Dissipation in Transmon Muqing Xu, Exchange in, ETH, Tsinghua University Muqing Xu 8 April 2016 1
Highlight The large E J /E C ratio and the low energy dispersion contribute to Transmon s most significant advantage---insensitivity to charge noise. E C = e 2 /2(C g + C B + C J ) This presentation will concentrate on the dissipation analysis performed by Koch et al., that is, noise performance. Introduce the basic ideas of dissipation and present their results. Koch et al. - 2007 - Charge-insensitive qubit design derived from the Cooper pair box Muqing Xu 8 April 2016 2
Highlight Transmon could be proved to have identical Hamiltonian as Cooper Pair Box. The distinguishing difference is the shunt capacitor which significantly increase the ratio E J /E C. Koch et al. - 2007 - Charge-insensitive qubit design derived from the Cooper pair box Muqing Xu 8 April 2016 3
Outline General idea of dissipation and decoherence Relaxation time---amplitude damping channel Dephasing time---phase damping channel Koch et al. - 2007 - Charge-insensitive qubit design derived from the Cooper pair box Muqing Xu 8 April 2016 4
General Ideas Loss of entanglement or the leak of quantum information. From quantum realm to classical realm, like a projection. We will include the state of the environment for convenience. T Sar et al. Nature 484, 82-86 (2012) doi:10.1038/nature10900 Muqing Xu 8 April 2016 5
General Ideas Depolarization channel 3 ψ A 0 E 1 p ψ A 0 E + p 3 i=1 σ i ψ A i E Stefan Kröll, Andreas Walther, Göran Johansson, Peter Samuelsson Quantum Information Lecture http://www.matfys.lth.se/education/quantinfo/ Muqing Xu 8 April 2016 6
Depolarization channel Bit flip error Phase flip error Both ψ σ x ψ ψ σ z ψ ψ σ y ψ I = 1 0 0 1, σ x = 0 1 1 0, σ y = 0 i i 0, σ z = 1 0 0 1 Muqing Xu 8 April 2016 7
General Ideas Bloch sphere representation of Density matrix ρ = 1 2 I + σ r Muqing Xu 8 April 2016 8
General Ideas Dephasing channel 0 A 0 E 1 p 0 A 0 E + p 0 A 1 E Stefan Kröll, Andreas Walther, Göran Johansson, Peter Samuelsson Quantum Information Lecture http://www.matfys.lth.se/education/quantinfo/ Muqing Xu 8 April 2016 9
General Ideas Relaxation channel 1 A 0 E 1 p 1 A 0 E + p 0 A 1 E Stefan Kröll, Andreas Walther, Göran Johansson, Peter Samuelsson Quantum Information Lecture http://www.matfys.lth.se/education/quantinfo/ Muqing Xu 8 April 2016 10
Relaxation time---amplitude damping channel Spontaneous Emission Purcell Effect Dielectric losses Quasiparticle tunneling Flux coupling General discussion: other sources Channel addition rule Γ = Γ α + Γ β + 1 T = 1 T α + 1 T β + Muqing Xu 8 April 2016 11
Relaxation time---amplitude damping channel Spontaneous Emission Spontaneous emission power P = 1 d 2 ω 4 4πε 0 3c 3 The oscillation is caused by cooper pairs, whose traveling (tunneling) distance is 15um In this case we have T se 1 = ħω 1 P = 12πε 0ħc 3 d 2 3 ~0.3ms ω 01 Muqing Xu 8 April 2016 12
Relaxation time---amplitude damping channel Purcell Effect When the system is put into a resonator, a cavity for instance, its Spontaneous Emission rate would be altered. The Purcell effect is a modification as the ratio of cavity modes density of state to free space modes density of state F = ρ c = 1 ρ f ΔνV / 8πn3 ν 2 c 3 Muqing Xu 8 April 2016 13
Relaxation time---amplitude damping channel Purcell Effect We define the transition rate γ k f,i = 2π ħ p ω k 1 k, f ħ λ k b + k a + a + b k k In the dispersive limit the transmon states acquire only a small photonic component, it means we could treat it as a perturbation After first order perturbation calculation, we get We thus get modified T 1 ~16μs γ k f,i = 2π ħ p ω k λ k 2 2 g f,i ω f,i ω r 2 0, i 2 Muqing Xu 8 April 2016 14
Relaxation time---amplitude damping channel Dielectric losses In insulating materials, especially amorphous SiO 2, it would affect the electric fields associated with the qubits and cause energy relaxation. In particular, in the transmon design the shunting capacitance C B offers the possibility to accumulate a large percentage of the electric fields in a well controlled spatial region with favorable substrates, Muqing Xu 8 April 2016 15
Relaxation time---amplitude damping channel Quasiparticle tunneling Following the arguments by Lutchyn et al. N qp = 1 + 3 2 Δk B T e Δ/k BT 2 N e E F valid for temperatures small compared to the superconducting gap Δ T 1 ~1s in this channel when temperature is below 100mK Muqing Xu 8 April 2016 16
Relaxation time---amplitude damping channel There is an intentional coupling between the SQUID loop and the flux bias (allowing for the E J tuning) through a mutual inductance M In addition, the entire transmon circuit couples to the flux bias via a mutual inductance M We choose to assume that the applied flux could be decomposed into external flux and a small noise term, Φ = Φ e + Φ n, Φ n Φ e Koch et al. - 2007 - Charge-insensitive qubit design derived from the Cooper pair box Muqing Xu 8 April 2016 17
Relaxation time---amplitude damping channel Flux coupling Then we Taylor expand the Josephson Hamiltonian H J H J + Φ n A, A = H J Φ ቚ Φ e In this way, we could write the dissipation rate (just transition rate) Γ 1 = 1 ħ 2 1 A 0 2 S Φn (ω) = 1 ħ 2 1 A 0 2 M 2 S In (ω) T 1 : 20ms~1s However, new research found that there is capacitive coupling that contribute much more to the decoherence time μs Muqing Xu 8 April 2016 18
Relaxation time---amplitude damping channel Other sources Coupling to spurious resonator modes Pinning and unpinning of vortices Bulk piezoelectricity Muqing Xu 8 April 2016 19
Dephasing time---phase damping channel Analytical discussion Charge noise & Flux noise Critical current noise Quasiparticle tunneling E C noise Muqing Xu 8 April 2016 20
Dephasing time---phase damping channel The energy shift of the eigenstates would cause a change of transition frequency. Then it would lead to a decay in the off diagonal elements In formal theory, H q = 1 2 u=x,y,z h u λ i σ u, λ i = λ i 0 + δλ i Then do Taylor Expansion, calculate the noise power by autocorrelation. We focus on the dephasing aspect, low-frequency noise in the σ z component. H q = ħω 01 2 σ z + 1 2 h z λ i δλ λ j σ z + O(δλ 2 ) j j Muqing Xu 8 April 2016 21
Dephasing time---phase damping channel Then do Taylor Expansion, calculate the noise power by autocorrelation, we get the law of off diagonal elements ρ 01 (t) = e iω01t exp 1 2 න dω 2π S v ω sin2 ωt/2 ω/2 2 When t t c, ρ 01 (t) = e iω01t exp 1 2 t S v(ω = 0) Dephasing time T 2 ~2/S v (ω = 0), Lorentzian lineshape Muqing Xu 8 April 2016 22
Dephasing time---phase damping channel Charge noise Small fluctuations regime T 2 ~ ħ E A n g Slow charge fluctuations with large amplitudes T 2 ~ 4ħ e 2 π ε 1 ~0.4ms, T 2 ~ π2 A 2 ħ 2 2 E n g 2 1 ~8s H q = 1 2 ħω 01 + ε 1 2 cos 2πn g + 2πδn g t σ z 1 = ħ2 π 2 A 2 E J 64E C ~1μs, Transmon in worst case Cooper Pair Box at sweet point Muqing Xu 8 April 2016 23
Dephasing time---phase damping channel Flux noise Noise in the externally applied flux translates into fluctuations of the effective Josephson coupling energy E J. T 2 ~ π2 A 2 ħ 2 2 E Φ 2 1 ~3.6ms, at integer flux quanta, sweet point Muqing Xu 8 April 2016 24
Dephasing time---phase damping channel Critical current noise Critical current noise is likely to be the limiting dephasing mechanism. Such rearrangements in the junction directly influence the critical current and hence the Josephson coupling energy E J = I c ħ/2e. An improvement of a factor of 2 T 2 ~ ħ A E I c 1 ~35μs Muqing Xu 8 April 2016 25
Dephasing time---phase damping channel Quasiparticle tunneling In CPB case, Γ 2 qp = Γ qp N qp, consider a jump from sweet point to half charge point But in transmon regime, due to large ratio of E J /E C, the charge dispersion is exponentially flat so that transition frequency variations due to a single charge are minimal. E C noise A loophole? Fluctuations in the effective capacitances of the circuit. No current reports of this noise source. Muqing Xu 8 April 2016 26
Dephasing time---phase damping channel Koch et al. - 2007 - Charge-insensitive qubit design derived from the Cooper pair box Muqing Xu 8 April 2016 27
References [1] Koch, Jens, et al. "Charge-insensitive qubit design derived from the Cooper pair box." Physical Review A 76.4 (2007): 042319. [2] Van der Sar, T., et al. "Decoherence-protected quantum gates for a hybrid solid-state spin register." Nature 484.7392 (2012): 82-86. [3] Preskill, John. "Lecture notes for physics 229: Quantum information and computation." California Institute of Technology (1998): 16. [4] Stefan Kröll, Andreas Walther, Göran Johansson, Peter Samuelsson Quantum Information Lecture http://www.matfys.lth.se/education/quantinfo/ [5] Girvin, Steven M. "Superconducting qubits and circuits: Artificial atoms coupled to microwave photons." Lectures delivered at Ecole d Eté Les Houches (2011). Muqing Xu 8 April 2016 28
Thank You! Muqing Xu 8 April 2016 29