introduction to superconducting qubits - useqip · adrian lupascu institute for quantum computing...

Introduction to superconducting qubitsUSEQIP

Adrian Lupascu

Institute for Quantum ComputingDepartment of Physics and Astronomy

University of Waterloo, Canada

June 3, 2010

Introduction tosuperconducting

qubits

A. Lupascu

Superconductingsystems in QIP

The L-C resonator

Canonicalquantization

Limitation of standardresonators

Superconductivity

Structure of thesuperconducting state

Important properties:no dissipation andphase coherence

The Josephsonjunction

The Josephsonrelations

Inductive behavior

Charging energy

Josephson junction vsLC resonator

Energy level structure:anharmonicity

The phase qubit

Circuit

”Mesoscopic”Hamiltonian

Two-state systemapproximation

Quantum stateinitialization andcontrol

Quantum statemeasurement

Superconducting qubits among other types of quantumhardware I

Various types of physical systems are presently being investigated asqubits.A first class of qubits is formed of the ”traditional” quantum systems.This class contains physical systems such as single atoms and ions [1],nuclear spins [2], and single photons [3]. The quantum behavior of suchsystems had already been observed in the laboratory and well understoodat the time when the new ideas in quantum computing emerged.

nuclear spins

Physical systems for QCphotons atoms

Blatt and Steane, "Quantum information

processing with trapped ions" (2005)

O’Brien et al. Nature 426, 264 (2003)

Vandersypen et al. Nature 414, 883 (2001)

impurities solid – state systems

Koppens et al. 442, 766 (2006)

Nakamura et al. Nature 398, 786 (1999)

Gurudev Dutt et al. Science 316, 1312 (2007)


These quantum systems are well characterized. They have the advantageof long quantum coherence times, which are in general well understood.Their drawback is the difficulty to scale the system up to a large numberof qubits.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





Superconducting qubits among other types of quantumhardware II

A second class of qubits is formed by ”artificial” quantum systems.These are solid state structures, based on semi-conductors orsuper-conductors, which are patterned at the micrometer scale usingmodern lithography techniques.Important features of solid-state qubits:

I large flexibility in design of quantum properties

I the excitation energy is higher than achievable cryogenictemperatures ⇒ initialization is easy

I good prospects for scalability

I decoherence related to the solid-state environment is difficult toremove

Review articles: superconducting qubits [4] and quantum dot basedqubits [5].


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





Superconducting qubits among other types of quantumhardware III

nuclear spins

Physical systems for QCphotons atoms

Blatt and Steane, "Quantum information


O’Brien et al. Nature 426, 264 (2003)

Vandersypen et al. Nature 414, 883 (2001)

impurities solid – state systems

Koppens et al. 442, 766 (2006)

Nakamura et al. Nature 398, 786 (1999)

Gurudev Dutt et al. Science 316, 1312 (2007)


The first demonstration of quantum coherence of a singlesuperconducting qubit dates from 1999 [6]. This field of research ispresently quite vast, and it evolved in many directions, some of whichexceed the framework of quantum information processing:

I quantum design issues

I characterization of decoherence

I quantum operations on one and two qubits

I interaction of superconducting qubits with microwave fields (circuitQED)


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





Superconducting qubits among other types of quantumhardware IV

I quantum measurement

The goal of this lecture is to introduce the main physical concepts onwhich superconducting qubits are based. A specific type ofsuperconducting qubit, the phase qubit, will be described in detail.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





Table of contents

Superconducting qubits among other types of quantum hardware

The L-C resonatorCanonical quantization starting from Kirchhoff’s lawsLimitation of standard resonators

SuperconductivityStructure of the superconducting stateImportant properties: no dissipation and phase coherence

The Josephson junctionThe Josephson relationsInductive behaviorCharging energyJosephson junction vs LC resonatorEnergy level structure: anharmonicity

The phase qubitCircuit”Mesoscopic” HamiltonianTwo-state system approximationQuantum state initialization and controlQuantum state measurement


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ I

In (most) superconducting qubits, the lowest excitations are characterizedby the dynamics of the electromagnetic fields and of the charges insuperconductors at frequencies ω that correspond to free spacepropagation wavelengths λ = c/ν much longer than the size of thecircuit.This corresponds to the lumped approximation for electrical circuits. Weanalyze a simple (model for a) lumped circuit, namely the LC resonator.

C L


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ Canonical quantization starting fromKirchhoff’s laws I

C L

ILIC

VLVC

Kirchhoff’s laws applied to the LC circuitI current conservation

IC + IL = 0 (1)

I zero voltage sumVC − VL = 0 (2)


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ Canonical quantization starting fromKirchhoff’s laws II

It is convenient for later use to describe two-port circuit elements interms of the branch flux Φb and branch charge Qb, which are related tothe branch voltage Vb and current Ib by

Qb(t) =∫ t

−∞ Ib(t′)dt′ (3)

Φb(t) =∫ t

−∞ Vb(t′)dt′.

The relations between flux/charge and current/voltage variables for acapacitor and inductor respectively are:

QC = CVC (4)

IL = ΦL/L.

After combining 1,2,3,4 we obtain

d( ˙CΦL)

dt= −ΦL

L. (5)

This suggests writing a Lagrangian for the LC resonator

L =1

2C ΦL

2 − 1

2

1

LΦ2

L, (6)


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ Canonical quantization starting fromKirchhoff’s laws III

for which 5 is the Lagrange equation [7]

d

dt

(dLdΦL

)=

dLdΦL

. (7)

The Lagrangian description of the equation of motion provide a startingpoint for a quantum description of the LC circuit using canonicalquantization.The Hamiltonian is expressed in terms of the ”coordinate” ΦL, and thecanonically conjugate ”momentum” pΦL

pΦL =dLdΦL

(8)

asH (pΦL ,ΦL) = pΦLΦL − L (9)

resulting in

H (pΦL ,ΦL) =1

2

1

Cp2

ΦL+

1

2

1

LΦ2

L (10)


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ Canonical quantization starting fromKirchhoff’s laws IV

The procedure for canonical quantization involves the following elements

I Classical Hamiltonian (function of phase space variables) ⇒Quantum Hamiltonian (operator in Hilbert space)

H → H

I Classical variables ⇒ Hermitian operators

ΦL → Φ

pΦL → pΦ

I Poisson brackets ⇒ Commutators

{a, b} → [a, b]

i~


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ Canonical quantization starting fromKirchhoff’s laws V

The quantum Hamiltonian of the LC circuit becomes

H =1

2

1

Cp2Φ +

1

2

1

LΦ2 (11)

where pΦ and Φ are operators satisfying the commutation relation

[pΦ, Φ] = i~. (12)

The operators pΦ and Φ are analogous to the x and p variables of thefamiliar harmonic oscillator formed by a mass M attached to a spring ofconstant k. The correspondence between the variables and theparameters in the two models is:

Harmonic oscillator LC resonatorx Φp pΦM Ck L−1


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ Canonical quantization starting fromKirchhoff’s laws VI

The quantum HO is studied in many basic textbooks on quantummechanics (see e.g. [8]). The energy eigenstates of 11 are

En = (n + 1/2) ~ωLC , (13)

withωLC = 1/

√LC (14)

(the frequency corresponds to the classical resonance frequency).F1

E0

E1

E2

E3

E4

~wLC


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ Limitation of standard resonators I

Genuine quantum effects are usually not observed in regular LC circuits,due to thermal population and dissipation.

1. Thermal populationFor preparation in the ground state, the temperature T needs tosatisfy

kBT � ~ωLC . (15)

This condition is not satisfied for microwave resonators at 300 K.By cooling using a dilution refrigerator, temperatures of T ∼ 10 mKcan be attained. Resonators in the microwave range cooled at thesetemperature are thermalized very effectively. For exampleh × 2 GHz = 10× kB × 10 mK.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The L-C resonator/ Limitation of standard resonators II

2. DissipationNormal metal LC resonators with ωLC = 2π × (1− 10) GHz havequality factors Q ∼ 101−2 → decoherence sets in fast, over timescales of the order of Q/ωLC .Using superconducting materials leads to a large reduction ofdissipation:Qnormal ∼ 101−2 → Qsuperconducting ∼ 104−6

In addition, superconductors with Josephson junctions are non-linear→ anharmonic energy level structure → effective two-level system orquantum bit


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





Superconductivity/ Structure of the superconducting state IPhysics of superconducting qubitsNormal-Superconducting transition

Normal state• finite conductivity

• no electron order

• no energy gap

@T=Tc (1.2 K for Aluminum)

Superconducting state• infinite conductivity

• electron order

• energy gap

The gap for electron-hole excitations survives in small electromagnetic fields.

Lower energy excitations,

with anharmonic spectrum

Josephson energy

charging energy


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





Superconductivity/ Important properties: no dissipation andphase coherence I

Dissipationless currents The ability to carry a dissipationless current isthe most important characteristic of a superconductor. It can beunderstood by considering the nature of the current carrying states in asuperconductor. These states are separated by a significant energy gapfrom normal-metal-like states in which dissipation is effective. Thecritical current density of low-temperature superconductors of the orderof MA/cm2, much higher than the currents that are relevant forsuperconducting qubits.Phase coherence The wavefunction of superconductors displays phasecoherence on a macroscopic scales. Phase gradients are associated withthe flow of current. In a bulk superconductor the current density is givenby

−→J =

~m

(∇γ − 2e

~−→A

)ρ, (16)

where γ is the phase, ρ is the particle density, and−→A is the vector

potential associated with a magnetic field. In this lecture we will notconsider the effect of applied magnetic fields, which are important onlyfor multiply connected superconductors, as applicable to e.g. theso-called flux qubits.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The Josephson junction/ I

A Josephson junction is a structure formed by two superconductingelectrodes separated by a thin tunnel barrier.

g1

g2

Transport of electrical charge through the barrier takes place throughquantum tunneling. The remarkable property of the Josephson junction isthe flow of current without dissipation, which is due to tunneling ofCooper pairs through the barrier.The phase difference δ = γ1 − γ2 is the important variable characterizingthe junction. In 1962, Josephson discovered the relations between thephase difference and the current and voltage across the junction. Theserelations form the basis for the description of superconducting circuitswith Josephson junctions.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The Josephson junction/ The Josephson relations I

The first Josephson relation (the current-phase relation)

I = Ic sin δ, (17)

Ic is the critical current of the junction, which depends on thesuperconducting material, barrier thickness and construction, and isproportional to the barrier surface. For typical junctions used for qubits,made of aluminum with a very thin (∼ nm) barrier, and surface rangingfrom 100x100 nm2 to 1x1 µm2, Ic = 0.1÷ 10µA.The second Josephson relation (the AC Josephson effect)

V =Φ0

2π

dδ

dt, (18)

This equation introduces an important constant, the flux quantum Φ0,which is dual to the Cooper pair charge 2e. It is given by

Φ0 =h

2e, (19)

which equals 2.067x10−15.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The Josephson junction/ Inductive behavior I

From the 2nd Josephson relation (18) and the definition of the branchflux, we find that

δ = 2πΦ

Φ0. (20)

Using the 1st Josephson relation (17):

I = Ic sin 2πΦ

Φ0. (21)

We introduce a new parameter, the Josephson inductance:

LJ =Φ0

2πIc. (22)

Including this definition in 21

I =Φ0

2πLJsin 2π

Φ

Φ0. (23)

The relation above is similar to 4. In general, a relation of the typeI = I (Φ) corresponds to an inductor. In the limit |Φ/Φ0| � 1

I ' Φ

LJ, (24)


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The Josephson junction/ Inductive behavior II

and the Josephson junction behaves like a linear inductor. For arbitraryvalues of Φ the inductance is non-linear.

For typical Josephson junctions used in qubit circuits: LJ = 0.03÷ 3 nH.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The Josephson junction/ Charging energy I

The Josephson relations correctly describe JJs the current flowingthrough the barrier. A second contribution to the overall current appearsdue to the accumulation of charge:

The total currentI = IJ + IC , (25)

with IJ given by 23 and ICdescribed in terms of a capacitance C :

VC = QcC

(26)

Q(t) =∫ t

−∞ I (t′)dt′.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The Josephson junction/ Josephson junction vs LC resonator I

For an isolated junction the total current I = 0, so 25 is equivalent to 1.Therefore, all the relations used to derive the dynamics of the LCresonators (Kirchhoff’s laws, and relations between flux/charge andvoltage/current) remain unchanged, except for the current-flux relation 4which is replaced by 23.The Josephson junction is described by the LJ -C model, analogous to theLC resonator:

C LC LJ

nonlinear

inductance

Using the steps that led from 1, 2, 3, and 4 to 11, we can find thequantum Hamiltonian of the Josephson junction

H =1

2

1

Cp2Φ +

(Φ0/(2π))2

LJ

(1− cos 2π

Φ

Φ0

)(27)


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The Josephson junction/ Energy level structure: anharmonicityI

-3.1415 0.0000 3.1415

F1

F2

0 p-pF F/ 0

U ( )LC F

U ( )JJ F

~wLC

E0

JJ

E1

JJ

E2

JJ

E3

JJ

E0

LC

E1

LC

E2

LC

E3

LC

Comparison between energy levels of the Josephson junction and the LCresonator

I E JJn ≤ E LC

n , due to the ”softer” potential

I(E JJn+2 − E JJ

n+1

)<(E JJn+1 − E JJ

n

), anharmonic energy level structure


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The phase qubit/ Circuit I

The phase qubit is a single Josephson junction, to which a controllablecurrent source is added.

C LJ

Ib

The role of the added current source

I augmentation of anharmonicity

I control of the quantum state


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The phase qubit/ ”Mesoscopic” Hamiltonian I

The classical description of the phase qubit circuit involves themodification of the Kirchhoff current law for a single junction by theaddition of a bias current Ib. Eq. 25 is thus used

IJ + IC = Ib. (28)

By using the Kirchhoff laws and the appropriate branch relations, weobtain the classical equation of motion

d( ˙CΦ)

dt= Ib −

Φ0

2πLJsin

(2π

Φ

Φ0

). (29)

The potential energy is the tilted washboard potential

U(Φ) = −IbΦ +1

2

(Φ0

2π

)21

LJ

(1− cos

(2π

Φ

Φ0

)). (30)

Eq. 29 can be used to build a Lagrangian (see 7) and finally leads to themesoscopic quantum Hamiltonian for the phase qubit

H =1

2

1

Cp2Φ +

Φ20

(2π)2LJ

(1− cos 2π

Φ

Φ0

)− IbΦ (31)


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The phase qubit/ Two-state system approximation I

Consider a fixed bias current Ib0 . Ic . For typical phase qubits, withIc ∼ 10µA and C ∼ 10 fF, the bias current is Ib0 ' 99%Ic . In thissituation, typically two-three levels are present in the potential well.

-3.1415 0.0000 3.1415

F2

###

0 p-pF F/ 0

U( )F

~w01E0

E1

E2

The qubit is the truncation of the Hilbert space to the subspace formedby the ground state |0〉 and the excited state |1〉. Note that both thestates |0〉 and |1〉, and the energy eigenvalues E0 and E1 depend on thebias current Ib. It is assumed that they are defined at Ib = Ib0.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The phase qubit/ Two-state system approximation II

The (controlled) physical evolution has to be such that the systemquantum state remains at all times in the computational subspace,spanned by states |0〉 and |1〉.The qubit Hamiltonian (at Ib = Ib0) is given by

H0 =~ω01

2Z, (32)

where Z is one of the three Pauli matrices

X =

[0 11 0

], (33)

Y =

[0 −ii 0

], (34)

Z =

[1 00 −1

](35)

. (36)


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The phase qubit/ Quantum state initialization and control I

For a bias current Ib = Ib0 + δIb, the Hamiltonian is

H (δIb) = H0 + H (Ib0 + δIb)− H (Ib0) . (37)

Using 31H (δIb) = H0 − δIbΦ. (38)

The control term can be written like

− δIbΦ = −δIbTr(XΦ)/2× X− δIbTr(ZΦ)/2× Z. (39)

InitializationAfter setting Ib to Ib0, a waiting time is allowed for energy relaxation.Due to the typical energy levels splitting (10 GHz) being much larger thatthe thermal energy (25 mK, which corresponds to 0.5 GHz is common ina dilution refrigerator), the ground state is prepared with large fidelity.ControlThe availability of the terms X and Z provides the means for arbitrarysingle qubit operations.


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





The phase qubit/ Quantum state measurement I

Measurement of the quantum state proceeds in two steps.

I The bias Ib is increase from Ib0 to Ib,readout adiabatically → theprobability of occupation for each energy eigenstates is not changed

I After a suitable waiting time: quantum tunneling takes place if thequbit was in state |1〉 and does not take place if the qubit was instate |0〉. For superpositions of |0〉 and |1〉, the probability ofno-tunneling/tunneling is given by the probabilities of the respectivecorresponding eigenstate. A tunneling event produces an avalancheeffect which can be easily detected by electrical means.

-3.1415 0.0000 3.1415

F2

F3

0 p-pF F/ 0

U( )F

during quantummanipulation: Ib0

at readout: Ib,readout


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





References I

Garcia-Ripoll, J. J., Zoller, P., and Cirac, J. I.Journal of Physics B: Atomic, Molecular and Optical Physics 38(9),S567–S578 (2005).

Vandersypen, L. M. K., Steffen, M., Breyta, G., Yannoni, C. S.,Sherwood, M. H., and Chuang, I. L.Nature 414, 883 (2001).

J. L. O’Brien, G. J. Pryde, A. G. W. T. C. R. and Branning, D.Nature 426, 264 (2003).

Clarke, J. and Wilhelm, F. K.Nature 453, 1031 (2008).

Loss, D. and DiVincenzo, D.Phys. Rev. A 57(1), 120 January (1998).

Nakamura, Y., Pashkin, Y. A., and Tsai, J. S.Nature 398, 786 (1999).


qubits

A. Lupascu


The L-C resonator



Superconductivity





Inductive behavior

Charging energy



The phase qubit

Circuit





References II

Goldstein, H., Poole, C., and Safko, J.Classical mechanics.Classical mechanics (3rd ed.) by H. Goldstein, C. Poolo, andJ. Safko. San Francisco: Addison-Wesley, 2002., (2002).

Cohen-Tannoudji, C., Diu, B., and Laloe, F.Quantum Mechanics, Volume 1.(1986).

introduction to superconducting qubits - useqip · adrian lupascu institute for quantum computing...

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