DPG_Talk_March2011_AlexandraM_Liguori

Probing quantum coherence in arrays of superconducting qubits

Alexandra M. Liguori, Susana F. Huelga, Martin B. Plenio

Institut fur Theoretische Physik, Universitat Ulm

15th March 2011, DPG, Dresden

A. Liguori Probing quantum coherence in arrays of superconducting qubits

Outilne

1 Original dynamic localisation effect

2 Dynamic localisation in superconducting qubit chains as tool to evaluatecoherence in system


Dynamic localisation on infinite chain

Time-dependent Hamiltonian (~ = 1)

H(t) = Vm=+∞∑

m=−∞

(σ+

mσ−m+1 + σ−mσ

+

m+1) −m=+∞∑

m=−∞

(E0 + E1) cos(ωt)σ+

mσ−m ,

V coupling strength between nearest-neighboursσ±m = (σx

m ± iσym)/2 acting on m-th site (σx and σy Pauli matrices)

E0 energy difference between adjacent sites, ω frequency

Mean-square displacement as function of E1 oscillates sinusoidally

Interaction picture→ effective Hamiltonian with J E0ω

Bessel function

HeffI = V

m=+∞∑

m=−∞

J E0ω

(E1

ω)(σ+

mσ−m+1 + σ−mσ

+

m+1) ,

RESULTS [Dunlap&Kenkre, PRB (1986); Holthaus&Hone, Phil. Mag. B (1996)]

Argument of Bessel function is oscillatory function of time, with oscillationfrequency proportional to magnitude of electric field, i.e.if E0 = nω, n ∈ R, and E1/ω is a zero of Bessel function Jthen mean-square displacement oscillates sinusoidally ⇒ initially localisedparticle remains localised⇔ DYNAMICAL LOCALISATION


Dynamic localisation on finite chain

Finite chain interacting with environment (~ = 1)

RESULTS [Vaziri&Plenio, New J. Phys. (2010)

if E0 = nω, n ∈ R, and E1/ω is a zero of Bessel function J⇒ mean-square displacement oscillates sinusoidally → suppression oftransport for some values of modulation E1 & initially localised particleremains localised;

effective coupling rates from averaging over transition amplitudes⇒suppression of transport is coherence effect due to destructive interference;

with dephasing noise oscillations still exist but amplitude decreases withincreasing dephasing.


Superconducting qubits

Effective two-level systems with a controllable transition frequency between theireigenstates → general superconducting qubit Hamiltonian

Hqubit = −Ezσz + Xcontrolσ

x (1)

→ depending on form of Ez and Xcontrol , superconducting qubit can be of charge,phase or flux type.

Flux or persistent current qubit

A superconducting loop interrupted by three Josephson junctions, two withcapacitance C1 and the third with C2

Josephson junctions coupling constants J1 and J2

Value of external magnetic flux Φ = 0.5Φ0 (Φ0 = h/2e superconducting fluxquantum)⇒ either in the right-hand or in the left-hand current state

Appropriate choice of parameters J1,2 and C1,2 ⇒ tunneling between the twoclassical states can occur

Xcontrol from (1) is tunneling amplitude ∆

Energy splitting Ez = 2Ip(Φ − 0.5Φ0)proportional to detuning Φ − 0.5Φ0, with Ip circulating current.


Dynamic localisation in a chain of superconducting qubits

OUR MODEL: chain of interacting superconducting qubits

H(t) = E1 sin(ωt)N∑

i=1

σzi +

N∑

i=1

∆iσxi +

N−1∑

i=1

Ji,i+1σzi ⊗ σ

zi+1 , (2)

Lsource() = γ1(2σ+

1 σ−1σ

+

1 σ−1 − {σ

+

1 σ−1 , }+ 2σ−1σ

+

1 σ−1σ

+

1 − {σ−1σ

+

1 , }) ,

Lsink () = γN(2σ−Nσ+

N − {σ+

Nσ−N , }+ 2σ+

N σ−N − {σ

−Nσ

+

N , }) ,

N qubits in chain

E1 field modulation

∆i tunneling amplitude for each qubit

Ji,i+1 coupling between qubits i and i + 1

γ1, γN , rates of source and sink respectively


Dynamic localisation in a chain of superconducting qubits

Quantum dynamical eq. for finite chain interacting with environment (~ = 1)

ddt

= −i[H, ] +Lsource() +Lsink () +Ldeph()

with

Ldeph() = γdeph

N∑

i=1

(2σ+

i σ−i σ

+

i σ−i − {σ

+

i σ−i , })

γdeph rate of dephasing noise.

Study current I as function of field modulation E1 in (2):

I = limt→∞

dpsink

dt(t)

with psink (t) =∫ T

02γNN,N(t)dt , N,N reduced density matrix of last site of chain.


Results

N = 3 superconducting qubits with experimental parameters given by group ofProf. J. E. Mooij in Delft:

tunnelings ∆1 = 8.9 GHz, ∆2 = 18.2 GHz, ∆3 = 6.4GHz;

persistent currents Ip1 = 411 nA, Ip

2 ≃ 350 nA, Ip3 = 456 nA;

inter-qubit coupling J12 = J23 = J = 200 MHz.

Current I as function of modulation E1:

100 102 104 106 108 1105

6

7

8x 10−10

E1 (*1010 sec−1)

I (se

c−1 )

γdeph

=0

γdeph

=0.01

γdeph

=0.02

DYNAMIC LOCALISATIONEFFECT: oscillating behaviour ofcurrent⇒ strongly suppressedtransport for some values of E1;

amplitudes of oscillations decreasewith increasing dephasing ratesγdeph ⇒ use current variations toestimate coherence or incoherencein the system.


5 5.2 5.4 5.6 5.8 66.7

6.8

6.9

7

7.1

7.2

I (sec*10−10)

C (

*10−

4 )Incoherence measure C [Vaziri&Plenio,New J. Phys. (2010)]:

C =∑

k,l

|k ,kl,l − k ,ll,k |

⇒ C as function of I for given value of E1

at which first resonance can be found

Conclusions

1 DYNAMIC LOCALISATION IN SUPERCONDUCTING QUBIT CHAIN:oscillating behaviour of current⇒ strongly suppressed transport for somevalues of E1;

2 amplitudes of oscillations decrease with increasing dephasing ⇒ use currentvariations to estimate coherence or incoherence in the system;

3 (2)⇒ incoherence measure C can be used as effective tool to estimatepresence of coherence in superconducting qubit chain by measuring currentI at fixed E1.


DPG_Talk_March2011_AlexandraM_Liguori

Science

Transcript of DPG_Talk_March2011_AlexandraM_Liguori