Electromagnetic radiation detectors based on Josephson junctions: Effective Hamiltonian

We theoretically analyze two setups of low-energy single-photon counters based on Josephson junctions (JJs). For this, we propose two simple and general models based on the macroscopic quantum tunneling formalism (MQT). The first setup is similar to the photon counter based on the “cold-electron bolometer” (CEB), where the JJ replaces the CEB in the center of the superconducting antenna. In the second setup, the JJ is capacitively coupled to the antenna. We derive the Hamiltonians for the two setups, and we write the Schrödinger equations, taking into account both the antenna and the JJ. The quantum particles of the MQT models move in two-dimensional potential landscapes, which are parabolic along one direction and may have the form of a washboard potential along another direction. Such a potential landscape has a series of local minima, separated by saddle points. If the particle is prepared in the initial state in the metastable “ground state” of a local minimum, then the photon absorption causes it to jump into an excited state. If the excitation energy is bigger than the potential barrier seen by the quantum particle (the difference between the ground state and the saddle point), the photon is detected. The models are simple and allow us to do mostly analytical calculations. We show that the two setups are equivalent from the MQT point of view since one Hamiltonian can be transformed into the other by changes in variables. For typical values of the JJ and antenna parameters, the setups may work as counters of photons of wavelengths up to at least 1 cm. Dark count rates due to the phase particle tunneling directly from the ground state into the running state have also been evaluated.

Figure 1

The equivalent circuit of the detector in the first setup. The current through the JJ is $I_J$, and the capacitance is $C_J$. The superconducting antenna of impedance $L_A$ and capacitance $C_A$ is connected in series with the JJ.

Figure 2

The equivalent circuit of the detector in the second setup. The current through the JJ is $I_J$, and the capacitance is $C_J$. The superconducting antenna of impedance $L_A$ and capacitance $C_A$ is capacitively connected with the JJ.

Figure 3

The potential energy $U_d^{\left(2\right)}$ in the rotated system of coordinates $({\varphi }_1,{\varphi }_2)$. The parameters take typical values $I_c=1\mu \mathrm{A},C_A=C_J=1$ pF [25, 26], and ${\omega }_A/\left(2\pi \right)=30\mathrm{GHz}$. The value $I/I_c\approx 0.85$ will be justified later. One can observe that the potential is very asymmetric in the variables ${\varphi }_1$ and ${\varphi }_2$. The plot in (b) is a detail of the plot in (a), emphasizing a local minimum.

Figure 4

The ratio ${\lambda }_1/{\lambda }_2$ for $I_c=1\mu \mathrm{A}$ (solid line), $I_c=13.7\mu \mathrm{A}$ (dashed line), $C_A=C_J=1$ pF, and ${\omega }_A/\left(2\pi \right)=30\mathrm{GHz}$ (notice that ${\lambda }_2\to 0$ when $I/I_c\nearrow 1)$.