{1}[\varSigma ^{E}_{L}(t,t^{}_{1})G^{}_{dd}(t^{}_{1},t)-G^{}_{dd}(t,t^{}_{1}\varSigma ^{E}_{L}(t^{}_{1},t)]^{<}_{}\ , \end{aligned}$$

(A.22)

where $\varSigma ^{E}_{L}(t,t')$ is

$$\begin{aligned} \varSigma ^{E}_{L}(t,t')=\sum _\mathbf{k}\epsilon ^{}_{k}|V^{}_\mathbf{k}|^{2}g^{}_\mathbf{k}(t,t'). \end{aligned}$$

(A.23)

In the wide-band limit, the Fourier transforms of $\varSigma ^E_L$ are

$$\begin{aligned} \varSigma ^{E,r(a)}_L(\omega )=\mp i\omega \varGamma ^{}_{L(R)},\ \ \ \varSigma ^{E,<}_{L(R)}(\omega )=2i\omega \varGamma ^{}_{L(R)}f^{}_{L(R)}(\omega )\ . \end{aligned}$$

(A.24)

Using the Langreth rule in Eq. (A.22), we have

$$\begin{aligned} I^{E}_{L}(t)&=\int dt_{1}\Big [\varSigma _{L}^{E,r}(t,t_{1})G^{<}_{dd}(t_{1},t)+\varSigma _{L}^{E,<}(t,t_{1})G^{a}_{dd}(t_{1},t)\nonumber \\&\quad -G^{r}_{dd}(t,t_{1})\varSigma _{L}^{E,<}(t_{1},t)-G_{dd}^{<}(t,t_{1})\varSigma _{L}^{E,a}(t_{1},t)\Big ]. \end{aligned}$$

(A.25)

The second and third term give

$$\begin{aligned}&\int dt_{1}\Big [\varSigma _{L}^{E,<}(t,t_{1})\overline{G^{a}_{dd}(t_{1},t)}-\overline{G^{r}_{dd}(t,t_{1})}\varSigma _{L}^{E,<}(t_{1},t)\Big ]\nonumber \\&\quad =2i\varGamma _{L}\int \frac{d\omega }{2\pi }\omega f_{L}(\omega )\Big (\overline{G^{a}_{dd}(\omega )}-\overline{G^{r}_{dd}(\omega )}\Big ). \end{aligned}$$

(A.26)

The first and last terms yield

$$\begin{aligned}&\int dt_{1}[\varSigma ^{E,r}_{L}(t,t_{1})G^{<}_{dd}(t_{1},t)-G^{<}_{dd}(t,t_{1})\varSigma ^{E,a}_{L}(t_{1},t)]\nonumber \\&\quad =\varGamma _{L}\int d\tau \left( \frac{\partial \delta (\tau )}{\partial \tau }G_{dd}^{<}(t-\tau ,t)-G_{dd}^{<}(t,t-\tau )\frac{\partial \delta (\tau )}{\partial \tau }\right) \nonumber \\&\quad = -\varGamma _{L}\int d\tau \delta (\tau )\frac{\partial }{\partial \tau } \left( G_{dd}^{<}(t-\tau ,t)-G_{dd}^{<}(t,t-\tau )\right) \nonumber \\&\quad =\varGamma ^{}_{L}\int d\tau \delta (\tau )\int dt_{1}\int dt_{2}\frac{\partial }{\partial \tau }\left[ G^{r}_{dd}(t-\tau ,t_{1})\varSigma ^{<}(t_{1},t_{2})G_{dd}^{a}(t_{2},t)\right. \nonumber \\&\qquad \left. -G^{r}_{dd}(t,t_{1})\varSigma ^{<}(t_{1},t_{2})G_{dd}^{a}(t_{2},t-\tau )\right] \nonumber \\&\quad =-i\varGamma _{L}\int dt_{1}\big (\varSigma ^{<}(t,t_{1})G_{dd}^{a}(t_{1},t)+G_{dd}^{r}(t,t_{1})\varSigma ^{<}(t_{1},t)\big )\nonumber \\&\qquad -2i\varGamma _{L}\epsilon _{d}(t)G_{dd}^{<}(t,t), \end{aligned}$$

(A.27)

where the last two steps used Eqs. (A.18) and (A.17). Finally, we find Eq. (30).

The energy fluxes which result from the temporal variation of the (left and right) tunneling Hamiltonians, Eq. (16) (with an analogous expression for $I^{E}_{\mathrm{tun},R}$) require the Green’s functions $G_{\mathbf{kp}}(t,t')$. Using the Dyson equations

$$\begin{aligned} G_{\mathbf{kp}}(t,t')= & {} V_{\mathbf{k}}V_{\mathbf{p}}^{*}\int dt_{1}\int dt_{2}g_{\mathbf{k}}(t,t_{1})G_{dd}(t_{1},t_{2})g_{\mathbf{p}}(t_{2},t')\nonumber \\ G_{\mathbf{pk}}(t,t')= & {} V_{\mathbf{p}}V_{\mathbf{k}}^{*}\int dt_{1}\int dt_{2}g_{\mathbf{p}}(t,t_{1})G_{dd}(t_{1},t_{2})g_{\mathbf{k}}(t_{2},t')\ , \end{aligned}$$

(A.28)

the Lagreth rules and Eq. (A.7) we find

$$\begin{aligned}&I^{E}_{\mathrm{tun},L}(t)-\epsilon ^{}_{d}(t)I^{}_{L}(t)+I^{E}_{L}(t) \nonumber \\&\quad =\int dt_{1}\int dt_{2}\big [\varSigma _{L}(t,t_{1})G_{dd}(t_{1},t_{2})\varSigma _{R}(t_{2},t)-\varSigma _{R}(t,t_{1})G_{dd}(t_{1},t_{2})\varSigma _{L}(t_{2},t)\big ]^{<}\nonumber \\&\quad =\int dt_{1}\int dt_{2}\big [\varSigma _{L}^{r}(t,t_{1})\big (G_{dd}^{r}(t_{1},t_{2})\varSigma _{R}^{<}(t_{2},t)+G_{dd}^{<}(t_{1},t_{2})\varSigma _{R}^{a}(t_{2},t)\nonumber \\&\qquad +G_{dd}^{a}(t_{1},t_{2})\varSigma _{R}^{a}(t_{2},t)\big ) - \varSigma _{R}^{r}(t,t_{1})\big (G_{dd}^{r}(t_{1},t_{2})\varSigma _{L}^{<}(t_{2},t)\nonumber \\&\qquad +G_{dd}^{<}(t_{1},t_{2})\varSigma _{L}^{a}(t_{2},t)\big )+G_{dd}^{a}(t_{1},t_{2})\varSigma _{L}^{a}(t_{2},t)\big )\big ]\nonumber \\&\quad =2\varGamma _{L}\varGamma _{R}\int \frac{d\omega }{2\pi }\big (f_{R}(\omega )-f_{L}(\omega )\big ) \int dt_{1}\big [e^{-i\omega (t_{1}-t)}G_{dd}^{r}(t,t_{1})\nonumber \\&\qquad +e^{-i\omega (t-t_{1})}G_{dd}^{a}(t_{1},t)\big ], \end{aligned}$$

(A.29)

Hence

$$\begin{aligned} \overline{I^{E}_{\mathrm{tun},L}(t)}=\overline{\epsilon ^{}_{d}(t)I^{}_{L}(t)}-\overline{I^{E}_{L}(t)} + 2\varGamma _{L}\varGamma _{R}\int \frac{d\omega }{2\pi }\big (f_{R}(\omega )-f_{L}(\omega )\big ) \Big [\overline{G_{dd}^{r}(\omega )}+\overline{G_{dd}^{a}(\omega )}\Big ]. \nonumber \\ \end{aligned}$$

(A.30)

For the two types of average, presented in Sects. 3 and 4, this combination of averages vanishes.