1. Introduction
Semiconductor transistors lie in a fundamental position in modern electronics. Analogous to them, superconducting (SC) transistors, which can control the SC current [1-7], are attracting more and more attention, since they have lower power consumption and may provide potential applications for quantum information processing [8,9]. Recently, SC transistors based on Josephson junction microstructures [5,10-12] and superconducting quantum interference devices (SQUID) [13-15] have already been implemented to rectify (significant non-reciprocal supercurrent), switch [6,16-19] and reverse [20-22] the supercurrent [7,9,23]. Usually, such devices need to be controlled by an external magnetic field and/or electrostatic voltage. To our knowledge, these are the only two kinds of available control methods thus far [3,6,24,25]. However, besides the above two tunable methods, other schemes for the SC transistors based on the Josephson junction to control the supercurrent have yet to be proposed.
The ability of microwaves to manipulate superconducting currents has long been recognized, such as the alternating current Josephson effect and the Shapiro steps [26-29]. In contrast, visible light is generally not employed for this purpose, primarily because its frequency is significantly higher than the superconducting energy gap. The integration of superconductors with wide-band gap semiconductors or semiconducting quantum dots offers a potential pathway for controlling superconducting currents using visible light [30-32]. In the past, hybrid systems of semiconductor quantum dots proximity-coupled to SCs (also referred to as superconducting p-n junctions) have predominantly been employed to investigate radiative emission, particularly focusing on the generation of polarization-correlated photons [31-36]. Despite the significant research dedicated to emission phenomena, the reverse process—controlling superconducting current using coherent light—remains relatively underexplored in the existing literature.
In this paper, we propose a theoretical scheme for an SC transistor that is coherently controlled by light. The system is composed of two SC leads weakly linked by a two-level quantum dot (QD) driven by coherent light [figure 1(a)], which is similar to a single-photon transistor based on SC transmission line resonators and qubits [37]. We discover that the supercurrent can be switched off or on by properly controlling the driving light intensity. More importantly, the supercurrent passing through the transistor depends not only on the driving light intensity, but also on the phase Φd of the coherent light, and the SC phase difference Δφs between the two leads. This gives rise to the relation ${I}_{{\rm{s}}}={I}_{c}({\rm{\Phi }})\,\sin {\rm{\Phi }}$, with Φ=Δφs+2Φd. In this sense, we refer to it as the light-controlled dc Josephson effect.
Figure 1. (a) Demonstration of the SC transistor setup. The width and depth of the QD can be controlled by the gate voltages, which determine the positions of the two fermionic levels. Coherent light is injected into the QD, driving the electron between the levels. (b) Typical energy values setup: the chemical potentials μL,R, the QD levels Ωe,g, and the driving light frequency ωd. Here, we set μL ≡ Ωg ≡ 0 as the energy reference, and focus on the case where the voltage difference equals to the driving light frequency μR-μL ≡ ωd. (c) The change in critical current Ic with driving strength ${{ \mathcal E }}_{{\rm{d}}}$ and light detuning Δω. (d) The critical current Ic as the driving strength ${{ \mathcal E }}_{{\rm{d}}}$ increases shows a Fano profile with the fixed light detuning Δω=0, 0.4, 0.8, 1.2Δs. Hereafter, the same coupling strength is set as γL=γR=γ=0.1Δs. |
Moreover, it is intriguing that the critical current Ic can achieve complete reversal by adjusting the detuning and driving intensity of light, thereby implementing a light-controlled π-junction in figure 1(c) [20-23,38-40]. Remarkably, the critical current Ic exhibits a Fano-like dependence on the driving light intensity with fixed light detuning [see figure 1(d)] [41,42]. Specifically, as the driving light intensity increases, the critical current Ic first increases in the positive direction, but then decreases to zero and increases in the negative direction, finally Ic drops back and approaches zero. This behavior can be understood by comparing the resonance of the QD levels subjected to driving light with the SC band spectrum.
Furthermore, an interference loop is formed by two such SC transistors, and the light fields applied to the two QDs generate a phase difference depending on their spatial distance. As a result, the supercurrents passing through the interference loop in the forward and backward directions are asymmetric, thereby achieving a light-controlled Josephson diode. The optimal non-reciprocity of such a Josephson diode reaches 54% by properly adjusting the light intensity, which exceeds the best record reported to date (∼40%) [15,24].
2. Theoretical model of the superconducting transistor setup
The setup of the SC transistor is demonstrated in figure 1(a), which consists of a two-level QD in contact with two s-wave SC leads. Denoting ${{\rm{\Delta }}}_{{\rm{n}}}\equiv | {{\rm{\Delta }}}_{{\rm{n}}}| {{\rm{e}}}^{{\rm{i}}{\varphi }_{{\rm{n}}}}$ and μn as the pairing strength and chemical potential of the two leads (n=L, R), the Hamiltonian of lead-n is described by [10,43]
$\begin{eqnarray}\begin{array}{rcl}{H}_{{\rm{n}}} & =& \displaystyle \sum _{{\boldsymbol{k}}\sigma }({\epsilon }_{{\rm{n}},{\boldsymbol{k}}}+{\mu }_{{\rm{n}}}){c}_{{\rm{n}},{\boldsymbol{k}}\sigma }^{\dagger }{c}_{{\rm{n}},{\boldsymbol{k}}\sigma }\\ & & +[{{\rm{e}}}^{2{\rm{i}}{\mu }_{{\rm{n}}}t}{{\rm{\Delta }}}_{{\rm{n}}}^{* }{c}_{{\rm{n}},-{\boldsymbol{k}}\downarrow }{c}_{{\rm{n}},{\boldsymbol{k}}\uparrow }+{\rm{H.c.}}],\end{array}\end{eqnarray}$
with the index σ=↑, ↓ for spin. The chemical potential μn can be tuned by external bias voltages. Hereafter, we assume that the two leads have ∣ΔR∣=∣ΔL∣=Δs, but different phases φR,L and chemical potentials μR,L.The QD contains two spin-degenerate fermionic levels, described by ${H}_{0}={\sum }_{\sigma }\,{{\rm{\Omega }}}_{{\rm{e}}}\,{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{e}}\sigma }+{{\rm{\Omega }}}_{{\rm{g}}}\,{d}_{{\rm{g}}\sigma }^{\dagger }{d}_{{\rm{g}}\sigma }$ (Ωe > Ωg). Here, we consider that the level spacing of the two QD levels is much larger than SC gap (e.g., Ωe-Ωg is at least 10-100 times of Δs) [33], and the intradot Coulomb interactions are neglected for simplicity [44,45]. The electrons tunnel between the QD and the two SC leads via the following interaction, which is widely adopted in the SC p-n junction model [32-36,46]:
$\begin{eqnarray}{H}_{{\rm{t}}}=-\displaystyle \sum _{{\boldsymbol{k}},\sigma }\left({{\rm{t}}}_{{\rm{L}},{\rm{g}},{\boldsymbol{k}}}{d}_{{\rm{g}}\sigma }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\sigma }+{{\rm{t}}}_{{\rm{R}},{\rm{e}},{\boldsymbol{k}}}{d}_{{\rm{e}}\sigma }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}}\sigma }+{\rm{H.c.}}\right).\end{eqnarray}$
We consider a monochromatic coherent light with frequency ωd injected into the QD, which drives the electron between the two QD levels [47-49]:
$\begin{eqnarray}{V}_{{\rm{d}}}=-\displaystyle \sum _{\sigma }\,{{ \mathcal E }}_{{\rm{d}}}\,{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{g}}\sigma }{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}}t-{\rm{i}}{\phi }_{{\rm{d}}}}+{\rm{H.c.}}\end{eqnarray}$
Here, Φd is the phase of the coherent light, ${{ \mathcal E }}_{{\rm{d}}}$ is the driving strength, and $| {{ \mathcal E }}_{{\rm{d}}}{| }^{2}$ is proportional to the light intensity (hereafter we set it as a positive real number). We focus on the situation where the driving light frequency ωd is nearly resonant with the level spacing of the QD (Ωe-Ωg ≃ ωd > Δs), and the driving strength is taken as ${{ \mathcal E }}_{{\rm{d}}}\sim {{\rm{\Delta }}}_{{\rm{s}}}$, which is much smaller than the level spacing Ωe-Ωg and thus well within the weak driving regime.3. Light-controlled dc Josephson effect
In the rotating frame applied by the unitary transformation U(t)=$\exp [{\rm{i}}({N}_{{\rm{R}}}+{\sum }_{\sigma }{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{e}}\sigma })({\omega }_{{\rm{d}}}t+{\phi }_{{\rm{d}}})]$ (here ${N}_{{\rm{n}}}={\sum }_{{\boldsymbol{k}}\sigma }{c}_{{\rm{n}},{\boldsymbol{k}}\sigma }^{\dagger }{c}_{{\rm{n}},{\boldsymbol{k}}\sigma }$ is the total electron number operator in lead-n), the Hamiltonian of the QD under coherent driving becomes 1 ) for the two SC leads is also modified [see equation (B7 ) in Appendix B ], where the chemical potentials are replaced by the effective ones ${\bar{\mu }}_{{\rm{L}}}={\mu }_{{\rm{L}}}$ and ${\bar{\mu }}_{{\rm{R}}}={\mu }_{{\rm{R}}}-{\omega }_{{\rm{d}}}$, and the SC phases are corrected as ${\bar{\varphi }}_{{\rm{R}}}={\varphi }_{{\rm{R}}}+2{\phi }_{{\rm{d}}}$ and ${\bar{\varphi }}_{{\rm{L}}}={\varphi }_{{\rm{L}}}$ [figure 2(a)].
$\begin{eqnarray}{\bar{H}}_{\,\rm{qd}\,}=\displaystyle \sum _{\sigma }\Space{0ex}{2.5ex}{0ex}[\displaystyle \sum _{{\rm{i}}={\rm{g}},{\rm{e}}}{\bar{{\rm{\Omega }}}}_{{\rm{i}}}{d}_{{\rm{i}}\sigma }^{\dagger }{d}_{{\rm{i}}\sigma }-({{ \mathcal E }}_{{\rm{d}}}{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{g}}\sigma }+{\rm{H.c.}})\Space{0ex}{2.5ex}{0ex}],\end{eqnarray}$
with ${\bar{{\rm{\Omega }}}}_{{\rm{e}}}={{\rm{\Omega }}}_{{\rm{e}}}-{\omega }_{{\rm{d}}}$ and ${\bar{{\rm{\Omega }}}}_{{\rm{g}}}={{\rm{\Omega }}}_{{\rm{g}}}$ for the upper and lower levels, respectively. Similarly, in this rotating frame, the Hamiltonian (
Figure 2. (a) Level distribution of the SC transistor setup in the rotating frame. (b) Splitting of energy E+ (dashed line) and E- (dash-dotted line) of QD levels with increasing driving strength ${{ \mathcal E }}_{{\rm{d}}}$. The intersections of the purple and gray dotted lines correspond to E±=Δs. The Fano profile of critical current arises because the flow direction of Cooper pairs reverses the back [from lead-R to lead-L in (c)] and forth [from lead-L to lead-R in (d)] between two SC leads with increasing driving strength. |
Further to this, if we set μL ≡ Ωg ≡ 0 as the energy reference and only focus on the situation that μR-μL ≡ ωd, namely, the voltage difference is equal to the driving light frequency ωd, the time-dependent coefficients in the Hamiltonian (1 ) for the two leads would disappear. In this case, the Hamiltonian for the full system becomes time-independent, and the effective chemical potentials of the two SC leads become ${\bar{\mu }}_{{\rm{L}}}={\mu }_{{\rm{L}}}=0$ and ${\bar{\mu }}_{{\rm{R}}}={\mu }_{{\rm{R}}}-{\omega }_{{\rm{d}}}=0$, which are equal to each other [figure 2(a)].
Based on this time-independent model in the rotating frame, the electric current flowing from the left lead to the QD is defined by the rate of change of the total electron number in the left lead, i.e., Is=-e∂t⟨NL⟩, which can be further obtained by calculating the evolution of the Green function based on the Heisenberg equations of motion [50,51].
As a result, in the steady state, we find that the total current through the SC transistor induced by the driving light is exactly obtained as [Appendix B ]
$\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{s}}} & =& \frac{e}{h}\displaystyle \int {\rm{d}}\nu \frac{256{{\rm{\Gamma }}}^{3}{{\rm{\Delta }}}_{{\rm{s}}}^{2}{\nu }^{2}{{ \mathcal E }}_{{\rm{d}}}^{2}\left({{\rm{\Omega }}}_{{\rm{\Gamma }}}^{2}-4{\nu }^{2}\right)\,\rm{Sgn}\,[\nu ]\Re [{\gamma }_{{\rm{s}}}(\nu )]{f}_{{\rm{L}}}(\nu )\sin ({\rm{\Phi }})}{64{{\rm{\Gamma }}}^{2}{\nu }^{4}{\gamma }_{{\rm{s}}}^{2}(\nu ){({{\rm{\Omega }}}_{{\rm{\Gamma }}}^{2}-4{\nu }^{2})}^{2}+{\left(4{{\rm{\Gamma }}}^{2}\left({{\rm{\Delta }}}_{{\rm{s}}}^{2}{{\rm{\Omega }}}_{\delta }^{2}({\rm{\Phi }})+{\nu }^{2}(4{\nu }^{2}-{{\rm{\Omega }}}^{2})\right)-{\gamma }_{{\rm{s}}}^{2}(\nu )[{({{\rm{\Gamma }}}^{2}-4{\nu }^{2})}^{2}-16({\nu }^{2}{{\rm{\Omega }}}^{2}-{{ \mathcal E }}_{{\rm{d}}}^{4})]\right)}^{2}}\\ & := & {I}_{c}({\rm{\Phi }})\sin ({\rm{\Phi }}).\end{array}\end{eqnarray}$
Here, γ is the tunneling strength between the QD and the two leads (we set γL=γR ≡ γ), and the following notations are adopted: ${\gamma }_{{\rm{s}}}(\nu )=\sqrt{{\nu }^{2}-{{\rm{\Delta }}}_{{\rm{s}}}^{2}}$, ${\rm{\Omega }}:=\sqrt{\delta {\omega }^{2}+2{{ \mathcal E }}_{{\rm{d}}}^{2}}$, ${{\rm{\Omega }}}_{{\rm{\Gamma }}}\,=\sqrt{2{{\rm{\Omega }}}^{2}+{{\rm{\Gamma }}}^{2}}$ and ${{\rm{\Omega }}}_{\delta }^{2}({\rm{\Phi }})=\delta {\omega }^{2}+2{{ \mathcal E }}_{{\rm{d}}}^{2}\cos {\rm{\Phi }}$, with Δω=(Ωe-Ωg)-ωd as the QD-light detuning.Especially, here Φ is a phase difference defined by 5 ) follows a Josephson-like relation, but here Φ incorporates not only the phase difference from the two SC leads, Δφs ≡ φR-φL, but also an additional correction from the light phase Φd, which was previously attainable only by passing a magnetic field through the junction [52].
$\begin{eqnarray}{\rm{\Phi }}={\bar{\varphi }}_{{\rm{R}}}-{\bar{\varphi }}_{{\rm{L}}}={\varphi }_{{\rm{R}}}-{\varphi }_{{\rm{L}}}+2{\phi }_{{\rm{d}}}.\end{eqnarray}$
It is worth noting that the above supercurrent (In the large detuning limit, $\delta \omega \gg {{ \mathcal E }}_{{\rm{d}}}$ ($\delta \omega ={{\rm{\Omega }}}_{{\rm{e}}}-{{\rm{\Omega }}}_{{\rm{g}}}\,-{\omega }_{{\rm{d}}}\equiv {\bar{{\rm{\Omega }}}}_{{\rm{e}}}-{\bar{{\rm{\Omega }}}}_{{\rm{g}}}$), the above critical current becomes Ic(Φ) ≃ Ic, which is a constant independent of the phase difference Φ. Consequently, the above supercurrent (5 ) simplifies to ${I}_{{\rm{s}}}={I}_{c}\sin {\rm{\Phi }}$ [Appendix B ]. Namely, in this case, the result well returns the same form as the dc Josephson effect [26], except that Φ is corrected by the light phase Φd.
On the other hand, it is evident that the critical current Ic(Φ) also strongly depends on the driving strength ${{ \mathcal E }}_{{\rm{d}}}$, or the driving light intensity. As shown in figure 1(c), the critical current Ic depending on the driving strength ${{ \mathcal E }}_{{\rm{d}}}$ as well as the detuning Δω (here the total phase difference is set as Φ=π/2) exhibits a butterfly-like shape. Notably, when the detuning is fixed at Δω=0.4Δs [solid red line in figure 1(c)], the critical current Ic shows a Fano profile depending on the driving strength ${{ \mathcal E }}_{{\rm{d}}}$ [figure 1(d)], which can be fully reversed in both positive and negative directions. In this case, this SC transistor realizes a π-junction controlled by the light intensity [20-23,38-40].
4. Fano dependence of the critical current
Unexpectedly, as the driving light intensity increases, the critical current does not always increase, but exhibits a Fano profile as depicted in figure 2(b), which has been observed in controllable single-photon transistors as well [37]. This Fano profile is elucidated in the weak coupling limit γ < Δs, which allows us to neglect high-order terms γ/Δs. In this limit, the critical current (5 ) is reduced to 4 ) of light-driven QD as ${E}_{\pm }=(1/2)(\delta \omega \pm \sqrt{\delta {\omega }^{2}+4{{ \mathcal E }}_{{\rm{d}}}^{2}})$. By considering the main contribution of the integral (7 ) near ν ≃-Δs-ϵ (ϵ is infinitesimal) due to ${({\nu }^{2}-{{\rm{\Delta }}}^{2})}^{-\frac{1}{2}}\to \infty $, the critical current is approximated as 8 ) exhibits two resonance peaks corresponding to E±=Δs, the positive or negative of which is determined by the sign of $2{{\rm{\Delta }}}_{{\rm{s}}}^{2}-(\delta {\omega }^{2}+2{{ \mathcal E }}_{{\rm{d}}}^{2})$.
$\begin{eqnarray}{I}_{c}\simeq \int {\rm{d}}\nu \frac{e{\rm{\Gamma }}{(\Re [{\gamma }_{{\rm{s}}}(\nu )])}^{-\frac{1}{2}}{{\rm{\Delta }}}_{{\rm{s}}}^{2}{{ \mathcal E }}_{{\rm{d}}}^{2}}{h\left[2{\nu }^{2}-{{\rm{\Omega }}}^{2}\right]{\nu }^{2}}\frac{{{\rm{\Gamma }}}^{2}}{{{\rm{\Gamma }}}^{2}+\frac{{\gamma }_{{\rm{s}}}^{2}(\nu ){({\nu }^{2}-{E}_{+}^{2})}^{2}{({\nu }^{2}-{E}_{-}^{2})}^{2}}{{\nu }^{4}{\left(({\nu }^{2}-{E}_{+}^{2})+({\nu }^{2}-{E}_{-}^{2})\right)}^{2}}},\end{eqnarray}$
where ${\gamma }_{{\rm{s}}}(\nu )=\sqrt{{\nu }^{2}-{{\rm{\Delta }}}_{{\rm{s}}}^{2}}$ and ${{\rm{\Omega }}}^{2}=(\delta {\omega }^{2}+2{{ \mathcal E }}_{{\rm{d}}}^{2})$. Here, the QD levels are obtained by diagonalizing the Hamiltonian ( $\begin{eqnarray}{I}_{c}\simeq \frac{2e{\rm{\Gamma }}}{h}\frac{{{ \mathcal E }}_{{\rm{d}}}^{2}{(2\varepsilon {{\rm{\Delta }}}_{{\rm{s}}})}^{-\frac{1}{2}}}{2{{\rm{\Delta }}}_{{\rm{s}}}^{2}-{{\rm{\Omega }}}^{2}}\frac{{{\rm{\Gamma }}}^{2}}{{{\rm{\Gamma }}}^{2}+\frac{2\varepsilon {({{\rm{\Delta }}}_{{\rm{s}}}^{2}-{E}_{+}^{2})}^{2}{({{\rm{\Delta }}}_{{\rm{s}}}^{2}-{E}_{-}^{2})}^{2}}{{{\rm{\Delta }}}_{{\rm{s}}}^{3}{\left(({{\rm{\Delta }}}^{2}-{E}_{+}^{2})+({{\rm{\Delta }}}^{2}-{E}_{-}^{2})\right)}^{2}}}.\end{eqnarray}$
And numerical computations reveal that the aforementioned approximation is reasonable [see figure 4(c)]. Note that as the driving strength increases, the critical current (
Figure 3. The supercurrent through SC quantum interference device controlled by coherent light (a) and a magnetic field (b). (c) Total phase-asymmetric current (red solid line) as the sum of the phase-symmetric current (blue dotted line) through QD-1 and the phase-symmetric current (light blue dashed line) through QD-2 with the driving strengths ${{ \mathcal E }}_{{\rm{d}},1}=1.0{{\rm{\Delta }}}_{{\rm{s}}}$ and ${{ \mathcal E }}_{{\rm{d}},2}=1.1{{\rm{\Delta }}}_{{\rm{s}}}$. (d) Diode efficiency η versus driving strengths ${{ \mathcal E }}_{{\rm{d}},1}$ and ${{ \mathcal E }}_{{\rm{d}},2}$ for QD-1 and QD-2. The optical phase difference and light detuning are set to: ΔΦd=π/10 and Δω=0. |
Figure 4. The energy level (a) of the two superconducting leads connected by a light-controlled quantum dot in equation ( |
Specifically, when the driving light intensity is weak, the two QD levels are positioned beneath the SC gap [figure 2(c)], effectively creating Andreev bound states [53]. These two states serve as a pathway for supercurrent transport: the Cooper pair flows from the lead-R into the upper QD level through the Andreev reflection (AR), then transitions to the lower QD level via light emission, and finally flows into lead-L via AR again [figure 2(c)]. Meanwhile, the critical current is positive due to $2{{\rm{\Delta }}}_{{\rm{s}}}^{2}-(\delta {\omega }^{2}+2{{ \mathcal E }}_{{\rm{d}}}^{2})\gt 0$.
As the intensity of the driving light further increases, one of the two QD states approaches the SC continuum spectrum, resulting in the emergence of a resonant peak of positive critical current. However, when one of the two QD levels surpasses the SC gap, the positive critical current gradually decreases and even reverses. Once both QD levels surpass the SC gap [figure 2(d)], the supercurrent increases negatively, and another resonance peak of negative critical current appears. At this point, the Cooper pair flows from lead-L to the lower QD level, is driven by light to the upper QD level and finally transports into the lead-R [figure 2(d)].
The above physical picture has also been justified. As shown in figure 2(b), the driving strength ${{ \mathcal E }}_{{\rm{d}}}^{\pm }$ corresponding to the intersection points of ${E}_{\pm }({{ \mathcal E }}_{{\rm{d}}})=\pm {{\rm{\Delta }}}_{{\rm{s}}}$ [two dotted purple lines in figure 2(b)] and the position ${{ \mathcal E }}_{{\rm{peak}}}^{\pm }$ of two resonant peaks of the critical current are approximately equal within the range of γ error, i.e., $| {{ \mathcal E }}_{{\rm{d}}}^{\pm }-{{ \mathcal E }}_{{\rm{peak}}}^{\pm }| \leqslant {\rm{\Gamma }}$. The deviation between the approximate theory using equation (8 ) and the accurate numerical result can be attributed to the weak coupling condition γ < Δs and the exclusive consideration of the primary contribution from the integral near ν ∼-Δs.
5. Light-controlled SQUID
Furthermore, if two such SC transistors are placed nearby and form a loop, they constitute a SQUID that is also coherently controlled by light [32,33,54] [figure 3(a)]. The current through the SQUID sums the contributions from each QD, Is=Is,1+Is,2, where the current Is,i through QD-i is given by the modified Josephson relation (5 ). Under a proper driving light intensity (${{ \mathcal E }}_{{\rm{d}}}\ll \delta \omega $), the current through the SQUID is given by [see Appendix C ]
$\begin{eqnarray}{I}_{{\rm{s}}}\simeq {I}_{c,1}\sin ({\rm{\Delta }}{\varphi }_{{\rm{s}}}+2{\phi }_{{\rm{d}},1})+{I}_{c,2}\sin ({\rm{\Delta }}{\varphi }_{{\rm{s}}}+2{\phi }_{{\rm{d}},2}),\end{eqnarray}$
with the phase difference between the two superconductors Δφs ≡ φR-φL and the phase Φd,i of driving light applied to QD-i. Here, we consider the light fields applied to the two QDs originate from the same point source, thus they have a deterministic phase difference, which is determined by their optical paths ΔΦd ≡ Φd,2-Φd,1 [see figure 3(a)].If the light intensities applied to the two QDs are equal, they yield equal critical currents Ic,1=Ic,2 ≡ Ic, and then the supercurrent through the SQUID becomes ${I}_{{\rm{s}}}=2{I}_{c}\sin ({\rm{\Delta }}{\varphi }_{{\rm{s}}}+{\phi }_{{\rm{d}},1}+{\phi }_{{\rm{d}},2})\cos {\rm{\Delta }}{\phi }_{{\rm{d}}}$. This is similar to the SQUID controlled by magnetic flux [figure 3(b)] [52]. The sinusoidal oscillation of the supercurrent, adjusted by changing the optical path difference, provides a method to set a comparison reference between the SC phase and the light phase.
More generally, if the light intensities applied to the two QDs are not equal and generate different critical currents ${I}_{c,1}({{ \mathcal E }}_{{\rm{d}},1},{\phi }_{{\rm{d}},1})\ne {I}_{c,2}({{ \mathcal E }}_{{\rm{d}},2},{\phi }_{{\rm{d}},2})$, the SQUID exhibits an asymmetric current-phase relation [55], which can be utilized to realize a Josephson diode [15]. From the Josephson relation (5 ), it can be verified that the current through QD-1 ${I}_{{\rm{s}},1}={I}_{c,1}({{\rm{\Phi }}}_{1})\sin {{\rm{\Phi }}}_{1}$ is an odd function of Φ1=Δφs+2Φd,1, since Ic,1(Φ1)=Ic,1(-Φ1) is even; in contrast, because of the optical phase difference ΔΦd, the current through QD-2 Is,2(Φ2 ≡ Φ1+2ΔΦd), is asymmetric with respect to Φ1 [see figure 3(c)].
As a result, summing Is,1 and Is,2, the maximum currents through the SQUID in the two directions are different, $| {I}_{{\rm{s}},{\rm{\max }}}^{+}| \ne | {I}_{{\rm{s}},{\rm{\max }}}^{-}| $ [figure 3(c)], which makes the SQUID behave as a diode. To quantify the non-reciprocity of the diode, we adopt the following diode efficiency [15,24]
$\begin{eqnarray}\eta =\frac{\left|| {I}_{{\rm{s}},{\rm{\max }}}^{+}| -| {I}_{{\rm{s}},{\rm{\max }}}^{-}| \right|}{| {I}_{{\rm{s}},{\rm{\max }}}^{+}| +| {I}_{{\rm{s}},{\rm{\max }}}^{-}| }.\end{eqnarray}$
We find that, when the optical path difference is set to ΔΦd ≃ 0.1π, and the driving strengths applied on the two QDs are set as ${{ \mathcal E }}_{{\rm{d}},1}\simeq 0.99{{\rm{\Delta }}}_{{\rm{s}}}$, ${{ \mathcal E }}_{{\rm{d}},2}\simeq 1.16{{\rm{\Delta }}}_{{\rm{s}}}$, the above diode efficiency achieves its maximum η ≃ 53.9% [see figure 3(d)]. This exceeds the best record reported in literature ( ∼ 40%), which was controlled by magnetic field [15,24].
6. Conclusion
We have theoretically proposed a novel scheme for a coherently controlled SC transistor that operates under properly driven light. The supercurrent through this transistor adheres to a Josephson-like relation, ${I}_{{\rm{s}}}={I}_{c}({\rm{\Phi }})\sin {\rm{\Phi }}$, where the phase Φ=Δφs+2Φd here incorporates not only the phase difference between the two SC leads Δφs, but also the phase of the driving light Φd. The critical current Ic in our Josephson relation depends on the phase, detuning, and intensity of driving light, and it shows a Fano profile with the increase of the driving light intensity.
Furthermore, a light-controlled SQUID can be implemented by using two such SC transistors. Such a light-controlled SQUID can realize the Josephson diode effect, and the optimized non-reciprocal efficiency achieves up to 54%, which surpasses the maximum record reported in the recent literature.
For the experimental implementation of such light-controlled SC transistors, three promising platforms are: (i) the SC p-n junction system, which is composed a p-n junction in contact with two superconductors on both sides [7,30,32]; (ii) the superconductor-semiconductor hybrid system, where the semiconductor nanowire is in contact with two SC leads [16,18,56,57]; (iii) the single or double QD system, made from 2D semiconductor heterostructure in contact with two SC leads [6,7,23,48,58-60]. In principle, these platforms are achievable under current laboratory conditions.
Therefore, our new scheme provides a promising platform for flexibly manipulating supercurrents by coherent light in SC circuits or hybrid superconductor-semiconductor systems, and it may also promote further research on the integration of light with hybrid semiconductor-superconductor systems and the development of semiconductor-superconductor optoelectronic devices [7,30-33].
Acknowledgments
This study is supported by NSF of China (Grant Nos. 12088101 and 11905007), and NSAF (Grants Nos. U1930403 and U1930402).
Appendix A The generalized Landauer formula
In this section, we derive the generalized Landauer formula for the current measurement by connecting the normal or superconducting leads. Usually, the current measurement of the system is achieved by connecting with the local electron leads. Here, we consider the Hamiltonian of the measured system as the quadratic fermions
$\begin{eqnarray}{H}_{s}=\frac{1}{2}{{\boldsymbol{d}}}^{\dagger }\cdot {\bf{H}}\cdot {\boldsymbol{d}},\end{eqnarray}$
where the vector operation is ${\boldsymbol{d}}={[{d}_{{\rm{1}}},{d}_{{\rm{2}}},\ldots ,{d}_{{\rm{N}}}]}^{{\rm{T}}}$ with the site-n electron operator ${d}_{{\rm{n}}}=[{d}_{{\rm{n}}\uparrow },{d}_{{\rm{n}}\downarrow },{d}_{{\rm{n}}\uparrow }^{\dagger },{d}_{{\rm{n}}\downarrow }^{\dagger }]$ in Nambu notation. The site-n of the system is connected with the electron reservoir α, which is described by $\begin{eqnarray}{H}_{{\rm{n}}}^{\alpha }=\frac{1}{2}\displaystyle \sum _{{\boldsymbol{k}}}{{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}^{\dagger }\cdot {{\bf{H}}}_{{\rm{n}},{\boldsymbol{k}}}^{\alpha }\cdot {{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}},\end{eqnarray}$
with the vector operation of the electron lead ${{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}={({c}_{{\rm{n}},{\boldsymbol{k}}\uparrow },{c}_{{\rm{n}},-{\boldsymbol{k}}\downarrow },{c}_{{\rm{n}},{\boldsymbol{k}}\uparrow }^{\dagger },{c}_{{\rm{n}},-{\boldsymbol{k}}\downarrow }^{\dagger })}^{{\rm{T}}}$. The electron reservoir can be an s-wave superconductor, regarded as the measured lead or providing the superconducting proximity effect, and the Hamiltonian matrix for α=s is $\begin{eqnarray}{{\bf{H}}}_{{\rm{n}},{\boldsymbol{k}}}^{\,\rm{s}\,}:=\left[\begin{array}{cccc}{\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & & & {{\rm{\Delta }}}_{{\rm{n}}}\\ & {\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & -{{\rm{\Delta }}}_{{\rm{n}}} & \\ & -{{\rm{\Delta }}}_{{\rm{n}}}^{* } & -{\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & \\ {{\rm{\Delta }}}_{{\rm{n}}}^{* } & & & -{\epsilon }_{{\rm{n}},{\boldsymbol{k}}}\end{array}\right].\end{eqnarray}$
When the pairing strength is zero, i.e. Δn=0, the Hamiltonian matrix of metal leads α=e is obtained as ${{\bf{H}}}_{{\rm{n}},{\boldsymbol{k}}}^{\rm{e}\,}\,=\,\rm{diag}\{{\epsilon }_{{\rm{n}},{\boldsymbol{k}}},{\epsilon }_{{\rm{n}},{\boldsymbol{k}}},-{\epsilon }_{{\rm{n}},{\boldsymbol{k}}},-{\epsilon }_{{\rm{n}},{\boldsymbol{k}}}\}$. The single electron tunneling interaction between site-n of the system and the lead is written as $\begin{eqnarray}{H}_{t}=-\displaystyle \sum _{{\boldsymbol{k}}\sigma }({{\rm{t}}}_{{\rm{n}},{\boldsymbol{k}}}\,{d}_{{\rm{n}},\sigma }^{\dagger }{c}_{{\rm{n}},{\boldsymbol{k}}\sigma }+{\rm{H.c.}}).\end{eqnarray}$
Here, tn,k is tunneling strength between the site-n and the connected electron lead.The current operation for the time t is defined by the change of total electron number ${N}_{{\rm{n}}}={\sum }_{{\boldsymbol{k}}\sigma }{c}_{{\rm{n}},{\boldsymbol{k}}\sigma }^{\dagger }{c}_{{\rm{n}},{\boldsymbol{k}}\sigma }$ in the connected reservoir n as A5 ) is obtained by the time evolution of the Green's function ${\partial }_{t}{{\boldsymbol{G}}}_{ij}(t,{t}^{{\prime} })={\rm{i}}\langle {\partial }_{t}{[{{\boldsymbol{d}}}^{\dagger }(t)]}_{i}{[{{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}({t}^{{\prime} })]}_{j}\rangle $ with $t={t}^{{\prime} }$, which is equivalent to calculating the time evolution of the vector operator d(t) and cn,k(t) [61,62]. Then, it follows from the Heisenberg equation that the operator's time evolution is obtained as
$\begin{eqnarray}{I}_{{\rm{n}}}(t)=-e\langle {\partial }_{t}{N}_{{\rm{n}}}\rangle =-2\frac{e}{\hslash }\displaystyle \sum _{{\boldsymbol{k}}}\Im [\langle {{\boldsymbol{d}}}^{\dagger }(t)\cdot {{\bf{P}}}^{+}\cdot {{\bf{T}}}_{{\rm{n}},{\boldsymbol{k}}}\cdot {{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}(t)\rangle ].\end{eqnarray}$
Here, the projection operator is ${{\bf{P}}}^{+}=\,\rm{diag}\,\{{{\mathsf{P}}}^{+},{{\mathsf{P}}}^{+},\ldots ,{{\mathsf{P}}}^{+}\}$ with 4 × 4 diagonal matrix ${{\mathsf{P}}}^{+}=\,\rm{diag}\,\{1,1,0,0\}$, and ${{\bf{T}}}_{{\rm{n}},{\boldsymbol{k}}}={\left[{\bf{0}},\ldots ,{{\mathsf{T}}}_{{\rm{n}},{\boldsymbol{k}}},\ldots ,{\bf{0}}\right]}^{{\rm{T}}}$ are the 4N × 4 tunneling matrices with 4 × 4 blocks ${{\mathsf{T}}}_{{\rm{n}},{\boldsymbol{k}}}:={\rm{diag}}\left\{{{\rm{t}}}_{{\rm{n}},{\boldsymbol{k}}},{{\rm{t}}}_{{\rm{n}},{\boldsymbol{k}}},-{{\rm{t}}}_{{\rm{n}},-{\boldsymbol{k}}}^{* }-{{\rm{t}}}_{{\rm{n}},-{\boldsymbol{k}}}^{* }\right\}$, which indicates the coupling between the system and the reservoir n. The current ( $\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}{\boldsymbol{d}}(t) & =& \delta (t){\boldsymbol{d}}(0)-{\rm{i}}{\bf{H}}\cdot {\boldsymbol{d}}(t)+{\rm{i}}\displaystyle \sum _{{\rm{m}}}\displaystyle \sum _{{\boldsymbol{k}}}{{\bf{T}}}_{{\rm{m}},{\boldsymbol{k}}}\cdot {{\boldsymbol{c}}}_{{\rm{m}},{\boldsymbol{k}}}(t),\\ {\partial }_{t}{{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}(t) & =& \delta (t){{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}(0)-{\rm{i}}{{\bf{H}}}_{{\rm{n}},{\boldsymbol{k}}}^{\alpha }\cdot {{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}(t)+{\rm{i}}{{\bf{T}}}_{{\rm{n}},{\boldsymbol{k}}}^{\dagger }\cdot {\boldsymbol{d}}(t).\end{array}\end{eqnarray}$
Here, the summation m contains all site positions of the system connected by electron lead. The above linear equations of the vector operator cn,k(t) are solvable by the Fourier transform as follow $\begin{eqnarray}{{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}(\omega )={{\mathsf{G}}}_{{\rm{n}},{\boldsymbol{k}}}^{\alpha }(\omega )\cdot [{{\boldsymbol{c}}}_{{\boldsymbol{k}}}(0)+{\rm{i}}{{\bf{T}}}_{{\rm{n}},{\boldsymbol{k}}}^{\dagger }\cdot {\boldsymbol{d}}(\omega )],\end{eqnarray}$
where the Green's function of the lead is ${{\mathsf{G}}}_{{\rm{n}},{\boldsymbol{k}}}(\omega )\,={\rm{i}}{[{\omega }^{+}-{{\bf{H}}}_{{\rm{n}},{\boldsymbol{k}}}^{\alpha }]}^{-1}$ with ω+ ≡ ω+iϵ (ϵ is infinitesimal). Similarly, the dynamic evolution of the system in ω-space is exactly solved as $\begin{eqnarray}{\boldsymbol{d}}(\omega )={\bf{G}}(\omega )\cdot \left[{\boldsymbol{d}}(0)+{\rm{i}}{\boldsymbol{\xi }}(\omega )\right],\end{eqnarray}$
Here, the Green's function of the system is defined as ${\bf{G}}(\omega )={\rm{i}}{[{\omega }^{+}-{\bf{H}}+{\rm{i}}{\bf{D}}(\omega )]}^{-1}$. The random force of all the connected electron lead in Fourier space is $\begin{eqnarray}{\boldsymbol{\xi }}(\omega )=\displaystyle \sum _{{\rm{m}}}{{\boldsymbol{\xi }}}_{{\rm{m}}}(\omega )=\displaystyle \sum _{{\rm{m}}}\displaystyle \sum _{{\boldsymbol{k}}}{{\bf{T}}}_{{\rm{m}},{\boldsymbol{k}}}\cdot {{\mathsf{G}}}_{{\rm{m}},{\boldsymbol{k}}}^{\alpha }(\omega )\cdot {{\boldsymbol{c}}}_{{\rm{m}},{\boldsymbol{k}}}(0),\end{eqnarray}$
and the dissipation kernel caused by the electron reservoir is $\begin{eqnarray}\begin{array}{rcl}{\bf{D}}(\omega ) & =& \displaystyle \sum _{{\rm{m}}}{{\bf{D}}}_{{\rm{m}}}(\omega )=\displaystyle \sum _{{\boldsymbol{k}}}{{\bf{T}}}_{{\rm{m}},{\boldsymbol{k}}}\cdot {{\mathsf{G}}}_{{\rm{m}},{\boldsymbol{k}}}^{\alpha }(\omega )\cdot {{\bf{T}}}_{{\rm{m}},{\boldsymbol{k}}}^{\dagger }\\ & \equiv & \displaystyle \sum _{{\rm{m}}}\frac{1}{2}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}(\omega )+{\rm{i}}{{\bf{V}}}_{{\rm{m}}}(\omega ).\end{array}\end{eqnarray}$
Under the local tunneling approximation of the electron exchange, i.e. ${{\rm{t}}}_{{\rm{n}},{\boldsymbol{k}}}{{\rm{t}}}_{{\rm{m}},{\boldsymbol{k}}}^{* }\simeq {\delta }_{{\rm{n}},{\rm{m}}}| {{\rm{t}}}_{{\rm{n}},{\boldsymbol{k}}}{| }^{2}$, the dissipation kernel of the site-n connected by the lead is simplified as ${{\bf{D}}}_{{\rm{n}}}(\omega )=\,\rm{diag}\,\{{\bf{0}},\ldots ,\,{{\mathsf{D}}}_{{\rm{n}}}(\omega ),\ldots ,\,{\bf{0}}\}$ with 4 × 4 blocks ${{\mathsf{D}}}_{{\rm{n}}}(\omega )\,=\frac{1}{2}{{\mathsf{\Gamma }}}_{{\rm{n}}}(\omega )+{\rm{i}}{{\mathsf{V}}}_{{\rm{n}}}(\omega )$, where the real part ${{\mathsf{\Gamma }}}_{{\rm{n}}}(\omega )={{\mathsf{\Gamma }}}_{{\rm{n}}}^{+}(\omega )+{{\mathsf{\Gamma }}}_{{\rm{n}}}^{-}(\omega )$ leads to dissipation and the imaginary part ${{\mathsf{V}}}_{{\rm{n}}}(\omega )$ provides an effective interaction for the site-n by the superconducting proximity effect [51]
$\begin{eqnarray}\begin{array}{rcl}{{\mathsf{\Gamma }}}_{{\rm{n}}}^{\pm }(\omega ) & =& \pm \frac{{\rm{\Theta }}(\pm \omega -| {{\rm{\Delta }}}_{{\rm{n}}}| ){{\rm{\Gamma }}}_{{\rm{n}}}}{\sqrt{{\omega }^{2}-| {{\rm{\Delta }}}_{{\rm{n}}}{| }^{2}}}{{\rm{\Sigma }}}_{{\rm{n}}}(\omega ),\,{{\mathsf{V}}}_{{\rm{n}}}(\omega )\\ & =& -\frac{1}{2}\frac{{\rm{\Theta }}(| {{\rm{\Delta }}}_{{\rm{n}}}| -| \omega | ){{\rm{\Gamma }}}_{{\rm{n}}}}{\sqrt{| {{\rm{\Delta }}}_{{\rm{n}}}{| }^{2}-{\omega }^{2}}}{{\rm{\Sigma }}}_{{\rm{n}}}(\omega ),\\ {{\rm{\Sigma }}}_{{\rm{n}}}(\omega ) & =& \left(\begin{array}{cccc}\omega & 0 & 0 & -{{\rm{\Delta }}}_{{\rm{n}}}\\ 0 & \omega & {{\rm{\Delta }}}_{{\rm{n}}} & 0\\ 0 & {{\rm{\Delta }}}_{{\rm{n}}}^{* } & \omega & 0\\ -{{\rm{\Delta }}}_{{\rm{n}}}^{* } & 0 & 0 & \omega \end{array}\right).\end{array}\end{eqnarray}$
Here, the spectral density of the coupling strength have been introduced γn(ω)=2π∑k∣tn,k∣2Δ(ω-εn,k) ≃ γn in wide-band approximation. When the electron reservoir connecting with site-n is metal lead, i.e. Δn=0, the metal lead only brings the dissipation ${{\mathsf{V}}}_{{\rm{n}}}(\omega )=0$ and ${{\mathsf{D}}}_{{\rm{n}}}(\omega )=\frac{1}{2}{{\mathsf{\Gamma }}}_{{\rm{n}}}\equiv \frac{1}{2}({{\mathsf{\Gamma }}}_{{\rm{n}}}^{+}+{{\mathsf{\Gamma }}}_{{\rm{n}}}^{-})$ with ${{\mathsf{\Gamma }}}_{{\rm{n}}}^{+}={{\rm{\Gamma }}}_{{\rm{n}}}\,\rm{diag}\,\{1,1,0,0\}$ and ${{\mathsf{\Gamma }}}_{{\rm{n}}}^{-}={{\rm{\Gamma }}}_{{\rm{n}}}\,\rm{diag}\,\{0,0,1,1\}$ respectively.Then, the current In(ω) in Fourier space by (A5 ) is obtained as A7 ), (A8 ), the current (A12 ) is further simplified as [51]
$\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{n}}}(\omega ) & =& -\frac{2e}{\hslash }\displaystyle \int \frac{{\rm{d}}\nu }{2\pi }\displaystyle \sum _{{\boldsymbol{k}}}\langle {{\bf{d}}}^{\dagger }(\nu )\\ & & \cdot {{\bf{P}}}^{+}\cdot {{\bf{T}}}_{{\rm{n}},{\boldsymbol{k}}}\cdot {{\bf{c}}}_{{\rm{n}},{\boldsymbol{k}}}(\omega +\nu )\rangle .\end{array}\end{eqnarray}$
Together with the vector operators cn,k(ω) and d(ω) in equations ( $\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{n}}}(\omega ) & =& \frac{2{\rm{i}}e}{h}\displaystyle \int {\rm{d}}\nu \{\,\rm{tr}\,[{{\bf{G}}}^{\dagger }(\nu )\cdot {{\bf{P}}}^{+}\cdot {{\mathbb{C}}}_{{\rm{n}}}(\nu ,\omega +\nu )\\ & & -\displaystyle \sum _{{\rm{m}}}{{\bf{G}}}^{\dagger }(\nu )\cdot {{\bf{P}}}^{+}\cdot {{\bf{D}}}_{{\rm{n}}}(\omega +\nu )\\ & & \cdot {\bf{G}}(\nu +\omega )\cdot {{\mathbb{C}}}_{{\rm{m}}}(\nu ,\omega +\nu )]\}.\end{array}\end{eqnarray}$
Here, the items, including the initial states of the system d(0), have been ignored since we will discuss the steady-state current after the long time relaxation in the following. And, the correlation matrices of the random force are defined as ${[{{\mathbb{C}}}_{{\rm{n}}}(\nu ,\omega +\nu )]}_{{\rm{ij}}}=\langle {[{{\boldsymbol{\xi }}}_{{\rm{n}}}^{\dagger }(\nu )]}_{{\rm{j}}}{[{{\boldsymbol{\xi }}}_{{\rm{n}}}(\omega +\nu )]}_{{\rm{i}}}\rangle $, by which the relation like fluctuation-dissipation theorem is given as [51] $\begin{eqnarray}\begin{array}{c}\mathop{{\rm{lim}}}\limits_{\omega \to 0}\left[(-{\rm{i}}\omega ){{\mathbb{C}}}_{{\rm{n}}}(\nu ,\omega +\nu )\right]\\ =\,{f}_{{\rm{n}}}(\nu ){{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{+}(\nu )+{\bar{f}}_{{\rm{n}}}(-\nu ){{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{-}(\nu ).\end{array}\end{eqnarray}$
Here, fn(ν) is the electron Fermi distribution in the initial state and ${\bar{f}}_{{\rm{n}}}(\nu )=1-{f}_{{\rm{n}}}(\nu )$ is hole distribution, respectively. Then, after a long time relaxation and by the final value theorem $\mathop{\mathrm{lim}}\limits_{t\to \infty }{I}_{{\rm{n}}}(t)=\Im \{\mathop{\mathrm{lim}}\limits_{\omega \to 0}(-{\rm{i}}\omega ){I}_{{\rm{n}}}(\omega )\}$, the generalized Landauer formula is obtained as In ≔ In,t+In,p, which includes the usual transport current In,t from the lead-n to the lead-m: $\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{n}},{\rm{t}}} & =& \frac{e}{2\hslash }\displaystyle \sum _{{\rm{m}}}{\displaystyle \int }_{-\infty }^{+\infty }\frac{{\rm{d}}\nu }{2\pi }\,\,\rm{tr}\,{[{{\bf{G}}}^{\dagger }{\{{{\bf{P}}}^{+},{{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{+}\}}_{+}{\bf{G}}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}^{+}]}_{(\nu )}\\ & & \times [{f}_{{\rm{n}}}(\nu )-{f}_{{\rm{m}}}(\nu )]\\ & & +\,\rm{tr}\,{[{{\bf{G}}}^{\dagger }{\{{{\bf{P}}}^{+},{{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{+}\}}_{+}{\bf{G}}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}^{-}]}_{(\nu )}[{f}_{{\rm{n}}}(\nu )-{\bar{f}}_{{\rm{m}}}(-\nu )]\\ & & +\,\rm{tr}\,{[{{\bf{G}}}^{\dagger }{\{{{\bf{P}}}^{+},{{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{-}\}}_{+}{\bf{G}}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}^{+}]}_{(\nu )}[{\bar{f}}_{{\rm{n}}}(-\nu )-{f}_{{\rm{m}}}(\nu )]\\ & & +\,\rm{tr}\,{[{{\bf{G}}}^{\dagger }{\{{{\bf{P}}}^{+},{{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{-}\}}_{+}{\bf{G}}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}^{-}]}_{(\nu )}[{\bar{f}}_{{\rm{n}}}(-\nu )-{\bar{f}}_{{\rm{m}}}(-\nu )],\end{array}\end{eqnarray}$
and the proximity current In,p caused by the superconducting proximity effect: $\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{n}},{\rm{p}}} & =& {\rm{i}}\frac{e}{\hslash }{\displaystyle \int }_{-\infty }^{+\infty }\frac{{\rm{d}}\nu }{2\pi }\\ & & \times \,\,\rm{tr}\,{\left\{{{\bf{G}}}^{\dagger }{[{{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{+},{{\bf{P}}}^{+}]}_{-}{\bf{G}}[\nu -({\bf{H}}+\displaystyle \sum _{{\rm{m}}}{{\bf{V}}}_{{\rm{m}}})]\right\}}_{(\nu )}\\ & & \times {f}_{{\rm{n}}}(\nu )+\,\rm{tr}\,\{{{\bf{G}}}^{\dagger }{[{{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{-},{{\bf{P}}}^{+}]}_{-}{\bf{G}}\\ & & \times [\nu -({\bf{H}}+\displaystyle \sum _{{\rm{m}}}{{\bf{V}}}_{{\rm{m}}})]{\}}_{(\nu )}{\bar{f}}_{{\rm{n}}}(-\nu )\\ & & +\displaystyle \sum _{{\rm{m}}}\,\rm{tr}\,{\{{{\bf{G}}}^{\dagger }{[{{\bf{V}}}_{{\rm{n}}},{{\bf{P}}}^{+}]}_{-}{\bf{G}}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}^{+}\}}_{(\nu )}{f}_{{\rm{m}}}(\nu )\\ & & +\displaystyle \sum _{{\rm{m}}}\,\rm{tr}\,{\{{{\bf{G}}}^{\dagger }{[{{\bf{V}}}_{{\rm{n}}},{{\bf{P}}}^{+}]}_{-}{\bf{G}}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}^{-}\}}_{(\nu )}{\bar{f}}_{{\rm{m}}}(-\nu ).\end{array}\end{eqnarray}$
In the above current formula (A15 ), the four terms indicate that the electron or hole from the lead-n flows to the lead-m, in which the electron or holes transforms into electron or holes. Moreover, the four terms of (A16 ) contribute to the average current due to the virtual process of electron exchange brought by the superconducting proximity effect.
When the site-n is connected by metal leads α=e, the projection operation P+ is commutative to the diagonal dissipation matrix ${[{{\bf{D}}}_{{\rm{n}}}(\nu ),{{\bf{P}}}^{+}]}_{-}=0$, and no effective proximity interaction caused by the normal electrode for system exist Vn(ν)=0. Therefore, the proximity current will not contribute to the total current, i.e., In,p=0 and the transport current is simplified as A17 ) is a generalized form of the Landauer formula [61,63], which has been used successfully for the current measurement caused by the edge state in the Kitaev model [50] and the nanowire-superconductor system [51].
$\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{n}}} & =& \frac{e}{h}\displaystyle \sum _{{\rm{m}}}\displaystyle \int {\rm{d}}\nu \,\,\rm{tr}\,{[{{\bf{G}}}^{\dagger }{{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{+}{\bf{G}}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}^{+}]}_{(\nu )}[{f}_{{\rm{n}}}(\nu )-{f}_{{\rm{m}}}(\nu )]\\ & & +\,\rm{tr}\,{[{{\bf{G}}}^{\dagger }{{\boldsymbol{\Gamma }}}_{{\rm{n}}}^{+}{\bf{G}}{{\boldsymbol{\Gamma }}}_{{\rm{m}}}^{-}]}_{(\nu )}[{f}_{{\rm{n}}}(\nu )-{\bar{f}}_{{\rm{m}}}(-\nu )].\end{array}\end{eqnarray}$
This simplified transport current (As an illustration for applying the generalized Landauer formula (A15 ) and (A16 ), we calculate the supercurrent of the light-controlled Josephson effect in Appendix B and light-controlled superconducting quantum interference devices in Appendix C .
Appendix B Direct current Josephson effect induced by the coherent light
In this section, we consider the Hamiltonian of the two s-wave superconductors, which is linked by the two-level quantum dot (QD) driven by the classical light
$\begin{eqnarray}{H}_{{\rm{tot}}}={H}_{0}+{V}_{{\rm{d}}}+{H}_{{\rm{L}}}+{H}_{{\rm{R}}}+{H}_{t}.\end{eqnarray}$
Here, the Hamiltonian of the QD with the two-level is $\begin{eqnarray}{H}_{0}={{\rm{\Omega }}}_{{\rm{e}}}\displaystyle \sum _{\sigma }{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{e}}\sigma }+{{\rm{\Omega }}}_{{\rm{g}}}\displaystyle \sum _{\sigma }{d}_{{\rm{g}}\sigma }^{\dagger }{d}_{{\rm{g}}\sigma }.\end{eqnarray}$
The single mode classical light drives electrons from the lower energy level Ωg to the upper energy level Ωe, which is described by $\begin{eqnarray}{V}_{{\rm{d}}}=-{{ \mathcal E }}_{{\rm{d}}}{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{g}}\sigma }{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}}t-{\rm{i}}{\phi }_{{\rm{d}}}}-{{ \mathcal E }}_{{\rm{d}}}{d}_{{\rm{g}}\sigma }^{\dagger }{d}_{{\rm{e}}\sigma }{{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{d}}}t+{\rm{i}}{\phi }_{{\rm{d}}}},\end{eqnarray}$
with the frequency ωd and the phase Φd of the driving light. And the real driving strength is defined as ${{ \mathcal E }}_{{\rm{d}}}\equiv {{\boldsymbol{E}}}_{{\rm{d}}}\cdot {\wp }_{{\rm{eg}}}$ by the transition dipole moment of quantum dot ℘eg. The two superconducting leads are described by Bardeen-Cooper-Schrieffer Hamiltonian [10,43] $\begin{eqnarray}\begin{array}{rcl}{H}_{{\rm{L}}} & =& \displaystyle \sum _{{\boldsymbol{k}}}({\epsilon }_{{\rm{L}},{\boldsymbol{k}}}+{\mu }_{{\rm{L}}})({c}_{{\rm{L}},{\boldsymbol{k}}\uparrow }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\uparrow }+{c}_{{\rm{L}},{\boldsymbol{k}}\downarrow }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\downarrow })\\ & & +{{\rm{e}}}^{-2{\rm{i}}{\mu }_{{\rm{L}}}t+{\rm{i}}{\varphi }_{{\rm{L}}}}{{\rm{\Delta }}}_{{\rm{L}}}{c}_{{\rm{L}},{\boldsymbol{k}}\uparrow }^{\dagger }{c}_{{\rm{L}},-{\boldsymbol{k}}\downarrow }^{\dagger }\\ & & +{{\rm{e}}}^{2i{\mu }_{{\rm{L}}}t-{\rm{i}}{\varphi }_{{\rm{L}}}}{{\rm{\Delta }}}_{{\rm{L}}}{c}_{{\rm{L}},-{\boldsymbol{k}}\downarrow }{c}_{{\rm{L}},{\boldsymbol{k}}\uparrow },\\ {H}_{{\rm{R}}} & =& \displaystyle \sum _{{\boldsymbol{k}}}({\epsilon }_{{\rm{R}},{\boldsymbol{k}}}+{\mu }_{{\rm{R}}})({c}_{{\rm{R}},{\boldsymbol{k}}\uparrow }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}}\uparrow }+{c}_{{\rm{R}},{\boldsymbol{k}}\downarrow }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}}\downarrow })\\ & & +{{\rm{e}}}^{-2{\rm{i}}{\mu }_{{\rm{R}}}t+{\rm{i}}{\varphi }_{{\rm{R}}}}{{\rm{\Delta }}}_{{\rm{R}}}{c}_{{\rm{R}},{\boldsymbol{k}}\uparrow }^{\dagger }{c}_{{\rm{R}},-{\boldsymbol{k}}\downarrow }^{\dagger }\\ & & +{{\rm{e}}}^{2{\rm{i}}{\mu }_{{\rm{R}}}t-{\rm{i}}{\varphi }_{{\rm{R}}}}{{\rm{\Delta }}}_{{\rm{R}}}{c}_{{\rm{R}},-{\boldsymbol{k}}\downarrow }{c}_{{\rm{R}},{\boldsymbol{k}}\uparrow }.\end{array}\end{eqnarray}$
Here, the chemical potential and the phase of the superconducting leads are φn and μn with n=L, R respectively. And, the tunneling interaction between QD and the superconducting leads are $\begin{eqnarray}{H}_{{\rm{t}}}=-\displaystyle \sum _{{\boldsymbol{k}}\sigma }\left({{\rm{t}}}_{{\rm{L}},{\rm{g}},{\boldsymbol{k}}}{d}_{{\rm{g}}\sigma }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\sigma }+{{\rm{t}}}_{{\rm{R}},{\rm{e}},{\boldsymbol{k}}}{d}_{{\rm{e}}\sigma }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}}\sigma }+{\rm{H.c.}}\right),\end{eqnarray}$
with the tunneling strength between QD and the superconducting lead tn,i,k.Then, we can calculate the superconducting current of the light-controlled Josephson effect in the rotating frame with respect to the unitary transformation B7 ) that the phase of superconducting leads become ${\bar{\varphi }}_{{\rm{L}}}={\varphi }_{{\rm{L}}}$ and ${\bar{\varphi }}_{{\rm{R}}}={\varphi }_{{\rm{R}}}+2{\phi }_{{\rm{d}}}$, and the chemical potential of superconducting leads are corrected as ${\bar{\mu }}_{{\rm{L}}}={\mu }_{{\rm{L}}}-{\omega }_{{\rm{d}}}$ and ${\bar{\mu }}_{{\rm{R}}}={\mu }_{{\rm{R}}}-{\omega }_{{\rm{d}}}$ in comparison with (B4 ). Moreover, the energy level of the light-controlled quantum dot connected by two superconducting leads from figure 4(a) effectively become figure 4(b) in the rotated representation.
$\begin{eqnarray}{\bf{U}}(t)\equiv {\bf{S}}(t)=\exp [{\rm{i}}({N}_{{\rm{R}}}+\displaystyle \sum _{\sigma }{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{e}}\sigma })({\omega }_{{\rm{d}}}t+{\phi }_{{\rm{d}}})].\end{eqnarray}$
According to the above transformation, the specific form of the Hamiltonian of light-driven QD, superconducting lead and tunneling interaction are respectively $\begin{eqnarray}\begin{array}{rcl}{\bar{H}}_{{\rm{QD}}} & =& {\bar{{\rm{\Omega }}}}_{{\rm{e}}}\displaystyle \sum _{\sigma }{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{e}}\sigma }+{\bar{{\rm{\Omega }}}}_{{\rm{g}}}\displaystyle \sum _{\sigma }{d}_{{\rm{g}}\sigma }^{\dagger }{d}_{{\rm{g}}\sigma }\\ & & -\displaystyle \sum _{\sigma }[{{ \mathcal E }}_{{\rm{d}}}{d}_{{\rm{e}}\sigma }^{\dagger }{d}_{{\rm{g}}\sigma }+{{ \mathcal E }}_{{\rm{d}}}{d}_{{\rm{g}}\sigma }^{\dagger }{d}_{{\rm{e}}\sigma }],\\ {H}_{{\rm{L}}} & =& \displaystyle \sum _{{\boldsymbol{k}}}({\epsilon }_{{\rm{L}},{\boldsymbol{k}}}+{\mu }_{{\rm{L}}})({c}_{{\rm{L}},{\boldsymbol{k}}\uparrow }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\uparrow }+{c}_{{\rm{L}},{\boldsymbol{k}}\downarrow }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\downarrow })\\ & & +{{\rm{e}}}^{-2{\rm{i}}{\mu }_{{\rm{L}}}t+{\rm{i}}{\varphi }_{{\rm{L}}}}{{\rm{\Delta }}}_{{\rm{L}}}{c}_{{\rm{L}},{\boldsymbol{k}}\uparrow }^{\dagger }{c}_{{\rm{L}},-{\boldsymbol{k}}\downarrow }^{\dagger }\\ & & +{{\rm{e}}}^{2{\rm{i}}{\mu }_{{\rm{L}}}t-{\rm{i}}{\varphi }_{{\rm{L}}}}{{\rm{\Delta }}}_{{\rm{L}}}{c}_{{\rm{L}},-{\boldsymbol{k}}\downarrow }{c}_{{\rm{L}},{\boldsymbol{k}}\uparrow },\\ {H}_{{\rm{R}}} & =& \displaystyle \sum _{{\boldsymbol{k}}}({\epsilon }_{{\rm{R}},{\boldsymbol{k}}}+{\bar{\mu }}_{{\rm{R}}})({c}_{{\rm{R}},{\boldsymbol{k}}\uparrow }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}}\uparrow }+{c}_{{\rm{R}},{\boldsymbol{k}}\downarrow }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}}\downarrow })\\ & & +{{\rm{e}}}^{-2{\rm{i}}{\bar{\mu }}_{{\rm{R}}}t+{\rm{i}}{\bar{\varphi }}_{{\rm{R}}}}{{\rm{\Delta }}}_{{\rm{R}}}{c}_{{\rm{R}},{\boldsymbol{k}}\uparrow }^{\dagger }{c}_{{\rm{R}},-{\boldsymbol{k}}\downarrow }^{\dagger }\\ & & +{{\rm{e}}}^{2{\rm{i}}{\bar{\mu }}_{{\rm{R}}}t-{\rm{i}}{\bar{\varphi }}_{{\rm{R}}}}{{\rm{\Delta }}}_{{\rm{R}}}{c}_{{\rm{R}},-{\boldsymbol{k}}\downarrow }{c}_{{\rm{R}},{\boldsymbol{k}}\uparrow },\\ {H}_{t} & =& -\displaystyle \sum _{{\boldsymbol{k}},\sigma }[{{\rm{t}}}_{{\rm{R}},{\rm{e}},{\boldsymbol{k}}}{d}_{{\rm{e}}\sigma }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}}\sigma }+{{\rm{t}}}_{{\rm{L}},{\rm{g}},{\boldsymbol{k}}}{d}_{{\rm{g}}\sigma }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\sigma }+{\rm{H.c.}}].\end{array}\end{eqnarray}$
Here, the reduced upper and lower level are defined as ${\bar{{\rm{\Omega }}}}_{{\rm{e}}}:={{\rm{\Omega }}}_{{\rm{e}}}-{\omega }_{{\rm{d}}}$ and ${\bar{{\rm{\Omega }}}}_{{\rm{g}}}:={{\rm{\Omega }}}_{{\rm{g}}}$. And it is seen from the Hamiltonian of the right superconducting lead in the third row of equation (Further, we set μR ≥ μL=Ωg ≡ 0 as the energy reference and consider that the frequency of the driving light is equal to the difference between the chemical potentials of the left and right superconducting leads i.e. μR-μL=ωd.
At this time, the Hamiltonian for the two leads becomes time-independent since the effective chemical potential are ${\bar{\mu }}_{{\rm{L}}}={\mu }_{{\rm{L}}}=0$ and ${\bar{\mu }}_{{\rm{R}}}={\mu }_{{\rm{R}}}-{\omega }_{{\rm{d}}}=0$, and the Hamiltonian of the total system also becomes time-independent. Then, we rewrite the Hamiltonian of the light-driven QD, superconducting leads, and tunneling interaction as the form of equations (A1 ), (A2 ) and (A4 ),
$\begin{eqnarray}\begin{array}{rcl}{\bar{H}}_{{\rm{QD}}} & =& \frac{1}{2}{{\boldsymbol{d}}}^{\dagger }\cdot {\bf{H}}\cdot {\boldsymbol{d}}\equiv \frac{1}{2}{{\boldsymbol{d}}}^{\dagger }\\ & \cdot & \left[\begin{array}{cccccccc}{\bar{{\rm{\Omega }}}}_{{\rm{e}}} & 0 & 0 & 0 & -{{ \mathcal E }}_{{\rm{d}}} & 0 & 0 & 0\\ 0 & {\bar{{\rm{\Omega }}}}_{{\rm{e}}} & 0 & 0 & 0 & -{{ \mathcal E }}_{{\rm{d}}} & 0 & 0\\ 0 & 0 & -{\bar{{\rm{\Omega }}}}_{{\rm{e}}} & 0 & 0 & 0 & {{ \mathcal E }}_{{\rm{d}}} & 0\\ 0 & 0 & 0 & -{\bar{{\rm{\Omega }}}}_{{\rm{e}}} & 0 & 0 & 0 & {{ \mathcal E }}_{{\rm{d}}}\\ -{{ \mathcal E }}_{{\rm{d}}} & 0 & 0 & 0 & {\bar{{\rm{\Omega }}}}_{{\rm{g}}} & 0 & 0 & 0\\ 0 & -{{ \mathcal E }}_{{\rm{d}}} & 0 & 0 & 0 & {\bar{{\rm{\Omega }}}}_{{\rm{g}}} & 0 & 0\\ 0 & 0 & {{ \mathcal E }}_{{\rm{d}}} & 0 & 0 & 0 & -{\bar{{\rm{\Omega }}}}_{{\rm{g}}} & 0\\ 0 & 0 & 0 & {{ \mathcal E }}_{{\rm{d}}} & 0 & 0 & 0 & -{\bar{{\rm{\Omega }}}}_{{\rm{g}}}\end{array}\right]\cdot {\boldsymbol{d}},\\ \end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{H}_{{\rm{n}}}^{{\rm{s}}} & =& \frac{1}{2}{{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}^{\dagger }\cdot {{\bf{H}}}_{{\rm{n}}}\cdot {{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}\equiv \frac{1}{2}{{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}}^{\dagger }\\ & \cdot & \left[\begin{array}{cccc}{\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & & & {{\rm{\Delta }}}_{{\rm{n}}}\\ & {\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & -{{\rm{\Delta }}}_{{\rm{n}}} & \\ & -{{\rm{\Delta }}}_{{\rm{n}}}^{* } & -{\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & \\ {{\rm{\Delta }}}_{{\rm{n}}}^{* } & & & -{\epsilon }_{{\rm{n}},{\boldsymbol{k}}}\end{array}\right]\cdot {{\boldsymbol{c}}}_{{\rm{n}},{\boldsymbol{k}}},{\rm{n}}={\rm{L}},{\rm{R}},\end{array}\end{eqnarray}$
$\begin{eqnarray}{H}_{t}=-\displaystyle \sum _{{\boldsymbol{k}},\sigma }[{{\rm{t}}}_{{\rm{R}},{\rm{e}},{\boldsymbol{k}}}{d}_{{\rm{e}}\sigma }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}}\sigma }+{{\rm{t}}}_{{\rm{L}},{\rm{g}},{\boldsymbol{k}}}{d}_{{\rm{g}}\sigma }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\sigma }+{\rm{H.c.}}],\end{eqnarray}$
where the vector operator is ${\boldsymbol{d}}={[{d}_{{\rm{e}}\uparrow },{d}_{{\rm{e}}\downarrow },{d}_{{\rm{e}}\uparrow }^{\dagger },{d}_{{\rm{e}}\downarrow }^{\dagger },{d}_{{\rm{g}}\uparrow },{d}_{{\rm{g}}\downarrow },{d}_{{\rm{g}}\uparrow }^{\dagger },{d}_{{\rm{g}}\downarrow }^{\dagger }]}^{{\rm{T}}}$ and the pairing strength of the left and right superconducting lead are ${{\rm{\Delta }}}_{{\rm{L}}}={{\rm{\Delta }}}_{{\rm{s}}}{{\rm{e}}}^{{\rm{i}}{\varphi }_{{\rm{L}}}}$ and ${{\rm{\Delta }}}_{{\rm{R}}}={{\rm{\Delta }}}_{{\rm{s}}}{{\rm{e}}}^{{\rm{i}}({\varphi }_{{\rm{R}}}+2{\phi }_{{\rm{d}}})}$. Notice that the superconducting phase of the pairing strength for the right superconducting lead can be adjusted by the phase of coherence light Φd.Thus, according to equations (A15 ) and (A16 ), the transport current (A15 ) is zero due to the same chemical potentials of the left and right superconducting lead, and the proximity current (A16 ) induced by the light simplified as B8 ) and the dissipation kernel D(ω)=∑m=L,RDm(ω) caused by the two superconducting leads in equation (A11 ), the Green's function ${\bf{G}}(\omega )={\rm{i}}{[{\omega }^{+}-{\bf{H}}+{\rm{i}}{\bf{D}}(\omega )]}^{-1}$ is obtained. Further, the dc current induced by the light, which is completely driven by the superconducting phase difference ${\rm{\Phi }}:={\bar{\varphi }}_{{\rm{R}}}-{\varphi }_{{\rm{L}}}=2{\phi }_{{\rm{d}}}+{\varphi }_{{\rm{R}}}-{\varphi }_{{\rm{L}}}$, are exactly calculated as B13 ) is simplified as ${I}_{{\rm{s}}}={I}_{c}\sin {\rm{\Phi }}$. And, it can be seen from equation (B13 ) that the integrand is given as
$\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{s}}} & =& \frac{e}{2\hslash }{\displaystyle \int }_{-\infty }^{+\infty }\frac{{\rm{d}}\nu }{2\pi }\,\,\rm{tr}\,\{[{\bf{G}}(\nu )-{{\bf{G}}}^{\dagger }(\nu )]{[{{\boldsymbol{\Gamma }}}_{{\rm{L}}}^{+}(\nu ),{{\bf{P}}}^{+}]}_{-}\}\\ & & \times {f}_{{\rm{L}}}(\nu )+\,\rm{tr}\,\{[{\bf{G}}(\nu )-{{\bf{G}}}^{\dagger }(\nu )]{[{{\boldsymbol{\Gamma }}}_{{\rm{L}}}^{-}(\nu ),{{\bf{P}}}^{+}]}_{-}\}{\bar{f}}_{{\rm{L}}}(-\nu ).\end{array}\end{eqnarray}$
In deriving the above current formula, the following relation is needed $\begin{eqnarray}{{\bf{G}}}^{\dagger }(\nu )-{\bf{G}}(\nu )=-2{\rm{i}}{\bf{G}}(\nu )(\nu -[{\bf{H}}+\displaystyle \sum _{{\rm{m}}}{{\bf{V}}}_{{\rm{m}}}(\nu )]){{\bf{G}}}^{\dagger }(\nu ).\end{eqnarray}$
Together with the Hamiltonian H of the measured system ( $\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{s}}} & =& \frac{e}{h}\displaystyle \int {\rm{d}}\nu \frac{256{{\rm{\Gamma }}}^{3}{{\rm{\Delta }}}_{{\rm{s}}}^{2}{\nu }^{2}{{ \mathcal E }}_{{\rm{d}}}^{2}\left({{\rm{\Omega }}}_{{\rm{\Gamma }}}^{2}-4{\nu }^{2}\right)\,\rm{Sgn}\,[\nu ]\Re [{\gamma }_{{\rm{s}}}(\nu )]{f}_{{\rm{L}}}(\nu )\sin ({\rm{\Phi }})}{64{{\rm{\Gamma }}}^{2}{\nu }^{4}{\gamma }_{{\rm{s}}}^{2}(\nu ){({{\rm{\Omega }}}_{{\rm{\Gamma }}}^{2}-4{\nu }^{2})}^{2}+{\left(4{{\rm{\Gamma }}}^{2}\left({{\rm{\Delta }}}_{{\rm{s}}}^{2}{{\rm{\Omega }}}_{\delta }^{2}({\rm{\Phi }})+{\nu }^{2}(4{\nu }^{2}-{{\rm{\Omega }}}^{2})\right)-{\gamma }_{{\rm{s}}}^{2}(\nu )[{({{\rm{\Gamma }}}^{2}-4{\nu }^{2})}^{2}-16({\nu }^{2}{{\rm{\Omega }}}^{2}-{{ \mathcal E }}_{{\rm{d}}}^{4})]\right)}^{2}}\\ & := & {I}_{c}({\rm{\Phi }})\sin ({\rm{\Phi }}).\end{array}\end{eqnarray}$
Here, we have considered the same coupling strength γL=γR=γ, and the light-atom detuning is defined as Δω=Ωe-Ωg-ωd. At the same time, the following notations are also used for simplifying the above formula $\begin{eqnarray}\begin{array}{l}{\rm{\Omega }}=\sqrt{\delta {\omega }^{2}+2{{ \mathcal E }}_{{\rm{d}}}^{2}},{{\rm{\Omega }}}_{{\rm{\Gamma }}}=\sqrt{2{{\rm{\Omega }}}^{2}+{{\rm{\Gamma }}}^{2}},\\ {{\rm{\Omega }}}_{\delta }^{2}({\rm{\Phi }})=\delta {\omega }^{2}+2{{ \mathcal E }}_{{\rm{d}}}^{2}\cos {\rm{\Phi }},\,{\gamma }_{{\rm{s}}}(\nu )=\sqrt{{\nu }^{2}-{{\rm{\Delta }}}_{{\rm{s}}}^{2}}.\end{array}\end{eqnarray}$
In the large detuning situation ($\delta \omega \gg {{ \mathcal E }}_{{\rm{d}}}$), since ${{\rm{\Omega }}}_{\delta }^{2}({\rm{\Phi }})\,=\delta {\omega }^{2}+2{{ \mathcal E }}_{{\rm{d}}}^{2}\cos {\rm{\Phi }}\simeq \delta {\omega }^{2}$, the above critical current Ic(Φ) ≃ Ic becomes a constant independent of the phase difference Φ, then the above supercurrent ( $\begin{eqnarray}\rho (\nu )=\frac{{{\rm{\Delta }}}_{{\rm{s}}}^{2}{\nu }^{2}{{ \mathcal E }}_{{\rm{d}}}^{2}\left({{\rm{\Omega }}}_{{\rm{\Gamma }}}^{2}-4{\nu }^{2}\right)\,\rm{Sgn}\,[\nu ]\Re [{\gamma }_{{\rm{s}}}(\nu )]{f}_{{\rm{L}}}(\nu )}{64{{\rm{\Gamma }}}^{2}{\nu }^{4}{\gamma }_{{\rm{s}}}^{2}(\nu ){({{\rm{\Omega }}}_{{\rm{\Gamma }}}^{2}-4{\nu }^{2})}^{2}+{\left(4{{\rm{\Gamma }}}^{2}\left({{\rm{\Delta }}}_{{\rm{s}}}^{2}{{\rm{\Omega }}}_{\delta }^{2}({\rm{\Phi }})+{\nu }^{2}(4{\nu }^{2}-{{\rm{\Omega }}}^{2})\right)-{\gamma }_{{\rm{s}}}^{2}(\nu )[{({{\rm{\Gamma }}}^{2}-4{\nu }^{2})}^{2}-16({\nu }^{2}{{\rm{\Omega }}}^{2}-{{ \mathcal E }}_{{\rm{d}}}^{4})]\right)}^{2}},\end{eqnarray}$
the main contribution of which to the integral comes from near ν ∼-Δs due to ${\left({\nu }^{2}-{{\rm{\Delta }}}_{{\rm{s}}}^{2}\right)}^{-\frac{1}{2}}\to \infty $ when ν →-Δs-ϵ (ϵ is infinitesimal), which can be proved exactly by numerical computation, as shown in figure 4(c).Appendix C The supercurrent through SQUID with two quantum dots
In this section, we consider that the two same QDs embedded in a superconducting quantum interference devices (SQUID) loop is driven by the coherence of light. According to the Hamiltonian of the QD coupled by the light in equations (B2 ) and (B3 ), the two QDs coupled through the coherence light is described by
$\begin{eqnarray}\begin{array}{rcl}{H}_{s} & =& \displaystyle \sum _{i=1,2}[{{\rm{\Omega }}}_{{\rm{e}}}\displaystyle \sum _{\sigma }{d}_{i,{\rm{e}}\sigma }^{\dagger }{d}_{i,{\rm{e}}\sigma }+{{\rm{\Omega }}}_{{\rm{g}}}\displaystyle \sum _{\sigma }{d}_{i,{\rm{g}}\sigma }^{\dagger }{d}_{i,{\rm{g}}\sigma }\\ & & -{{ \mathcal E }}_{i,{\rm{d}}}{d}_{i,{\rm{e}}\sigma }^{\dagger }{d}_{i,{\rm{g}}\sigma }{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}}t-{\rm{i}}{\phi }_{i,{\rm{d}}}}\\ & & -{{ \mathcal E }}_{i,{\rm{d}}}{d}_{i,{\rm{g}}\sigma }^{\dagger }{d}_{i,{\rm{e}}\sigma }{{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{d}}}t+{\rm{i}}{\phi }_{i,{\rm{d}}}}].\end{array}\end{eqnarray}$
Here, we have considered the same the frequency of the driving light ωd,1=ωd,2=ωd. Similarly, the tunneling interaction between QD and the superconducting leads under the rotating wave approximation are rewritten as $\begin{eqnarray}\begin{array}{rcl}{H}_{t} & =& -\displaystyle \sum _{i=1,2}\displaystyle \sum _{{\boldsymbol{k}},\sigma }[{{\rm{t}}}_{{\rm{R}},{\rm{e}},{\boldsymbol{k}},}{d}_{i,{\rm{e}}\sigma }^{\dagger }{c}_{{\rm{R}},{\boldsymbol{k}},\sigma }\\ & & +{{\rm{t}}}_{{\rm{L}},{\rm{g}},{\boldsymbol{k}},}{d}_{i,{\rm{g}}\sigma }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}},\sigma }+{\rm{H.c.}}].\end{array}\end{eqnarray}$
In the case where the light frequency is equal to the difference of chemical potential between two superconducting leads, i.e. ωd=μR-μL, we can introduce unitary transform B2 ). Respectively, by defining the vector operator of the right superconducting lead
$\begin{eqnarray}\begin{array}{rcl}{\bf{U}}(t) & =& \exp [{\rm{i}}({N}_{{\rm{R}}}+\displaystyle \sum _{i\sigma }{d}_{i,{\rm{e}}\sigma }^{\dagger }{d}_{i,{\rm{e}}\sigma }){\mu }_{{\rm{R}}}t\\ & & +{\rm{i}}\displaystyle \sum _{i\sigma }{d}_{i,{\rm{e}}\sigma }^{\dagger }{d}_{i,{\rm{e}}\sigma }{\phi }_{i,{\rm{d}}}],\end{array}\end{eqnarray}$
to obtain the transformed Hamiltonian: ${\bar{H}}_{s}=\frac{1}{2}{{\boldsymbol{d}}}^{\dagger }\cdot {\bf{H}}\cdot {\boldsymbol{d}}$, where the Hamiltonian matrix in vector operator basis ${\boldsymbol{d}}={[{d}_{1,{\rm{e}}},{d}_{1,{\rm{g}}},{d}_{2,{\rm{e}}},{d}_{2,{\rm{g}}}]}^{{\rm{T}}}$ is H=diag{HQ,1, HQ,2} with the block matrix HQ,i defined in equation ( $\begin{eqnarray}\begin{array}{rcl}{\bar{{\boldsymbol{c}}}}_{i,{\rm{R}},{\boldsymbol{k}}} & =& {[{\bar{c}}_{i,{\rm{R}},{\boldsymbol{k}}\uparrow },{\bar{c}}_{i,{\rm{R}},-{\boldsymbol{k}}\downarrow },{\bar{c}}_{i,{\rm{R}},{\boldsymbol{k}}\uparrow }^{\dagger },{\bar{c}}_{i,{\rm{R}},-{\boldsymbol{k}}\downarrow }^{\dagger }]}^{{\rm{T}}}\\ & \equiv & [{{\rm{e}}}^{{\rm{i}}{\phi }_{i,{\rm{d}}}}{c}_{{\rm{R}},{\boldsymbol{k}}\uparrow },{{\rm{e}}}^{{\rm{i}}{\phi }_{i,{\rm{d}}}}{c}_{{\rm{R}},-{\boldsymbol{k}}\downarrow },\\ & & {{\rm{e}}}^{-{\rm{i}}{\phi }_{i,{\rm{d}}}}{c}_{{\rm{R}},{\boldsymbol{k}}\uparrow }^{\dagger },{{\rm{e}}}^{-{\rm{i}}{\phi }_{i,{\rm{d}}}}{c}_{{\rm{R}},-{\boldsymbol{k}}\downarrow }^{\dagger }{]}^{{\rm{T}}},\end{array}\end{eqnarray}$
we can obtain the tunneling Hamiltonian between QD and the superconducting leads $\begin{eqnarray}\begin{array}{rcl}{H}_{t} & =& -\displaystyle \sum _{i=1,2}\displaystyle \sum _{{\boldsymbol{k}}\sigma }[{{\rm{t}}}_{{\rm{R}},{\rm{e}},{\boldsymbol{k}}}{d}_{i,{\rm{e}}\sigma }^{\dagger }{\bar{c}}_{i,{\rm{R}},{\boldsymbol{k}}\sigma }\\ & & +{{\rm{t}}}_{{\rm{L}},{\rm{g}},{\boldsymbol{k}}}{d}_{i,{\rm{g}}\sigma }^{\dagger }{c}_{{\rm{L}},{\boldsymbol{k}}\sigma }+{\rm{H.c.}}].\end{array}\end{eqnarray}$
and the Hamiltonian of the superconducting lead-n: $\begin{eqnarray}\begin{array}{rcl}{\bar{H}}_{i,{\rm{n}}}^{{\rm{s}}} & =& \frac{1}{2}{\bar{{\boldsymbol{c}}}}_{i,{\rm{n}},{\boldsymbol{k}}}^{\dagger }\cdot {{\bf{H}}}_{i,{\rm{n}}}\cdot \,{\bar{{\boldsymbol{c}}}}_{i,{\rm{n}},{\boldsymbol{k}}}\equiv \frac{1}{2}{\bar{{\boldsymbol{c}}}}_{i,{\rm{n}},{\boldsymbol{k}}}^{\dagger }\\ & & \cdot \left[\begin{array}{cccc}{\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & & & {{\rm{\Delta }}}_{i,{\rm{n}}}\\ & {\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & -{{\rm{\Delta }}}_{i,{\rm{n}}} & \\ & -{{\rm{\Delta }}}_{i,\alpha }^{* } & -{\epsilon }_{{\rm{n}},{\boldsymbol{k}}} & \\ {{\rm{\Delta }}}_{i,{\rm{n}}}^{* } & & & -{\epsilon }_{{\rm{n}},{\boldsymbol{k}}}\end{array}\right]\cdot \,{\bar{{\boldsymbol{c}}}}_{i,{\rm{n}},{\boldsymbol{k}}},\,{\rm{n}}={\rm{L}},{\rm{R}},\end{array}\end{eqnarray}$
where the pairing term of the left and right superconducting lead are ${{\rm{\Delta }}}_{i,{\rm{L}}}={{\rm{\Delta }}}_{{\rm{s}}}{{\rm{e}}}^{{\rm{i}}{\varphi }_{{\rm{L}}}}$ and ${{\rm{\Delta }}}_{i,{\rm{R}}}={{\rm{\Delta }}}_{{\rm{s}}}{{\rm{e}}}^{{\rm{i}}({\varphi }_{{\rm{R}}}+2{\phi }_{i,{\rm{d}}})}$.As the block diagonal Hamiltonian H=diag{HQ,1, HQ,2} of the measured system and the dissipation kernel D(ω)=∑m=L,RDm(ω) in equation (A11 ) are block diagonal, the Green's function ${\bf{G}}(\omega )={\rm{i}}{[{\omega }^{+}-{\bf{H}}+{\rm{i}}{\bf{D}}(\omega )]}^{-1}$ is also rewritten as 8 × 8 block diagonal matrix. Therefore, also using equation (B11 ), the total current is the sum of the current flowing through two QDs respectively B13 ).
$\begin{eqnarray}{I}_{{\rm{s}}}={I}_{{\rm{s}},1}+{I}_{{\rm{s}},2}.\end{eqnarray}$
Here, the current through each quantum dot is ${I}_{{\rm{s}},i}={I}_{c,i}({{\rm{\Phi }}}_{i})\sin {{\rm{\Phi }}}_{i}$ with total phase Φ1=φR-φL+2Φ1,d and Φ2=φR-φL+2Φ2,d, which is given in equation (