# Circuit quantum electrodynamics

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Circuit quantum electrodynamics (circuit QED) provides a means of studying the fundamental interaction between light and matter. 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. [1]

Cavity quantum electrodynamics 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 light photons is significant. It could in principle be used to construct a quantum computer.

An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric oscillators and some interferometers. Light confined in the cavity reflects multiple times producing standing waves for certain resonance frequencies. The standing wave patterns produced are called modes; longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns across the cross section of the beam.

Mesoscopic physics is a subdiscipline of condensed matter physics that deals with materials of an intermediate length. The scale of these materials can be described as being between the nanoscale size of a quantity of atoms and of materials measuring micrometres. The lower limit can also be defined as being the size of individual atoms. At the micrometre level are bulk materials. Both mesoscopic and macroscopic objects contain a large number of atoms. Whereas average properties derived from its constituent materials describe macroscopic objects, as they usually obey the laws of classical mechanics, a mesoscopic object, by contrast, is affected by fluctuations around the average, and is subject to quantum mechanics.

## Resonator

The resonant devices used for circuit QED are superconducting coplanar waveguide microwave resonators, [2] [3] 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 ${\displaystyle \Omega }$ of the peripheric microwave equipment to avoid partial reflection of the signal. [4] The electric field is basically confined between the center conductor and the ground planes resulting in a very small mode volume ${\displaystyle V_{m}}$ which gives rise to very high electric fields per photon ${\displaystyle E_{0}}$ (compared to three-dimensional cavities). Mathematically, the field ${\displaystyle E_{0}}$ can be found as

Superconductivity is a phenomenon of exactly zero electrical resistance and expulsion of magnetic flux fields occurring in certain materials, called superconductors, when cooled below a characteristic critical temperature. It was discovered by Dutch physicist Heike Kamerlingh Onnes on April 8, 1911, in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum mechanical phenomenon. It is characterized by the Meissner effect, the complete ejection of magnetic field lines from the interior of the superconductor during its transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the idealization of perfect conductivity in classical physics.

Coplanar waveguide is a type of electrical planar transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals. On a smaller scale, coplanar waveguide transmission lines are also built into monolithic microwave integrated circuits. Conventional coplanar waveguide (CPW) consists of a single conducting track printed onto a dielectric substrate, together with a pair of return conductors, one to either side of the track. All three conductors are on the same side of the substrate, and hence are coplanar. The return conductors are separated from the central track by a small gap, which has an unvarying width along the length of the line. Away from the central conductor, the return conductors usually extend to an indefinite but large distance, so that each is notionally a semi-infinite plane.

Microwaves are a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter; with frequencies between 300 MHz (1 m) and 300 GHz (1 mm). Different sources define different frequency ranges as microwaves; the above broad definition includes both UHF and EHF bands. A more common definition in radio engineering is the range between 1 and 100 GHz. In all cases, microwaves include the entire SHF band at minimum. Frequencies in the microwave range are often referred to by their IEEE radar band designations: S, C, X, Ku, K, or Ka band, or by similar NATO or EU designations.

${\displaystyle E_{0}={\sqrt {\frac {\hbar \omega _{r}}{2\varepsilon _{0}V_{m}}}}}$,

where ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle \omega _{r}}$ is the angular frequency, and ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space.

One can distinguish between two different types of resonators: ${\displaystyle \lambda /2}$ and ${\displaystyle \lambda /4}$ resonators. Half-wavelength resonators are made by breaking the center conductor at two spots with the distance ${\displaystyle \ell }$. The resulting piece of center conductor is in this way capacitively coupled to the input and output and represents a resonator with ${\displaystyle E}$-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

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

A capacitor is a passive two-terminal electronic component that stores electrical energy in an electric field. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. The capacitor was originally known as a condenser or condensator. The original name is still widely used in many languages, but not commonly in English.

${\displaystyle \lambda /2:\quad \nu _{n}={\frac {c}{\sqrt {\varepsilon _{\text{eff}}}}}{\frac {n}{2\ell }}\quad (n=1,2,3,\ldots )\qquad \lambda /4:\quad \nu _{n}={\frac {c}{\sqrt {\varepsilon _{\text{eff}}}}}{\frac {2n+1}{4\ell }}\quad (n=0,1,2,\ldots )}$

with ${\displaystyle \varepsilon _{\text{eff}}}$ being the effective dielectric permittivity of the device.

In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

## Artificial atoms, Qubits

The first realized artificial atom in circuit QED was the so-called Cooper-pair box, also known as the charge qubit. [5] 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 (${\displaystyle N}$ Cooper pairs for the ground state ${\displaystyle \mid g\rangle }$ and ${\displaystyle N+1}$ for the excited state ${\displaystyle \mid e\rangle }$). By controlling the Coulomb energy (bias voltage) and the Josephson energy (flux bias) the transition frequency ${\displaystyle \omega _{a}}$ 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 [6] (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 ${\displaystyle d}$ (up to 103 times that of large ${\displaystyle n}$ Rydberg atoms), which qualifies them as extremely suitable coupling counterparts for the light field in circuit QED.

In condensed matter physics, a Cooper pair or BCS pair is a pair of electrons bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper. Cooper showed that an arbitrarily small attraction between electrons in a metal can cause a paired state of electrons to have a lower energy than the Fermi energy, which implies that the pair is bound. In conventional superconductors, this attraction is due to the electron–phonon interaction. The Cooper pair state is responsible for superconductivity, as described in the BCS theory developed by John Bardeen, Leon Cooper, and John Schrieffer for which they shared the 1972 Nobel Prize.

The Josephson effect is the phenomenon of supercurrent, a current that flows indefinitely long without any voltage applied, across a device known as a Josephson junction (JJ), which consists of two or more superconductors coupled by a weak link. The weak link can consist of a thin insulating barrier, a short section of non-superconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-s-S).

In quantum computing, a qubit  or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include: the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states/levels simultaneously, a property which is fundamental to quantum mechanics and quantum computing.

## Theory

The full quantum description of matter-light interaction is given by the Jaynes–Cummings model. [7] 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.

${\displaystyle {\mathcal {H}}_{\text{JC}}=\underbrace {\hbar \omega _{r}\left(a^{\dagger }a+{\frac {1}{2}}\right)} _{\text{cavity term}}+\underbrace {{\frac {1}{2}}\hbar \omega _{a}\sigma _{z}} _{\text{atomic term}}+\underbrace {\hbar g\left(\sigma _{+}a+a^{\dagger }\sigma _{-}\right)} _{\text{interaction term}}}$

In this formulation ${\displaystyle \omega _{r}}$ is the resonance frequency of the cavity and ${\displaystyle a^{\dagger }}$ and ${\displaystyle a}$ are photon creation and annihilation operators, respectively. The atomic term is given by the Hamiltonian of a spin-½ system with ${\displaystyle \omega _{a}}$ being the transition frequency and ${\displaystyle \sigma _{z}}$ the Pauli matrix. The operators ${\displaystyle \sigma _{\pm }}$ are raising and lowering operators (ladder operators) for the atomic states. For the case of zero detuning (${\displaystyle \omega _{r}=\omega _{a}}$) the interaction lifts the degeneracy of the photon number state ${\displaystyle \mid n\rangle }$ and the atomic states ${\displaystyle \mid g\rangle }$ and ${\displaystyle \mid e\rangle }$ and pairs of dressed states are formed. These new states are superpositions of cavity and atom states

${\displaystyle \mid n,\pm \rangle ={\frac {1}{\sqrt {2}}}\left(\mid g\rangle \mid n\rangle \pm \mid e\rangle \mid n-1\rangle \right)}$

and are energetically split by ${\displaystyle 2g{\sqrt {n}}}$. If the detuning is significantly larger than the combined cavity and atomic linewidth the cavity states are merely shifted by ${\displaystyle \pm g^{2}/\Delta }$ (with the detuning ${\displaystyle \Delta =\omega _{a}-\omega _{r}}$) depending on the atomic state. This provides the possibility to read out the atomic (qubit) state by measuring the transition frequency.[ citation needed ]

The coupling is given by ${\displaystyle g=E\cdot d}$ (for electric dipolar coupling). If the coupling is much larger than the cavity loss rate ${\displaystyle \kappa ={\frac {\omega _{r}}{Q}}}$ (quality factor ${\displaystyle Q}$; the higher ${\displaystyle Q}$, the longer the photon remains inside the resonator) as well as the decoherence rate ${\displaystyle \gamma }$ (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.

## Related Research Articles

Spontaneous emission is the process in which a quantum mechanical system transitions from an excited energy state to a lower energy state and emits a quantised amount of energy in the form of a photon. Spontaneous emission is ultimately responsible for most of the light we see all around us; it is so ubiquitous that there are many names given to what is essentially the same process. If atoms are excited by some means other than heating, the spontaneous emission is called luminescence. For example, fireflies are luminescent. And there are different forms of luminescence depending on how excited atoms are produced. If the excitation is affected by the absorption of radiation the spontaneous emission is called fluorescence. Sometimes molecules have a metastable level and continue to fluoresce long after the exciting radiation is turned off; this is called phosphorescence. Figurines that glow in the dark are phosphorescent. Lasers start via spontaneous emission, then during continuous operation work by stimulated emission.

In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator and hence, the coherent states arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.

In atomic physics, a dark state refers to a state of an atom or molecule that cannot absorb photons. All atoms and molecules are described by quantum states; different states can have different energies and a system can make a transition from one energy level to another by emitting or absorbing one or more photons. However, not all transitions between arbitrary states are allowed. A state that cannot absorb an incident photon is called a dark state. This can occur in experiments using laser light to induce transitions between energy levels, when atoms can spontaneously decay into a state that is not coupled to any other level by the laser light, preventing the atom from absorbing or emitting light from that state.

Resolved sideband cooling is a laser cooling technique allowing cooling of tightly bound atoms and ions beyond the Doppler cooling limit, potentially to their motional ground state. Aside from the curiosity of having a particle at zero point energy, such preparation of a particle in a definite state with high probability (initialization) is an essential part of state manipulation experiments in quantum optics and quantum computing.

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

Superconducting quantum computing is an implementation of a quantum computer in superconducting electronic circuits. Research in superconducting quantum computing is conducted by Google, IBM, BBN Technologies, Rigetti, and Intel. as of May 2016, up to nine fully controllable qubits are demonstrated in a 1D array, up to sixteen in a 2D architecture.

The Jaynes–Cummings model 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, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.

The Kramers–Heisenberg dispersion formula is an expression for the cross section for scattering of a photon by an atomic electron. It was derived before the advent of quantum mechanics by Hendrik Kramers and Werner Heisenberg in 1925, based on the correspondence principle applied to the classical dispersion formula for light. The quantum mechanical derivation was given by Paul Dirac in 1927.

In spectroscopy, the Autler–Townes effect, named after American physicists Stanley Autler and Charles Townes, is a type of the dynamical Stark effect, corresponding to the case when an oscillating electric field is tuned in resonance to the transition frequency of a given spectral line, and resulting in a change of the shape of the absorption/emission spectra of that spectral line.

Resonance fluorescence is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency of the atom.

A vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single-mode electromagnetic cavity and reabsorbs them. The atom interacts with a single-mode field confined to a limited volume V in an optical cavity. Spontaneous emission results as a consequence of coupling between the atom and the vacuum fluctuations of the cavity field. The vacuum Rabi frequency is given by

In quantum computing, and more specifically in superconducting quantum computing, the phase qubit is a superconducting device based on the superconductor-insulator-superconductor (SIS) Josephson junction, designed to operate as a quantum bit, or qubit. The phase qubit is closely related, yet distinct from, the flux qubit and the charge qubit, which are also quantum bits implemented by superconducting devices.

In physics, quantum beats are simple examples of phenomena that cannot be described by semiclassical theory, but can be described by fully quantized calculation, especially quantum electrodynamics. In semiclassical theory (SCT), there is an interference or beat note term for both V-type and -type atoms. However, in the quantum electrodynamic (QED) calculation, V-type atoms have a beat term but -types do not. This is strong evidence in support of quantum electrodynamics.

In ion trapping experiments, the Lamb Dicke regime is a quantum regime in which the coupling between the ion's internal qubit's states and its motional states is sufficiently small so that transitions that change the motional quantum number by more than one, are strongly suppressed.

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. Schoelkopf, Michel Devoret, Steven M. Girvin and their colleagues at Yale University in 2007. Its name is an abbreviation of the term transmission line shunted plasma oscillation qubit; one which consists of a Cooper-pair box "where the two superconductors are also capacitatively shunted in order to decrease the sensitivity to charge noise, while maintaining a sufficient anharmonicity for selective qubit control".

In quantum optics, a superradiant phase transition is a phase transition that occurs in a collection of fluorescent emitters, between a state containing few electromagnetic excitations and a superradiant state with many electromagnetic excitations trapped inside the emitters. The superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters.

Ramsey interferometry, also known as Ramsey–Bordé interferometry or the separated oscillating fields method, is a form of atom interferometry that uses the phenomenon of magnetic resonance to measure transition frequencies of atoms. It was developed in 1949 by Norman Ramsey, who built upon the ideas of his mentor, Isidor Isaac Rabi, who initially developed a technique for measuring atomic transition frequencies. Ramsey's method is used today in atomic clocks and in the S.I. definition of the second. Most precision atomic measurements, such as modern atom interferometers and quantum logic gates, have a Ramsey-type configuration. A modern interferometer using a Ramsey configuration was developed by French physicist Christian Bordé and is known as the Ramsey–Bordé interferometer. Bordé's main idea was to use atomic recoil to create a beam splitter of different geometries for an atom-wave. The Ramsey–Bordé interferometer specifically uses two pairs of counter-propagating interaction waves, and another method named the "photon-echo" uses two co-propagating pairs of interaction waves.

Cavity optomechanics is a branch of physics which focuses on the interaction between light and mechanical objects on low-energy scales. It is a cross field of optics, quantum optics, solid-state physics and materials science. The motivation for research on cavity optomechanics comes from fundamental effects of quantum theory and gravity, as well as technological applications.

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