Circuit Quantum Electrodynamics
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Circuit quantum electrodynamics (circuit QED) provides a means of studying the fundamental interaction between light and matter (
quantum optics Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
). As in the field of
cavity quantum electrodynamics Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. It could in principle be u ...
, a single photon within a single mode
cavity Cavity may refer to: Biology and healthcare * Body cavity, a fluid-filled space in many animals where organs typically develop ** Gastrovascular cavity, the primary organ of digestion and circulation in cnidarians and flatworms * Dental cavity or ...
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 Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
and a promising candidate for future
quantum computation Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thoug ...
. In the late 2010s decade, experiments involving cQED in 3 dimensions have demonstrated deterministic gate teleportation and other operations on multiple
qubit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
s.


Resonator

The resonant devices used for circuit QED are
superconducting Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
coplanar waveguide
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequency, frequencies between 300 MHz and 300 GHz respectively. Different sources define different fre ...
resonators, which are two-dimensional microwave analogues of the
Fabry–Pérot interferometer In optics, a Fabry–Pérot interferometer (FPI) or etalon is an optical cavity made from two parallel reflecting surfaces (i.e.: thin mirrors). Optical waves can pass through the optical cavity only when they are in resonance with it. It is n ...
. Coplanar waveguides consist of a signal carrying centerline flanked by two
grounded Grounding or grounded may refer to: Science and philosophy * Grounding (metaphysics), a topic of wide philosophical interest * Grounding (psychology), a strategy for coping with stress or other negative emotions * Grounding in communication, th ...
planes. This planar structure is put on a dielectric substrate by a photolithographic process.
Superconducting Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
materials used are mostly
aluminium Aluminium (aluminum in AmE, American and CanE, Canadian English) is a chemical element with the Symbol (chemistry), symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately o ...
(Al) or
niobium Niobium is a chemical element with chemical symbol Nb (formerly columbium, Cb) and atomic number 41. It is a light grey, crystalline, and ductile transition metal. Pure niobium has a Mohs hardness rating similar to pure titanium, and it ha ...
(Nb). Dielectrics typically used as substrates are either surface oxidized
silicon Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic ...
(Si) or
sapphire Sapphire is a precious gemstone, a variety of the mineral corundum, consisting of aluminium oxide () with trace amounts of elements such as iron, titanium, chromium, vanadium, or magnesium. The name sapphire is derived via the Latin "sap ...
(Al2O3). The line impedance is given by the geometric properties, which are chosen to match the 50 \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 which gives rise to very high electric fields per photon E_0 (compared to three-dimensional cavities). Mathematically, the field E_0 can be found as E_0=\sqrt, where \hbar is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
, \omega_r is the angular frequency, and \varepsilon_0 is the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
. One can distinguish between two different types of resonators: \lambda/2 and \lambda/4 resonators. Half-
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
resonators are made by breaking the center conductor at two spots with the distance \ell. The resulting piece of center conductor is in this way capacitively coupled to the input and output and represents a resonator with 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 \lambda/2: \quad \nu_n=\frac\frac \quad (n=1,2,3,\ldots) \qquad \lambda/4:\quad \nu_n=\frac\frac \quad (n=0,1,2,\ldots) with \varepsilon_ being the effective dielectric
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more i ...
of the device.


Artificial atoms, Qubits

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 pair In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Coope ...
s is coupled via Josephson junctions to a gated superconducting island. The state of the Cooper-pair box (
qubit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
) is given by the number of Cooper pairs on the island (N Cooper pairs for the ground state \mid g \rangle and N+1 for the excited state \mid e \rangle). By controlling the Coulomb energy ( bias voltage) and the Josephson energy (flux bias) the transition frequency \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 In quantum computing, and more specifically in superconducting quantum computing, a transmon is a type of superconducting charge qubit that was designed to have reduced sensitivity to charge noise. The transmon was developed by Robert J. Schoelk ...
qubits (more charge noise insensitive compared to the Cooper-pair box) and
flux qubit In quantum computing, more specifically in superconducting quantum computing, flux qubits (also known as persistent current qubits) are micrometer sized loops of superconducting metal that is interrupted by a number of Josephson junctions. These ...
s (whose state is given by the direction of a
supercurrent A supercurrent is a superconducting current, that is, electric current which flows without dissipation in a superconductor. Under certain conditions, an electric current can also flow without dissipation in microscopically small non-superconductin ...
in a superconducting loop intersected by Josephson junctions). All of these devices feature very large dipole moments d (up to 103 times that of large n Rydberg atoms), which qualifies them as extremely suitable
coupling A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end mov ...
counterparts for the light field in circuit QED.


Theory

The full quantum description of matter-light interaction is given by the
Jaynes–Cummings model The Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with or without the prese ...
. The three terms of the Jaynes–Cummings model can be ascribed to a cavity term, which is mimicked by a harmonic oscillator, an atomic term and an interaction term. \mathcal_=\underbrace_+\underbrace_+\underbrace_ In this formulation \omega_r is the resonance frequency of the cavity and a^\dagger and a are photon creation and annihilation operators, respectively. The atomic term is given by the Hamiltonian of a
spin-½ In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one ful ...
system with \omega_a being the transition frequency and \sigma_z the
Pauli matrix In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
. The operators \sigma_\pm are raising and lowering operators (
ladder operator In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising ...
s) for the atomic states. For the case of zero detuning (\omega_r=\omega_a) the interaction lifts the degeneracy of the photon number state \mid n \rangle and the atomic states \mid g \rangle and \mid e \rangle and pairs of dressed states are formed. These new states are superpositions of cavity and atom states \mid n,\pm \rangle=\frac 1\left(\mid g\rangle \mid n \rangle\pm \mid e\rangle \mid n-1\rangle\right) and are energetically split by 2g\sqrt n. If the detuning is significantly larger than the combined cavity and atomic
linewidth A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to ident ...
the cavity states are merely shifted by \pm g^2/\Delta (with the detuning \Delta=\omega_a-\omega_r) depending on the atomic state. This provides the possibility to read out the atomic (qubit) state by measuring the transition frequency. The coupling is given by g=E \cdot d (for electric dipolar coupling). If the coupling is much larger than the cavity loss rate \kappa=\fracQ (quality factor Q; the higher Q, the longer the photon remains inside the resonator) as well as the decoherence rate \gamma (rate at which the qubit relaxes into modes other than the resonator mode) the strong coupling regime is reached. Due to the high fields and low losses of the coplanar resonators together with the large dipole moments and long decoherence times of the qubits, the strong coupling regime can easily be reached in the field of circuit QED. Combination of the Jaynes–Cummings model and the coupled cavities leads to the Jaynes–Cummings–Hubbard model.


See also

* Superconducting radio frequency


References

{{Quantum computing Quantum information science