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NTT-qubit/860504-2004-04 Coherent control of a flux qubit by phase-shifted resonant microwave pulses Tatsuya Kutsuzawa1,2,3, Hirotaka Tanaka1,3, Shiro Saito1,3, Hayato Nakano1,3, Kouichi Semba1,3, and Hideaki Takayanagi1,2,3 1NTT Basic Research Laboratories, NTT Corporation, Atsugi, 243-0198, Japan 2Department of Physics, Tokyo University of Science, 5 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan 0 3CREST, Japan Science and Technology Agency, Japan 0 (Dated: February 2, 2008) 2 Thequantumstateofafluxqubitwassuccessfullypulse-controlledbyusingaresonantmicrowave. n WeobservedRamsey fringes byapplyingapairofphase-shifted π microwave pulseswithout intro- a 2 J ducing detuning. With this method, the qubit state can be rotated on an arbitrary axis in the x-y planeoftheBlochsphereinarotatingframe. Weobtainedaqubitsignalfromacoherentoscillation 5 with an angular velocity of up to 2π×11.4G rad/s. In combination with Rabi pulses, this method 2 enables us to achieve full control of single qubit operation. It also offers the possibility of orders of ] magnitudeincreases in thespeed of the arbitrary unitary gate operation. l l a PACSnumbers: 03.67.Lx,85.25.Dq,85.25.Cp,03.65.Yz h - s e A superconducting circuit containing Josephson junc- (((aaa)))OOOnnn ccchhhiiippp 222555mmmKKK 444...222KKK 333000000KKK m tions is a promising candidate as a quantum bit (qubit), . which is an essential building block for future quantum VVV t computers [1]. In the superconducting circuit, the qubit ooouuuttt a MMMeeeaaannndddeeerrr LLLooowww PPPaaassssss m is represented by two quantized states which are collec- ΦΦΦeeexxxttt fffiiilllttteeerrrsss fffiiilllttteeerrrsss tive states of a “macroscopic” number of Cooper pairs. - III d Recently, the NMR-like coherentcontrolofseveraltypes bbbiiiaaasss n oftheseJosephsonqubitshasbeenreported[2,3,4,5,6]. III o In addition to Rabi, Ramsey, and spin echo type exper- MMMWWW c (((bbb))) (((ccc))) iments on a single qubit, conditional gate control with a πππ///222 PPPuuulllssseeeπππ///222 PPPuuulllssseee [ more complex pulse sequence has been demonstratedfor IIIMMMWWW v1 ainttweroacqtuinbgitwsyistthema [h7a]romroinniacsoyssctiellmatcoorn[s8i,st9in].goAflathqouubgiht IIIMMMWWW UUU111RRRFFF IIIbbbiiiaaasss ttt111222 DDDCCC PPPuuulllssseee 92 thecontrollabilityofthesequbitshasbeenrapidlydevel- UUU222RRRFFF tttddd oped, the typical coherence time of the Josephson qubit 5 remains rather short compared with other qubit candi- 1 FIG.1: (a)Schematicdrawingsofthemeasurementcircuit. The 0 dates. This has made the effective use of this precious squares with crosses represent Josephson junctions. Microwaves 5 coherence time and the realization of a shorter gate op- (MW) were applied to the qubit from the on-chip strip line. (b) 0 erating time desirable. Schematic drawings of the phase-shifted double pulse generation. / The relative phase shift between two pulses was precisely con- at Inquantumcomputation,itisessentialtocontroleach trolled with two synchronous MW generators U1RF and U2RF, m qubitbyperformingarbitraryunitary operationsatwill. eachMW-pulsewasre-shapedbymultiplyingwitharecutangular For one qubit, Rabi oscillation and Ramsey fringes ex- pulse from a synchronous driven pulse generator, then a pair of - phase-controlledMW-pulsesweredeliveredtothesamplethrough d periments provide information related to the control of π amicrowaveadder. (c)Timingchartoftheresonant MW-pulses n the qubitstate|Ψi=cosθ|0i+eiφsinθ|1i. Inthe Bloch 2 2 2 separatedbytimet12 andareadoutDC-pulse. o sphere notation, Rabi oscillations give us both informa- c tion and the ability to control the polar angle θ. The : v Ramseyfringesgiveustheabilitytocontroltheazimuth withthreeJosephsonjunctionsofcriticalcurrentIqubit ≃ c i angle φ. However,the observationofthe Ramsey fringes X 0.6µA for larger junctions) and an under-damped dc- of a flux-qubit usually involves a few hundred MHz de- SQUID (an outside loop with two small Josephsonjunc- r a tuning from the qubit resonant frequency. In this pa- tions of critical current ISQ ≃ 0.15µA for each junc- c per, we propose a new method for observing Ramsey tion) as a detector. The qubit contains three Josephson fringes, the phase shift method, which can control the junctions, two of which have the same Josephson cou- phase of microwave (MW) pulses at the resonant fre- pling energy E = ~Iqubit/2e. The third has αE , with J c J quency of the qubit. The advantage of this method is 0.5<α<1. The α value can be controlledby designing that it provides nearly two orders of magnitude faster the ratio of the area of the smallest Josephson junction azimuthangleφ controlofaqubit thanthe conventional to the other two larger junctions in the qubit. We used detuning method. a sample with α ≃ 0.7 and the areas of the larger and Figure 1(a) shows schematic drawings of the measure- smaller junctions were 0.1×0.3 µm2 and 0.1×0.2 µm2, ment circuit; a superconducting qubit (an inside loop respectively. The loop area ratio of the qubit to the 2 ((aa)) 00 zz ((bb)) 00 zz ((cc)) 00 zz (cid:3)(cid:2) yy yy yy ϕϕ (cid:12) (cid:11) xx xx xx (cid:10) (cid:6)(cid:2) (cid:9) (cid:7)(cid:8) 11 11 11 FIG. 2: Schematic diagram of qubit vector motion induced by π (cid:5)(cid:2) the phase-shifted double 2 on-resonance pulses (ω = ω0). It is (cid:0)(cid:2) (cid:0)(cid:3) (cid:4)(cid:2) d(ae)scTrihbeedquibnitthveecrtootratinintghferainmiteiaolfstthateeq.uTbhiteLquarbmitoirsfirneqthueengcryouωn0d. (cid:1) (cid:13)(cid:14)(cid:15)(cid:1)(cid:16)(cid:17)(cid:18)(cid:19) (cid:1) π state |0i. (b) The first resonant pulse (ϕ = 0) tips the qubit 2 vectortotheequator. Thequbitvectorremainsthere,becausethe FIG. 3: On-resonance Ramsey fringes observed by using the on-resonance pulse is used. (c) The second resonant π2 pulse, in phase-shifted double π2 pulse techniπque. The Larmorfrequency is whichthephase-shiftϕ6=0isintroduced,tipsthequbitvectoron ω/2π=11.4GHz. Thewidthofthe 2 pulse,5ns,isdeterminedby another axis,whichisatanangleϕfromthex-axis. Rabi oscillation. Anexponentially damped sinusoidal curvefitted withthedecaytimeconstantT2=0.84nsisalsoshown. SQUID was about 3:5. The dc-SQUID with two Joseph- twopulseswaspreciselycontrolledwithtwosynchronous son junctions was under-damped with no shunt resistor. microwave generators. The bias current line delivers the The qubit and the dc-SQUID were coupledmagnetically readoutpulses,andtheswitchingeventisdetectedonthe viamutualinductanceM ≃7pH.ThealuminumJoseph- signalline (V ). Asregardsthe shapeofthe biaspulse, sonjunctionswerefabricatedbyusingsuspendedbridges out itspeakwidthof150ns,islimitedbytherisetimeofthe andtheshadowevaporationtechniques[10]. Bycarefully filtersinstalledinthe bias currentline. The widthofthe designing the junction parameters [11, 12, 13], the inner trailingplateauof600ns,andthetrailingheightratioof loopcanbemadetobehaveasaneffectivetwo-statesys- 0.7,wereselectedinordertooptimizethe discrimination tem [14, 15]. In fact, the readout result of the qubit of the switching voltage of the dc-SQUID detector [6]. changes greatly with the qubit design, ranging from the The qubit is described by the Hamiltonian Hˆ = purely classical to the quantum regime [16, 17]. 0 ~ (ε σˆ +∆σˆ ),whereσˆ arethePaulimatrices. Wees- The sample was cooled to 25 mK with a dilution re- 2 f z x x,z frigeratoranditunderwentasuperconductingtransition timated the qubit tunnel splitting ∆ ≃1 GHz from the 2π at ∼1.2 K. In order to reduce external magnetic field spectroscopy. The two eigenstates of σˆz are macroscop- fluctuations, both the sample holder and the operating ically distinct states with the qubit supercurrent circu- magnetweremountedinsideathree-foldµ-metalcan. As lating in opposite directions, i.e., the clockwise state |Ri schematicallyshowninFig.1(a), the qubitis biasedwith andthecounter-clockwisestate|Li. Thedc-SQUIDpicks a static magnetic flux Φext using an external coil. A mi- up a signal that is proportional to σˆz. To control the crowaveon-chip strip line was placed at 60 µm from the qubit state we employ a microwave current burst, which qubit to induce oscillating magnetic fields in the qubit induces oscillating magnetic fields in the qubit loop. loop. TheswitchingvoltageoftheSQUIDwasmeasured ThateventisdescribedbytheperturbationHamiltonian by the four-probe method. The generation of phase- Vˆ = ~2a(t)σˆz, where a(t) = acos(ωt+ϕ) has an ampli- shifted operating pulses is shown schematically in Fig. tude correlated to the power of the applied microwave 1(b). The relative phase shift between two pulses was pulse, ω is the angular frequency and ϕ represents the precisely controlled with two synchronous MW genera- origin of the phase of the applied microwave pulse. To tors U1 and U2 . In order to trim the pulse shape maintain the analogy between the spin 1 and the flux RF RF 2 and reduce the noise level, each pulse-modulated MW qubit, we write the Hamiltonian in the rotating frame burst will be re-shaped by multiplying it with a rectan- approximation,Hˆrot = ~2ωσˆz+e−iω2tσˆz(Hˆ0+Vˆ)eiω2tσˆz,in gular pulse from a synchronous driven pulse generator, terms of the energy eigenstates (|0i, |1i) and obtain thenapairofphase-controlledpulsesaredeliveredtothe sample through a microwave adder. Figure 1(c) shows ~ ω−ω −aeiϕsinθ tohuet tpiumlsine.g Tchhaertono-fchthipe MopWeraltiinnegppruovlsiedsesanadmtihcerorweaavde- Hˆrot = 2 (cid:18)−ae−iϕs0inθ ω0−ω (cid:19) (1) currentburst which induces an oscillating magnetic field where ω = 1 ε 2+∆2 is the qubit Larmor frequency in the qubit loop. We adjusted the timing of a read- 0 ~ f out DC-pulse, which is delivered to the detector SQUID at the measurepd flux bias and θ ≡arctan ∆. εf throughthe currentbias line (I )just after the second Figure2showsschematicdiagramsdescribinghowthe bias π controlpulse. Thewidthandamplitudeofthe π pulse qubit vector is operated during the Ramsey fringe ex- 2 2 are determined by the Rabi oscillation at the resonant periment with the phase shift technique in a rotating frequency(ω ). The relative phase shift between these frame. We assume that the initial state of the qubit is 0 3 the ground state |0i. The first resonant π pulse (ϕ=0) ns is required for every 2π azimuth angle rotation of the 2 tipsthequbitvectortowardstheequatorwiththex-axis qubitvector. Thisoperatingtimecannotbeasshortas1 as the rotating axis (Hˆ ∝ σˆ ). The qubit vector re- ns,because a detuning of1 GHz doesnotworkproperly. rot x mains there because we introduce no detuning (ω =ω ). However, with the phase shift technique with the reso- 0 After a time t , the second resonant π pulse with a nantfrequency,aswehaveshown,itispossibletorevolve 12 2 given phase shift ϕ6=0 tips the qubit vector on another therotationalaxisofthequbitvectorwithinthexy-plane axis at an angle ϕ from the x-axis. The resulting qubit with the frequency above 11 GHz. Using the following vector does not reach the south pole (|1i) of the Bloch relation Z(φ) = X(π)Y(φ)X(−π), the azimuth angle φ 2 2 sphere. The detector SQUID switches by picking up the rotationonthez-axiscanbe decomposedintothree suc- z-component of the final qubit vector after the trigger cessive rotational operations such that −π rotation on 2 readout pulse. Repeating this sequence typically 10,000 the x-axis, φ rotation on the y-axis, and π rotation on 2 times, with a fixed t , we obtain the switching proba- thex-axis. Ifthequbitisdrivenstronglyenough,each π- 12 2 bility. Figure 3 shows the damped sinusoidal oscillation pulse width can be as shortas 0.1 ns, therefore the total obtained by changing the pulse interval t . The phase compositeoperationX(π)Y(φ)X(−π)canbecompleted 12 2 2 shift of the second pulse was programmed from the fol- in ∼1 ns. lowing relation ; ϕ=ω t mod 2π. This equation gives 0 12 Comparedwiththeconventionaldetuningmethod,the a 2π phase change to the resonant microwavepulse dur- phaseshifttechniqueprovidesuswiththeopportunityto ing a period of T = 2π. 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