Circuit quantum electrodynamics

Quantum-mechanical effects at the macroscopic level were first explored in Josephson-junction-based superconducting circuits in the 1980s. In recent decades, the emergence of quantum information science has intensified research toward using these circuits as qubits in quantum information processors. The realization that superconducting qubits can be made to strongly and controllably interact with microwave photons, the quantized electromagnetic fields stored in superconducting circuits, led to the creation of the field of circuit quantum electrodynamics (QED), the topic of this review. While atomic cavity QED inspired many of the early developments of circuit QED, the latter has now become an independent and thriving field of research in its own right. Circuit QED allows the study and control of light-matter interaction at the quantum level in unprecedented detail. It also plays an essential role in all current approaches to gate-based digital quantum information processing with superconducting circuits. In addition, circuit QED provides a framework for the study of hybrid quantum systems, such as quantum dots, magnons, Rydberg atoms, surface acoustic waves, and mechanical systems interacting with microwave photons. Here the coherent coupling of superconducting qubits to microwave photons in high-quality oscillators focusing on the physics of the Jaynes-Cummings model, its dispersive limit, and the different regimes of light-matter interaction in this system are reviewed. Also discussed is coupling of superconducting circuits to their environment, which is necessary for coherent control and measurements in circuit QED, but which also invariably leads to decoherence. Dispersive qubit readout, a central ingredient in almost all circuit QED experiments, is also described. Following an introduction to these fundamental concepts that are at the heart of circuit QED, important use cases of these ideas in quantum information processing and in quantum optics are discussed. Circuit QED realizes a broad set of concepts that open up new possibilities for the study of quantum physics at the macro scale with superconducting circuits and applications to quantum information science in the widest sense.

Figure 1

Left panel: harmonic potential vs flux of the $\mathit{LC}$ circuit with ${\mathrm{\Phi }}_0=h/2e$ the flux quantum. Right panel: response of the oscillator to an external perturbation as a function of the detuning $\delta$ of the perturbation from the oscillator frequency. Here $\kappa ={\omega }_r/Q$, with $Q$ the oscillator’s quality factor, is the full width at half maximum (FWHM) of the oscillator response. Equivalently, $1/\kappa$ is the average lifetime of the single-photon state $|1⟩$ before it decays to $|0⟩$. Inset: lumped-element $\mathit{LC}$ oscillator of inductance $L$ and capacitance $C$.

Figure 2

(a) Schematic layout of a $\lambda /2$ coplanar waveguide resonator of length $d$, center conductor width $w$, and ground plane separation $s$, together with its capacitively coupled input and output ports. The cosine shape of the second mode function ($m=1$) is illustrated with pink arrows. Also shown is the equivalent lumped-element circuit model. Adapted from [42]. (b) Cross-section cut of the coplanar waveguide resonator showing the substrate (dark blue), the two ground planes, and the center conductor (light blue) as well as schematic representations of the $E$ and $B$ field distributions. (c) Transmission vs frequency for an overcoupled resonator. The first three resonances of frequencies $f_m=(m+1)f_0$ are illustrated with $f_0=v_0/2d\sim 10\mathrm{GHz}$ and linewidth ${\kappa }_m/2\pi =f_m/Q$.

Figure 3

Telegrapher model of an open-ended transmission-line resonator of length $d$. $L_0$ and $C_0$ are, respectively, the inductance and capacitance associated to each node $n$ of flux ${\mathrm{\Phi }}_n$. The resonator is coupled to external transmission lines (not shown) at its input and output ports via the capacitors $C_{\kappa }$.

Figure 4

(a) Photograph of a 3D rectangular superconducting cavity showing the interior volume of the waveguide enclosure housing a sapphire chip and transmon qubit, with two symmetric coaxial connectors for coupling signals in and out. (b)–(e) First four ${\mathrm{TE}}_{mnl}$ modes of a 3D rectangular superconducting cavity obtained from comsol. (f) Schematic representation of a coaxial $\lambda /4$ cavity with electric field (solid line) pointing from the inner conductor to the sidewalls and evanescent field (dashed line) rapidly decaying from the top of the inner conductor. Adapted from [391].

Figure 5

(a) Cosine potential well of the transmon qubit (solid line) compared to the quadratic potential of the $LC$ oscillator (dashed lines). The spectrum of the former has eigenstates labeled $\{|g⟩,|e⟩,|f⟩,|h⟩,\dots \}$ and is characterized by an anharmonicity $-E_C$. (b) Circuit for the fixed-frequency transmon qubit. The square with a cross represents a Josephson junction with Josephson energy $E_J$ and junction capacitance $C_J$. (c) By using a SQUID rather than a single junction, the frequency of the transmon qubit becomes flux tunable.

Figure 6

Frequency difference ${\omega }_j-{\omega }_0$ of the first three energy levels of the transmon Hamiltonian obtained from numerical diagonalization of Eq. (22) expressed in the charge basis $\{|n⟩\}$ for different $E_J/E_C$ ratios and a fixed plasma frequency ${\omega }_p/2\pi =5\mathrm{GHz}$. For large values of $E_J/E_C$ the energy levels become insensitive to the offset charge $n_g$.

Figure 7

Schematic representation of a transmon qubit (green) coupled to (a) a 1D transmission-line resonator, (b) a lumped-element $LC$ circuit, and (c) a 3D coaxial cavity. (a) Adapted from [42]. (c) Adapted from [391].

Figure 8

Box: energy spectrum of the uncoupled (gray lines) and dressed (blue lines) states of the Jaynes-Cummings Hamiltonian at zero detuning: $\mathrm{\Delta }={\omega }_q-{\omega }_r=0$. Transmon states are labeled $\{|g⟩,|e⟩\}$, while photon numbers in the cavity are labeled $|n=0,1,2,\dots ⟩$ and plotted vertically. The degeneracy of the two-dimensional manifolds of states with $n$ quanta is lifted by $2g\sqrt{n}$ by the electric-dipole coupling. The light blue line outside of the main box represents the third excited state of the transmon, labeled $|f⟩$. Although it is not illustrated here, the presence of this level shifts the dressed states in the manifolds with $n\ge 2$ quanta ([151]).

Figure 9

Energy spectrum of the uncoupled (gray lines) and dressed states in the dispersive regime (blue lines). The two lowest transmon states are labeled $\{|g⟩,|e⟩\}$, while photon numbers in the cavity are labeled $|n=0,1,2,\dots ⟩$ and are plotted vertically. In the dispersive regime, the $g\text{−}e$ transition frequency of the qubit in the absence of resonator photons is Lamb shifted and takes the value ${\omega }_q+\chi$. Moreover, the cavity frequency is pulled by its interaction with the qubit and takes the qubit-state-dependent value ${\omega }_r\pm \chi$.

Figure 10

(a) Transmon qubit coupled to an arbitrary impedance, such as that realized by a 3D microwave cavity. (b) The Josephson junction can be represented as a capacitive element $C_J$ and a linear inductive element $L_J$ in parallel with a purely nonlinear element that is indicated by the spiderlike symbol. Here $C_{\mathrm{\Sigma }}=C_S+C_J$ is the parallel combination of the Josephson capacitance and the shunting capacitance of the transmon. (c) Normal mode decomposition of the parallel combination of the impedance $Z(\omega )$ together with $L_J$ and $C_{\mathrm{\Sigma }}$ represented by effective $LC$ circuits.

Figure 11

$LC$ circuit capacitively coupled to a semi-infinite transmission line used to model both damping and driving of the system. Here ${\widehat{b}}_{\mathrm{in}}(t)$ and ${\widehat{b}}_{\mathrm{out}}(t)$ are the oscillator’s input and output fields, respectively.

Figure 12

Transmon qubit coupled capacitively to a semi-infinite transmission line and inductively to a flux line. These ports are used to control the qubit state and to change its transition frequency. They also lead to qubit decay into the transmission line and to dephasing due to flux noise.

Figure 13

Because the dressed states in the dispersive regime are entangled qubit-cavity states, cavity damping at the rate $\kappa$ leads to qubit relaxation at the Purcell rate ${\gamma }_{\kappa }$. Conversely, qubit relaxation leads to cavity decay at the inverse Purcell rate ${\kappa }_{\gamma }$. Adding a Purcell filter (not shown) reduces the cavity density of states at the qubit frequency and therefore suppresses Purcell decay.

Figure 14

Schematic representation of the microwave measurement chain for field detection in circuit QED, with the resonator depicted as a Fabry-Perot cavity. The signal (RF) from a microwave source is applied to the input port of the resonator first passing through attenuators to reduce the level of thermal radiation. After passing through a circulator, the resonator’s output field is first amplified by a quantum-limited amplifier, such as a JPA or a JTWPA, and then by a HEMT amplifier. The signal is then mixed with a local oscillator (LO). The signal at the output of the mixer is digitized with an analog-to-digital converter (ADC) and can be further processed by a field-programmable gate array (FPGA). The two lines at the output of the mixer correspond to the two quadratures of the field. The temperature at which the different components are operated is indicated.

Figure 15

(a) Amplification chain with amplifiers of gain $G_{1,2}$ and noise mode ${\widehat{h}}_{1,2}$ with attenuation modeled by beam splitters of transmittivity ${\eta }_{1,2}$. The beam splitters each have a vacuum port with vacuum mode ${\widehat{v}}_{1,2}$ such that $⟨{\widehat{v}}_{1,2}^{†}{\widehat{v}}_{1,2}⟩=0$. The quantum efficiency derived from this model is $\eta =1/(N_T+1)\le 1$, with $N_T=⟨{\widehat{h}}_T^{†}{\widehat{h}}_T⟩$ the total added noise number given in Eq. (91). (b) Alternative model where a noisy amplifier is modeled by a noiseless amplifier preceded by a beam splitter of transmittivity $\overline{\eta }$. The quantum efficiency derived from this model is $\overline{\eta }=1/(2\mathcal{A}+1)\le 1/2$, with $\mathcal{A}$ the added noise given in Eq. (94).

Figure 16

Schematic representation of an IQ mixer. The rf signal ${\widehat{a}}_{\mathrm{amp}}$ is split into two parts at a power divider, here illustrated as a beam splitter to account for added noise due to internal modes. Ideally, only vacuum noise $\widehat{v}$ is introduced at that stage. The two outputs are combined with a LO at mixers. By phase shifting the LO by $\pi /2$ in one of the two arms, it is possible to simultaneously measure the two quadratures of the field.

Figure 17

Pictorial phase-space distribution of a coherent state and its marginal along an axis $X_{\varphi }$ rotated by $\varphi$ from $X$.

Figure 18

(a) Path in phase space leading up to steady state of the intracavity pointer states ${\alpha }_g$ and ${\alpha }_e$ for $2\chi /\kappa =1$, a measurement drive at the bare cavity frequency with an amplitude leading to one measurement photon at steady state, and assuming infinite qubit relaxation time (top panel). Corresponding marginals along $x$ with the signal, noise, and error defined in the text (bottom panel). The circles of radius $1/\sqrt{2}$ represent vacuum noise. Path in phase space for (b) $2\chi /\kappa =10$ and (c) $2\chi /\kappa =0.2$. (d) Signal-to-noise ratio as a function of $2\chi /\kappa$ for an integration time ${\tau }_m/\kappa =200$ (dark blue) and ${\tau }_m/\kappa =10$ (light blue). The maximum of the SNR at short integration time is shifted away from $2\chi /\kappa =1$. The letters correspond to the ratio $2\chi /\kappa$ of (a)–(c).

Figure 19

Resonator transmission (dashed lines) and corresponding phase shifts (solid lines) for the two qubit states (blue, ground; red, excited). When driving the resonator close to its pulled frequencies, the resonator response strongly depends on the state of the qubit. Adapted from [40].

Figure 20

Numerical simulations of the qubit-oscillator master equation for (a),(c),(e) the time evolution starting from the bare state $|0,e⟩$ (light blue lines) or $|1,g⟩$ (blue lines), and (b),(d),(f) steady-state response $A^2=|⟨\widehat{a}⟩{|}^2$ as a function of the cavity drive frequency (dark blue lines) for the three coupling regimes. $P_e$: qubit excited state population. (a),(b) Bad-cavity limit: $(\kappa ,{\gamma }_1,g)/2\pi =(10,0,1)\mathrm{MHz}$. (c),(d) Bad-qubit limit: $(\kappa ,{\gamma }_1,g)/2\pi =(0,10,1)\mathrm{MHz}$. (e),(f) Strong coupling: $(\kappa ,{\gamma }_1,g)/2\pi =(0.1,0.1,100)\mathrm{MHz}$ [dashed lines in (e) and solid lines in (f)] and $(\kappa ,{\gamma }_1,g)/2\pi =(1,1,100)$ [solid lines in (e)]. The light blue line in (f) is computed with a thermal photon number of ${\overline{n}}_{\kappa }=0.35$, as opposed to ${\overline{n}}_{\kappa }=0$ for all other results.

Figure 21

(a) Transmission-line resonator transmission vs probe frequency in the first observation of vacuum Rabi splitting in circuit QED (solid blue line). The qubit is a Cooper pair box qubit with $E_J/h\approx 8\mathrm{GHz}$ and $E_C/h\approx 5.2\mathrm{GHz}$. The solid red line is a calculated spectrum with $2g/2\pi \approx 11.6\mathrm{MHz}$, $\kappa /2\pi \approx 0.8\mathrm{MHz}$, and ${\gamma }_2/2\pi \approx 0.7\mathrm{MHz}$. As a reference, the dashed light blue line is the measured transmission with the qubit strongly detuned from the resonator. Adapted from [493]. (b) Resonator transmission with a transmon qubit. The vacuum Rabi splitting is even more resolved with $2g/2\pi =350\mathrm{MHz}$, $\kappa /2\pi \sim 800\mathrm{kHz}$, and ${\gamma }_2/2\pi \sim 200\mathrm{kHz}$. Notice the change in probe frequency range from (a). Adapted from [430].

Figure 22

Vacuum Rabi splitting revealed in numerical simulations of the cavity transmission $A^2=|⟨\widehat{a}⟩{|}^2$ as a function of probe frequency and qubit transition frequency for the same parameters as in Fig. 20 The bare cavity and qubit frequencies are indicated by the horizontal and diagonal dashed lines, respectively. The vacuum Rabi splitting of Fig. 20 is obtained at resonance (${\omega }_r={\omega }_q$) along the vertical dotted line.

Figure 23

Ground state and first two doublets of the Jaynes-Cummings ladder. The dark blue arrows correspond to the transitions that are probed in a vacuum Rabi experiment. The transitions illustrated with light blue arrows lead to additional peaks at transition frequencies lying inside the vacuum Rabi doublet at elevated temperature or increased probe power. On the other hand, the matrix element associated with the red transitions would lead to a response at transition frequencies outside of the vacuum Rabi doublet. Those transitions, however, are suppressed and are not observed ([390]).

Figure 24

Power broadening of the qubit line. (a) Excited qubit population (left vertical axis) and phase (right vertical axis) as a function of the drive detuning ${\delta }_q$ for the Rabi amplitudes ${\mathrm{\Omega }}_R/2\pi =0.1$ (light blue line), 0.5 (blue line), and 1 MHz (dark blue line). The phase is obtained from $\varphi =\arctan (2\chi ⟨{\widehat{\sigma }}_z⟩/2)$, with $2\chi /\kappa =1$. (b) Excited qubit population and phase at ${\delta }_q=0$ and as a function of ${\mathrm{\Omega }}_R^2$. The horizontal dashed gray line corresponds to qubit saturation $P_e=1/2$. (c) Qubit linewidth as a function of ${\mathrm{\Omega }}_R^2$. All three panels have been obtained from numerical simulations of the dispersive qubit master equation with ${\gamma }_1/2\pi =0.1\mathrm{MHz}$ and ${\gamma }_{\phi }/2\pi =0.1\mathrm{MHz}$, with the exception of the dashed blue lines in (b) and (c) that correspond to the analytical expressions found in the text.

Figure 25

Excited state population as a function of the qubit drive frequency. (a) Dispersive regime with $\chi /2\pi =0.1\mathrm{MHz}$ and (b) strong dispersive limit with $\chi /2\pi =5\mathrm{MHz}$. The resolved peaks correspond to different cavity photon numbers $|n⟩$. The spectroscopy drive amplitude is fixed to ${\mathrm{\Omega }}_R/2\pi =0.1\mathrm{MHz}$ and the damping rates to ${\gamma }_1/2\pi =\kappa /2\pi =0.1\mathrm{MHz}$. In (a) the measurement drive is on resonance with the bare cavity frequency, with amplitude $\epsilon /2\pi =0,0.2,\text{and}\hspace{0.17em}\hspace{0.17em}0.4\mathrm{MHz}$ for the light blue, blue, and dark blue lines, respectively. In (b) the measurement drive is at the pulled cavity frequency ${\omega }_r-\chi$ with amplitude $\epsilon /2\pi =0.1\mathrm{MHz}$.

Figure 26

Change of effective resonator frequency ${\omega }_{r\sigma }$ with increasing measurement drive power for the different states $\sigma$ of the transmon qubit (dotted blue line, ground state; solid red line, first excited state; dashed gray line, second excited state). The horizontal green dashed line is the bare resonator frequency. (a) Two-level artificial atom taking into account (b) three levels of the transmon and (c) six levels of the transmon. The system parameters are chosen such that $({\omega }_{01},{\omega }_{12},g)/2\pi =(6,5.75,0.1)\mathrm{GHz}$. Adapted from [49].

Figure 27

False colored optical microscope image of a four-transmon device. The transmon qubits are shown in yellow, the coupling resonators are shown in cyan, the flux lines for single-qubit tuning are shown in green, the charge lines for single-qubit manipulation are shown in pink, and a common feedline for multiplexed readout is shown in purple, with transmission-line resonators for dispersive readout (red) employing Purcell filters (blue). Adapted from [7].

Figure 28

Single-qubit gate errors extracted from randomized benchmarking for Gaussian and DRAG pulses as a function of total gate time and pulse width $\sigma$ for Gaussian pulses. The experimental results (symbols) are compared to numerical simulations (lines) with two or three transmon levels. Adapted from [96].

Figure 29

Schematic illustration of some of the two-qubit gate schemes discussed in the text. Exchange interaction between two qubits (a) from direct capacitive coupling and (b) mediated by a coupler such as a bus resonator. The qubits are tuned in and out of resonance with each other to activate and deactivate the interaction, respectively. (c) All-microwave gates activated by microwave drives on the qubits and/or a coupler such as a bus resonator. In this scheme, the qubits can have a fixed frequency. (d) Parametric gates involving modulation of a system parameter, such as a tunable coupler. Adapted from [519].

Figure 30

Spectrum of two transmon qubits coupled to a common resonator as a function of the frequency of the second qubit in the (a) two-excitation and (b) one-excitation manifolds. The solid lines are obtained by numerical diagonalization of Eq. (138) in the charge basis with five transmon levels and five resonator levels, and with parameters adapted from [129]: $E_{J1(2)}/h=28.48(42.34)\mathrm{GHz}$, $E_{C1(2)}/h=317(297)\mathrm{MHz}$, and $g_{1(2)}/2\pi =199(190)\mathrm{MHz}$. In the one-excitation manifold, both the $2g$ anticrossing of the first qubit with the resonator and the $2J$ anticrossing of the two qubits are visible. In the two-excitation manifold, the 11-02 anticrossing of magnitude $\zeta$ can be seen. Notice the change in horizontal scale between the two panels. The states are labeled as $|1\text{st qubit},2\text{nd qubit},\text{resonator}⟩$. The dashed light blue lines are guides for the eye following the bare frequency of the first qubit.

Figure 31

Wigner function $W(\beta )$ obtained numerically for (a) four-component and (b) two-component cat states with $\alpha =4$. Red is positive and blue is negative.

Figure 32

Wigner function of the intracavity field for Fock state superpositions $|0⟩+e^{i\phi }|3⟩+|6⟩$ for five values of the phase $\phi$; see the panel titles. Top row: theory. Bottom row: experimental data. Adapted from [208].

Figure 33

(a) Wigner function of a squeezed vacuum state $S(\zeta )|0⟩$ with $r=0.75$ and $\theta =\pi /2$. The white contour line is an ellipse of semiminor axis $e^{-r}/2$ and semimajor axis $e^r/2$. (b) Squeezing level vs $\varphi$ for $r=\mathrm{0.5.1.0},1.5$ and $\theta =\pi$. The horizontal line corresponds to vacuum noise $\mathrm{\Delta }{X}_{\mathrm{vac}}^2=1/4$. (c) Experimental setup used by [345] to prepare a squeezed vacuum state using a Josephson parametric amplifier and to send, via a circulator (gray box), this state to a transmon qubit in a 3D cavity (blue box). Adapted from [345].

Figure 34

(a) Voltage-biased transmon qubit with the three relevant flux branches. (b) Replacement of the classical voltage source with an $LC$ oscillator. The dashed arrows indicate the sign convention.