Floquet formulation for the investigation of multiphoton quantum interference in a

superconducting qubit driven by a strong field

Son, Sang-Kil and Chu, Shih-IDepartment of Chemistry, University of Kansas

AbstractWe present a Floquet investigation of multiphoton quantum interference in a strongly driven superconducting qubit. The procedure involves a transformation of a time-dependent problem into an equivalent time-independent infinite-dimensional Floquet matrix eigenvalue problem. The results of a two-level qubit system show quantum interference fringes around multiphoton resonance positions in agreement with the experimental results1). We further explore the interference patterns in terms of quasienergies and the resonance position shifts as the tunneling strength increased. The Floquet formulation promises a new and accurate approach for the investigation of quantum interference phenomenon in the qubits.

Introduction

• SQUID: Superconducting Quantum Interference Devices

• Superconducting qubit which has two magnetic flux states is a promising candidate for quantum computing.

• Recent experiments demonstrate quantum interference fringes around multiphoton resonance positions in a strongly driven superconducting qubit1,2).

Computational DetailsTwo-level qubit without RF field

Two-level qubit driven by RF field

H0 = −

1

2

(

ε0 ∆

∆ −ε0

)

H(t) = −

1

2

(

ε(t) ∆∆ −ε(t)

)

where ε(t) = ε0 + Arf cos ωt

∆: tunneling strength, ε0: flux detuning, Arf: RF field amplitude

ε0

∆

Arf

i!∂

∂tΨ(t) = H(t)Ψ(t) where H(t) = H(t + τ)

and Φ(t) = Φ(t + τ)

⇒ Ψ(t) = e−iεt/!Φ(t) where ε is the quasienergy

H(t) ≡ H(t) − i!∂

∂t⇒ H(t)Φ(t) = εΦ(t)

H(t) =∑

n

H[n]

e−inωt and Φ(t) =

∑

n

Φ[n]e−inωt

Floquet state: |αn⟩ = |α⟩ ⊗ |n⟩

⇒ ⟨αn|HF |βm⟩ = H[n−m]αβ + nωδαβδnm

Floquet theorem3)

Equivalent time-independent eigenvalue problem

+Arf

4

−

∆

2−

ε0

2− ω

+ε0

2− ω

+ε0

2+ ω

−

ε0

2+ ω

−

ε0

2

+ε0

2

−

∆

2

−

∆

2

−

∆

2

−

∆

2

−

∆

2

+Arf

4+

Arf

4

+Arf

4

−

Arf

4

−

Arf

4−

Arf

4

−

Arf

4

...

... [+1][−1] [0]

[−1]

[0]

[+1]

Floquet matrix structure Time-dependent Hamiltonian

Fourier components

H(t) = −

1

2

(

ε(t) ∆∆ −ε(t)

)

where ε(t) = ε0 + Arf cos ωt

H[0]

= −

1

2

(

ε0 ∆

∆ −ε0

)

H[+1]

= −

1

4

(

Arf 0

0 −Arf

)

H[−1]

= −

1

4

(

Arf 0

0 −Arf

)

Time-dependent wavefunction:

Time-averaged switching probability:

Ψ(t) = |α⟩∑

nanei(nω−λ)t + |β⟩

∑n

bnei(nω−λ)t

Pα→β = 2

(

∑

na2

n

) (

∑

nb2

n

)

-4

-3

-2

-1

0

1

2

3

4

Ener

gy (E

/hν)

Quasienergies of Δ/hν=0.5 and Arf/hν=5.0

0

1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Popu

latio

n

Detuning flux (ε0/hν)

Results

-4

-3

-2

-1

0

1

2

3

4

En

erg

y (

E/hν)Quasienergies of Δ/hν=0.5 and Arf/hν=1.0

energy(g) without RFenergy(e) without RF

quasienergy(g) with RFquasienergy(e) with RF

0

1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Po

pu

latio

n

Detuning flux (ε0/hν)

Plots of quasienergies as a function of ε0

uses two beamsplitters: The first divides anoptical signal into two coherent waves thattravel along paths with different effectivelengths, and the second recombines andsuperposes these waves, leading to quantuminterference fringes in the measured outputsignal. In a driven qubit, according to anidea discussed by Shytov et al. (19), thebeamsplitters can be realized by Landau-Zener (LZ) transitions at a level avoidedcrossing. Over one oscillation period of thedriving field, the qubit is swept through theavoided crossing twice (Fig. 1A). Startingfrom the marker, at the first LZ transition(time t1), the ground state k0À is split into acoherent superposition of the ground and ex-cited states, k1À and k0À, which, after evolvingindependently and accumulating a relativephase Dq12, interfere at the second LZtransition (time t2). The corresponding qubit-state energy evolution (first period, Fig. 1B)between the recurrent LZ transitions (shadedregion) provides a phase-space analog tothe two arms and the beamsplitters of anoptical MZ interferometer (top left, Fig. 1B).The interference phase

Dq12 01

I

Z t2

t1

eðtÞdt; eðtÞ 0 ek0ÀðtÞ j ek1ÀðtÞ

ð1Þ

where I 0 h /2p, h is the Planck constant, ande is the energy difference between states k0Àand k1À, depends on the magnitude of the qubitenergy detuning excursion for times t1 G t G t2.The interference fringes in the occupation prob-ability correspond to integer and half-integervalues of Dq12/2p. Known as St[ckelberg os-cillations with Rydberg atoms (20, 21), this

mechanism can be applied to quantum con-trol (22).

The qubit MZ interferometer differs in anumber of ways from an optical interfer-ometer. First, instead of a photon, the in-terferometry is performed with the use ofthe quantum state of a qubit. Second, in thequbit, we have the interference of paths inphase space rather than in coordinate space;the phase Dq12 (Eq. 1) is determined by thequbit level splitting, which plays the roleof the optical path length. Finally, becausethey are more fragile than photons and easierto decohere, qubit states can be manipulatedin a coherent fashion only at relatively shorttime scales.

We used a periodic driving signal, a har-monic variation of the qubit detuning e(t)

H 0 Y1

2ðDs x þ eðtÞs zÞ;

eðtÞ 0 e0 þ Arf coswt ð2Þ

where D is the tunnel splitting, sx and sz

are Pauli matrices, e0 is the detuning pro-portional to dc flux bias, and Arf is theradio frequency (rf) field amplitude pro-portional to the rf flux bias (23). In this case(Fig. 1B), we have cascaded LZ transitionswhich occur when the driving amplitude ex-ceeds detuning, giving rise to the interferencefringes at Arf 9 ke0k (Fig. 1C). Although thephase Dq12 equals the shaded area in Fig.1B and is dependent on Arf , the total phasegained over one period, q 0 E1/I^Xe(t)dt 02pe0/Iw, equals the difference of theshaded and unshaded areas and is in-dependent of Arf . As consecutive pairs ofLZ transitions (consecutive MZ interfer-

ometers) interfere constructively when q 02pn, the fringes will appear around theresonance detuning values

e0;n 0 nh n ð3 Þ

where n 0 0, 1, 2, I and n 0 w/2p. Anotherinterpretation of this condition is that thesequential LZ transitions excite multipho-ton resonances.

Although coherent multiphoton reso-nances between discrete states of an rf-driven charge qubit have been reported (5, 24)and multiphoton transitions used to driveRabi oscillations in a flux qubit (25, 26), inthese works as well as in the earlier workon quantum dot systems (27, 28), only afew photon transitions could be observed,with coherence quickly weakening as rf am-plitude increased (29). In contrast, we wereable to observe coherent resonances of veryhigh order, up to n 0 20, which requiresdriving the system at a high rf amplitude.The fringes for high n are as clear as thosefor n ; 1, indicating that the qubit preserves asubstantial amount of coherence even in thestrongly driven regime.

We realized a tunable artificial atom witha niobium persistent-current qubit (Fig. 2A),a superconducting loop interrupted by threeJosephson junctions (30). When the qubitloop is threaded with a magnetic flux fq ,F0/2, the system exhibits a double-wellpotential-energy landscape (fig. S1). Theclassical states of the wells are persistentcurrents Iq with opposing circulation, de-scribed by energy bands Te0/2 0 TIqF0dfqlinear in the flux detuning dfq K fq j F0/2.The double-well barrier allows quantumtunneling of strength D, opening the avoided

Fig. 1. MZ interference in a strongly drivenqubit. (A) Starting at the dot marker, the qubitstate is swept by an rf field. After an LZtransition at the first avoided crossing (time t1),the resulting superposition state of k0À and k1À(dashed lines) accumulates a phase Dq12 (shadedregion) and interferes at the return LZ transition(time t2). The qubit state is subsequently drivenaway from the avoided crossing and then returnsto the starting flux position. This single period ofqubit evolution is a single MZ interferometer.Depending on the interference phase Dq12,amplitude may build in the excited state. a.u.,arbitrary units. (B) The corresponding qubitenergy variation induced by a periodic rf field,Eq. 2, results in an equivalent optical cascade ofMZ interferometers (MZ#1 to #3, top) withresonance condition Eq. 3. (C) The population ofthe qubit excited state, Eq. 6 , as a function of rfamplitude Arf and detuning e0. Note theinterference fringes (I to VI) at Arf 9 e0 and themultiphoton resonances at e0 0 nhn.

R E P O R T S

9 DECEMBER 2005 VOL 310 SCIENCE www.sciencemag.org1654

Mean energy4) is computed from eigenvectors:

Analogue of Mach-Zehnder interference*

Constructive interference condition with the phase:

∆θ = 2πn

* Picture taken from Oliver et al., Science 310, 1653 (2005)1)

Constructive interference condition with the mean energy:

E

hν=

n

2(n: integer)

E ≡

1

τ

∫

τ

0

ε(t)dt = λ +

⟨⟨

Φ(t)

∣

∣

∣

∣

i∂

∂t

∣

∣

∣

∣

Φ(t)

⟩⟩

= λ −

∑

n

(

a2

n+ b

2

n

)

nω

(λ: quasienergy)

∆θ =1

!

∫ τ

0

[εe(t)−εg(t)] dt =2Eτ

!=

4πE

hν

Arf/hν

ε 0/h

ν

Switching probability Mean energy

Plots of mean energies show the same interference fringes.

Comparison with switching probability and mean energy

∆/hν = 0.1 ∆/hν = 0.1

10 200

10

20

Arf/hν10 200

Pα→β

Plots of switching probabilities are agreed with experimental results1).

(

E +ε0

2

)

/hν

ε 0/h

ν

Arf/hν10 200

10

20

∆/hν = 0.1 ∆/hν = 0.5

∆/hν = 1.0 ∆/hν = 1.5

Plots of switching probability changing Pα→β ∆/hν

Conclusion

• The plots of switching probabilities show quantum interferences due to accumulated phase difference in agreement with the experimental results.

• The phase difference can be derived from the mean energy (quasi-energy), and the mean energy plot shows the same interference patterns.

• Resonance position shifts are observed as the tunneling strength increases.

• The Floquet formalism is extended to investigate quantum interference phenomenon in the qubits.

References

1. W. D. Oliver, et al., Science 310, 1653 (2005).

2. D. M. Berns, et al., Phys. Rev. Lett. 97, 150502 (2006).

3. S. I. Chu and D. A. Telnov, Phys. Rep. 390, 1 (2004).

4. S. I. Chu, Adv. Chem. Phys. 73, 739 (1989).