Lasing in circuit quantum electrodynamics with strong noise

We study a model which can describe a superconducting single-electron transistor or a double quantum dot coupled to a transmission-line oscillator. In both cases the degree of freedom is given by a charged particle, which couples strongly to the electromagnetic environment or phonons. We consider the case where a lasing condition is established and study the dependence of the average photon number in the resonator on the spectral function of the electromagnetic environment. We focus on three important cases: a strongly coupled environment with a small cutoff frequency, a structured environment peaked at a specific frequency, and $1/f$ noise. We find that the electromagnetic environment can have a substantial impact on the photon creation. Resonance peaks are in general broadened and additional resonances can appear.

Figure 1

A schematic depiction of the lasing cycle for the SSET and for the double dot. (a) A Cooper pair tunnels trough the left junction onto the island and via two consecutive single-charge tunneling events into the right lead. The energy for the process is provided by the transport voltage $V$, and the gate voltage $V_G$ can be used to change the gate charge $\delta N_{\mathrm{G}}$. The energy of the island with two additional charges is tuned to be smaller than the state with zero additional charges. The energy difference is approximately in resonance with a cavity coupled to the island. (b) An electron tunnels from the left lead into the left dot. Then the electron tunnels from left to right dot, emits a photon, and finally tunnels into the right lead.

Figure 2

(a) Energy diagram for the SSET and the electromagnetic resonator with $n$ (left panel) and $n+1$ photons (right panel) at the resonance condition $\delta \omega ={\omega }_0-\mathrm{\Delta }E=0$. Two states $|\uparrow ,n\rangle$ and $|\downarrow ,n+1\rangle$ are energetically degenerate. The odd charge states $|1,n\rangle$ and $|1,n+1\rangle$ are shown at the energy $-E_C+eV$. The longitudinal coupling to the bosonic bath $V_{\mathrm{z}}$, which causes dephasing, increases away from the symmetry point $\delta N_{\mathrm{G}}=0$. It may spoil the transverse coupling to the oscillator $V_{\mathrm{g}}$ crucial for coherent lasing. The impact of the longitudinal coupling to the bosonic bath $V_{\mathrm{ch}}$, which causes relaxation, will be small since the states $|\downarrow ,n\rangle$ and $|\uparrow ,n\rangle$ are energetically separated by the gap $\mathrm{\Delta }E$. (b) Creation of population inversion by quasiparticle tunneling. For $\delta N_G>0$, the condition $cos(\xi /2)>sin(\xi /2)$ is satisfied. The probability for the pumping-up process $|\downarrow ,n\rangle \to |1,n\rangle \to |\uparrow ,n\rangle$ (left panel) is higher than that for the pumping-down process $|\uparrow ,n\rangle \to |1,n\rangle \to |\downarrow ,n\rangle$ (right panel). At the symmetry point, $\delta N_G=0$, both processes occur equally, $cos(\xi /2)=sin(\xi /2)=1/\sqrt{2}$, and thus no population inversion is created.

Figure 3

(a) The propagator describing the time evolution of the off-diagonal component of the reduced density matrix $\langle \uparrow ,n\left|\rho \right|\downarrow ,n+1\rangle$. (b) The leading diagrams for the self-energy. The two diagrams are second-order expansions in ${\tilde{V}}_{\mathrm{qp}}$.

Figure 4

The correlator $S_{\mathrm{ch}}\left(\mathrm{\Delta }E\right)$ as a function of the energy splitting $\mathrm{\Delta }E$ for a fixed noise width $\sqrt{k_{B}T{\epsilon }_C}=0.07$ (all energies are given in units of the charging energy $E_C$). Blue, ${\epsilon }_C=0.05$; magenta, ${\epsilon }_C=0.03$; brown, ${\epsilon }_C=0.01$. For the parameters $E_J=0.2, {\mathrm{\Gamma }}_{\mathrm{qp}}=0.0325$.

Figure 5

The average number of photons $\langle n\rangle$ and the Fano factor as a function of the energy splitting $\mathrm{\Delta }E$ (all energies are given in units of the charging energy $E_C$). Blue, ${\epsilon }_C=0.005$; magenta, ${\epsilon }_C=0.01$; brown, ${\epsilon }_C=0.015$. For the parameters $E_J=0.2, k_{B}T=0.5, {\omega }_0=0.4, {\mathrm{\Gamma }}_{\mathrm{qp}}=0.0325, \kappa =0.0006, g=0.004$.

Figure 6

The correlators $S_{\mathrm{g}}\left(\delta \omega \right)$ and $S_{\mathrm{ch}}\left(\mathrm{\Delta }E\right)$ as a function of the energy splitting $\mathrm{\Delta }E$, for coupling to a single mode (all energies are given in units of the charging energy $E_C$). Brown, ${\epsilon }_C=0.05$; magenta, ${\epsilon }_C=0.075$; blue, ${\epsilon }_C=0.1$. For the parameters $E_J=0.2, k_{B}T=0.02, {\mathrm{\Gamma }}_{\mathrm{qp}}=0.0325, {\omega }_0=0.4, {\omega }_L=0.075, g=0.004$.

Figure 7

The average number of photons $\langle n\rangle$ and the current $I$ as a function of the energy splitting (all energies in units of the charging energy $E_C$). Blue, ${\epsilon }_C=0.05$; magenta, ${\epsilon }_C=0.075$; brown, ${\epsilon }_C=0.1$. For the parameters ${\omega }_L=0.1, E_J=0.2, g=0.004, {\mathrm{\Gamma }}_{\mathrm{qp}}=0.0325, \kappa =0.00005, k_{B}T=0.02, {\omega }_0=0.4$.