Phase diagram andfrustration of decoherence inYshaped Josephson - - PowerPoint PPT Presentation

Phase diagram andfrustration of decoherence inYshaped Josephson junction networks

Firenze,October 2008 Firenze,October 2008

Main idea YShaped networkofJosephson junction chains (YJJN)with amagnetic frustration⇒ ⇒ ⇒ ⇒Finitecoupling fixed point (FFP)inthephase diagram; YJJNworkingnear theFFP⇒ ⇒ ⇒ ⇒ Frustration of decoherence intheemerging twolevel quantum system(2LQS); Application:engineering ofareduceddecoherence 2LQS. Technology:renormalization group+boundary conformal field theory.

Planofthetalk: 1.TheYJJNas ajunction ofcharged,onedimensional, bosonic systems; 2.Theparameters andthephase diagram oftheYJJN: weakly coupled andstrongly coupled fixed points; 3.Emergence ofaFFPinthephase diagram; 5.Spectral densityandfrustration ofdecoherence inthe YJJNworkingnear theFFP; 6.Conclusions,possibleapplications,perspectives. 4.Current’spatter intheYJJNnear thefixed points:the YJJNas a“quantumswitch”;

1.TheYJJNandits fieldtheoretical description

ϕ2 ϕ3

EJ EJ EJ

Φ

1

ϕ

λ λ λ

Central region Hamiltonian

[ ]

∑ ∑

= + − =

• +

−       − ∂ ∂ − =

+

• ϕ

φ φ

φ

EC>>EJ ⇒

⇒ ⇒ ⇒ Effective (3)spin Hamiltonian

• +

+ =

• −

− ∂ ∂ − =

• φ
• φ

=

+

{ }

∑ ∑

= − + + =

• +

− − =

• ϕ

Lowenergy eigenstates (h>EJ)

↑↑↑

• −

= ε

• ↓↑↑

+ ↑↓↑ + ↑↑↓

• ϕ

ε

• −

− =

• ↓↑↑

− ↑↓↑ − ↑↑↓

− π π

• π

ϕ ε − − − =

• ↓↑↑

− ↑↓↑ − ↑↑↓

− π π

• π

ϕ ε + − − =

• Only these states will be kept intheeffective theory

Charge tunneling atthe“inner boundaries”

• φ

φ λ∑

=

− − =

“Weak tunneling”limit:λ<<h,EJ ⇒ ⇒ ⇒ ⇒SchriefferWolff transformation ⇒ ⇒ ⇒ ⇒Boundary interactionterm

∑

= −

+ − =

+

• γ

φ φ

• ϕ

ϕ λ + ≈

• ϕ

γ =

Effective field theory ofaJJchain

(L. I.Glazman andA.I.Larkin,PRL79,3736(1997), D.GiulianoandP.Sodano,NPB711,480(2005))

[ ]

∑ ∑ ∑ ∑

= + +

             − + − −         − ∂ ∂ − =

• φ

φ φ

Mapping onto spin chain+JordanWigner fermions+Bosonization ⇒ ⇒ ⇒ ⇒Luttinger liquid (LL)effective Hamiltonian

∑ ∫

=

                          ∂ Φ ∂ +         ∂ Φ ∂ =

• π

(N=n+1/2)

LLparameters andboundary conditions Weak boundary coupling (EW/EJ <<1)⇒ ⇒ ⇒ ⇒Neumann boundary conditions attheinner boundary (∂Φ(k)(0)/∂x=0); Coupling to thebulksuperconductors ⇒ ⇒ ⇒ ⇒Dirichlet boundary conditions attheouter boundary Φ(k)(L)=√2(2πn(k)+φ(k));

• +

+ − + =

• −

+ =

• −
• =

= π

• −

=

“Normal”fields

∑

=

Φ =

• Φ

− Φ = χ

• Φ

− Φ + Φ = χ

• π

ϕ ϕ χ + − =

• π

φ φ φ χ + − + =

Boundary Hamiltonian

• +

+

• −

=

∑

=

γ χ α

• 

       − − = − = =

• α

α α

(Dynamical)boundary conditions attheinner boundary

• =

+

• −

∂ ∂

∑

γ χ α α χ π

• Weakly coupled FP
• =

∂ ∂

• χ
• Strongly coupled FP

Minimumof HBou

2.Phase diagram oftheYJJN:weakly andstrongly coupled fixed points Weakly coupled fixed point

• +

+ + + =

∑

π

α π ξ χ

Modeexpansion for theplasmon fields

• ϕ

ϕ ξ − =

• φ

φ φ ξ − + =

• τ

χ α τ χ α τ χ α

τ τ

• −

−

• −

≈

• ≠

≠

O.P.E. between boundary vertices: Dimensionless boundary coupling G(L)=LEW(a/L)1/g

Secondorder renormalization group equations

( )

γ γ γ

• −

+ − =

• γ

+ − =

• γ

γ − =

Phase diagram

γ g<1 1<g<9/4 G γ 9/4<g

2π/3 π/3 2π/3 π/3 π/3 2π/3

γ

*g<1:stable fixed point atG=γ=0;fixed lines atγ=0,π/3,2π/3.

*1<g<9/4:strongly coupled fixed point for γ≠π/3;finitecoupling fixed point for γ=π/3. *9/4<g:strongly coupled stable FP

Strongly coupled fixed point G>∞ ⇒

⇒ ⇒ ⇒Dirichlet boundary conditions attheinner boundary,as well.(χ1(0), χ2(0))span atriangular lattice, depending onthevalue ofγ

sublattice A sublattice B sublattice C

For γ=π/3theminima lie onahoneycomb lattice(merging oftwo triangular sublattices)

Modeexpansion oftheplasmon fields attheSFP

      + + + =

−

∑

• π

α π π π θ ψ

• Dual fields

      − − + =

∑

• π

α π π ξ χ

For γ≠kπ+π /3theminimaspan only oneofthethree sublattices :inthis case,theleading boundary perturbation is given by acombination of“long”V instantons.

• 

    

• ±

= τ ψ ρ τ

• ∑

=

+ − =

• !
• The“Vinstanton” operators have conformal dimension

hS(g)=4g/3:for ¾<g<1(andfor γ≠kπ+π/3)both the weakly coupled andthestrongly coupled fixed point is stable (repulsiveFFP).

• −

− = − = = ρ ρ ρ

3.Emergence ofastable finitecoupling fixed point For γ=kπ+π /3two triangular sublattices become degenerateinenergy:they merge to form ahoneycomb lattice.Inthis case,theleading boundary perturbation is given by acombination of“short”Winstanton.

• 

    

• ±

= τ ψ α τ

• +

− =

∑

= +

τ ς

τ+,τ areeffective isospin operators,connecting sites on inequivalent sublattices

The“Winstanton” operators have conformal dimension hF(g)=4g/9:for 1<g<9/4neither theweakly coupled,

• rthestrongly coupled fixed point is stable:theIR

behavior ofthesystemis driven by an attractive FFP Perturbative renormalization group equation for the running coupling strength

• ζ

ζ ζ −       − =

Lattice B Lattice C W−instanton Lattice A V−instanton

4.Current patternnear the(WandS)fixed points

• β

β β β β β β β ∂ ∂ − =       ∂ ∂ + ∂ ∂ − =       ∂ ∂ + ∂ ∂ =

• "
• "
• "
• ϕ

ϕ β − =

• ϕ

ϕ ϕ β − + =

Current:logarithmic derivatives ofthepartition function Z

Weakly coupled fixed point

Perturbative calculation:theresult is the“typical” sinusoidal behavior,as afunction oftheapplied phase differences

[ ] [ ] [ ]

• γ

β α γ β α γ β α γ β α γ β α γ β α +

• −

+

• =

+

• −

+

• =

+

• −

+

• =
• "
• "
• "

Zeromode contribution to theenergy eigenvalues

                + + +       + + = + =

• π

β ε π β π

• #

$

• −

= = =

• %

ε ε ε

Strongly coupled fixed point

Onafinitesize systemthis breaks thedegeneracy between theminimaoftheboundary potential (labelled by then’s)

Themain contribution to thetotalcurrent comes from the zeromodeterm inthetotalenergy:this implies abrupt jumps (perturbatively rounded by Vinstantons)atthe degeneracy between two eigenvalues

      + + − =             + + +       + − =             + + +       + =

• "
• "
• "

π β π β π β π β π β

Tuning two eigenstates ofthezeromodeoperatornear by adegeneracy ⇒

⇒ ⇒ ⇒ effective twolevel quantumdevice For instance:setting

• <

< − π β

δ π β + − =

• 

       <<      

• π

δ

Thefollowing two states define an effective 2LQD

↓> ↑> ≡ > >

• %

%

Operating thesystemas aquantumswitch

δ ∗

2 3 1

=| >

Strong coupling

a) b)

1

I =I =−2I

2 3

δ2 (δ =0)

1

2 3 1

=| >

δ2 measures thedetuning

• ffthe

degeneracy: acting onthis parameter

• nemakes

thesystem “switch” between the two states

5.Thesystemworkingnear theFFP:current patternandfrustration ofdecoherence

Though it is possible to setupaselfconsistent formalism to formally derivethecurrent patternnear theFFP,a closedformula canbe given only for g=9/4ε,with ε<<1. Inthis case,onemay settheparameters as

π α

• =

δ π δ β β + − = + ≈

• ε

ζ ≈

Thecurrent patternnear theFFP

• β

β ζ β α α π δ π β ζ β α α π δ π

• "
• "
• "

=         −         + + − − =         −         + + − =

Current across thethree arms

Again,this is a smoothened sawtoothlike behavior but, now,it is associated to a stable FP

δ ∗ δ1 (δ =0)

2

1 2

I =−I

2 3 1

=| >

Finite coupling

a) b)

2 3 1

=| >

We relate thedecoherence to theentanglement ofthe systemwith theplasmon bath ⇒

⇒ ⇒ ⇒ spectral densityof

states oftheeffective 2LQD,Χ”(\)/\

(E.Novais et al.,Phys.Rev. B72,014417(2005))

(Ω) (Ω) (Ω) (Ω)

+−

χ [ ]

RPA

(Ω)

+−

χ [ ]

RPA

• =

+

[ ] χ +−

(0)

[ ] χ +− = [ ] χ +−

(0)

a) b) ζ ζ

ε

ε ς β χ

• +

⊥

> − − − Γ −

• −

> = >

• &%
• ς

β β + =

Using theRPAapproximation sketched above yields

(Ω)/Ω

χ ’’

NFP DFP FFP

Ω

Near theSFP:noentanglement between the2LQDandthe bath,but noquantumtunneling between thestates either (noenergy renormalization); Near theWFP:fullentanglement between the2LQDand thebath (fulldecoherence); NeartheFFP:consistent(androbust)tunnelsplittingof thetwostates,withan accettable level of(frustrated) decoherence

6.Conclusions

• 1. AYshaped JJnetwork may exhibit aFFPinits phase

diagram;

• 2. AttheFFPan effective 2levelquantumdevices

emerges,with enhanced quantumcoherence;

• 3. Relevant issues for apractical realization:stabilizing

γ,applying theexternal phases,controlling other sources ofnoise…