Coherent oscillations in a charge qubit

Transcription

1 Coherent oscillations in a charge qubit The qubit The read-out Characterization of the Cooper pair box Coherent oscillations Measurements of relaxation and decoherence times Tim Duty, Kevin Bladh, David Gunnarsson Rob Schoelkopf Yale University

2 A Single Cooper-pair pair Box Qubit Integrated with an RF-SET Read-out system single Cooper-pair box RF-SET V 1> = One extra Cooper-pair in the box > = No extra Cooper-pair in the box Bouchiat et al. Physica Scripta (99) Nakamura et al., Nature (99) Makhlin et al. Rev. Mod. Phys. (1) Aassime, et al., PRL (1) Vion et al. Nature (2)

3 The Qubit The Single Cooper-pair pair Box (SCB)

4 The Single Cooper-pair pair box (SCB) Cg Q=n2e E / EC 1..8 E J Vg C n g H = Q2 2C Σ E J cosθ = 4 E C (n n g ) 2 E J cosθ E C = e 2 2C, n = Q 2e, n g = C 1 V g > E C > E J > T 2.5K 1.5K.5K 2mK <n> n g

5 The SCB as a two level system and qubit H = 4E C (n n g ) 2 E J cosθ using ket representation we get ( ) 2 n H =4E C n n g n 1 E 2 J n n +1 + n +1 n n n ( ) If we assume that E J << E C and we stay close to the degeneracy point, only two states, and 1, matters. Thus we get : H =4E Ch (V g ) E (B) 1 J 1 = 4E Ch ( V )σ z 1 E g 2 J ( B )σ x Ψ = α + β 1 = α β α and β are complex numbers = 1, 1 = 1

6 The read-out The Radio-Frequency Single-Electron Electron-Transistor (RF-SET)

7 The Radio-Frequency Single Electron Transistor Very high speed: 137 MHz R. Schoelkopf, et al. Science (98) Very high sensitivity: Limited by cold amplifiers A. Aassime, et al., APL (1) 3.2 µe/ Hz Typical values Q= 3 µe/ Hz Bw= = 15 MHz

8 SET IV-characteristics R SET =44.1k½ C =37 af C g Å2aF I I (na) 5-5 Vg = Qg Cg C g C -1 C V (mv) Vg Operation point double JQP V bias

9 The sample holder 5GHz pulse line, Nb-coax Combined powder and LC filter Shot noise power (nw) Shot noise from the SET, I=1µA P(1µA)-P(,no Yoko) Q 17.4 f c =33.8 MHz f BW =9.5MHz Frequency (MHz) Magnet for Tuning Ej Bias-Tee Sample holder

10 Controlling the charge on the box The Coulomb Staircase

11 Temperature dependence of the staircase in the normal state mk 29 mk 446 mk <n> [e] C g V g [e]

12 The Coulomb staircase comparing the normal and the superconducting state Box Charge [e] Normal state Superconducting state: Sample 1 Superconducting state: Sample 2 Artificial two level system Very good control Very sensitive read-out C g V g [e]

13 What would you expect in the superconducting state E/E C n g ² ~ k B T ln( N) 2.4 K for Al L S L + S, L = size of long step S = size of short step Using E C <1.2K pure 2e periodicity is obtained Tuominen et al. PRL (93) Lafarge et al. Nature (93)

14 Spectroscopy Determining E C and E J

15 H = Chalmers University of Technology Energy Levels of the Cooper-Pair Box 2ECσ z (1 Q ) EJσ x / ν 1 E J >E C E [Ec] 2 1 ν Q box [e] Box Charge N box [e] 1 1 > > 1 2 Gate Charge Q [e] 1.5 E J <E C 1 2 Q

16 Characterization of the Cooper-pair pair box: Determining E J and E C 4 35 Spectroscopy data E C =42.GHz, E J =2.2GHz 16 Modulation of E J with perpendicular B-field f HF [GHz] E J /h [GHz] Spectroscopy Q [e] perpendicular B-field (φ/φ )

17 2e-periodic staircase: Experiment vs. theory E C and E J extracted from spectroscopy Stair case fit with no free parameters Derivative of the staircase

18 Staircase with Tunable E J

19 Coherent oscillations

20 Manipulation with dc-pulses E/E C ²t n g t< Starting at n g t= Go to n g + n g t= t Go back to n g The probability to find the qubit in the exited state oscillates as a function of t. After t the charge is measured by the RF-SET

21 Continuous measurement with dc-pulses T r =59ns, amplitude 1e pulse train 3 Pulse train off Pulse train on pulse train off pulse train on t 1ps t rise 3ps Q Box [e] 2 1 T =59 ns r Amplitude up to 7%, Difference from 1% can be explained by finite rise time of the pulses (3 ps) 1 2 Q g [e] Difference between these two curves = excess charge Q box

22 Coherent oscillations excess charge Qbox vs. n g and t Note dead zones Oscillation frequency = E J /h Agrees well with E J from spectroscopy Seen in 5 different samples

23 Oscillations at the charge degeneracy Q box [e] f 1 =4.3GHz, T 2 1 ns Bad news: T 2 ~1 ns pulse duration t [ns] Good news: We observe oscillations A very high fidelity! >7% Deviation from 1. e due to finite risetime (~3ps) of pulses, i.e. no missing amplitude

24 Comparison: E J from Spectroscopy and from coherent oscillations E J is modulated with perpendicular B-field E J [GHz] 8 4 Spectroscopy Coherent oscillation perpendicular B-field

25 Possible sources of decoherence The SET: The continuous measurement can of course decohere the system, pulsed measurements should improve the situation. Non-equilibrium quasi particles may be present in the system. Transition between exited state and qp state (Zorin, condmat/312225) Back ground charges are known as an important source of decoherence. At the degeneracy point, that decoherence should be drastically reduced. However, if the dc-pulse is not perfectly square, the system is not exactly at the degeneracy point during the evolution. Then background charge noise couples stronger to the system. DC-pulses may shake up background charges or other resonant modes (environment, cavity etc.) Flux-noise: Less likely, we will test a box without squid-loop

26 Measurements of T 1 and T 2

27 Decoherence and mixing The qubit can be disturbed in two different ways. Relaxation or mixing The environment can exchange energy with the qubit, mixing the two states by stimulated emission or absorption. This has the characteristic time T 1 Describes the diagonal elements in the density matrix Fluctuations at resonance, S(ω 1 ) Decoherence The environment can create loss of phase memory by smearing the energy levels, thus changing the phase velocity. This process requires no energy exchange, and it has the characteristic time T 2 Describes the decay of the off-diagonal elements in the density matrix Fluctuations at low frequencies, S() T 1 T 2

28 Determining a T 1 that is smaller than T meas Q box (t) 2 1 Tr The average charge <Q box > depends both on T 1 and T R t 1.8 T 1 =72ns T 1 =87ns n depends on the pulse rise time <Q box > [e] < n > (t R ) = 2n t 1 t R 1 e t R / t1 1+ e t R / t T [ns] R

29 T 1 Measurements vs Q and E J provide info on S(ω) ) and form of coupling Γ relax T 1 1 = κ sin 2 η S( ω = E) 6 5 E J 5GHz T 1 [ns] E J 8GHz E J 9GHz gate charge Q [e]

30 We find T 1 short and independent of SET bias in 6 different samples. Chalmers University of Technology 15 1 T1 [ns] 5 DJQP bias JQP bias I bias [pa]

31 Dephasing far away from charge degeneracy.3 data T 2 =18ps, n =.8e.2.1 twin pulse: t=ps to1ns Q box [e] Rotation in x-y plane (Nakamura et al. 22) Time [ns] T 2 =18ps and large initial amplitude

32 Measurements of T 2 vs. gate charge Q 15 4 dephasing time T 2 [ns] 1 5 E =6.5 GHz twin pulse J E =9.1GHz twin pulse J E =9.1GHz single pulse J E =4.3GHz single pulse J dephasing time T 2 [ps] E J =6.5 GHz twin pulse E J =9.1GHz twin pulse gate charge Q [e] gate charge Q [e] Q dependence coupling to charge Very similar to data from NEC

33 Summary 2e periodic staircase for E C <1.2 K Characterized Cooper-pair box, E C and E J determined with good accuracy Observed coherent oscillations in 5 samples, oscillation period agrees well with E J T 1 1ns, scales with sin 2 η intermediate gate charge T 2 1ns, due to charge noise T 2 T 1 at the degeneracy point

34 Microwave irradiation of the Cooper-pair pair box By irradiating the atom with microwaves, we can change the population of the levels. This also allows us to determine the parameters of the artificial atom: E C and E j. 3 E J >E C 4 data E C =1.5GHz, E J =2.2GHz Q box [e] ν HF [GHz] E J <E C Q N g

35 The Single-Cooper- Pair Box: E Chalmers University of Technology >> E C >> E J (B) >> T 2.4K.9-1.7K 1.-.5K 2mK C g C,E J V g e> QP state 4E C g> E J H = 2ECσ z (1 Q ) EJσ x / Q =C g V g /e <n> Q =C g V g /e 4 6

36 Determining T GHz.2 data T 2 =15ps E [GHz] GHz Q box [e] microwave amplitude [a. u.] T 2 =11ps, from spectroscopic peak width t [ns] T 2 =15ps, from twin π/2 pulses