SmallSmall JosephsonJosephson JunctionsJunctions inin ResonantResonant CavitiesCavitiesDavid G. StroudDavid G. Stroud, Ohio State Univ., Ohio State Univ.

Collaborators: W. A. Al-Saidi,Collaborators: W. A. Al-Saidi,

Ivan Tornes, E. AlmaasIvan Tornes, E. Almaas

Work supported by NSF DMR01-Work supported by NSF DMR01-0498704987

H2.005

Motivation: to develop models for controllable two-level superconducting systems for possible use in quantum computation

What is a Josephson What is a Josephson Junction?Junction?

A Josephson junction is a weak link A Josephson junction is a weak link between two strong superconductorsbetween two strong superconductors

Small junction: typically region between Small junction: typically region between superconductors is insulating (i); junction superconductors is insulating (i); junction underdamped underdamped

ICurrent I

I

ssi

HamiltonianHamiltonian ofof a smalla small JosephsonJosephson junctionjunction

Let be the phase difference across a junctionLet be the phase difference across a junction Let n be the difference in the number of Cooper Let n be the difference in the number of Cooper

pairs on the two superconducting islandspairs on the two superconducting islands Then the Hamiltonian of the junction isThen the Hamiltonian of the junction is

where U= /C is the charging energy and J is where U= /C is the charging energy and J is the Josephson coupling energy.the Josephson coupling energy. (C=capacitance) (C=capacitance)

cos)2/1( 2 JUnH

2)2( e

CommutationCommutation relationsrelationsin ],[

EH

cos/)2/( 22 JUH

Schrodinger equation is

where

Wave function which solves this equation is that of a particle of mass moving in a cosine potential

U/2

Voltage-biasedVoltage-biased JosephsonJosephson junctionjunction

If junction is voltage-biased,the kinetic If junction is voltage-biased,the kinetic energy term in the Hamiltonian energy term in the Hamiltonian becomesbecomes

2/)(2/ 22 nnUUn If Josephson energy vanishes, energy eigenvalues are just

2/)( 2nnUEn With n = 0, 1, 2,…..

Schematic of voltage-biasedSchematic of voltage-biasedjunctionjunction

Junction Junction Hamiltonian is Hamiltonian is

gC JJ EC ,

gV

)(2

)2(

cos)2

(

2

2

JgC

Jgg

C

CC

eE

Ee

VCnEH

Cg=gate capacitance; Vg=gate voltage

SchrodingerSchrodinger equationequation forfor voltage-biasedvoltage-biased smallsmall junctionjunction

cos)/)(2/( 2 JniUH

EH

Solve to obtain eigenvalues and wave functions as functions of n

Experiment: junction can be placed in macroscopic superposition of two distinct eigenstates [e. g. Bouchiat et al, Phys. Scr. T76, 165 (1998); Nakamura et al, PRL 79, 2328 (1997); Nature 398, 786 (1999); Zorin et al, cond-mat/0105211]

Single-Mode Resonant Single-Mode Resonant cavitycavity

)2/1( aaH

E,B

Hamiltonian of

Resonant Cavity Mode is

Here is the cavity frequency, a and a+ are photon annihilation/creation operators. Energies are (m+1/2)

Cavity fields

Cavity-junction interactionCavity-junction interaction

In presence of a vector potential, In presence of a vector potential, phasephase

difference in expression for difference in expression for Josephson energy is replaced by a Josephson energy is replaced by a gauge-invariant phase difference:gauge-invariant phase difference:)](cos[cos aag

Here g represents the junction-cavity coupling strength, and is related to the cavity vector potential in the junction

aa

)(xE

0

StrengthStrength ofof junction-cavityjunction-cavity couplingcoupling

Normalized cavity electric field

= Cavity frequency

Flux quantum ( = hc/2e)

Note: line integral taken across junction

Thus, maximum coupling occurs when E-field is concentrated in as small a volume as possible

22

0

322 ])([

)2(dsxE

cg

Supercon Supercon

IIII S S

Microcavity

Josephson junction is two superconductors (S and S) separated by insulating region

Schematic illustration of geometry for a voltage-biased Josephson junction in a microcavity

S

JJ EC ,

gCgV

SolutionSolution ofof combinedcombined junction-junction-cavitycavity problemproblem

Compute Hamiltonian matrix in product basis Compute Hamiltonian matrix in product basis |km> of junction and cavity states (k=0,1,2,|km> of junction and cavity states (k=0,1,2,…= junction states; m=0,1,2…= cavity …= junction states; m=0,1,2…= cavity states)states)

Diagonalize to obtain eigenvalues and Diagonalize to obtain eigenvalues and eigenfunctions as functions of g, offset eigenfunctions as functions of g, offset voltage, cavity frequency, etc.voltage, cavity frequency, etc.

At certain values of offset voltage, states of At certain values of offset voltage, states of cavity and junction are cavity and junction are highlyhighly entangledentangled – i. – i. e., cannot be written as products of cavity e., cannot be written as products of cavity and junction states.and junction states.

Energy level diagram for Energy level diagram for junction-cavity systemjunction-cavity system

n

Lowest eigenvalues E/U of the junction-cavity system, plotted as a function of for the following parameters: /U=0.3, J/U=0.7, and several values of g as indicated. Two arrows indicate pairs of degenerate states [Al-Saidi and Stroud, PRB65, 014512 (2002)].

n

Time-dependentTime-dependent behaviorbehavior ofof Josephson-cavityJosephson-cavity system, system,

=0.26=0.26

n

Left: energy in junction (solid line) and in cavity (dashed line) versus time. Right: probability that junction is in first excited state (full line) and that the cavity is in m=1 (one-photon) state (dashed line). Calculation includes ALL the quantum states of junction and cavity. Cavity oscillates between m =0 and m = 1.

n[Obtained by solving time-dep’t Schrodinger equation starting from |k=0,m=1>]

Josephson-cavity system, = Josephson-cavity system, = 0.060.06 n

Same as previous picture, except that offset voltage is tuned so that junction oscillates between ground and first excited state, and cavity oscillates between m=0 and m=2 states.

Two-level approximationTwo-level approximation

Calculate Rabi frequency including only the Calculate Rabi frequency including only the two nearly degenerate states.two nearly degenerate states.

For = 0.26, these are |For = 0.26, these are |k=0;m=1k=0;m=1> and> and||k=1;m=0k=1;m=0>. For = 0.06, they are |>. For = 0.06, they are |

k=0,m=2k=0,m=2> and |> and |k=2;m=0k=2;m=0>.>. For the first case, 2-state approx. gives a For the first case, 2-state approx. gives a

Rabi frequency within Rabi frequency within 1%1% of exact of exact calculation. For second case, it gives a Rabi calculation. For second case, it gives a Rabi frequency within frequency within 20%20% of exact calc. of exact calc.

n

n

Full curve: Time-dependent probability that junction in first excited state, given resonator initially in a “coherent state” (coherent superposition of different number states) and junction in ground state. Dashed curve: junction treated in two-level approximation, but all photon states included. [Al-Saidi and Stroud, Phys. Rev. B65, 014512 (2002).]

”Collapse and revival”[coined by J. H. Eberley et al, PRL 44, p. 1323 (1980)]

Jaynes-Cummings ModelJaynes-Cummings Model

Hamiltonian for a Hamiltonian for a two-statetwo-state systemsystem coupled coupled to a to a singlesingle modemode of a quantized of a quantized

electromagnetic field (or any single mode electromagnetic field (or any single mode oscillator):oscillator):

)()( 12 aSSaaaSEEH z

12 EE

zSSS ,,

Energy splitting of two lowest levels

are components of spin-1/2 operators

Our numerical results using full Hamiltonian for junction-cavity system very close to Jaynes-Cummings results including only two lowest junction levels.

[E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89

(1963)]

RemarksRemarks aboutabout singlesingle junctionjunction coupledcoupled toto single-modesingle-mode cavitycavity

Quantum behavior closely resembles that Quantum behavior closely resembles that of of two-statetwo-state systemsystem coupled to coupled to cavitycavity modemode, at certain values of offset voltage, at certain values of offset voltage

Junction does Junction does NOTNOT have to be a charging- have to be a charging-energy dominated*; energy dominated*; chargingcharging and and JosephsonJosephson energy can be energy can be comparablecomparable

*Regime studied by Shnirman et al, PRL79, 2371 (1997) and by Buisson and Hekking, cond-mat/0008275.

SeveralSeveral JunctionsJunctions in ain a ResonantResonant CavityCavity

Is it possible to couple several Josephson Is it possible to couple several Josephson junctions to the junctions to the SAMESAME resonant cavity? resonant cavity?

YesYes! The mathematical model is very similar ! The mathematical model is very similar to one used to treat to one used to treat severalseveral identicalidentical two-two-levellevel atomsatoms placed in the placed in the samesame resonantresonant cavitycavity: the Dicke model [R. H. Dicke, 1954]: the Dicke model [R. H. Dicke, 1954]

The models would be identical if the The models would be identical if the junctions were truly two-level systems, but junctions were truly two-level systems, but this is true only approximately this is true only approximately

ModelModel HamiltonianHamiltonian forfor Several-Several-JunctionJunction ProblemProblem

intHHHH jcavity

cavityH

jH

intH

= cavity

Hamiltonian= Hamiltonian for jth junction

=Hamiltonian for junction-cavity

interaction

DickeDicke modelmodel

TheThe HamiltonianHamiltonian forfor thethe DickeDicke modelmodel isis

)(])2/1[( * jjjz aSSaSaaH

Here a and are the annihilation and creation operators for the photon mode, S’s are components of the spin-1/2 operators, and = strength of coupling between the two-level system and the cavity mode. Junction-cavity system differs from the Dicke model because junctions are non-identical and non 2-level systems.

a

Rabi oscillations in a system of Rabi oscillations in a system of several junctions in a cavityseveral junctions in a cavity

This Figure plots the time-dependent “inversion” S(t) (sum of probabilities that each junction is in its excited state) for systems of 2 and 4 junctions in a resonant cavity, given that initially exactly one junction is excited. Rabi frequency is approximately proportional to square root of number of junctions. [Calculated using full Hamiltonian] (Al-Saidi and Stroud, Phys. Rev. B65, 224512 (2002).)

S(t) for coherent initial state, S(t) for coherent initial state, one and two junctionsone and two junctions

Time-dependent “inversion” S(t) (=sum of probabilities that each junction is in its excited state) for (a) N = 1 and (b) N = 2 junctions, given that resonator is initially in a “coherent” state (eigenstate of annihilation operator), and junctions in their ground state. Solid lines: full numerical solution; dashed lines, two-level approximation. The two junctions are assumed identical.

RemarksRemarks aboutabout systemsystem ofof severalseveral junctionsjunctions in ain a cavitycavity

Two-level (Dicke) approximation works quite well.Two-level (Dicke) approximation works quite well. Influence of higher levels can be incorporated into an Influence of higher levels can be incorporated into an effectiveeffective

dipole-dipoledipole-dipole interactioninteraction between the “qubits”: between the “qubits”:

If If junctionsjunctions areare non-identicalnon-identical, there are still Rabi oscillations , there are still Rabi oscillations (frequency proportional to square root of junction number) if (frequency proportional to square root of junction number) if junctions slightly different. For larger differences, each junctions slightly different. For larger differences, each junction couples independently to the cavity. junction couples independently to the cavity.

In “In “classicalclassical limitlimit” (many photons), dynamical equations ” (many photons), dynamical equations predict “predict “self-inducedself-induced resonantresonant stepssteps” in the IV characteristics ” in the IV characteristics (Almaas and Stroud, 2002) in agreement with expt (Barbara et (Almaas and Stroud, 2002) in agreement with expt (Barbara et al, PRL, 1999).al, PRL, 1999).

)(' )()()()( kjk

j k

j SSSSH

LongLong JosephsonJosephson “ “ringring” ” withwith crossedcrossed staticstatic andand cavitycavity

magneticmagnetic fieldsfields

HH

Cavity

magnetic

field H’

Vortex in ring behaves like Cooper pair in small junction

Vortex in ring is like a magnetic moment interacting with magnetic fields

v

III

QubitQubit fromfrom vortexvortex coupledcoupled toto cavitycavity magneticmagnetic fieldfield

MagneticMagnetic momentmoment ofof vortexvortex couplescouples toto applied applied static magnetic field +magnetic fieldstatic magnetic field +magnetic field of cavity.of cavity.

When current is applied across the ring, vortex is When current is applied across the ring, vortex is driven around the ring. Applied static magnetic driven around the ring. Applied static magnetic field produces cosine potential.* field produces cosine potential.*

With magnetic field of cavity mode, quantum With magnetic field of cavity mode, quantum Hamiltonian of system is formally identical (for Hamiltonian of system is formally identical (for small vortex velocities) to that of small junction small vortex velocities) to that of small junction

Hence, should be able to make qubit analogous to Hence, should be able to make qubit analogous to that produced by small junction coupled to a that produced by small junction coupled to a cavity mode (but via cavity mode (but via magneticmagnetic, , notnot electricelectric, , fields. fields.

Quantum mechanics in case of static field already seen in experiments, and calculated by Wallraff et al (Nature vol.425, p. 6954 2003).

Hamiltonian for vortex-cavity Hamiltonian for vortex-cavity systemsystem

Hamiltonian is (for slow vortex)Hamiltonian is (for slow vortex)

)2/(

)sin('cos)2/(2

eI

aaaaHHmp v

Form of Hamiltonian is identical to that of

small junction coupled to electric field of

cavity

H=

SummarySummary Have computed quantum states of states of Have computed quantum states of states of

JosephsonJosephson junctionsjunctions coupledcoupled toto single-modesingle-mode cavitiescavities..

Junctions may behave like Junctions may behave like two-leveltwo-level systemssystems which can be which can be stronglystrongly entangledentangled with cavity with cavity states.states.

SeveralSeveral junctionsjunctions in ain a cavitycavity, can behave , can behave like groups of controllable two-level systems like groups of controllable two-level systems which can be entangled via the cavity.which can be entangled via the cavity.

Work in progress: (i) Work in progress: (i) OtherOther geometriesgeometries for for two level systems; (ii) two level systems; (ii) couplingcoupling viavia magneticmagnetic fieldsfields of cavity modes; (iii) calculation of of cavity modes; (iii) calculation of dissipationdissipation..