Sagnac interference in carbon nanotubes

The Sagnac interference mode arises when two interfering counterpropogating beams traverse a loop, but with their velocities detuned by a small amount $2u$, with $v_{R/L}=v_F\pm u$. In this paper we perform a perturbative nonequilibrium calculation of Sagnac interference in single-channel wires as well as armchair nanotube loops. We study the dependence of the Sagnac conductance oscillations on temperature and interactions. We find that the Sagnac interference is not destroyed by strong interactions, but becomes weakly dependent on the velocity detuning $u$. In armchairs nanotubes with typical interaction strength, $0.25\le g\le 0.5$, we find that the necessary temperature for observing the interference effect, $T_{\text{SAG}}$ is also only weakly dependent on the interaction, and is enhanced by a factor of 8 relative to the temperature necessary for observing Fabry-Pérot interference in the same system, $T_{\text{FP}}$.

Figure 1

(a) In a Mach-Zehnder interferometer the input beam is split into two beams which traverse independent paths before being recombined. (b) In a Fabry-Pérot interferometer a beam is split into a deflected ray, which is recombined at the output with a ray that traverses a loop. (c) The Sagnac interferometer splits the beam into a two beams which traverse the loop in two opposite orientations, and get recombined at the output. This allows a very sensitive measurement of the angular velocity of the interferometer, as it results in a different relative speed in the clockwise and counterclockwise rays. Clear rectangles represent beam splitters, and patterned rectangles represent mirrors.

Figure 2

(Color online) Schematic of a nanotube loop reported in Ref. 19. This geometry allows electrons to tunnel from the point $X$ on the loop to a distant point $X^{\prime }$ on the other end of the loop, and vice versa. We refer to this process as cross-loop tunneling. An electron entering from the left can traverse the loop moving right with velocity $v_R$, without scattering, or tunnel from $X$ to $X^{\prime }$ and traverse the loop moving left with velocity $v_L$.

Figure 3

(Color online) (a) The energy spectrum of an armchair nanotube. When the chemical potential is tuned away from the degeneracy points by a gate voltage, the left and right movers in each node will have different velocities, which leads to the Sagnac interference in the loop geometry. (b) This figure shows right- and left-moving electrons in node 1. The loop was unraveled in this figure, so the tunneling appears to be nonlocal, from point $X$ to point $X^{\prime }$, the two ends of the loop. The scattering shown in figure is from a right-moving electron from a given node at point $X$ to a left-moving electron, of the same node, at point $X^{\prime }$, and vice versa. This scattering gives rise to the Sagnac interference. (c) Sagnac interference can also arise without the loop geometry through internode tunneling, since right movers at node 2 have the same velocity as left movers at node 1.

Figure 4

The Keldysh contour used in the nonequilibrium calculation. The Keldysh time-ordering operator $T_K$ orders operators along the contour, so fields on the $(+)$ branch are always at an earlier time than fields on the $(-)$ branch.

Figure 5

(Color online) Differential conductance oscillations of a single channel of fermions, as a function of bias voltage $V_{\text{sd}}$, for velocity detuning $u/v_F=0.1$ and interaction strength $g=0.5$. The beating is due to the only two voltage oscillation frequencies in the problem, ${\Omega }_1=\frac{eL}{\hslash \left(v+u\right)}$ and ${\Omega }_2=\frac{eL}{\hslash \left(v-u\right)}$, where $v=v_F/g$. The voltage is in units of $\hslash v_F/eL$. The shorter voltage oscillation period is $\Delta V_{\text{sd}}=2\pi {\left(\frac{{\Omega }_1+{\Omega }_2}{2}\right)}^{-1}\approx 12.5\hslash v_F/eL$, and the large oscillation period is $\Delta V_{\text{sd}}=2\pi {\left(\frac{{\Omega }_1-{\Omega }_2}{2}\right)}^{-1}\approx 250\hslash v_F/eL$.

Figure 6

Coherent Sagnac oscillation amplitude vs temperature for different values of $\left(v_F/u\right)$, for noninteracting electrons $\left(g=1\right)$. The slowest decaying plot corresponds to $v_F/u=100$, and the fastest decaying plot corresponds to $v_F/u=10$. Temperature is given in units of $\hslash v_F/k_{B}L$.

Figure 7

(Color online) $T^{\ast }$ vs $u/v_F$, where $T^{\ast }$ is the temperature at which the coherent differential conductance (the part of the conductance which oscillates with gate voltage) reaches $e^{-1}$ of its zero-temperature value. For a noninteracting system, $g=1$, the single-channel case gives the same temperature dependence as the case with spin and node degeneracies, $T^{\ast }\propto v_F/u$. The single-channel temperature dependence is given for $g=0.5$ (squares, dashed), and $g=0.25$ (diamonds, dashed). The carbon nanotube temperature dependence is given for $g=0.5$ (triangles), and $g=0.25$ (inverted triangles). Temperature is given in units of $\hslash v_F/k_{B}L$. For reference, the $T^{\ast }$ corresponding to the $g=1$ Fabry-Pérot oscillations is also plotted.

Figure 8

(Color online) (a) Differential conductance oscillations for a nanotube, i.e., including both spins and both nodes in the spectrum, for velocity detuning $u/v_F=0.1$, and interaction strength $g=0.5$. In the nanotube case, the beating is due to the four voltage frequencies in the problem, ${\Omega }_i=\frac{eL}{\hslash v_i}$, where $v_{1/2}=v_F\pm u$, $v_3\approx v_F$, and $v_4\approx v_F/g$. (b) The voltage Fourier transform of the oscillations in (a) clearly displays the four dominant frequencies, ${\Omega }_{1-4}$, corresponding to the four velocities in the problem, and encode the nanotube parameters $v_F$, $g$, and $u/v_F$.