NTT Basic Research Laboratories, NTT Corporation, Japan Superconducting Flux Qubit as a Macroscopic Artificial Atom Hideaki Takayanagi NTT Basic Research Laboratories, NTT Corporation, Japan NTT物性科学基礎研究所 髙 柳 英 明 Outline Quantum Information Research at NTT Fux Qubit Single-Shot Measurement Multi-Photon Absorption Rabi Oscillation Conclusions

QIT Project in NTT Basic Research Laboratories Head: H. Takayanagi About 20 researchers participate to the project which consists of five sub-projects. Four qubit-research projects and a quantum cryptography one.

Solid-State Qubits Four Kinds of Qubit Coupled QDs (artificial molecule) Exciton in QDs SQUID Single-shot measurement Multi-photon absorption Rabi oscillation Rabi oscillation Quantum gate operation cooled atom Atom Chip

Quantum cryptography with a single photon 電気光学 変調器 Amp Gene- rator 時間間隔 解析器 Alice Bob Helium Cryostat Quantum dot lens Pin-hole Lens Single-mode fiber Grating Space filter Beam Splitter Half-wavelength ¼ wavelength 50%-50% Detector 1 Detector 2 Detector 3 Detector 4 waveguide Counter Photon 0 Mirror Attenuator Titanium-Sapphire Laser Testing Nature, 420 (2002) 762

Josephson persistent current Qubit J. E. Mooij et al.,Science 285, 1036 (1999). aEJ , g3 Phase difference Fqubit = f F0 B + 2 f p = g 1 2 3 q f = Fqubit / F0 EJ , g1 EJ , g2 Josephson Energy : cos ( 2 a - + = EJ U ) ) cos ( g 1 2 2 f p j p =0.6 U  =0.8  =1.0 j p m j m f = Fqubit / F0 = 0.5 g 2 ) ( 2 1 j = p + g m - g 1

Schematic qubit energy spectrum 15 100 10 5 D Energy (GHz) Energy (GHz) -5 -100 -10 0.4 0.5 0.6 F / F 0.49 0.50 0.51 qubit Fqubit / F0 80 . eV, 44 30 1055 = a m C J E

Three-Josephson-junction Loop: Description EJ ; C ext= f 0 3 1 2 0< <1.0 Josephson Energy (1 junction): Coupling energy (1 junction): EC = e2/ 2C Flux quantization: Josephson Energy: Our system consists in 3 JJ in series. 2 J are similar, the third one is smaller. It differs from a factor alpha, which should be between 0.5 and 1. Thus if EJ is the NRG of the 2 similar J, the NRG of the last one is alpha EJ. A magnetic flux =f0 penetrates the loop. f is called frustration. The arrows define the direction of the phases. Flux quantization gives: Using this equation and the NRG for one J, we find the JE of the loop. This energy is periodic in 1 and 2 with period 2. J.E. Mooij､et al (1999)

Three-Josephson-junction Loop: Energy Diagram f=0.5 2 minima in each unit cell. We introduce two new phase variables m and p defined by these equations. This graph is the NRG diagram of the qubit as a function of m and p. We note that there are 2 minima in each unit cell. Top View

Three-Josephson-junction Loop:  Dependence of the Potential =0.6  =0.8  =1.0 Now let’s see the dependence of the potential on the parameters alpha and f. We note that When  increases, the barrier height between the two minima in one unit cell increases while the barrier height between a minima of one cell and a minima of the next cell decreases. Note if alpha is less that 0.5, there is only one minima in each cell (for faille m and faille p equal 0 modulo 2 paille) If  increases, the barrier height : increases between the two minima of one unit cell decreases between the minima of adjacent cells

Three-Josephson-junction Loop: Flux Dependence of the States Quantum ground state |0> Classical states Quantum first excited state |1> <Iq>/Ip E0 (1) E Level splitting /0 Classical states = persistent currents of opposite sign. Degenerated at f = 0.5 Quantum tunnelling “anti-crossing” Symmetric and antisymmetric superposition of the macroscopic persistent currents Let’s see the consequences on the states. When the applied flux  is close to 0.50, the Josephson energy becomes a double potential well. The classical states in each well correspond to persistent currents of opposite sign. Classically they are degenerated at half quantum flux. These two classical states are coupled via quantum tunnelling through the barrier. quantum tunnelling leads to an “anti-crossing” with symmetric and antisymmetric superposition of the macroscopic persistent currents. This figure shows the energy of the classical states (in dashed lines) and of the ground state and the first excited state as a function of the applied flux faille. We can note the level splitting due to quantum tunnelling. The bottom plot shows the quantum mechanical expectation value of the persistent current in the loop, for the ground state and the first excited state, plotted in unit of IP. This insert shows the potential when f is smaller than 0.5, equal to 0.5 and greater than 0.5. The horizontal axis is the Josephson phase coordinate, faille m. This is a schematic of the position of the energy levels.

Mutual inductance M ~ 7 pH Sample Fabrication e-beam lithography Shadow evaporation Lift-off Josephson junctions Al / Al2O3 / Al Junction area SQUID : 0.1 x 0.08 m2 Qubit : 0.1 x L m2, ( a = 0.8 ) L = 2 ~ 0.2 Loop size SQUID ~ 7 x 7 m2 Qubit ~ 5 x 5 m2 Mutual inductance M ~ 7 pH Qubit and a detector dc-SQUID NTT Atsugi

suspended-bridge & shadow evapolation e-beam lithography suspended-bridge & shadow evapolation

Sample and Cavity Cavity To mixing chamber Microwave line Thermometer DC measurement To mixing chamber Microwave line Thermometer Vm line Ibias line NTT Atsugi Samples A loop Cavity

1 2 3 4 5 Twisted Constantan wire 100  Sample box RF line DC measurement 1 2 3 4 5 10 nF HP 20dB connectors R.T. 2.4mm connectors 4.2K Through capacitor Flexible coaxial cable HP 10dB 1.2K attenuator 0.8K resistance 0.4K 10mK Heat anchor for outer shield Twisted Constantan wire 100  No on-chip capacitor and resistor No on-chip control line Change twisted wires to thin coaxial cables to introduce dc-pulse 200  200  200  Sample box Loop antenna ~ 1mm above the sample

Readout through a dc-SQUID DC measurement Readout through a dc-SQUID Vm I b qubit Record each switching when Vm = Vth~ 30 mV as a function of Isw Sweep Ib ( 140 Hz ) Tilt SQUID potential I b ~ 100 nA Isw Isw Isw 4~6 nA Isw t 70 ~ 100 μsec Vm ~ 7 ms Vth(~30μV) t

Readout with a dc-SQUID DC measurement Readout with a dc-SQUID Current is swept I(V) curve Isw(/ 0) curve Magnetic field is swept The current flowing in the SQUID is swept upwards. We measure the voltage to obtain the I-V characteristic of the SQUID. This graph is an example of I(V) curve. From this curve we deduce the value of the switching current. The applied magnetic field is also swept and thus we can plot the switching current of the SQUID as a function of the magnetic field or as a function of the filling. Several I-V curves are measured for each value of the magnetic field. This graph is an example of such a curve. The unit of the horizontal axis is the filling of the qubit. Data are averaged. Theoretical calculation leads to this equation. The switching current is a periodic function of the magnetic flux. Its period is equal to the flux quantum 0 . The results we obtain are in agreement with this equation apart from some rounding at the bottom. Furthermore, we observe additional steps when the filling fo the qubit is equal to 0.5.

Qubit step in the SQUID Isw DC measurement Qubit step in the SQUID Isw Qubit switches its current sign Flux in SQUID changes through M SQUID Isw changes Step on the Isw(/ 0) M dc-SQUID Qubit Φ Why do we observe these steps ? At half quantum flux, the qubit switches its current sign, for example from clockwise to anticlockwise. Thus the flux in the SQUID changes by mutual inductance, leading to a change of the switching current. As a consequence, a step appears on the curve. This graph shows an example of step. Fqubit / F0

Parameter dependence of the qubit step ( D, Ej, Ec ) SQUID I Lqubit LSQUID Qubit Josephson junctions : Al / Al2O3 / Al Junction area : SQUID 0.2 x 0.2 m2 qubit 0.2 x L m2, L=0.3, 0.5, 1.0 Now I present you the measurements we have performed. a SQUID is used to measure the flux of the qubit. It is fabricated around the qubit loop, on the same layer. This SEM picture also shows the system qubit ( the inner loop) and Squid (the outer loop). The superconductor we use is Al. The insulator is Al oxide. The geometrical parameters are indicated here : square micrometer…. Loop size : Lqubit = 5.1, 9.7, 19.0 (m) LSQUID = 6.3, 10.9, 20.2

Number of tunneled pair n Two energy scale Ec, EJ energy energy Pair tunneling superconductor -p p superconductor Number of tunneled pair n Phase difference Tunnel barrier H = Ec - EJ cos g - Iex g [n,g]=i Josephson energy : EJ charging energy : Ec =(2ne)2/(2CJ ) kBT << EJ << Ec < D →　charge qubit 　　　　　 kBT << Ec << EJ < D →　phase、flux qubit

D kBT~25mK Qubit energy splitting QB# 6 Junction area = 0.2 m2 QB# 5 Loop size : Lqubit = 9.7 m LSQUID = 10.9 m QB# 5 Junction area = 0.1 m2 Loop size : Lqubit = 9.7 m LSQUID = 10.9 m QB# 4 Junction area = 0.06 m2 Loop size : Lqubit = 9.7 m LSQUID = 10.9 m ( D ~ 2MHz << kBT ) ( D ~ 0.4GHz ~ kBT ) ( D ~ 2GHz > kBT ) Fqubit / F0 Fqubit / F0 Fqubit / F0 QB# 3 Junction area = 0.2 m2 Loop size : Lqubit = 5.1 m LSQUID = 6.3 m QB# 8 Junction area = 0.1 m2 Loop size : Lqubit = 19.0 m LSQUID = 20.2 m QB# 7 Junction area = 0.06 m2 Loop size : Lqubit = 19.0 m LSQUID = 20.2 m Fqubit / F0 Fqubit / F0 Fqubit / F0 D kBT~25mK Qubit energy splitting

Calculated qubit energy level D=0.4 GHz D=2 GHz D=2 MHz Ej=544 GHz Ec=1.6 GHz Ej/Ec=338 Ej=280 GHz Ec=3.2 GHz Ej/Ec=87 Ej=130 GHz Ec=5.4 GHz Ej/Ec=24

Optimal operation point for SQUID Qubit signals appear at half-integer Sensitivity of dc-SQUID depends on magnetic fields We can achieve excellent resolution at f = 1.5 ↓ ↑

Spectroscopy D = 2.6 GHz 0.001 F0 M 2.4 GHz after averaging EJ = 312 GHz, EC = 3.8, a = 0.7 D = 2.6 GHz after averaging w/o averaging 0.001 F0 M 2.4 GHz

Qubit signals at different SQUID modulation DC measurement Qubit signals at different SQUID modulation S/N depends on SQUID Isw design qubit and SQUID to be crossed at small Isw |> |> |> |> T = 25 mK

f= <Iq>/Ip E0 (1) E /0 Level splitting Classical states Quantum ground state |0> Classical states Quantum first excited state |1> <Iq>/Ip E0 (1) E Level splitting /0 f=

Boltzman Distribution Quantum ground state |0> Classical states Quantum first excited state |1> <Iq>/Ip E0 (1) E Level splitting /0

Schematic qubit energy spectrum 15 100 10 5 D Energy (GHz) Energy (GHz) -5 -100 -10 0.4 0.5 0.6 F / F 0.49 0.50 0.51 qubit Fqubit / F0 80 . eV, 44 30 1055 = a m C J E

Spectroscopy D excited state ground state DC measurement Pulse measurement D excited state ground state

Readout without averaging DC measurement Single shot measurement into { l0>, l1> } bases The <Iq> step shape does not change. Only the population changes. Fqubit / F0

Close-up of Isw, T=25 mK Fqubit / F0 Histogram is well separated ! DC measurement Histogram is well separated ! counts counts Fqubit / F0 f f = 1.50102 0.001 F0 M 2.4 GHz

( canonical ensemble average ) Readout after averaging DC measurement Expected Current ( canonical ensemble average ) Fqubit / F0

Experimental setup SLP-1.9 RF line R.T. 4.2K RFin : ２ attenuators Pulse measurement Experimental setup 1 2 3 4 5 SLP-1.9 RF line R.T. 4.2K RFin : ２ attenuators RFout : terminator + attenuator DC : LP filter + Meander filter Flexible coaxial cable HP 10dB 1.2K 0.8K 0.4K I - I + V + V - 29mK Weinschell 10dB Thin coaxial cable f 0.33 mm Meander filters Sample cavity RF in RF in Terminator 50 W Sample cavity On-chip strip line

Multi-photon transition between superposition of macroscopic quantum states ｰ ( ) /√2　1st excited state ＋ ( ) /√2　ground state 3 3 2 2 1 1 3 1 2 2 1 3

Multi-photon spectroscopy Multi-photon transition Multi-photon spectroscopy SQUID readout -2 -1 1 2 d I SW (nA) 1.504 1.502 1.500 1.498 1.496 F qubit / RF : 3.8 GHz -10 dBm １ 3 D=0.86GHz 1-photon 2 -photon 2 1 -1 -2 d I SW (nA) 1.504 1.502 1.500 1.498 1.496 F qubit / RF : 3.8 GHz 0 dBm １ 3

Multiphoton absorption at 9.1 GHz RF Power dependence triple double single off off off PRF = - 21 dBm 0 dBm 9.6 dBm 13.2 dBm 10 dBm 12 dBm 12 dBm

Peak width vs MW intensity Multi-photon transition Peak width vs MW intensity Bloch Kinetic Equation 9100MHz ----- (3) ------------------ (4)

Pulse measurement scheme repetition: 3kHz ( 333 ms) 180 ns ~1μs SQUID switch Ib DC pulse Non-switching resonant microwave time MW discrimination of the switching event Vout + Vout - V th I bias Fext Non-switching events Switching events Ibias + Ibias - SQUID Non-switching event Switching event

Relaxation time T1 Probability [%] D Delay Time [ m s] data exp-fit T Pulse measurement 030304_1 (1,2)FQB2 Relaxation time T1 15 9.1 GHz 1 ms pulse 1st excited sate 10 55 50 45 30 25 Probability [%] 1 2 data MW 5 D exp-fit Energy (GHz) T = 1.6 m s 1 -5 40 35 Ground state -10 0.49 0.50 0.51 Fqubit / F0 500 ns 3 ms Delay Time [ m s] 1 ms delay time Ib pulse height 1.474 V, Trailing height ratio 0.6

Quantum Oscillation : Rabi oscillations Pulse measurement Quantum Oscillation : Rabi oscillations 11.4 GHz 150 ns Dephasing time ~ 30 ns 600 s pulse width ( ns ) switching probability ( % ) Resonant MW pulse width Trailing height ratio 0.7 MW amplitude (a.u.) Rabi frequency ( MHz ) NTT Atsugi

Summary Future plan Spectroscopy of MQ artificial 2-level system Qubit readout without averaging (DC) Multi-photon transition between superposed MQ states Coherent quantum oscillation ( Rabi oscillation ) T1 ~1.6 ms, T2 Rabi ~ 30 ns Future plan Ramsey, Spin echo Two qubit fabrication and operation MQC with single shot resolution

collaborators NTT Basic Research Laboratories Hirotaka Tanaka                                                       collaborators NTT Basic Research Laboratories Hirotaka Tanaka Shiro Saito Hayato Nakano Frank Deppe Takayoshi Meno Kouich Semba Tokyo Institute of Technology Masahito Ueda Yokohama National University Yoshihiro Shimazu Tomoo Yokoyama Tokyo Science University Takuya Mouri Tatsuya Kutsuzawa

エネルギー固有状態をone-shot measurementで見た。 の時、 を測っている。 を測っているのではない。 これを測ると､ 0.5 と のsuperpositionは、生きている。 と の間のsuperpositionは死んでいる。

0.5 time domain で真ん中に出る理由 Qubitの磁場の量子力学的平均値を取っているから Qubitの磁場はszのはず（projection)。 を使って､time-dependentなSchrödinger 方程式を解き､SQUIDのswitching current を求めると、EJ/ECが小さくなると、ピークは １つ、反対にEJ/ECが大きくなると、ピークは ２つになる。 ピーク１つ ピーク２つ 0.5