This is the published version, also available here: http://dx.doi.org/10.1063/1.3093823.Microwave resonances between discrete macroscopically distinct quantum states with single photon and multiphoton absorption are observed in a strongly driven radio frequency superconducting quantum interference device flux qubit. The amplitude of the resonant peaks and dips are modulated by the power of the applied microwave irradiation and a population inversion is generated at low flux bias. These results, which can be addressed with Landau–Zener transition, are useful to develop an alternative means to initialize and manipulate the flux qubit, as well as to do a controllable population inversion used in a micromaser

Chen, Jian

Cong, Shanhua

Han, Siyuan

Kang, Lin

Sun, Guozhu

Wang, Yiwen

Wen, Xueda

Wu, Peiheng

Xu, Weiwei

Yu, Yang

English

KU ScholarWorks

Population inversion induced by Landau–Zener transition in a stronglydriven rf superconducting quantum interference deviceGuozhu Sun,1,a Xueda Wen,2 Yiwen Wang,2 Shanhua Cong,1 Jian Chen,1 Lin Kang,1Weiwei Xu,1 Yang Yu,2,b Siyuan Han,1,3 and Peiheng Wu11Department of Electronic Science and Engineering, Research Institute of Superconductor Electronics,Nanjing University, Nanjing 210093, People’s Republic of China2Department of Physics, National Laboratory of Solid State Microstructures, Nanjing University,Nanjing 210093, People’s Republic of China3Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USAReceived 20 November 2008; accepted 7 January 2009; published online 9 March 2009Microwave resonances between discrete macroscopically distinct quantum states with single photonand multiphoton absorption are observed in a strongly driven radio frequency superconductingquantum interference device flux qubit. The amplitude of the resonant peaks and dips are modulatedby the power of the applied microwave irradiation and a population inversion is generated at lowflux bias. These results, which can be addressed with Landau–Zener transition, are useful to developan alternative means to initialize and manipulate the flux qubit, as well as to do a controllablepopulation inversion used in a micromaser. © 2009 American Institute of Physics.DOI: 10.1063/1.3093823As controllable artificial atoms, superconducting qubitshave received considerable attention because of providing anew paradigm of quantum solid state physics. So far, manyfantastic macroscopic quantum coherent phenomena1,2 havebeen demonstrated in superconducting qubits. In addition,recent experiments show that the interaction between super-conducting qubits and microwave MW resonant cavity canproduce single photon,2 which leads to a possible applicationof superconducting qubits as a micromaser. It is well knownthat population inversion, which maintains a majority of at-oms in excited states rather than in ground state, has to berealized in order to ensure the amplification of the light andthus the laser process. Previous work3,4 suggests that it ispossible to generate population inversion in the supercon-ducting quantum circuits subjected to MW radiation. In thisletter, we report a further step in this direction: a controllablepopulation inversion in a strongly driven radio frequencysuperconducting quantum interference device rf-SQUID byemploying Landau–Zener LZ transition.LZ transition is a celebrated quantum mechanical phe-nomenon in the quantum world.5–7 It has been found in vari-ous physical systems, such as atoms in accelerating opticallattices,8,9 superlattices,10 nanomagnets,11,12 quantum dots,13and Josephson junctions.14–16 Recently LZ transition is alsofound in the superconducting qubits,17–20 providing new in-sights into the fundamentals of quantum mechanics andholding promise for the superconducting qubits’ application.One may use LZ to enhance the quantum tunneling rate,21,22prepare the quantum state,23 control the qubit gateoperations13,24 effectively, and do the controllable populationinversion as demonstrated in this letter.Our design of the superconducting flux qubit, being im-mune to charge noise and comparatively easy to be read out,is based on rf-SQUID.25 An rf-SQUID consists of a super-conducting loop with inductance L interrupted by one Jo-sephson junction with capacitance C and critical current Ic.Its dynamics can be described in terms of the variable  andare identical to those of a particle of “mass” C with kineticenergy Ċ2 /2 moving in the potential U. HereU = U0 122 −  fq0 2 − L cos20 	 , 1where 0 is the flux quantum, U0=02 / 42L andL=2LIc /0. For 1L4.6, when a magnetic flux  fqclose to one half of a flux quantum is applied, the potential isa double well potential and the lowest states in each of thedouble wells serve as the qubit’s states. The states in differ-ent wells correspond to macroscopic current circulatingaround the loop with clockwise and counterclockwise direc-tions, which can be distinguished by a direct current SQUIDdc-SQUID magnetometer. Due to its large geometric size,the quantum phenomena found in the rf-SQUID are really“macroscopic.”Our samples are fabricated with Nb /AlOx /Nb trilayer onan oxidized Si wafer using a standard photolithography pro-cess. An optical micrograph and schematic of one of thedesigns is shown in Fig. 1. The qubit is essentially an rf-SQUID in second order gradiometric design. Two tunneljunctions in parallel are used instead of one in order to adjustIc with an applied flux  fCJJ, which consequently change thebarrier height of the well in situ. We use an on chip flux biasline to control the tilt of the potential with an applied flux fq. Additional flux bias line is used to produce  fdc thusbiasing the readout dc-SQUID at the maximum sensitivityregion. The mutual inductance between the qubit and dc-SQUID is well designed so that we can distinguish the dif-ferent flux generated by the circulating currents in the qubit.The sample is mounted on a chip carrier enclosed in a super-conducting aluminum sample cell. The detailed descriptionof the system is in Ref. 26. The device is thermalized atT=20 mK.The time profile of the manipulation and measurementsequence is shown in Fig. 2. After calibrating the sampleaElectronic mail: gzsun@nju.edu.cn.bElectronic mail: yuyang@nju.edu.cn.APPLIED PHYSICS LETTERS 94, 102502 20090003-6951/2009/9410/102502/3/$25.00 © 2009 American Institute of Physics94, 102502-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.237.46.100 On: Thu, 11 Sep 2014 16:08:23parameters, we keep the flux bias  fCJJ; hence the barrier ofthe well is at an optimal value. The qubit is biased at a flux fq near 0.50. A MW generates a sinusoidal electromag-netic flux in the qubit and produces transitions betweenquantum states. A readout current pulse is applied to dc-SQUID shortly after MW irradiation is shut down. By settingthe amplitude and duration of bias current pulse properly, thedc-SQUID either switches to finite voltage or stays at zerovoltage corresponding to the qubit states being in left wellL or in right well R. The switching probability which canbe obtained by repeating the trials 2104 times, therebyrepresents the population in R state. Changing  fq graduallyand repeating the above measurement, we obtain the popu-lation in R state as a function of the flux bias at fixed MWfrequency and power Fig. 2a. Then by tuning MW power,we get the dependence of the population in R state on  fqand MW power.Shown in Fig. 2a are example data for MW frequencyf =15.9 GHz and nominal MW power P=−20 dBm. With-out MW irradiation, the qubit relaxes to the ground state.There is a steplike transition near 0.50, indicating thechange in ground state from L to R at 0.50. When irra-diated by continuous MW 1 S width matching theenergy-level spacing, peaks dips appear in the curvesquares, corresponding to MW induced excitation from Lto R from R to L. The positions of all peaks and dipsagree with the energy-level diagram calculated using the pa-rameters L=1080 pH, C=80 pF, and L=1.39, which aredetermined from independent measurement Fig. 2b. Sincethe power of the MW is relatively large, transitions due totwo- and three-photon absorptions can also be observed.These multiphoton absorptions are reported before in persis-tent current qubits.17,18,27 However, the dimension of our rf-SQUID is much larger than the persistent current qubit. Inaddition, the driving frequency is large in our experiment.Therefore, large MW powers are needed to observe multi-photon transition.The population in R state versus qubit flux bias  fq andMW power at frequency f =15.9 GHz is shown in Fig. 3a.It is interesting that the population is not a monotonic func-tion of the MW power. The simple photon induced transitionpicture of a two level system predicts that the population onthe excited states will increase monotonically from 0 to thesaturate 0.5 with MW power. Therefore, our results suggestmore than two levels were involved into our rf-SQUID qubit.We use notations iR and iL to represent the ith states inFIG. 1. Color online a Optical micrograph of the sample. b Schematicof the manipulation and measurement of the qubit. c A schematic of thetime profile of the manipulation and measurement procedure.FIG. 2. Color online a Example data show the dependence of the popu-lation in R state on the qubit flux bias without MW irradiation dots andwith MW frequency f =15.9 GHz and nominal MW power P=−20 dBmsquares, respectively. b Energy-level diagram calculated using the pa-rameters in our experiments. The dashed arrows indicate the position of theresonant dips observed in the experiments as examples.FIG. 3. Color online a 3D view of the dependence of the population inR state on the bias flux and MW power. X, y, and z axes represent MWpower, the qubit flux bias  fq, and the population in R state, respectively.MW frequency is 15.9 GHz. Strong population inversion due to LZ transi-tion is clearly demonstrated. b The population in R state as a function offlux bias and MW power simulated using parameters in our experiments.102502-2 Sun et al. Appl. Phys. Lett. 94, 102502 2009 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.237.46.100 On: Thu, 11 Sep 2014 16:08:23right well and left well, respectively, and model the stronglydriven qubit by the four-level Hamiltonian in the basis ofdiabatic states 1R , 1L , 0R , 0LH0 = E1Rt 11 0 0111 E1Lt 01 00 01 E0Rt 0001 0 00 E0Lt , 2Ext=Ex0+kxrf sin t ,x 1R ,1L ,0R ,0L, where iji , j 0,1 is the interwell tunneling splitting Fig. 2b, Ex0 isthe detuning proportional to dc flux bias, rf is the amplitudeof the periodic driving signal i.e., MW,  is the MW fre-quency, and kx=dEx /df is the diabatic energy-level slope ofstate x. The parameters are shown in Fig. 2b. The quantumdynamics of the system, including the effects of various de-caying rates, is described by the Bloch equation of the timeevolution of the density operator28	̇ = − iĤ0,	 + 	 . 3The second term 	 describes the relaxation and dephasingprocesses due to the environment’s dissipation.Figure 3b shows the simulated population using Eqs.2 and 3 with parameters discussed above. Both singlephoton and multiphoton absorption due to LZ transition areclearly observed. The population is also modulated by MWpower with strong population inversion. The agreement withthe experimental results is remarkable.The Mach–Zehnder interference with a maximum mini-mum in the resonant peaks dips generated by LZ transitionare reported before.17,18 However, unlike the result in theprevious work, the interference fringes in our experiments donot exhibit a clear Bessel function dependence on MWpower. One reason of the difference is the short decoherencetime of our qubit since our qubit couples strongly with theenvironment due to the larger size. A rough estimation29,30gives intrawell relaxation rate 11 nS−1, interwell relax-ation rate between 0R and 0L inter1 S−1, anddephasing rate 20.5 nS−1, which are appropriate for ourqubit. Another reason of the difference is that we used anorder of magnitude higher MW frequency. The modulationperiod of the population is proportional to the ratio of MWamplitude and the frequency. For a larger frequency we needlarger amplitude to get the same modulation. However, withincreasing MW power, the nonlinear or other high powereffects will be dominant. This will degrade the perfect Besselfunction dependence on the MW amplitude.In addition, for a large MW frequency, dominant LZtransition occurs at the 01 crossing31 instead of the 00crossing because 0001 and the rate of LZ transitions isproportional to ij2 .18 By choosing a large MW frequency,one can easily reach the second diamond region in Ref. 31where the population inversion can be generated. In our ex-periments, strong population inversions are generated withthe maxima of 90% and 75% for n=1 and n=2, respectively,which are due to the competition between transitions to therespective excited states 1R or 1L combined with fastintrawell relaxation to 0L or 0R. Since the MW powerand frequency can be adjusted conveniently, one may realizea precise controllable population inversion in a supercon-ducting qubit.In summary, we have generated transitions betweenmacroscopic quantum states in a strongly driven supercon-ducting rf-SQUID flux qubit. The single photon and multi-photon absorption process have been observed. The popula-tion on the excited state is modulated with MW power. Inaddition, population inversion can also be realized, which isconsistent with the LZ transition theory. Therefore, usinghigh MW frequency and power, we may do the controllablepopulation inversion in superconducting flux qubit. This en-ables one to envision a micromaser based on macroscopicquantum transitions.This work is partially supported by NSFC Grant Nos.10704034 and 10534060, the State Key Program for BasicResearch of China Contract No. 2006CB921801, the Natu-ral Science Foundation of Jangsu Province Contract No.BK2007713.1Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 3572001.2J. Clarke and F. K. Wilhelm, Nature London 453, 1031 2008, andreferences therein.3S. Han, R. Rouse, and J. E. Lukens, Phys. Rev. Lett. 76, 3404 1996.4J. Q. You, Y.-x. Liu, C. P. Sun, and F. Nori, Phys. Rev. B 75, 1045162007.5L. D. Landau, Phys. Z. 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Population inversion induced by Landau--Zener transition in a strongly driven rf superconducting quantum interference device

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Population inversion induced by Landau--Zener transition in a strongly driven rf superconducting quantum interference device

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