An Introduction to Superconducting Qubits and Circuit Quantum Electrodynamics
aa r X i v : . [ qu a n t - ph ] A ug An Introduction to Superconducting Qubits andCircuit Quantum Electrodynamics
Nicholas Materise
Superconducting qubits and circuit quantum electrodynamics have enabled designof solid state sources of quantum information. The performance of these devices hasscaled exponentially over the last fifteen years, in terms of their energy relaxationand dephasing times, drawing interest from adjacent communities including the Ax-ion Dark Matter Experiment (ADMX). Recently, superconducting qubits have beentargeted for use as single photon detectors in the ADMX high frequency experi-ment, ADMX-HF [16]. The goal of this article is to give members of the ADMXcommunity an introduction to some of the models used to analyze and design super-conducting qubits. This review is not an exhaustive coverage of the field, but it aimsto guide the reader to relevant literature and analysis techniques that closely followexperiment.
A qubit is a two level system or a system whose controllable quantum dynam-ics involve its two lowest lying energy levels. Nature provides several forms ofqubits or carriers of quantum information including single photons, trapped ions,and atoms in high finesse cavities. Superconducting qubits realize artificial atoms with engineered energy levels using the non-linearity of Josephson junctions andsurrounding microwave circuitry [2]. The quantum dynamics of these systems fol-lows that of a damped and driven anharmonic oscillator whose anharmonicity iscontrolled by choice of circuit parameters, e.g. linear capacitance and inductance ofthe Josephson junction [15]. For experimental design and control, practitioners drawfrom the Jaynes-Cummings model and its variants from cavity quantum electrody-namics (QED) [9, 15]. Circuit quantum electrodynamics borrows the application ofsecond quantized Hamiltonians from atomic optics via a standardized procedure forquantizing passive circuit. This section will introduce simple models for Joseph-son junctions and their role in superconducting qubits. We will then discuss circuit
Nicholas MateriseLawrence Livermore National Laboratory7000 East AveLivermore, CA 94550e-mail: [email protected] quantization methods and Black box quantization techniques used to obtain secondquantized Hamiltonians.
The operational modes of superconducting qubits vary by their energy spectra,where non-linearity plays a role in realizing accessible and isolated states. If weconsider the lowest two levels of the quantum harmonic oscillator to be the groundand excited states of a qubit ( | g i , | e i ) , the energies for the two states are separatedby integer multiples of ¯ h ω . The classical electric circuit model for an oscillator isthe LC circuit, shown in Figure 1. We will refer to this model in Section 2.3 whenwe derive the second quantized form of the Hamiltonian for an LC circuit. Figure 1compares the LC oscillator circuit to an anharmonic qubit, the transmon. Notice thatthe spacing between the excited state | e i and the next highest state | f i is smallerthan the spacing between | g i and | e i . In more anharmonic oscillators, the spacingis larger, further isolating the qubit states from the other states of the oscillator. Thetransmon trades off its anharmonicity for reduced sensitivity to charge noise [2]. − . − . . . . φ E i genene r g i e s | i ¯ hω | i ¯ hω | i Harmonic Oscillator Energies (a) − . − . . . . φ E i genene r g i e s | g i ¯ hω | e i ¯ h ( ω − α ) | f i Anharmonic Oscillator Energies (b)(c) (d)
Fig. 1
Comparison of the quantum harmonic oscillator with anharmonic oscillator. (a) and (b) givethe eigenenergies of the two oscillators, where the horizontal lines are the eigenenergies and thedashed lines represent notional potentials. (c) and (d) are the corresponding circuit models for anLC circuit and a transmon qubit [11].ntroduction to Superconducting Qubits 3
An anharmonic oscillator-based superconducting qubit inherits its non-linearityfrom Josephson junctions, where the non-linearity is tunable through fabrication andmicrowave circuit design. To develop some intuition for the dynamics of Josephsonjunctions, we will discuss classical circuit models for the device and their role insuperconducting qubits.
There are several phenomenological models for Josephson junctions that are mo-tivated by the underlying device physics and limits of the electric circuit analogs.We will review the Resistive and Capacitively Shunted Junction (RCSJ) model asoutlined in [6]. I d ( t ) E J E C (a) L J G N C J I d ( t ) (b) G N ( V ) C J G J ( (cid:1) ) I d ( t ) (c) Fig. 2 (a) Circuit diagram for a current driven Josephson junction, (b) RCSJ circuit model, (c)equivalent circuit with current sources replacing the conductance G N and inductor L J In Figure 2 above, the left most circuit shows a current-driven Josephson withdrive current, I d . The junction is approximated as the parallel combination of an in-ductor L J , conductance G N , and capacitor C J . We replace the inductor and conduc-tance with two voltage controlled current sources (VCCS’s), G J ( ϕ ) , G N ( V ) , wherewe use the Gxxx
VCCS notation from SPICE [12]. Kirchhoff’s current law at thenode joining the three circuit elements with the drive current source reads [6] I d ( t ) = I c sin ϕ + V G N ( V ) + C J dVdt (1) G N ( V ) = (cid:26) , | V | ≤ ∆ / e / R N , | V | ≥ ∆ / e (2)All occurrences of V refer to the voltage across the three elements representingthe Josephson junction from the node of their intersection to ground. The super- Nicholas Materise conducting gap energy at zero temperature, ∆ , gives the voltage where the junctiontransitions from superconducting to a normal metal with a normal resistance R N , seeEq. 2. For finite temperatures,(Gross et al. 2016) gives the temperature dependentconductance in the RCSJ based on the the density of states of quasiparticles in theJosephson junction [6].The VCCS G J ( ϕ ) varies sinusoidally with the junction phase, ϕ , which is a func-tion of the voltage across the junction and given by the Josephson equation V = Φ π d ϕ dt (3) Φ = h e ≡ Magnetic flux quantumIf we substitute Eq. 3 into Eq. 1, we arrive at a second order linear differentialequation in the phase variable, ϕ I d ( t ) = Φ π C J d ϕ dt + Φ π d ϕ dt G N (cid:18) Φ π d ϕ dt (cid:19) + I c sin ϕ (4)This equation is analogous to a driven pendulum, where the capacitance andconductance are proportional to the mass and damping parameter for the pendulum,respectively [15]. For practical, classical simulations of Josephson junctions, the twoVCCS model shunted by the junction capacitance is sufficient to produce hysteresisin the current-voltage (IV) characteristic curve. Numerical simulation of the circuitin Figure 2 is well suited for SPICE [12] circuit solvers or coupled to geometries inmultiphysics codes such as COMSOL [5].The RCSJ model is an intuitive model for the behavior of a Josephson junctionwith an applied dc or ac drive current, though it is not as suitable for supercon-ducting qubit design and simulation. Circuit Quantum Electrodynamics providesa framework analyzing such systems with the language of atomic optics or cavityquantum electrodynamics. We will examine the key features of circuit QED and itsutility in the design and simulation of superconducting qubits. Circuit quantum electrodynamics (QED) combines microwave engineering, circuitanalysis, and quantum optics. Fabry-Perot cavities from optics are replaced by res-onant microwave cavities or lumped element microwave resonators in circuit QED.The procedure for obtaining the quantized Hamiltonian and subsequent dynamics ofthe system follows first from a classical treatment, then quantization of the classicalvariables as operators and relating those operators to bosonic single-mode raisingand lowering operators n ˆ a ( † ) i o . ntroduction to Superconducting Qubits 5 We return to the LC oscillator circuit in Figure 1 and write the Lagrangian for thecircuit in terms of the flux variable φ which is treated as the generalized coordinatefor the system [4]. L (cid:0) φ , ˙ φ (cid:1) = C ˙ φ − L φ (5)We treat the charge q on the capacitor as the conjugate momentum and performa Legendre transformation to obtain the Hamiltonian as a function of both q and φ . q = ∂ L ∂ ˙ φ = C ˙ φ = ⇒ ˙ φ = q / C H ( q , φ ) = ˙ φ q − L = C ˙ φ + L φ H = C q + L φ (6)Following the example in Chapter 3 of [15], the charge and flux variables arequantized by converting them to operators with the commutation relation (cid:2) ˆ φ , ˆ q (cid:3) = i ¯ h . If we take the resonance frequency of the LC circuit to be ω = ( LC ) − / andreplace 1 / L in the potential term of the Hamiltonian, we arrive at the familiar formfor a harmonic oscillator with mass C . H → ˆ H = ˆ q C + C ω ˆ φ (7)We define raising and lowering operators for this quantum harmonic oscillator inanalogy to those used in the one-dimensional model and write the second quantizedform of the Hamiltonian.ˆ q = − i r ¯ h ω C (cid:0) ˆ a − ˆ a † (cid:1) , ˆ φ = r ¯ h ω C (cid:0) ˆ a + ˆ a † (cid:1) (8)ˆ H = ¯ h ω (cid:0) ˆ a † ˆ a + / (cid:1) (9) In the previous section, we covered a procedure for quantizing an LC oscillator cir-cuit which leads to an approximate generalization for any device given its frequencydependent impedance function. This approach connects full wave electromagneticsimulations of microwave circuits to their quantum mechanical analogs in circuitQED. Given a single port S -parameter as a function of frequency, one can obtain theimpedance at the port by the transformation Nicholas Materise Z = ( + S )( − S ) − (10) ≡ identity matrix with same dimensions as S Following the
Black box quantization methods outlined in [13, 17], the impedancefunction, Z ( ω ) can be expressed as a pole-residue expansion in the complex fre-quency s = j ω , where j = −√−
1, following the electrical engineering convention. Z ( s ) = M ∑ k = r k s − s k + d + es (11) { r k = a k + jb k } ≡ residues , { s k = ξ k + j ω k } ≡ polesThe above rational function can be obtained by a least squares fit of the originalimpedance using the Vector Fit software outlined in [7] and available at [18]. If wetake the case where d = s → ∞ vanishes or e = k -th term in the seriesand we find the k -th term is the impedance for a parallel RLC oscillator circuit. Z k ( s ) = r k s − s k = r k s − s k + r ∗ k s − s ∗ k ≃ a k ss − ξ k s + ω k = ⇒ Z k ( s ) = ω k r k Q k ss + ω k Q k s + ω k (12) ω k = ( L k C k ) − / , Q k = ω k R k C k = − ω k / ξ k , R k = − a k / ξ k The total impedance, Z ( s ) is a series combination of RLC oscillators and if wetake the dissipationless limit by ignoring the resistances, we can treat Z ( s ) as a seriescombination of LC circuits and apply the same analysis from Section 2.3.1 to eachsubcircuit. If we shunt the resulting circuit, with a single Josephson junction, we canobtain a simple model for the Hamiltonian of a qubit coupled to a superconductingresonator with M -modes. For a full derivation of the non-linear components of theHamiltonian ˆ H nl , see [13]; we reproduce the salient features here.ˆ H = ˆ H + ˆ H nl (13)ˆ H = ∑ i ¯ h ω i ˆ a † i ˆ a i , ˆ H nl = E J (cid:18) − cos ˆ ϕ − ˆ ϕ (cid:19) (14)ˆ H nl ≈ − ∑ i α i ˆ a †2 i ˆ a i − ∑ i = j χ i j ˆ a † i ˆ a i ˆ a † j ˆ a j (15)ˆ ϕ = πΦ ∑ i ˆ φ i = πΦ ∑ i r ¯ h ω i C i (cid:16) ˆ a i + ˆ a † i (cid:17) (16) ntroduction to Superconducting Qubits 7 The Hamiltonian above is referred to as the dispersive Hamiltonian for a weaklyanharmonic qubit coupled to a series of harmonic modes. In the non-linear term,ˆ H nl , the first contribution describes the anharmonicities of those modes and the qubitmode or self-Kerr terms and the second term gives the cross-Kerr terms [13]. Both { α i } and { χ i j } are experimentally observable, tying this model for qubit-circuitinteractions to physical devices. L R C L R C L R C L M R M C M (cid:1) ∝ (cid:2) i (cid:3) i Fig. 3
Series combination of RLC circuits shunted by a single Josephson junction representing theblack box circuit from Eq. 12 and similar in design to the circuit in [3]
The models used to describe the operation of superconducting qubits follow intu-itive modifications to the familiar damped and driven oscillator systems from classi-cal and quantum mechanics. These models arise from careful application of circuitQED to incorporate the quantum effects of macroscopic structures in microwave cir-cuits. Although the dispersive Hamiltonian describes many superconducting qubitsystems in quantum information experiments, this article did not apply the model tothe problem single photon counting. For more resources related to circuit QED andsingle photon counting, please refer to [4, 15, 1, 10, 14, 8].
This work was performed under the auspices of the U.S. Department of Energy byLawrence Livermore National Laboratory under Contract DE-AC52-07NA27344and funded by the Laboratory Directed Research and Development programs atLLNL project numbers 15-ERD-051, 16-SI-004.
Nicholas Materise
References
1. L. Bishop.
Circuit Quantum Electrodynamics . PhD thesis, Yale University, 2010.2. A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf. Cavity quantum elec-trodynamics for superconducting electrical circuits: An architecture for quantum computation.
Phys. Rev. A , 69:062320, Jun 2004.3. J. Bourassa, F. Beaudoin, J. M. Gambetta, and A. Blais. Josephson-junction-embeddedtransmission-line resonators: From Kerr medium to in-line transmon.
Phys. Rev. A , 86:013814,Jul 2012.4. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf. Introduction toquantum noise, measurement, and amplification.
Rev. Mod. Phys. , 82:1155–1208, Apr 2010.5. COMSOL Multiphysics. .6. R. Gross, A. Marx, and F. Deppe.
Applied Superconductivity: Josephson Effect and Supercon-ducting Electronics . Walter de Gruyter, 2016.7. B. Gustavsen and A. Semlyen. Rational approximation of frequency domain responses byvector fitting.
IEEE Transactions on Power Delivery , 14(3):1052–1061, Jul 1999.8. E. Holland.
Cavity State Reservoir Engineering in Circuit Quantum Electrodynamics . PhDthesis, Yale University, 2015.9. E. T. Jaynes and F. W. Cummings. Comparison of quantum and semiclassical radiation theo-ries with application to the beam maser.
Proceedings of the IEEE , 51(1):89–109, Jan 1963.10. B. Johnson.
Controlling Photons in Superconducting Electrical Circuits . PhD thesis, YaleUniversity, 2011.11. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret,S. M. Girvin, and R. J. Schoelkopf. Charge-insensitive qubit design derived from the Cooperpair box.
Phys. Rev. A , 76:042319, Oct 2007.12. L. W. Nagel and D. Pederson. SPICE (Simulation Program with Integrated Circuit Emphasis).Technical Report UCB/ERL M382, EECS Department, University of California, Berkeley,Apr 1973.13. S. E. Nigg, H. Paik, B. Vlastakis, G. Kirchmair, S. Shankar, L. Frunzio, M. H. Devoret, R. J.Schoelkopf, and S. M. Girvin. Black-Box Superconducting Circuit Quantization.
Phys. Rev.Lett. , 108:240502, Jun 2012.14. M. Reed.
Entanglement and Quantum Error Correction with Superconducting Qubits . PhDthesis, Yale University, 2013.15. D. Schuster.
Circuit Quantum Electrodynamics . PhD thesis, Yale University, 2007.16. T. M. Shokair, J. Root, K. A. Van Bibber, B. Brubaker, Y. V. Gurevich, S. B. Cahn, S. K.Lamoreaux, M. A. Anil, K. W. Lehnert, B. K. Mitchell, A. Reed, and G. Carosi. Futuredirections in the microwave cavity search for dark matter axions.
International Journal ofModern Physics A , 29(19):1443004, 2014.17. F. Solgun, D. W. Abraham, and D. P. DiVincenzo. Blackbox quantization of superconductingcircuits using exact impedance synthesis.
Phys. Rev. B , 90:134504, Oct 2014.18. The Vector Fitting Web Site – SINTEF.