Second-order coherence of microwave photons emitted by a quantum point contact

Shot noise of electrons that are transmitted with probability $T$ through a quantum point contact (biased at a voltage $V_0$) leads to a fluctuating current that in turn emits radiation in the microwave regime. By calculating the Fano factor $F$ for the case where only a single channel contributes to the transport, it has been shown that the radiation produced at finite frequency ${\omega }_0$ close to $e{V}_0/\hslash$ and at low temperatures is nonclassical with sub-Poissonian statistics ($F<1$). The origin of this effect is the fermionic nature of the electrons producing the radiation, which reduces the probability of simultaneous emission of two or more photons. However, the Fano factor, being a time-averaged quantity, offers only limited information about the system. Here, we calculate the second-order coherence $g^{\left(2\right)}\left(\tau \right)$ for this source of radiation. We show that due to the interference of two contributions, two photon processes (leading to bunching) are completely absent at zero temperature for $T=50\%$. At low temperatures, we find a competition of the contribution due to Gaussian current-current fluctuations (leading to bunching) with the one due to non-Gaussian fluctuations (leading to antibunching). At slightly elevated temperatures, the non-Gaussian contribution becomes suppressed, whereas the Gaussian contributions remain largely independent of temperature. We show that the competition of the two contributions leads to a nonmonotonic behavior of the second-order coherence as a function of time. As a result, $g^{\left(2\right)}\left(\tau \right)$ obtains a minimal value for times ${\tau }^{*}\simeq {\omega }_0^{-1}$. Close to this time, the second-order coherence remains below 1 at temperatures where the Fano factor is already above 1. We identify realistic experimental parameters that can be used to test the sub-Poissonian nature of the radiation.

Figure 1

Source (left part) and analyzer (right part) of the nonclassical radiation: A quantum point contact (QPC) is biased at a constant voltage $V_0$ and produces current-current fluctuations due to shot and thermal noise. These fluctuations are filtered by an LC resonator and transmitted to the detector as electromagnetic radiation via a transmission line with characteristic impedance $Z_{\text{tl}}$.

Figure 2

Comparison of the photon rate of Eq. (7) (solid line) with the approximate expression Eq. (9) (dashed line) for $T=\frac{1}{2}$ and $k_B\vartheta /e{V}_0=0.1$ as a function of $\gamma /{\omega }_0$ normalized to the value $n_0=RT{G}_Q{Z}_0{k}_B\vartheta /4\hslash$ valid for $\gamma \to 0$.

Figure 3

Diagrams contributing to the photon-photon correlator $n^{\left(2\right)}$. The dashed lines denote the Keldysh contour and the current operators $I^{+}$ ($I^{-}$), denoted by black dots, are located on the lower (upper) branch. Each photon-number operator $n$ (gray boxes) consists of a current operator on each branch. (a) The contribution due to Gaussian current-current fluctuations: Each current operator consists of two fermionic operators and the solid lines with the arrows indicate their contractions. Although the diagram is reducible in terms of the current operators [as it decays into a product of two second-order correlators $S\left(\omega \right)$], it contributes the genuine term (14) to the irreducible correlator $n^{\left(2\right)}\left(\tau \right)$. (b) The three diagrams (nG1, nG2, nG3) of the non-Gaussian contributions to the photon-correlation function that originate from irreducible fourth-order current correlators. The sketches are analogous to (a): The current-operators are shown as black dots, the Keldysh contour as dashed lines, and the contractions as solid lines. We have omitted the arrows on the solid lines as each diagram contributes twice with the contributions differing by the direction of the arrows that define the contractions [20].

Figure 4

Photon correlation functions $n_G^{\left(2\right)}\left(\tau \right)$ due to Gaussian current-current fluctuations and $n_{\text{nG}}^{\left(2\right)}\left(\tau \right)$ due to non-Gaussian fluctuations at zero temperature for $Q=5$. It can be seen that both correlators decay on a characteristic scale set by the inverse of the cavity decay rate. The positive contribution $n_G^{\left(2\right)}$ and the negative contribution $n_{\text{nG}}^{\left(2\right)}$ add up to the photon-photon correlator $n^{\left(2\right)}\left(\tau \right)$ (dashed line). As $n^{\left(2\right)}$ is negative, the second-order coherence $g^{\left(2\right)}\left(\tau \right)$ is smaller than 1, indicating that the photons are antibunched.

Figure 5

Critical temperatures ${\vartheta }_c$ and ${\vartheta }_c^{*}$ as a function of the inverse quality factor $Q={\omega }_0/\gamma =e{V}_0/\hslash \gamma$. The dashed line is the evaluation of ${\vartheta }_c$ with all terms contributing to the diagrams in Fig. 3 taken into account. This result should be compared to the solid line that involves the approximate expressions (16). The results differ by at most a few percentages showing that (16) is a reasonable approximation for the relevant quality factors $Q\gtrsim 5$. Whereas ${\vartheta }_c$ indicates for which temperatures nonclassical signatures of the radiation can be detected in the time-averaged quantity (Fano factor), the dotted line shows the result for ${\vartheta }_c^{*}$, the critical temperature for which the second-order coherence $g^{\left(2\right)}\left(\tau \right)$ is below 1. The latter is evaluated with the approximate expressions of Eq. (16).

Figure 6

(a) The second-order coherence $g^{\left(2\right)}\left(\tau \right)$ as a function of the autocorrelation time $\tau$ and the temperature $\vartheta$ of the electronic system. The data have been evaluated for a quality factor $Q=5$ using the approximate expressions (16). (b) Line cuts (solid lines) for temperatures ${\vartheta }_c^{*}, {\vartheta }_c$, and $\vartheta =0.13e{V}_0/k_B$ from top to bottom. The dashed line is a calculation for $\vartheta ={\vartheta }_c$ using the exact expression for the diagrams in Fig. 3. It can be seen that the approximation (16) captures all essential features and even underestimates the nonclassical correlations.

Figure 7

(a) The plot shows the Fano factor $F$ as a function of $V_0$ for temperatures $\vartheta =(0,0.05,0.1,0.15)\hslash {\omega }_0/k_B$ from bottom to top. It can be seen that at low temperatures the minimal value of $F$ is achieved slightly above the resonant condition with $e{V}_0\approx (1.2-1.3)\hslash {\omega }_0$. (b) Second-order coherence $g^{\left(2\right)}\left(\tau \right)$ for the temperature $\vartheta =0.05\hslash {\omega }_0/k_B$ as a function of $\tau$. The different lines correspond to different bias voltages with $V_0=(1.0,1.1,1.2,1.3)\hslash {\omega }_0/e$ from bottom to top. The second-order coherence offers additional time-resolved information when compared to the Fano factor. For finite temperatures, $g^{\left(2\right)}$ is a nonmonotonous function with a minimum at time ${\tau }^{*}\approx 5{\omega }_0^{-1}$. All the results have been obtained using the realistic value $Q={\omega }_0/\gamma =5$ for the quality factor.