Circuit quantum electrodynamics

Circuit quantum electrodynamics (circuit QED) provides a means of studying the fundamental interaction between light and matter (quantum optics). As in the field of cavity quantum electrodynamics, a single photon within a single mode cavity coherently couples to a quantum object (atom). In contrast to cavity QED, the photon is stored in a one-dimensional on-chip resonator and the quantum object is no natural atom but an artificial one. These artificial atoms usually are mesoscopic devices which exhibit an atom-like energy spectrum. The field of circuit QED is a prominent example for quantum information processing and a promising candidate for future quantum computation. Circuit quantum electrodynamics (circuit QED) provides a means of studying the fundamental interaction between light and matter (quantum optics). As in the field of cavity quantum electrodynamics, a single photon within a single mode cavity coherently couples to a quantum object (atom). In contrast to cavity QED, the photon is stored in a one-dimensional on-chip resonator and the quantum object is no natural atom but an artificial one. These artificial atoms usually are mesoscopic devices which exhibit an atom-like energy spectrum. The field of circuit QED is a prominent example for quantum information processing and a promising candidate for future quantum computation. In the late 2010's decade, experiments involving cQED in 3 dimensions have demonstrated deterministic gate teleportation and other operations on multiple qubits. The resonant devices used for circuit QED are superconducting coplanar waveguide microwave resonators, which are two-dimensional microwave analogues of the Fabry–Pérot interferometer. Coplanar waveguides consist of a signal carrying centerline flanked by two grounded planes. This planar structure is put on a dielectric substrate by a photolithographic process. Superconducting materials used are mostly aluminium (Al) or niobium (Nb). Dielectrics typically used as substrates are either surface oxidized silicon (Si) or sapphire (Al2O3).The line impedance is given by the geometric properties, which are chosen to match the 50 Ω {displaystyle Omega } of the peripheric microwave equipment to avoid partial reflection of the signal.The electric field is basically confined between the center conductor and the ground planes resulting in a very small mode volume V m {displaystyle V_{m}} which gives rise to very high electric fields per photon E 0 {displaystyle E_{0}} (compared to three-dimensional cavities). Mathematically, the field E 0 {displaystyle E_{0}} can be found as E 0 = ℏ ω r 2 ε 0 V m {displaystyle E_{0}={sqrt {frac {hbar omega _{r}}{2varepsilon _{0}V_{m}}}}} , where ℏ {displaystyle hbar } is the reduced Planck constant, ω r {displaystyle omega _{r}} is the angular frequency, and ε 0 {displaystyle varepsilon _{0}} is the permittivity of free space. One can distinguish between two different types of resonators: λ / 2 {displaystyle lambda /2} and λ / 4 {displaystyle lambda /4} resonators. Half-wavelength resonators are made by breaking the center conductor at two spots with the distance ℓ {displaystyle ell } . The resulting piece of center conductor is in this way capacitively coupled to the input and output and represents a resonator with E {displaystyle E} -field antinodes at its ends. Quarter-wavelength resonators are short pieces of a coplanar line, which are shorted to ground on one end and capacitively coupled to a feed line on the other. The resonance frequencies are given by λ / 2 : ν n = c ε eff n 2 ℓ ( n = 1 , 2 , 3 , … ) λ / 4 : ν n = c ε eff 2 n + 1 4 ℓ ( n = 0 , 1 , 2 , … ) {displaystyle lambda /2:quad u _{n}={frac {c}{sqrt {varepsilon _{ ext{eff}}}}}{frac {n}{2ell }}quad (n=1,2,3,ldots )qquad lambda /4:quad u _{n}={frac {c}{sqrt {varepsilon _{ ext{eff}}}}}{frac {2n+1}{4ell }}quad (n=0,1,2,ldots )} with ε eff {displaystyle varepsilon _{ ext{eff}}} being the effective dielectric permittivity of the device. The first realized artificial atom in circuit QED was the so-called Cooper-pair box, also known as the charge qubit. In this device, a reservoir of Cooper pairs is coupled via Josephson junctions to a gated superconducting island. The state of the Cooper-pair box (qubit) is given by the number of Cooper pairs on the island ( N {displaystyle N} Cooper pairs for the ground state ∣ g ⟩ {displaystyle mid g angle } and N + 1 {displaystyle N+1} for the excited state ∣ e ⟩ {displaystyle mid e angle } ). By controlling the Coulomb energy (bias voltage) and the Josephson energy (flux bias) the transition frequency ω a {displaystyle omega _{a}} is tuned. Due to the nonlinearity of the Josephson junctions the Cooper-pair box shows an atom like energy spectrum. Other more recent examples for qubits used in circuit QED are so called transmon qubits (more charge noise insensitive compared to the Cooper-pair box) and flux qubits (whose state is given by the direction of a supercurrent in a superconducting loop intersected by Josephson junctions). All of these devices feature very large dipole moments d {displaystyle d} (up to 103 times that of large n {displaystyle n} Rydberg atoms), which qualifies them as extremely suitable coupling counterparts for the light field in circuit QED.