Finite frequency noise in a chiral Luttinger liquid coupled to phonons

We study transport between quantum Hall edge states at filling factor $\nu =1$ in the presence of electron-acoustic-phonon coupling. After a Bogoliubov-Valatin transformation, the low-energy spectrum of interacting electron-phonon system is presented. The electron-phonon interaction splits the spectrum into charged and neutral “downstream” and neutral “upstream” modes with different velocities. In the regimes of dc and periodic ac biases the tunneling current and nonequilibrium finite frequency nonsymmetrized noise are calculated perturbatively in tunneling coupling of the quantum point contact. We show that the presence of electron-phonon interaction strongly modifies noise and current relations compared to the free-fermion case.

Figure 1

The condition of validation of the BV transformation. The light orange region satisfies the inequality $g(\alpha ,\beta )<0$, and the light blue region is given by $g(\alpha ,\beta )>0$. The BV transformation is valid in the light blue region.

Figure 2

The renormalized spectrum of ${\epsilon }_1/{\epsilon }_s$ and ${\epsilon }_2/{\epsilon }_s$ as a function of renormalized Fermi energy ${\epsilon }_F/{\epsilon }_s$. For comparison the spectrum of the uncoupled phonon and plasmon system is shown as well. We set $\beta =0.1$.

Figure 3

The renormalized spectrum of ${\epsilon }_3/{\epsilon }_s$ as a function of renormalized Fermi energy ${\epsilon }_F/{\epsilon }_s$. We set $\beta =0.1$.

Figure 4

The squares of the first two matrix elements of the BV transformation as a function of renormalized Fermi energy ${\epsilon }_F/{\epsilon }_s$ [see Eq. (12)]. We set $\beta =0.1$.

Figure 5

The square of the third matrix element of the BV transformation as a function of renormalized Fermi energy ${\epsilon }_F/{\epsilon }_s$ [see Eq. (12)]. We set $\beta =0.1$.

Figure 6

The sum of squares of the first two BV matrix elements [see Eq. (12)]. We set $\beta =0.1$. Note that the sum is not equal to 1.

Figure 7

The relations between the first row elements of the BV matrix in Eq. (12). We set $\beta =0.1$.

Figure 8

Schematic representation of the setup: A QPC (dashed red line) with tunneling amplitude $\tau$ connects up ($u$) and down ($d$) chiral edge channels of integer QH at filling factor $\nu =1$. Each channel is coupled to acoustic phonons. The bias $\mu$ is applied to the upper incoming edge channel, while the down edge channel is grounded.

Figure 9

The tunneling current $I_{\mu }$ versus applied dc voltage $\mu$ at zero temperature for different even values of $b$ [see Eq. (24)].

Figure 10

The tunneling current $I_{\mu }$ versus applied dc voltage $\mu$ at zero temperature for different odd values of $b$ [see Eq. (24)].

Figure 11

The normalized nonsymmetrized finite frequency noise $S\left(\omega \right)/I_{\mu }$ is plotted versus the dimensionless frequency $\omega /\mu >0$ (emission noise) at zero temperature and fixed dc bias $\mu >0$ [see Eq. (31)]. We set $b=0.6$ and $b=1.2$.

Figure 12

The normalized tunneling current $I/I_{\mathrm{\Omega }}$ is plotted versus the dimensionless dc component of bias $\mu /\mathrm{\Omega }>0$ at zero temperature and fixed $\mathrm{\Omega }>0$ [see Eq. (40)]. Here we set $b=0.6, b=1.2, {\mu }_1/\mathrm{\Omega }=2$, and $I_{\mathrm{\Omega }}$ is given by Eq. (24) by simply replacing $\mu$ by frequency $\mathrm{\Omega }$.

Figure 13

The normalized unsymmetrized finite frequency noise $S\left(\omega \right)/I_{\mathrm{\Omega }}$ is plotted versus the dimensionless frequency $\omega /\mathrm{\Omega }>0$ (emission noise) at zero temperature and fixed $\mu >0$ and $\mathrm{\Omega }>0$ [see Eq. (43)]. Here we set $b=0.6, b=1.2, {\mu }_1/\mathrm{\Omega }=2, \mu /\mathrm{\Omega }=3$, and $I_{\mathrm{\Omega }}$ is given by Eq. (24) by simply replacing $\mu$ by frequency $\mathrm{\Omega }$.

Figure 14

The normalized unsymmetrized zero-frequency noise $S\left(0\right)/I_{\mathrm{\Omega }}$ is plotted versus the dimensionless dc component of bias $\mu /\mathrm{\Omega }>0$ at zero temperature and fixed $\mathrm{\Omega }>0$ [see Eq. (46)]. Here we set $b=0.6, b=1.2, {\mu }_1/\mathrm{\Omega }=2$, and $I_{\mathrm{\Omega }}$ is given by Eq. (24) by simply replacing $\mu$ by frequency $\mathrm{\Omega }$.