Quantum Jump Approach to Switching Process of a Josephson Junction Coupled to a Microscopic Two-Level System
Xueda Wen, Yiwen Wang, Ning Dong, Guozhu Sun, Jian Chen, Lin Kang, Weiwei Xu, Peiheng Wu, Yang Yu
aa r X i v : . [ qu a n t - ph ] J u l Quantum Jump Approach to Switching Process of a Josephson Junction Coupled to aMicroscopic Two-Level System
Xueda Wen, Yiwen Wang, Ning Dong, Guozhu Sun, JianChen, Lin Kang, Weiwei Xu, Peiheng Wu, and Yang Yu ∗ National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China Research Institute of Superconductor Electronics and Department of Electronic Science and Engineering,Nanjing University, Nanjing 210093, People’s Republic of China
With microwave irradiation, the switching current of a Josephson junction coupled to a micro-scopic two-level system jumps randomly between two discrete states. We modeled the switchingprocess of the coupled system with quantum jump approach that was generally used in quantumoptics. The parameters that affect the character of the quantum jumps between macroscopic quan-tum states are discussed. The results obtained from our theoretical analysis agree well with those ofthe experiments and provide a clear physical picture for the macroscopic quantum jumps in Joseph-son junctions coupled with two-level systems. In addition, quantum jumps may serve as a usefultool to investigate the microscopic two-level structures in solid-state systems.
PACS numbers: 74.50.+r, 85.25.Cp
I. INTRODUCTION
Recent progress on superconducting qubits basedupon Josephson Junction (JJ) unambiguously demon-strated the quantum behavior of the macroscopicvariables.
Moreover, quantum jumps, an inter-esting quantum phenomenon previously studied in quan-tum optics, was experimentally demonstrated forthe first time in a junction coupled with a microscopictwo-level system (TLS) recently. The JJ-TLS couplingsystem possesses Λ-type energy level structure and mi-crowave photons are used to generate transitions betweenquantum states. However, the state of the system is readout by detecting macroscopic quantum tunneling processrather than that by detecting photon emissions in quan-tum optics. Quantum jumps then manifests itself in theform of jumping randomly between upper branch andlower branch of the switching currents. In the languageof quantum measurement theory, the switching currentsin the upper branch or lower branch serve as a pointerfrom which the macroscopic quantum state of the JJ-TLS coupling system can be determined. In this situa-tion the ensemble description of the dynamics of junc-tions based on the master equation method failsin describing trajectories of a single quantum system.Since quantum jump approach developed in the 1980s hasmade great successes in describing fluorescence of singletrapped ions , in this paper we generalize the quantumjump approach to the switching process of JJ-TLS cou-pling system and make a systematic study of the param-eters that have effects on the process. The same methodhas also been used to investigate quantum jumps in Rabioscillations of a JJ-TLS coupling system. However, inthat work the biased current of the junction is fixed at anappropriate value while here it keeps changing during theswitching current measurement. Therefore, new mecha-nisms such as Landau-Zener transitions may involve inthe dynamics of the JJ-TLS coupling system. This article is organized as follows. In Sec. II, we de-scribe the physics of the current-biased Josephson junc-tion briefly and introduce the quantum jump approachfor simulating the switching process of a current-biasedjunction. In Sec. III we generalize the quantum jumpapproach to the switching process of JJ-TLS couplingsystem and discuss the parameters that have effects onthe process. In Sec.IV we compare our theoretical resultswith experimental data and make a conclusion in Sec.V.
II. QUANTUM JUMP APPROACH TOSWITCHING PROCESS OF ACURRENT-BIASED JOSEPHSON JUNCTION
The Hamiltonian of a current-biased Josephson junc-tion as shown in Fig.1(a) reads H JJ = 12 C ˆ Q − I Φ π cos ˆ δ − I Φ π ˆ δ, (1)where I is the critical current of the Josephson junc-tion, I is the bias current, C is the junction capacitance,Φ = h/ e is the flux quantum, ˆ Q denotes the chargeoperator and ˆ δ represents the gauge invariant phase dif-ference across the junction, which obeys the convectionalquantum commutation relation [ˆ δ, ˆ Q ] = 2 ei . The statesof the current-biased Josephson junction can be con-trolled through the bias current I ( t ) given by I ( t ) = I dc + ∆ I ( t ) = I dc + I µw cos ωt, (2)where the classical bias current is parameterized by a dccomponent I dc and an ac component with the magnitude I µw and frequency ω . For I dc < I , the effective po-tential of the system (shown in Fig.1(b)) has a series ofmetastable wells. At low temperature, the current-biasedjunction has quantized energy levels, with the two lowestenergy states labeled as | i and | i . Microwaves induce FIG. 1: (a) RCSJ equivalent circuit of a current-biasedJosephson tunnel junction. (b) Washboard potential of acurrent-biased Josephson junction showing various coherentand incoherent processes at low temperature. | i and | i areground state and the first excited state which are proposed todo quantum information process as a superconducting phasequbit. transitions between | i and | i at a frequency ω = E − E ¯ h = ω p (1 −
536 ¯ hω p ∆ U ) , (3)where ω p ( I dc ) = 2 / (2 πI / Φ C ) / (1 − I dc /I ) / isthe small oscillation frequency at the bottom of thewashboard potential and ∆ U ( I dc ) = (2 √ I Φ / π )(1 − I dc /I ) / is the barrier height. It is apparent from Eq.(3)that the energy spacing ω is a function of the bias cur-rent I dc . Therefore, if we ramp I dc from 0 to I , thebarrier ∆ U is decreasing. At certain I dc called switchingcurrent the system will tunnel out of the potential well.In addition, a microwave with frequency matching the en-ergy level spacing will generate a transition between | i and | i . As shown in the top panel of Fig.2(b), the mainpeak of switching current distribution corresponds to thetunneling from the ground state | i , and the resonantpeak corresponds to the tunneling from the first excitedstate | i . By plotting the frequency of microwave vs. theposition of the resonant peak we can obtain the energyspectrum of the junction.To simulate the switching process of current-biasedjunction, we firstly write the Hamiltonian of the junc-tion in subspace {| i , | i} H JJ = ¯ h (cid:18) m cos ωt Ω m cos ωt ω ( I dc ) (cid:19) , (4)where Ω m = I µw p / hω C is Rabi frequency. Con-sidering the dissipative effect of environment, the timeevolution of the system can be described by the non-Hermitian effective Hamiltonian H eff = H qb − i ¯ h γ + Γ ) | ih | − i ¯ h | ih | , (5)where γ is the energy relaxation rate from | i to | i ,and Γ i is the tunneling rate from state | i i (i = 0, 1) out ofthe potential (Fig. 1(b)). It is noticed that both γ and Simulation
Master Equation (b)
Simulation P r obab ili t y (a) N u m be r o f S w i t c h i ng E v en t s Switching Current ( A ) Master Equation
FIG. 2: (color online) Simulated switching currents (lowerpanel) of a junction obtained by quantum jump approach(a) without microwave and, (b) with microwave respectively.The parameters used for simulations are: I = 35 . µ A, C = 4pF, ω/ π =9.02GHz, and γ =0.6 µs − . The rampingrate is dI dc /dt = 4 . × − µ A/s. By making the histogramof the switching currents we obtained the switching currentdistribution, shown as the symbols in the top panel. The redlines are ensemble results obtained using the master equa-tions. Γ i are functions of the bias current I dc . At temperature T , the relaxation rate γ is given by γ = ω π R Q R [1 + coth( ¯ hω k B T )] × |h | ˆ δ | i| , (6)where R Q ≡ h/ e ≃ . k Ω is the natural quantum unitof resistance and R is the shunting resistance in RCSJmodel (Fig. 1(a)). The tunneling rate Γ i from the state | i i can be obtained with the WKB methodΓ i = 1 T ( E i ) exp( − S f ( E i )¯ h ) , (7)where T ( E i ) is the classical period of motion and S f ( E i )is the action across the classically forbidden region.Then the quantum jump approach for simulating theswitching process of junctions can be summarized as fol-lows:(i) At t = 0, initializing the junction in the groundstate: | Ψ( t = 0) i = | i .(ii) For I dc ( t + ∆ t ) = I dc ( t ) + ( dI/dt )∆ t , calculate thecorresponding energy spacing and various transition ratesaccording to Eq.(3-7).(iii) Determine whether the system evolves accordingto the schr¨odinger equation, or makes a ’jump’. (a) If a quantum tunneling escape happens, registerthe switching current I s = I dc ( t ), and then turn to step(v). FIG. 3: (a) (Color online) Schematic of a TLS locating insidethe Josephson tunnel barrier. Some particles can tunnel be-tween two lattice positions with different wavefunctions | L i and | R i , respectively. (b) Illustration of transition from | g i to | e i for the TLS. Assuming the system is initially in state | g i , at a certain biased current I dc , the microwave is resonantwith ω and a transition from | g i to | g i happens. Further-more, when the biased current is ramped through the avoidedenergy level crossing, the Landau-Zener transition may leadto a finite occupation probability in state | e i . (b) If a relaxation event happens, then the systemjumps to the ground state | i .(c) If no jumps happen, the system evolves under theinfluence of the non-Hermitian form. (iv) For the case (b) and (c), repeat from step (ii).(v) Repeat to obtain the switching current I s .(vi) Average switching current I s over many simulationruns.The numerical results obtained with quantum jumpapproach are shown in Fig.2. The parameters we usedin the simulation are from experiments. In addition,we calculate the switching current distribution with themaster equation method. The agreement between thequantum jump approach and the master equation indi-cates that quantum jump approach is valid to model theswitching process of junction. Furthermore, as discussedin Sec. III, quantum jump approach is more powerfulthan master equation method when stochastic charac-teristics of a single quantum system play an importantrole.
III. QUANTUM JUMP APPROACH TOSWITCHING PROCESS OF A JJ-TLSCOUPLING SYSTEM
Firstly we give a brief description of the physics of JJ-TLS coupling system. TLSs are extensively observed insuperconducting phase , charge and flux qubitsrecently. A TLS is understood to be a particle or a smallgroup of particles that tunnels between two lattice con-figurations, with different wave functions | L i and | R i cor-responding to different junction critical current I L and I R , respectively (Fig. 3(a)). The interaction Hamilto- nian between the junction and TLS can be written as: H int = − Φ I R π cos δ ⊗| R ih R |− Φ I L π cos δ ⊗| L ih L | . (8)For convenience, we transfer to the energy eigenstate ba-sis of TLS with | g i and | e i being the ground state and theexcited state, respectively. Then the total Hamiltonianof the JJ-TLS in the basis {| g i , | g i , | e i , | e i} is givenby: H = ¯ h m cos ωt m cos ωt ω ( I dc ) Ω c
00 Ω c ω T LS Ω m cos ωt m cos ωt ω ( I dc ) + ω T LS , (9)where ω T LS is the energy frequency of the TLS, and Ω c is the coupling strength between Josephson junction andTLS. In experiments, the coupling strength Ω c can becharacterized in spectroscopic measurements and usuallylies from 20MHz to 200MHz. The time evo-lution of the JJ-TLS coupling system under the dissipa-tive effect of environments can be described by the effec-tive Hamiltonian H eff = H − i ¯ h g | g ih g | − i ¯ h γ + Γ g ) | g ih g |− i ¯ h e | e ih e | − i ¯ h γ + Γ e ) | e ih e | , (10)where Γ i is the tunneling rate from state | i i . We empha-size that in the asymmetric double well model of TLS,the energy basis of TLS is approximated to the positionbasis. In this approximation, states | g i and | e i corre-spond to different critical currents. Therefore, the tun-neling rates from different states are different. With noloss of generality, suppose the state | e i corresponds to thesmaller critical current. Then the procedure for simulat-ing the switching process of the JJ-TLS coupling systemcan be summarized as follows:(i) Initializing the system in state | g i for f lag = 0, orin state | e i for f lag = 1, where f lag is a marker.(ii) For I dc ( t + ∆ t ) = I dc ( t ) + ( dI/dt )∆ t , calculatethe corresponding energy spacing ω ( I dc ) and varioustransition rates.(iii) Determine whether the system evolves accordingto the schr¨odinger equation, or makes a ’jump’.(a) If a quantum tunneling event happens, register theswitching current I s = I dc ( t ). Furthermore, if the systemtunnels from | g i or | g i , set f lag = 0; else if the systemtunnels from | e i or | e i , set f lag = 1; and then turn tostep (v).(b) If a relaxation event happens, then the systemjumps to the corresponding ground state | g i or | e i ,i.e., the system jumps from | g i to | g i , or from | e i to | e i .(c) If no jumps happen, the system evolves under theinfluence of the non-Hermitian form.(iv) For the case (b) and (c), repeat from step (ii).(v) Repeat to obtain the switching current I s . (a) S w i t c h i ng C u rr en t s ( A ) (b) Number of Switching Events
FIG. 4: Simulated trajectories of switching current of JJ-TLS coupling system under different microwave amplitude.The parameters used in the simulations are: ω/ π =9.02GHz, ω TLS / π = 8 . c / π = 200MHz, γ =0.6 µs − , dI dc /dt = 4 . × µ A/s and (a) Ω m =2MHz, (b) Ω m =10MHz.With the microwave amplitude increasing , the jumps betweenupper branch and lower branch become more frequent, result-ing in a shorter lifetime in each branch. The simulation results are shown in Fig. 4. It is ap-parent that the switching current jumps between upperbranch and lower branch randomly, which is the majorcharacteristic of macroscopic quantum jumps observedin experiments. In addition, it is found that the jumpsbecome more frequent with the microwave amplitude in-creasing (Fig.4). The underlying physics can be under-stood as follows. The jumps between upper branch andlower branch of the switching current are fulfilled throughthe coupling between state | g i and | e i . With increasingthe microwave amplitude, the system initialized in | g i has a larger transition rate to | g i in the expressionΓ = Ω m γ + γ ) , (11)where γ = ( γ + Γ g + Γ g ) / ω − ω . There-fore, it is much easier for the system to jump to state | e i ,i.e., jump from the upper branch to the lower branch, andvice versa. It is easier to understand for the extreme caseΩ m = 0. Then the system has no probability to occupystate | g i , thus no probability to transfer to | e i .Furthermore, it is noticed that the transition processfrom | g i to | e i is actually a Landau-Zener transitionas illustrated in Fig. 3(b). Disregarding all decay terms,the asymptotic probability of a Landau-Zener transition (b) S w i t c h i ng C u rr en t s ( A ) (a) Number of Switching Events
FIG. 5: Simulated trajectories of switching current of JJ-TLScoupling system under different bias current ramping rates.The parameters used in the simulations are: ω/ π =9.02GHz, γ =0.6 µs − , Ω m =10MHz and (a) dI dc /dt = 4 . × µ A/s,(b) dI dc /dt = 8 . × µ A/s. With the ramping rate in-creasing , the jumps between upper branch and lower branchbecome less frequent, resulting in a longer lifetime in eachbranch. In addition, the switching current in both branchesbecomes higher for the larger ramping rate. is given by P LZ = exp( − π ¯ h Ω c υ ) , (12)where 2¯ h Ω c is the magnitude of the energy splitting,and υ ≡ dε/dt denotes the variation rate of the en-ergy spacing for noninteracting levels. Notice that υ ≡ ( dε/dI dc )( dI dc /dt ), where dε/dI dc is determined by theintrinsic parameters of the junction and dI dc /dt is deter-mined by the ramping rate. It can be easily inferred fromEq.(12) that as the ramping rate increased, the transitionrate between | g i and | e i becomes smaller. Therefore,the jumps between upper branch and lower branch of theswitching currents become less frequent and the lifetimefor each branch is longer. To support this argument,we simulate the trajectories of the switching currents fordifferent ramping rates. As shown in Fig. 5, with theramping rate increasing , the jumps between differentbranches become less frequent, as expected from our the-oretical analysis. IV. EXPERIMENTS
We have compared the results of our theoretical anal-ysis with the experimental data. The sample used in our S w i t c h i ng C u rr en t s ( A ) Number of Switching Events (b) (a)
FIG. 6: Experimental trajectories of switching current of JJ-TLS coupling system under different microwave power with(a) -7.2dBm and (b) -5.8dBm. The ramping rate is dI dc /dt =4 . × µ A/s. As increasing the microwave power, the jumpsbecome much more frequent. experiments was a 10 µ m × µ m Nb/Al O x /Nb Joseph-son junction. The junction parameters are I ≈ µ Aand C ≈ > ∼ The junctionvoltage was amplified by a differential amplifier and theswitching current was recorded when a voltage greaterthan the threshold was first detected during every ramp.In the spectroscopy measurement of the junction, anavoided crossing caused by the coupling between junctionand TLS was observed at ω/ π = 8 . c / π = 400MHz. When a microwave fieldwith ω/ π = 9 .
02 GHz was applied, the coupling betweenjunction and TLS was turned on. In this case the macro-scopic quantum jumps between upper branch and lowerbranch of the switching current were observed. To inves-tigate the effect of microwave power as discussed in Sec.III, we fixed the microwave frequency at ω = 9 .
02 GHzand the ramping rate at dI dc /dt = 4 . × µ A/s. The
Number of Switching Events (b) S w i t c h i ng C u rr en t s ( A ) (a) FIG. 7: Experimental trajectories of switching current of JJ-TLS coupling system under different ramping frequency. Themicrowave power is -6.6dBm and the ramping rates are (a) dI dc /dt = 4 . × µ A/s and (b) dI dc /dt = 8 . × µ A/s,respectively. With the ramping rate increasing, the jumpsbetween different branches become less frequent. microwave power was adjusted from -20dBm to -3dBm.As shown in Fig. 6, with the microwave power increasing,quantum jumps between different branches become muchfrequent. Similarly, to investigate the effect of rampingrate, we adjusted the ramping rate dI dc /dt from 2 . × µ A/s to 16 . × µ A/s while keeping other parametersfixed. As expected, the increasing of ramping rate resultsin less frequent jumps between upper branch and lowerbranch of the switching current (Fig. 7). The agreementbetween our simulated results and the experimental dataconfirmed the validity of the quantum jump approach.
V. CONCLUSION
We have used the quantum jump approach to simu-late the switching process of the JJ-TLS coupling system.The mechanism that dominates the quantum jumps phe-nomenon was discussed. In addition, we investigated theparameters that have effects on the behavior of quan-tum jumps. It is found that a higher microwave poweror a smaller ramping rate can make the quantum jumpshappen more frequently, which has significance in con-trolling the state of TLS. Furthermore, our theoreticalresults agree with the experimental data, indicating thevalidity of our approach. The model and method we usedhere can be easily generalized to other solid-state systemssuch as flux and charge qubits, quantum dots, trappedirons and so on.
VI. ACKNOWLEDGMENTS
This work was partially supported by the NSFC (un-der Contracts No. 10674062 and No. 10725415), the State Key Program for Basic Research of China (underContract No. 2006CB921801), and the Doctoral Fundsof the Ministry of Education of the People’s Republic ofChina (under Contract No. 20060284022). ∗ Electronic address: [email protected] Y. Makhlin, G. Sch¨on, and A. Shnirman, Rev. Mod. Phys , 357 (2001). Y. Yu, S. Y. Han, X. Chu, S. I. Chu, and Z. Wang, Science , 889 (2002). J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina,Phys. Rev. Lett. , 117901 (2002); N. Katz, M. Ans-mann, Radoslaw C. Bialczak, Erik Lucero, R. McDermott,Matthew Neeley, Matthias Steffen, E. M. Weig, A. N. Cle-land, John M. Martinis, and A. N. Korotkov, Science ,1498 (2006); M. Hofheinz, H. Wang, M. Ansmann, R. C.Bialczak, E. Lucero, M. Neeley, A. D. OConnell, D. Sank,J. Wenner, John M. Martinis and A. N. Cleland, Nature , 546 (2009). T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura,and J. S. Tsai, Nature , 941 (2003); A. O. Niskanen,K. Harrabi, F. Yoshihara, Y. Nakamura, S. Lloyd, and J.S. Tsai, Science , 723 (2007). I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P.M. Harmans, J. E. Mooij, Nature , 159 (2004). W. D. Oliver, Y. Yu, J. C. Lee, K. K. Berggren, L. S.Levitov, and T. P. Orlando, Science , 1653 (2005); D.M. Berns, M. S. Rudner, S. O. Valenzuela, K. K. Berggren,W. D. Oliver, L. S. Levitov, and T. P. Orlando, Nature , 51 (2008). D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J.M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson,M. H. Devoret, S. M. Girvn and R. J. Schoelkopf, Nature , 515 (2007); L. S. Bishop, J. M. Chow, J. Koch, A. A.Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, andR. J. Schoelkopf, Nat. Phys. , 105 (2008); L. DiCarlo, J.M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson,D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvinand R. J. Schoelkopf, Nature , 240 (2009). M. O. Scully and M. S. Zubariry,
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