Circuit QED: A promising advance towards quantum computing

Transcription

1 Circuit QED: A promising advance towards quantum computing Himadri Barman Jawaharlal Nehru Centre for Advanced Scientific Research Bangalore, India. QCMJC Talk, July 10, 2012

2 Outline Basics of quantum computation. QED and cavity-qed. Superconducting qubits. Circuit QED (cqed). Quantum Rabi model and its analytical solution.

3 Basics of quantum computation Qubits: States of a two-level system or their superposition. Single qubit: a 0 + b 1 Two qubits: a 00 + b 01 + c 10 + d 11 In general they are entangled: cannot be written as a product of qubits. DiVincenzo s criteria (D. P. DiVincenzo, arxiv:quant-ph/ v3) 1 Scalability of physical system with well characterized qubits. 2 The ability to initialize the state of the qubits. 3 Decoherence (coherence) time >> gate operation time. 4 A universal set of quantum gates. 5 A qubit specific measurement capability. Here we shall mainly talk about the third criterion.

4 Basics of quantum computation Basic anatomy of a quantum algorithm

5 Quantum electrodynamics (QED): Semiclassical Matters interacting with light. Two-state problem with a potential oscillating in time (Chap. 5, J. J. Sakurai, Mod. Q.M.). H = H 0 +V (t) H 0 = E E (E 2 > E 1 ) Probability of finding in state 2 V (t) = γe iωt γe iωt 2 1 c 2 (t) 2 = γ2 / h 2 sin 2 (Ωt) where Ω γ 2 / h 2 + (ω ω 21 ) 2 /4 and ω 21 (E 2 E 1 )/ h. Ω 2

6 QED: Rabi oscillation From probability conservation law: c 1 (t) 2 = 1 c 2 (t) 2 Resonance condition: ω = ω 21, i.e. Ω = γ/ h. This principle has been applied in nuclear magnetic resonance (NMR) and masers. Nobel prizes: I. I. Rabi (1944); E. M. Purcell and F. Bloch (1952); C. H. Townes, N. Basov, and A. Prokhorov (1964), N. F. Ramsey (1989).

7 Cavity QED: Atom in a resonator A Rydberg atom: hydrogen-like atom in excited state with large principal q. no.: E n = Ry/(n δ) 2 Ry/n 2 for large n) placed inside a cavity resonator. 2g = vacuum Rabi frequency, κ= cavity decay rate, γ= transverse decay rate, t=transition time. Hamiltonian: Ĥ = hω(a a ) + hg(a σ + aσ + ) + h σ z + H κ + H γ where 2 = energy level spacing, ω=resonator s frequency. Raimond, Brune, and Haroche, Rev. Mod. Phys. 73, 565 (2001)

8 Dephasing and decoherence Coherence can be tested through interference. Interference pattern decays under repeated trials. T 2 =dephasing timescale, T 1=equilibration time, T 2 =decoherence timescale; T 1 T 2 /2. Laad et al., Nature 464, 45 (2010)

9 Jaynes-Cummings Hamiltonian (strong coupling limit: g >> γ,κ,1/t) Ĥ JC = hω(a a ) + hg(a σ + aσ + ) + σ z One can easily think that this operates only on two possible states: e, n and g, n + 1. e, n = excited state of the atom with n photons, n g, n + 1 = ground state of the atom with n + 1 photons, n + 1 Then in matrix form H = (n ) hω1 + [ δ g n + 1 g n + 1 δ where δ 1 2 hω. After diagonalizing the off-diagonal term we get the eigen energies ] E ± n = (n ) hω ± δ 2 + g 2 (n + 1) (n ) ± Ω n Jaynes and Cummings, Proc. IEEE 51, 89 (1963)

10 Vacuum Rabi oscillation We also can find the probability to find in the ground state g at time t, for an atom initially at the excited state e. P e (t) = p(n)sin 2 (g n + 1t) n where p(n) is the photon number distribution. In ν space, maxima should occur at hν, 2 hν, 3 hν,. Also T φ 10 ns. Brune et al. Phys. Rev. Lett. 76, 1800 (1996)

11 Quantum harmonic oscillator (QHO) on an electrical circuit LC circuit: In quantum version Ĥ = ˆφ 2 2L + ˆq2 2C. Analogous to a QHO: Ĥ = 1 2m ˆp2 + mω2 0 2 ˆx 2. So Ĥ = hω 0 (a a ); ω = 1/ LC. However, energy levels are equally spaced E = hω 0. We need transition to be restricted only between two levels. So we need non-linearity: anharmonic quantum oscillator.

12 Josephson junction: a non-linear inductor I = I c sinφ where I c = (2π/Φ 0 )E J ; Φ = h/(2e); and φ = φ L φ R. Also we have dφ/dt = (2π/Φ 0 )V Using the relation V = L J di/dt we get L J = Φ 0 /(2πI c cosφ) So a Josephson junction is equivalent to a non-linear inductor (which we want).

13 Superconducting qubits: Cooper pair box (CPB) Ĥ CPB = E C ( ˆN N g ) 2 E J cos ˆφ where E C = (2e) 2 /(2(C J +C g ))=charging energy; N g = C g V g /(2e). Just projecting on the space formed by the two states n and n + 1, we can rewrite in the matrix form H CPB = C gv g 2(C + 2C J ) σ z + E J 2 σ x In absence of tunneling, E = E C (1 2N g ), degeneracy at N g = 1/2, but lifted in presence of tunneling. This is a desirable region for qubit operation. Shnirman, Schon, and Hermon, Phys. Rev. Lett. 79, 2371 (1997); Bouchiat, Vion, Joyez, Esteve, and Devoret, Phys. Scr. T76, 165 (1998)

14 Superconducting qubits: Cooper pair box (CPB) Bouchiat et al. Phys. Scr. T76, 165 (1998) Vion et al. Science 296, 886 (2002) Degeneracy points (N g = 1/2-integer) zero slope least affected by noise. Shows optimal coherence ( sweet spots ). T φ 0.5 µs.

15 Circuit QED: On-chip realization of cavity QED

16 Circuit QED: On-chip realization of cavity QED Cavity replaced by a 1D superconducting transmission line. A transmission line is a chain of LC oscillators. This creates coplanar waves that get reflected in the gap. Use superconducting charge qubits with E J /E C >> 1: transmon. The oscillators generate a microwave photon coupled to the transmon. Blais et al., Phys. Rev. A 69, (2004)

17 Circuit QED: On-chip realization of cavity QED Everywhere are the sweets spots for large E J /E C. Koch et al. PRA 76, (2007) T φ = 5.5 ± 0.2 µs Schreier et al. PRB 77, (R) (2008)

18 Quantum Rabi model a σ + + aσ terms added to the Jaynes-Cumming Hamiltonian: Ĥ R = hωa a + hgσ x (a + a) + σ z Why Ĥ JC is solvable, Ĥ R seems not? Charge C = a a (σ z + 1) is conserved (good q. #), [C,H JC ] = 0. Braak (2011) pointed out that Ĥ R has a Z 2 (parity) symmetry can be decomposed into two subspaces.

19 Comparison with other candidates Source:

20 Some comments Advantages Can study open quantum systems in contrast to ultra-cold atoms. More practical for implementation since easy to fabricate on a chip. Can be tuned easily by suitably designing the circuits. Disdvantages Coherence time is still smaller. Can work at low temperature only. Presence of disorder is still a persisting problem.

21 Thanks for your kind attention!