Dynamcs of a Superconductng Qubt Coupled to an LC Resonator Y Yang Abstract: We nvestgate the dynamcs of a current-based Josephson juncton quantum bt or qubt coupled to an LC resonator. The Hamltonan of ths superconductor-based system s developed and the energes and egen-functons of the entangled states are studed by Harmonc approxmaton. Snce such a superconductng juncton behaves as a two-level artfcal atom coupled to a harmonc oscllator, ths system can be treated as a sold-state analog of an atom n a cavty whch s the fundamental system n the welldeveloped feld of Cavty Quantum Electrodynamcs..Josephson Juncton: A superconductng qubt The structure of a Josephson Juncton s shown n Fg.. As we already know, the relaton between the tunnelng current flowng through the juncton and the voltage across the juncton s gven by Josephson Relatons[]: I I snγ 0 V dγ π dt where, γ s the phase dfference between the two superconductors; s the crtcal current of the juncton and h 5 s the flux quantum. Φ 0.07 0 T m e. A Josephson Juncton coupled to a LC resonator I 0
The system to be consdered s shown n Fg., a Josephson Juncton couplng to a LC resonator n seres. The relaton between current passng through the resonator and current through the juncton can be obtaned by Krchhoff's Law: I I + C V + CV c j j j b Vc V j L I LC Plug Josephson Relatons nto these equatons and make a substtuton γ We have π L I LC Φ 0 Φ 0 (C j + C )( π ) γ C π γ + I c sn(γ ) I b 0 C Φ 0 (γ γ ) Φ 0 γ 0 π π L The Lagrangan of the system s easly obtaned from these two equatons and thus the Hamltonan: H qubt H H LC H couplng P P PP + U (γ ) + + mωγ ξ m m mm wth conjugate momenta:
and Φ 0 Φ 0 P CJ ( ) γ + C( ) ( γ γ ) π π P C( ) ( γ γ ) π Φ ( ) π 0 m C J m Φ CC ( ) ( ) mξ 0 J π CJ + C ω C + C L C C ξ J J C+ CJ C Ib U( γ) Ic(cos( γ) + Jγ) J π I 3. Decoupled momentum and coupled potental Let consder a transformaton defned by: c ξ P γ P± P ± γ± ξ γ ± ξ Under ths transformaton, the Hamltonan becomes H ( P+ + P ) + U'( γ +, γ ) m ξ Ths potental s shown below at J J 0, where the lowest two states of the Juncton have an energy space that equals the energy space of the LC resonator.
4. Egen energy and states of the system We can start drectly from the Hamltonan H H H qubt LC couplng P P PP H + U( γ) + + mωγ ξ m m mm to calculate the egen-value and states of the system usng numercal method. Ths nvolves a large matrx that can not be solved by general algorthms. Here I wll use harmonc oscllator approxmaton to solve ths problem. Near the lowest pont of the washboard potental, we can expand the potental and drop all the constants to get a new Hamltonan: Where, H ω P + x + ω P + x ξωω P P α β α β ( γ ( )) x m ω arcsn J x m ω γ ω P m P α ω P m P β ( J ) Ths Hamltonan can be wrtten n matrx form: T ω ξωω T H P P x x + ξω ω ω Dagonalzng gves us the energy: π Ic ω CJ 4 ( ( ) 4 ) ω ± ω ω ω ω ξ ω ω + ± + When we tune the bas current of the Josephson Juncton (change the frequency of the qubt), we can observe the entangled states between the LC resonator and the qubt, denoted by an avoded crossng n the plot of Energy VS bas current.
Energy of the system as a functon of J, the normalzed bas current n the Juncton. The state notaton s /Juncton, LC>. Crcles denote the uncoupled /0> to /> level spaces for Juncton (blue) and LC resonator (black). At the resonance pont where J 0.990, an avoded crossng occurs wth a splt of ξω In experment, the system s frst cooled down to ground state /00>. For low and hgh bas, the energy level transtons are from the ground state /00> to excted states /0> or /0>, dependng on the frequency of mcrowave we apply to the system. At the resonance pont, the frst two excted states become: 0 >± 0 > At the resonant pont, the energy of the system s: ω ( ± ± ξω ) are: P ± P P ± ( α β ) and the normal modes We can construct the frst few states of the system by usng the creaton operators: a ( a ) ± ± α aβ and the defnton: j k, (!!) + 00 jk jk a a
The frst few states are: 0,0 00 0, ( 0 0 ),0 ( 0 + 0 ) 0, ( 0 + 0 ), ( 0 0 ),0 ( 0 + 0 ) These are the egen-functons of the system under Harmonc Oscllator Approxmaton wth transformed coordnates. 5. Concluson and future work We derved the Hamltonan for the system of a Josephson phase qubt coupled to a LC resonator. Harmonc approxmaton s utlzed to calculate the energy of the system and the entangled states are verfed by an avoded crossng n the energy spectrum. Frst sx states of the system are obtaned.
Although harmonc approxmaton s suffcent for us to understand the energy and entanglement of the system, t s too rough for descrbng the energy and state explctly. Plus, t becomes less vald when the well of the washboard potental goes shallow, whch s n fact the rght case for a qubt. Therefore a more powerful technque s requred n future work to solve for the energes and states of the system. Consderng ths system as a sold-state analog of an atom n a cavty[], we shall explore the possblty of observng n such a system, quantum-mechancal effects such as the AC Stark shft, whch has been observed n an atom-cavty system[3] and more recently, n a Cooper par box-resonator system[4]. Ths mght be useful n developng non-demolton types of measurements for determnng the state of a qubt. References:. B. D. Josephson, Phys. Lett., 5 (96). J. M. Ramond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 (00) 3. A. Blas, R. Huang, A. Wallraff, et al. Cavty quantum electrodynamcs for superconductng electrcal crcuts: An archtecture for Quantum Computaton, Phys, Rev. A, 69, 0630 (004) 4. A. Wallraff, D. Schuster, A. Blas, et al. Crcut Quantum Electrodynamcs: Coherent Couplng of a Sngle Photon to a Cooper Par Box, cond-mat/040735 v 3 Jul 004 5. A. J. Berkley, H, Xu, R. C, Ramos, et al. Entangled Macroscopc Quantum States n Two Superconductng Qubts, Scence, 300, 548 (003) 6. H. Xu, F. W. Strauch, S. K. Dutta, et al. Spectroscopy of Three-Partcle Entanglement n a Macroscopc Superconductng Crcut, Phys. Rev. Lett. 94, 07003 (005)