Photon-Number Squeezing in Circuit Quantum Electrodynamics

A superconducting single-electron transistor (SSET) coupled to an anharmonic oscillator, e.g., a Josephson junction-$L\mathrm{\text{−}}C$ circuit, can drive the latter to a nonequilibrium photon-number distribution. By biasing the SSET at the Josephson quasiparticle cycle, cooling of the oscillator as well as a laserlike enhancement of the photon number can be achieved. Here, we show that the cutoff in the quasiparticle tunneling rate due to the superconducting gap, in combination with the anharmonicity of the oscillator, may create strongly squeezed photon-number distributions. For low dissipation in the oscillator, nearly pure Fock states can be produced.

Figure 1

A strongly squeezed distribution $p(n)$ of photon-number states in a Josephson junction-$L\mathrm{\text{−}}C$ oscillator driven by an SSET obtained for $\delta N_G=-0.1$, $eV=5.6$, $\Delta =2.2$, $g=0.01$, $\omega =0.4$, $\Omega =0.001$, $E_J=0.1$, $\kappa /\gamma =0.004$ (all energies in units of $E_C$). Inset: Energy dependence of the quasiparticle transition rate and energy differences for two transitions, $\Delta E_n=E_{-,n+1}-E_{1,n}-eV$. Because of the anharmonicity of the oscillator, they lie above and below the threshold.

Figure 2

An SSET with Josephson coupling $E_J$ and quasiparticle tunneling rate proportional to $\gamma$ is controlled by the transport voltage $V$ and the gate voltage $V_G$. It is coupled with strength $g$ to an anharmonic oscillator. The oscillator’s dissipation is characterized by the parameter $\kappa$. The sign of the anharmonicity can be controlled by the flux $\Phi$.

Figure 3

The charge states $|N=0⟩$ and $|N=2⟩$ are coupled by the Josephson effect $E_J$ to form the basis states $|\uparrow ⟩$ and $|\downarrow ⟩$. Tuned to resonance with the oscillator ($E_J\approx \omega$), they form the eigenstates $|\pm ,n⟩$ of the coherent part of the Hamiltonian.

Figure 4

Energy differences $\Delta E_{1,n\to \pm ,n+1}$ versus photon number $n$ for the parameters $\delta \omega$, $\Omega$, and $\overline{g}$ as used in Fig. 1. Also shown is a cycle of transitions, which increases the number of photons. Dashed arrows correspond to the rate ${\Gamma }_{1,n\to \pm ,n+1}^{\mathrm{qp}}$ and solid arrows to the rate ${\Gamma }_{\pm ,n\to 1,n}^{\mathrm{qp}}$. Vertical transitions are due to the dissipation decreasing the number of photons.

Figure 5

(a) Average photon number $⟨n⟩$ and (b) Fano factor $F$ versus oscillator frequency $\omega$ and gate charge $\delta N_G$ for transport voltage (in units of $E_C$) $eV=5.6$, $g=0.01$, $\Omega =0.0005$, $E_J=0.1$, and $\kappa /\gamma =0.02$.