Correlated Cooper pair transport and microwave photon emission in the dynamical Coulomb blockade

We study theoretically electromagnetic radiation emitted by inelastic Cooper-pair tunneling. We consider a dc-voltage-biased superconducting transmission line terminated by a Josephson junction. We show that the generated continuous-mode electromagnetic field can be expressed as a function of the time-dependent current across the Josephson junction. The leading-order expansion in the tunneling coupling, similar to the $P\left(E\right)$ theory, has previously been used to investigate the photon emission statistics in the limit of sequential (independent) Cooper-pair tunneling. By explicitly evaluating the system characteristics up to the fourth order in the tunneling coupling, we account for dynamics between consecutively tunneling Cooper pairs. Within this approach we investigate how temporal correlations in the charge transport can be seen in the first- and second-order coherences of the emitted microwave radiation.

Figure 1

(a) We consider a voltage $V$ biased Josephson junction in parallel with a junction capacitance $C$ and in series with a transmission line (TL), which is here represented as an infinite set of capacitances $C^{\prime }$ and inductances $L^{\prime }$ per unit length. (b) Charge tunneling is influenced by the possibility to emit radiation in the TL. In the superconducting state, this is the only way for a tunneling Cooper pair to dissipate the gained electrostatic energy. (c) In the considered theory circuit, the tunneled charge and the emitted photon(s) can be seen as excitations in the low- and high-frequency domains of the same electromagnetic environment, respectively.

Figure 2

Fluctuations at the Josephson junction always induce radiation to the transmission line $\left(Z_0\right)$. The spectrum of the junction-current and junction-voltage fluctuations, and the spectrum of the emitted radiation are equivalent up to a frequency-dependent constant. However, their relative magnitude might be essentially different, for example, due to the possibility of fast junction current fluctuations to be shunted by the parallel capacitor. Also, substantial enhancement of junction current fluctuations can be induced at certain frequencies by reflections in the transmission line, considered in Sec. 5.

Figure 3

Top: The two diagrams that contribute to current correlations in the leading order. The left-hand side diagram gives the contribution from forward Cooper-pair tunneling and the right-hand side diagram from backward tunneling. The connection lines visualize pair correlations after taking the ensemble average. Middle: Two diagrams contributing to the current in the fourth order. These diagrams can be reduced to a product of two first-order ones and a term describing interaction between the pairs (wiggly lines). This is done to pick out the finite asymptotic behavior for large $t-t^{\prime }$, since here the correlator does not decay to zero for finite $t-t^{\prime \prime }$ and $t^{\prime }-t^{\prime \prime \prime }$ (with increasing $t-t^{\prime }$). Bottom: These fourth-order diagrams decay to zero when $t-t^{\prime }$ increases and are not needed to be split into two.

Figure 4

(a) The imaginary part of $J\left(t\right)$ for an Ohmic transmission line with $R_{\mathrm{Q}}C=1$ ns and $\pi Z_0/R_{\mathrm{Q}}=0.1,1,2,3,4$ (from the bottom to top for $t<0$). The linear decrease near $t=0$ is connected to the finite charging energy $e^2/2C$ and is important if its decay, on time scale $Z_{0}C$, is slow compared to the induced oscillations, leading to a crossover at $Z_0\sim R_{\mathrm{Q}}$. (b) The form of the real part is determined by the relative magnitude of the cutoff frequency ${\omega }_{\mathrm{c}}=1/Z_{0}C$ and thermal energy ${\omega }_T=k_{\mathrm{B}}T/h$. For a typical low-Ohmic case, ${\omega }_{\mathrm{c}}$ dominates and the function is practically linear (green line, $Z_0/R_{\mathrm{Q}}=0.05,T=100$ mK, $C=10$ fF). At short times (top) a transient behavior occurs, similar to that at larger times (bottom) for a high-Ohmic environment at zero temperature (red line, $Z_0/R_{\mathrm{Q}}=4,C=10$ fF). When the temperature is higher than the energy cutoff, the short-time behavior is quadratic (blue line, $Z_0/R_{\mathrm{Q}}=4,T=20$ mK, $C=100$ fF).

Figure 5

Spectral density of the current fluctuations in the second- (left) and in the fourth-order (right) for three Ohmic transmission lines (TL) as a function of the bias voltage $V$ (a.u.). The positive frequencies describe emission and the negative absorption. (a) For the low-Ohmic TL ($Z_0/R_{\mathrm{Q}}=0.05,T=100$ mK) most of the fluctuations appear at the Josephson frequency $2eV/h$, seen as the V-shaped resonance lines. The fourth-order contribution describes their weak redirection to lower frequencies. (b) For increased TL resistance ($Z_0/R_{\mathrm{Q}}=0.5,T=10$ mK) lower-frequency fluctuations increase, due to the increased probability for multiphoton production during CP tunneling. The fourth-order spectral density mainly redirects fluctuations to lower frequencies. (c) For the high-Ohmic TL ($Z_0/R_{\mathrm{Q}}=4,T=10$ mK) the second-order fluctuations are shifted to lower frequencies, $\pm {\omega }_{\mathrm{J}}-2e^2/C\hslash$, corresponding to the energy loss when charging the junction capacitor by a Cooper pair. The fourth-order spectral density describes redirection of the fluctuations to lower frequencies but is also characterized by zero-frequency dip or peak, corresponding to blockade or enhancement of consecutive Cooper-pair tunneling. For all plots $C=10$ fF.

Figure 6

Left: The second-order coherence of Cooper-pair transport as defined in Eq. (41) for the setup of Fig. 5. Right: The corresponding $P\left(E\right)$ function and the studied voltage-bias points (arrows). The $g_{\mathrm{CP}}^{\left(2\right)}\left(t\right)$ function describes temporal correlations in the junction-current fluctuations originating in nonequilibrium between consecutively tunneling Cooper pairs. Coulomb blockade during recharging of the junction reduces the zero-frequency noise $[\int dt{g}_{\mathrm{CP}}^{\left(2\right)}\left(t\right)<0]$ whereas enhanced Cooper-pair tunneling rate increases the noise $[\int dt{g}_{\mathrm{CP}}^{\left(2\right)}\left(t\right)>0]$. At long time separations the correlations vanish, ${lim}_{t\to \infty }{g}_{\mathrm{CP}}^{\left(2\right)}\left(t\right)=0$.

Figure 7

(a) Second-order coherence $g^{\left(2\right)}\left(0\right)$ of photons at the Josephson frequency $2eV/\hslash$ for several low-Ohmic transmission lines and detection bandwidths. In the limit $Z_0\ll R_{\mathrm{Q}}$ the statistics are close to Poissonian, $g^{\left(2\right)}\left(0\right)\approx 1$. The second-order coherence decreases approximately linearly with increasing $Z_0/R_{\mathrm{Q}}$, meaning that photons become increasingly antibunched. Similar behavior is also found in the case of a single-mode cavity [8] (pink line). Here $\mathrm{BW}=\mathrm{\Delta }/\pi$. (b) The time dependence $g^{\left(2\right)}\left(t\right)$ for the largest bandwidth in (a). The antibunching vanishes in the time scale of the detector sensitivity $1/\mathrm{\Delta }$. The parameters used are ${\omega }_{\mathrm{J}}/2\pi =3.7$ GHz, $T=10$ mK, and $C=10$ fF. To obtain numerical convergence we have used an additional sharp cutoff of environmental modes above 13 GHz.

Figure 8

(a) We consider a transmission line with a stepped characteristic impedance providing a high zero-frequency impedance $\left(R=4R_{\mathrm{Q}}\right)$ and a resonance at ${\omega }_0/2\pi =5$ GHz. (b) When biased at $2eV=4e^2/2C+\hslash {\omega }_0$, the charge transport is expected to occur through repeated single-photon assisted Cooper-pair tunneling with a finite $RC$ charging time in between. (c) Second-order coherence of the photons emitted at the resonance frequency ${\omega }_0$ [11]. At temperatures roughly below $3E_C/k_{\mathrm{B}}\approx 60$ mK we observe antibunching within the $RC$ time scale. We measure photons at the resonance frequency within bandwidth $\mathrm{\Delta }/\pi =1$ GHz. (d) Second-order coherence of Cooper-pair transport as defined via low-frequency current noise, Eq. (41). The $g_{\mathrm{CP}}^{\left(2\right)}\left(t\right)$ function describes simultaneous changes in the low-frequency emission due to correlations between consecutively tunneling Cooper pairs.

Figure 9

In the more general analysis we consider an arbitrary transmission line section that is from one side terminated by the Josephson junction and from the other side connected to a semi-infinite transmission line with a constant characteristic impedance $Z_0$. In an experimental setup the outgoing high-frequency radiation can be measured independently from the direct current by using a bias T [5].

Figure 10

Visualization of full calculation of $G^{\left(2\right)}$. Each contribution in the summation (51) is given a four-branched diagram. The two upper branches correspond to photon annihilation operators, $\widehat{a}\left({\omega }_3\right)$ and $\widehat{a}\left({\omega }_4\right)$, and the two lower to creation operators, ${\widehat{a}}^{†}\left({\omega }_1\right)$ and ${\widehat{a}}^{†}\left({\omega }_2\right)$. The circles describe timings of tunneling events in operators ${\widehat{a}}_n \left(n\ge 1\right)$ (see Sec. 4a) and letters X are related to the vacuum noise $\left(n=0\right)$. Initially, only the last tunneling of ${\widehat{a}}_n \left(n\ge 1\right)$ is associated with Fourier frequency, marked as the blue shell, whereas circles with no extra shell describe previous (time-ordered) evolution on the same branch. There is no time-ordering between points at different branches. In the ensemble average, the vacuum-noise operators (X) contribute only when placed to inner branches. There they become connected to the circles of the same time direction, changing them either to blue (with Fourier frequency of the inner branch) or to green (with the sum of Fourier frequencies of the two branches). The total number of colored shells gives the order of the current moment in which this expression is related. After summation over all terms a Keldysh time ordering appears, which allows expressing the result using only two branches.