Floquet formulation for the investigation of multiphoton quantum interference in a superconducting qubit driven by a strong field

Transcription

1 Floquet formulation for the investigation of multiphoton quantum interference in a superconducting qubit driven by a strong field Son, Sang-Kil and Chu, Shih-I Department of Chemistry, University of Kansas 1

2 Abstract We 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 infinitedimensional 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 results 1). 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. 2

3 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 qubit 1,2). 3

4 Computational Details Two-level qubit without RF field Two-level qubit driven by RF field A rf H = 1 2 ε ( ε ε ) H(t) = 1 ( ) ε(t) 2 ε(t) where ε(t) = ε + A rf cos ωt : tunneling strength, ε : flux detuning, A rf : RF field amplitude

5 Floquet theorem 3) i Ψ(t) = Ĥ(t)Ψ(t) where Ĥ(t) = Ĥ(t + τ) t Ψ(t) = e iεt/ Φ(t) where ε is the quasienergy and Φ(t) = Φ(t + τ) Equivalent time-independent eigenvalue problem Ĥ(t) Ĥ(t) i t Ĥ(t)Φ(t) = εφ(t) Ĥ(t) = n Ĥ [n] e inωt and Φ(t) = n Φ [n] e inωt Floquet state: αn = α n αn ĤF βm = H [n m] αβ + nωδ αβ δ nm 5

6 Floquet matrix structure... [ 1] ε 2 ω 2 [ 1] [] 2 + ε 2 ω A rf + A rf [+1] Time-dependent Hamiltonian H(t) = 1 ( ε(t) 2 ε(t) where ε(t) = ε + A rf cos ωt ) [] [+1] A rf + A rf ε 2 2 A rf 2 + ε 2 + A rf A rf ε 2 + ω 2 + A rf 2 + ε 2 + ω... Fourier components H [] = 1 ( ) ε 2 ε H [+1] = 1 ( ) Arf A rf H [ 1] = 1 ( ) Arf A rf Time-dependent wavefunction: Time-averaged switching probability: Ψ(t) = α a ne i(nω λ)t + β b ne i(nω λ)t n n ( ) ( ) P α β = 2 n a2 n n b2 n 6

7 Results Plots of quasienergies as a function of ε Quasienergies of "/h!=.5 and A rf /h!=5. 3 Energy (E/h!) quasienergy(e) with RF quasienergy(g) with RF energy(e) without RF energy(g) without RF Population Detuning flux (# /h!) 7

8 Mean energy ) is computed from eigenvectors: Ē 1 τ τ Analogue of Mach- Zehnder interference * ε(t)dt = λ + Φ(t) i t Φ(t) = λ ( a 2 n + b 2 ) n nω n (λ: quasienergy) Constructive interference condition with the phase: θ = 2πn θ = 1 τ [ε e (t) ε g (t)] dt = 2Ēτ = πē hν Constructive interference condition with the mean energy: Ē hν = n 2 (n: integer) * Picture taken from Oliver et al., Science 31, 1653 (25) 1) 8

9 Comparison with switching probability and mean energy 2 Switching probability P α β Mean energy /hν =.1 /hν =.1 (Ē + ε 2 ) /hν ε/hν A rf /hν 1 2 A rf /hν Plots of switching probabilities are agreed with experimental results 1). Plots of mean energies show the same interference fringes. 9

10 Plots of switching probability P α β changing /hν /hν =.1 /hν =.5 /hν = 1. /hν = 1.5 ε /hν Arf /hν 1

11 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. 11

12 References 1. W. D. Oliver, et al., Science 31, 1653 (25). 2. D. M. Berns, et al., Phys. Rev. Lett. 97, 1552 (26). 3. S. I. Chu and D. A. Telnov, Phys. Rep. 39, 1 (2).. S. I. Chu, Adv. Chem. Phys. 73, 739 (1989). 12