Superconducting qubits Overview of Solid state qubits Superconducting qubits Detailed example: The Cooper pair box Manipulation methods Read-out methods Relaxation and dephasing Single qubits: Experimental status

1. Overview of solid state qubits

Solid State Qubits

Energy scales Atoms and ions Solid state systems NMR-systems Single system Identical systems possible ∆E=> ~500 THz optical frequencies High temperature Hard to scale Solid state systems Single system System taylored ∆E=> ~10 GHz microwave frequencies Low temperature 20mK Potentially scalable NMR-systems Ensemble system Identical systems possible ∆E=> ~100 MHz radio frequencies High temperature Impossible to scale ?

Things to keep in mind These is a macroscopic system, they typically contains 109 atoms Thus coherence is hard to keep, relatively short decoherence times Nanolithography makes the system (relatively) easy to scale Experimental research on solid state qubits started late compared to other types of qubits

Requirements for read-out systems Two different strategies: Single quantum systems quantum limited detectors Ensembles of systems normal detectors Photons Photon detectors (hard for IR and mw) Magnetic flux SQUIDs Charge Single Electron Transistors Single spin (Convert to charge)

2. Superconducting qubits

Advantages and drawbacks of superconducting systems Energy gap Protects against low energy exitations Good detectors SQUIDs, SETs Nano lithography Relatively easy scaling Drawbacks ”Large systems” Relatively short decoherence times Cooling The low energies require cooling to <<1K

Superconducting qubit characteristics Which degree of freedom charge, flux, phase True 2-level system quasi 2-level system Representation of one qubit single system Manipulation Microwave pulses, rectangular pulses Level splitting ~10GHz Type of read-out Squid, SET, dispersive Back action of read out Single shot possible Operation time 100 ps to 1ns Decoherence time 4 µs obtained Scalability potentially good Coupling between qubits static, tunable, via cavity

Carge versus phase/flux Fluxoid Quantisation in a superconducting ring Charge Quantisation on a metalic island (Josephson) junction Josephson junction Flux Charge Inductance Capacitance Josephson coupling energy EJ Charging energy EC Current Voltage Conductance Resistance

Superconducting qubits charge qubit EJ ~ 0.1 EC Quantronium EJ ~ EC flux qubit EJ ~ 30 EC phase qubit EJ ~ 100000 EC NEC, Chalmers, Yale Saclay, Yale Delft, NTT NIST/Santa Barbara Q and j are conjugate variables which obey the commutation relation: Q well defined  well defined

The NEC qubit I1〉 I0〉 superposition state 0.5 1 1.5 -1 Probe Tunnel junction SQUID loop Box Gate Single Cooper-pair tunneling Reservoir I0〉 I1〉 superposition state Y. Nakamura et al., Nature 398, 786 (1999).

The quantronium, charge-phase (Saclay) T2≈0.5 µs D. Vion et al., Science 296, 286 (2002)

Coupling a qubit to a resonator (Yale) Jaynes-Cummings Hamiltonian Vacuum Rabi splitting, A. Wallraff, Nature 431 162 (2004)

Persistent-current qubit (Delft) flux qubit with three junctions & small geometric loop inductance 0.5  Icirc E F/Fo 2D +Ip -Ip F H = hsz + Dsx with h=(F/Fo-0.5) FoIp Ibias Science 285, 1036 (1999)

The phase qubit (NIST) Large size ~100µm McDermott et al. Science 307, 1299 (2005)

3. Detailed example: The Cooper pair box

The Single Cooper-pair box (SCB) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 -0.2 E / E C n g E J Charge states degenerate ∆ > EC > EJ > T 2.5K ~1.5K ~0.5K 20mK Likharev and Zorin LT17 (84)

The Single Cooper-pair box (SCB) Which degree of freedom Charge Representation of qubit single circuit Levels multi but uses only the two lowest Manipulation mw- or rectangular- pulses Type of read-out Single Electron Transistor, Josephson junction, dispersive Back action of read out single shot possible, very low for dispersive RO Operation time 100 ps Decoherence time 1µs Scalability Yes Coupling capacitive, or via resonator

How does the SCB work as a Qubit Analogy to a single spin in a magnetic field Shnirman, Makhlin, Schön, PRL, RMP, Nature

Eigenstates and Eigenvalues E/4EC <n> ng Eigen values for two state system Eigen states The Coulomb staircase Expectation value for n

Energies, and the optimal point Energy levels E/4EC Optimal point <n> The Coulomb staircase Level splitting ng=CgVg/2e At the optimal point the system is insensitive (to first order) to fluctuations in the control parameters ng and 

Representation of the Qubit The basic building block of a quantum computer is called a Qubit Any two level system which acts quantum mechanically (having quantum coherence) can in principle be used as a Qubit. The Bloch Sphere Ground state Exited state

4. Manipulation methods

Manipulation with rectangular pulses t<0 Starting at ng0 t=0 Go to ng0+∆ng t=∆t Go back to ng0 The left sphere with two adjacent pure charge states at the north and south poles corresponds to a CPB with EJ /EC << 1, which is driven to the charge degeneracy point with a fast dc gate pulse. Nakamura et al. Nature (99)

Microwave pulses (NMR-style) The energy eigenstates at the poles described within the rotating wave approximation. The spin is represented by a thin arrow whereas fields are represented by bold arrows. The dotted lines show the spin trajectory, starting from the ground state. NMR-like Control of a Quantum Bit Superconducting Circuit E. Collin, et al. Phys. Rev. Lett., 93, 15 (2004).

Comparing the two methods Microwave pulses Slower, typical π pulse Timing is easier, NMR techniques can be used Smaller amplitude and monocromatc More gentle on the environment Works at the optimal point Rectangular pulses Faster, typical π pulse More accurate timing required Large amplitude and wide frequency content Shakes up the environment Can not stay at optimal point

5. Read-out methods

Read-out with a probing junction (NEC) Josephson-quasiparticle cycle Fulton et al. PRL ’89 -2e -e + probe Cooper-pair box detect the state as current initialize the system to Repeted measurement gives a current

The NEC qubit I1〉 I0〉 superposition state 0.5 1 1.5 -1 Probe Tunnel junction SQUID loop Box Gate Single Cooper-pair tunneling Reservoir I0〉 I1〉 superposition state Y. Nakamura et al., Nature 398, 786 (1999).

Read-out with SET sample and hold (NEC) Manipulation Trap quasi particles Measure trap with SET Single shot, but still fairly slow Measurement circuit is electrostatically decoupled from the qubit Final states are read out after termination coherent state manipulation

Switching SQUID readout scheme (Delft) pulsed bias current ~ 5 ns rise/fall time tmeas~5 ns, ttrail~500 ns time  quantum operations reference trigger I V Switching each readout only two possible outputs pulse height adjusted for ~50% switching SQUID switched to gap voltage probability measured with ~ 5000 readouts SQUID still at V=0 2 . 3 4 5 6 7 1 8 9 switching probability (%) p u l s e h i g t @ A W n r a o ( V ) pulse height  Room temp. output signal time  Vthr V

Switching Junction readout scheme (Saclay)

Read out with the RF-SET

A single Cooper-pair box qubit (Chalmers) integrated with an RF-SET Read-out system A two level systen based on the charge states |1> = One extra Cooper-pair in the box |0> = No extra Cooper-pair in the box Büttiker, PRB (86) Bouchiat et al. Physica Scripta (99) Nakamura et al., Nature (99) Makhlin et al. Rev. Mod. Phys. (01) Aassime, PD et al., PRL (01) Vion et al. Nature (02) ∆ >> EC >> EJ(B) >> T 2.5K 0.5-1.5K 0.05-1K 20mK

The Single Electron Transistor

The Radio-Frequency Single Electron Transistor (RF-SET) Very high speed: 137 MHz R. Schoelkopf, et al. Science 280 1238 (98) Charge sensitivity: ∂Q= 3.2 µe/√Hz A. Aassime et al. APL 79, 4031 (2001) Current Voltage characteristics Modulation of conductance and reflection Vdc Vac

Performance of the RF-SET Frequency domain: ∂Q=0.035erms, fg=2MHz Time domain: ∆Q=0.2e, inset 0.05e Results: Aassime et al. Charge sensitivity: ∂Q= 3.2 µe/√Hz APL 79, 4031 (2001) Energy sensitivity: ∂e = 4.8 h PRL 86, 3376 (2001)

Dispersive read-outs, parametric capacitance or inductance

Coupling a qubit to a resonator (Yale) A. Wallraff, Nature 431 162 (2004)

The Cooper-pair box as a Parametric Capacitance We define an effective capacitance which contains two parts At the degeneracypoint we get: Büttiker, PRB (86) Likharev Zorin, JLTP (85) c.f. the parametric Josephson inductance

Quadrature measurements with the RF-SET Disspative part, R Reactive part, C (or L) Cooper-pair transistor similar to Cooper-pair box High speed: 137 MHz Schoelkopf, et al. Science (98) Charge sensitivity: ∂Q= 3.2 µe/√Hz Aassime et al. APL 79, 4031 (2001)

The Josephson Quasiparticel cycle: JQP Phase response DJQP Drain -3, -1 -1, 1 Source 0, 2 J1 J2 Cooper-pair resonance Quasi part transition One junction is resonant part of the time Quasi part transition Cooper-pair resonance

The Double Josephson Quasiparticle cycle: DJQP Drain jcn in resonance Source jcn in Drain -3, 1 Source -1, 1 Drain -1, 1 Source 0, 2 One junction is always resonant but only one at the time This results in an average CQ

Quantitative Comparison of the Quantum Capacitance If the phase shift is small it can be approximated by: Assuming equal capacitances we can calculate CQ for a two level system and compare with the data. T=140 mK From spectroscopy EJ/EC=0.12 Temperature adds to the FWHM

Spectroscopy on the quantum capacitance By tuning only one junction into resonance we can excite the system to the exited state, which has a capacitance of the opposite sign Junction 1 Junction 2 P= Probability to be in the exited state EJ1=3.0 GHz EJ2=2.8 GHz

Quantum Cap Qubit Cc Cin CT Cg Operate and readout at optimal point No intrinsic dissipation Tank circuit protects qubit by filtering environment Lumped element version of Yale cavity experiment, Wallraff et al. Nature (04)

Measuring the charge on the box The Coulomb Staircase

The Coulomb Staircase Comparing the Normal and the Superconducting State 2e periodicity is achieved if Ec is sufficiently small <1K Bouchiat et al. Physica Scripta (98) Aumentado et al. PRL (04) Gunnarsson et al. PRB (04)

The Coulomb staircase comparing the normal and the superconducting state Small step occures due to quasi particle poisoning Bouchiat et al. Physica Scripta (98)

What would you expect in the superconducting state Using EC<1.0K pure 2e periodicity is obtained Tuominen et al. PRL (93) Lafarge et al. Nature (93)

Size of the odd step versus magnetic field Lower Ec is better ∆ is suppressed by parallell magntic field Faster suppression in the reservoir than in the box, due to film thickness Reservoir 40 nm Box 25 nm

Spectroscopy

Energy Levels of the Cooper-Pair Box 1 2 0.5 1.5 2.5 3 3.5 4 Q box [e] E J >E C <E 1 2 3 4 E [Ec] Gate Charge Q [e] Box Charge N box | 0 > | 1 > n 01 n01

Spectroscopy of the Cooper-Pair Box 1 2 0.5 1.5 2.5 3 3.5 4 Q box [e] E J >E C <E Level splitting ∆E(ng) B-field dependence of EJ 0.2 0.4 0.6 0.8 15 20 25 30 35 40 ng [e] fHF [GHz] data E C =42.0GHz, E J =20.2GHz

Relaxation and dephasing

Dephasing and mixing T1 T2 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 T1 Describes the diagonal elements in the density matrix Fluctuations at resonance, S(w01) Important during read-out Dephasing 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 Describes the decay of the off-diagonal elements in the density matrix Fluctuations at low frequencies, S(0) Important during “computation” T1 T2

Decoherence sources What are the major decoherence sources in your system? Active sources (stimulated) absorption and emision heat, noise,…. Passive sources (spontaneous) emission only quantum fluctuations external degrees of freedom photons, phonons, quasiparticles... Can they be controlled ? Cooling, shielding, filtering, tailoring the environment Qubit 50Ω Manipulation Read-out Material dependent Microscopic fluctuators

The spin boson model A single two level system (weakly) coupled to an environment described as a bath of harmonic oscillators. The effect of the harmonic oscillators can be described as a fluctuating gate voltage, or a fluctuating ng

Decoherence rates The slow longitudinal fluctuations leads to dephasing =>  The transversal fluctuations which are resonant with the levelsplitting causes mixing (relaxation) =>  The rates are directly proportional to the spectral densities Negative frequencies exitation Positive frequencies relaxation

Examples for spectral densities 1/f like charge fluctuations Log S(f) Shot noise from SET read-out Environmental circuit impedance 50Ω Log f [Hz] If these are the only contributions 1/f noise will be important for dephasing and environment will be important for relaxation (if SET is switched off)

Measurements of T1 and T2

Manipulation with dc-pulses T ≈100 ns r ∆t≈100ps trise≈30ps The probability to find the qubit in the exited state oscillates as a function of ∆t. The charge is measured continuously by the RF-SET Difference between these two curves = excess charge ∆Qbox c.f. Nakamura et al. (99) t<0 Starting at ng0 t=0 Go to ng0+∆ng t=∆t Go back to ng0

Oscillations at the charge degeneracy Oscillation frequency = EJ/h agrees well with EJ from spectroscopy. Good news: We observe oscillations in 6 samples A high fidelity! >70% Deviation from 1.0 e due to finite risetime (~30ps) of pulses, i.e. no missing amplitude 16 12 8 4 Spectroscopy Coherent oscillation perpendicular B-field EJ [GHz] Bad news: T2 only ~10 ns

Flux qubit : Rabi oscillations time trigger Ib pulse read-out operation AMW tMW Chuorescu, Bertet . 2 4 6 8 1 3 5 Rabi frequency (MHz) R F a m p l i t u d e , A / ( )

Rabi and Ramsey in the Quantronium (Saclay) Ramsey fringes Vion, Esteve, et al. Science 2002

Extracting T2 from free precession oscillations Note: relatively large visibility, here ~60% Oscillation period agrees well with level splitting T2 decreases rapidly as the gate charge is detuned from the degeneracy point.

Measurements of T2 vs. Gate Charge ng 0.2 0.4 0.6 0.8 1 1.2 10 8 9 gate charge n g [e] EJ/h=3.6GHz EJ/h=9.4GHz T2-1 Double pulse data agrees with single pulse data Q0 dependence  coupling to charge T2 limited by relaxation at the degeneracy point T2 limited by 1/f-niose away from the degeneracy point Very similar to data from NEC and JPL T. Duty et al., J. Low Temp. Phys. (04)

Determining a T1 that is smaller than Tmeas The average charge <Qbox> depends both on T1 and TR t Qbox(t) 1 2 500 1000 0.2 0.4 0.6 0.8 1 T R [ns] =72ns =87ns <Qbox> [e] n0 depends on the pulse rise time

T1 Measurements vs ng and EJ provide info on S(w) and form of coupling 10 20 30 40 50 60 0.2 0.4 0.6 0.8 1 gate charge ng [e] T [ns] E J 5GHz 8GHz 9GHz The dependence indicates that the qubit is coupling to charge. We can extrapolate the measurement to the degeneracy point and compare with T2 measurements

Measured samples K. Bladh, T. Duty, D. Gunnarsson, P. Delsing New Journal of Physics, 7, 180 (2005) Focussed issue on: Solid State Quantum Information

Summarizing T1 and T2 T2 away from degeneracy point T2 seems to be limited by charge noise whan the qubit is tuned away from Degeneracy point. The extracted value for the 1/f noise is almost an order of magnitude worse than standard values for SETs T2 at degeneracy point At degeneract T2 seems to be limited by relaxation, Best value 10ns. T1 T1 seems to be due to charge noise. Possibly due to the back ground charges

Back-action

The needed measurement time The mixing time depends on the shot noise in the SET Signal to noise ratio (SNR)

Spectral density of the voltage fluctuations of the SET island for our best SET Assuming readout at ∆E=2.4K G. Johansson et al.PRL 2001

Summarizing results for two samples Assume k=1% , EJ=0.1K, ∆EAl=2.4K, ∆Enb=15K and that the SET dominates the mixing Sample 1 : I= 6.7 nA, ∂Q= 6.3µe/√Hz [A.Aassime et al. PRL 86, 3376 (2001)] tm tmix SNR Al-qubit 0.40 µs 8.6 µs 4.6 Nb-qubit 0.40 µs 1.9 ms 68 Sample 2: I= 8 nA, ∂Q= 3.2µe/√Hz [A.Aassime et al. APL 79, 4031 (2001)] Al-qubit 0.10 µs 6.4 µs 8.0 Nb-qubit 0.10 µs 1.3 ms 114 Summarized in: K. Bladh et al. Physica Scripta T102, 167 (2002)

We find T1 short and independent of SET bias in 6 different samples. 150 100 T1 [ns] DJQP bias JQP bias 50 100 200 300 400 500 600 700 800 900 I [pA] bias

Summary Macroscopic systems that allow tailoring and scaling. Energy gap protects against low energy excitations. Different flavors depending on EJ/EC ratio Optimal point important to avoid decoherence T2/Top ≈ 1 µs / 1 ns = 1000 T1 and T2 can be estimated from spectral densities Dispersive read-out schemes promising -> QND Coupling to cavities allow coupling and cavity QED on-chip