1. Small Josephson Junctions in Resonant Cavities David G. Stroud, Ohio State Univ. Collaborators: W. A. Al-Saidi, Ivan Tornes, E. Almaas Work supported by NSF DMR01-04987 H2.005 Motivation: to develop models for controllable two-level superconducting systems for possible use in quantum computation

2. What is a Josephson Junction? A Josephson junction is a weak link between two strong superconductors A Josephson junction is a weak link between two strong superconductors Small junction: typically region between superconductors is insulating (i); junction underdamped Small junction: typically region between superconductors is insulating (i); junction underdamped I Current I I s s i

3. Hamiltonian of a small Josephson junction Let be the phase difference across a junction Let be the phase difference across a junction Let n be the difference in the number of Cooper pairs on the two superconducting islands Let n be the difference in the number of Cooper pairs on the two superconducting islands Then the Hamiltonian of the junction is Then the Hamiltonian of the junction is where U= /C is the charging energy and J is the Josephson coupling energy. (C=capacitance) where U= /C is the charging energy and J is the Josephson coupling energy. (C=capacitance)

4. Commutation relations Schrodinger equation is where Wave function which solves this equation is that of a particle of mass moving in a cosine potential

5. Voltage-biased Josephson junction If junction is voltage-biased,the kinetic energy term in the Hamiltonian becomes If junction is voltage-biased,the kinetic energy term in the Hamiltonian becomes If Josephson energy vanishes, energy eigenvalues are just With n = 0, 1, 2,…..

6. Schematic of voltage-biased junction Junction Hamiltonian is Cg=gate capacitance; Vg=gate voltage

7. Schrodinger equation for voltage- biased small junction Solve to obtain eigenvalues and wave functions as functions of Experiment: junction can be placed in macroscopic superposition of two distinct eigenstates [e. g. Bouchiat et al, Phys. Scr. T76, 165 (1998); Nakamura et al, PRL 79, 2328 (1997); Nature 398, 786 ( 1999 ); Zorin et al, cond-mat/0105211 ]

8. Single-Mode Resonant cavity E,B Hamiltonian of Resonant Cavity Mode is Here is the cavity frequency, a and a+ are photon annihilation/creation operators. Energies are (m+1/2) Cavity fields

9. Cavity-junction interaction In presence of a vector potential, phase difference in expression for Josephson energy is replaced by a gauge-invariant phase difference: Here g represents the junction-cavity coupling strength, and is related to the cavity vector potential in the junction

10. Strength of junction-cavity coupling Normalized cavity electric field = Cavity frequency Flux quantum ( = hc/2e) Note : line integral taken across junction Thus, maximum coupling occurs when E-field is concentrated in as small a volume as possible

11. Supercon I II SS Microcavity I I Josephson junction is two superconductors (S and S) separated by insulating region Schematic illustration of geometry for a voltage-biased Josephson junction in a microcavity S

12. Solution of combined junction- cavity problem Compute Hamiltonian matrix in product basis |km> of junction and cavity states (k=0,1,2,…= junction states; m=0,1,2…= cavity states) Compute Hamiltonian matrix in product basis |km> of junction and cavity states (k=0,1,2,…= junction states; m=0,1,2…= cavity states) Diagonalize to obtain eigenvalues and eigenfunctions as functions of g, offset voltage, cavity frequency, etc. Diagonalize to obtain eigenvalues and eigenfunctions as functions of g, offset voltage, cavity frequency, etc. At certain values of offset voltage, states of cavity and junction are highly entangled – i. e., cannot be written as products of cavity and junction states. At certain values of offset voltage, states of cavity and junction are highly entangled – i. e., cannot be written as products of cavity and junction states.

13. Energy level diagram for junction- cavity system Lowest eigenvalues E/U of the junction-cavity system, plotted as a function of for the following parameters: /U=0.3, J/U=0.7, and several values of g as indicated. Two arrows indicate pairs of degenerate states [Al-Saidi and Stroud, PRB65, 014512 (2002)].

14. Time-dependent behavior of Josephson-cavity system, =0.26 Left: energy in junction (solid line) and in cavity (dashed line) versus time. Right: probability that junction is in first excited state (full line) and that the cavity is in m=1 (one-photon) state (dashed line). Calculation includes ALL the quantum states of junction and cavity. Cavity oscillates between m =0 and m = 1. [Obtained by solving time-dep’t Schrodinger equation starting from |k=0,m=1>]

15. Josephson-cavity system, = 0.06 Same as previous picture, except that offset voltage is tuned so that junction oscillates between ground and first excited state, and cavity oscillates between m=0 and m=2 states.

16. Two-level approximation Calculate Rabi frequency including only the two nearly degenerate states. Calculate Rabi frequency including only the two nearly degenerate states. For = 0.26, these are |k=0;m=1> and For = 0.26, these are |k=0;m=1> and |k=1;m=0>. For = 0.06, they are |k=0,m=2> and |k=2;m=0>. For the first case, 2-state approx. gives a Rabi frequency within 1% of exact calculation. For second case, it gives a Rabi frequency within 20% of exact calc. For the first case, 2-state approx. gives a Rabi frequency within 1% of exact calculation. For second case, it gives a Rabi frequency within 20% of exact calc.

17. Full curve: Time-dependent probability that junction in first excited state, given resonator initially in a “coherent state” (coherent superposition of different number states) and junction in ground state. Dashed curve: junction treated in two-level approximation, but all photon states included. [Al-Saidi and Stroud, Phys. Rev. B65, 014512 (2002).] ”Collapse and revival” [coined by J. H. Eberley et al, PRL 44, p. 1323 (1980)]

18. Jaynes-Cummings Model Hamiltonian for a two-state system coupled to a single mode of a quantized electromagnetic field (or any single mode oscillator): Energy splitting of two lowest levels are components of spin-1/2 operators Our numerical results using full Hamiltonian for junction- cavity system very close to Jaynes-Cummings results including only two lowest junction levels. [ E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963 )]

19. Remarks about single junction coupled to single-mode cavity Quantum behavior closely resembles that of two-state system coupled to cavity mode, at certain values of offset voltage Quantum behavior closely resembles that of two-state system coupled to cavity mode, at certain values of offset voltage Junction does NOT have to be a charging- energy dominated*; charging and Josephson energy can be comparable Junction does NOT have to be a charging- energy dominated*; charging and Josephson energy can be comparable *Regime studied by Shnirman et al, PRL79, 2371 (1997) and by Buisson and Hekking, cond-mat/0008275.

20. Several Junctions in a Resonant Cavity Is it possible to couple several Josephson junctions to the SAME resonant cavity? Is it possible to couple several Josephson junctions to the SAME resonant cavity? Yes! The mathematical model is very similar to one used to treat several identical two-level atoms placed in the same resonant cavity: the Dicke model [R. H. Dicke, 1954] Yes! The mathematical model is very similar to one used to treat several identical two-level atoms placed in the same resonant cavity: the Dicke model [R. H. Dicke, 1954] The models would be identical if the junctions were truly two-level systems, but this is true only approximately The models would be identical if the junctions were truly two-level systems, but this is true only approximately

21. Model Hamiltonian for Several- Junction Problem = cavity Hamiltonian = Hamiltonian for jth junction = Hamiltonian for junction-cavity interaction

22. Dicke model The Hamiltonian for the Dicke model is The Hamiltonian for the Dicke model is Here a and are the annihilation and creation operators for the photon mode, S’s are components of the spin-1/2 operators, and = strength of coupling between the two-level system and the cavity mode. Junction-cavity system differs from the Dicke model because junctions are non-identical and non 2-level systems.

23. Rabi oscillations in a system of several junctions in a cavity This Figure plots the time-dependent “inversion” S(t) (sum of probabilities that each junction is in its excited state) for systems of 2 and 4 junctions in a resonant cavity, given that initially exactly one junction is excited. Rabi frequency is approximately proportional to square root of number of junctions. [Calculated using full Hamiltonian] (Al-Saidi and Stroud, Phys. Rev. B65, 224512 (2002).)

24. S(t) for coherent initial state, one and two junctions Time-dependent “inversion” S(t) (=sum of probabilities that each junction is in its excited state) for (a) N = 1 and (b) N = 2 junctions, given that resonator is initially in a “coherent” state (eigenstate of annihilation operator), and junctions in their ground state. Solid lines: full numerical solution; dashed lines, two-level approximation. The two junctions are assumed identical.

25. Remarks about system of several junctions in a cavity Two-level (Dicke) approximation works quite well. Two-level (Dicke) approximation works quite well. Influence of higher levels can be incorporated into an effective dipole-dipole interaction between the “qubits”: Influence of higher levels can be incorporated into an effective dipole-dipole interaction between the “qubits”: If junctions are non-identical, there are still Rabi oscillations (frequency proportional to square root of junction number) if junctions slightly different. For larger differences, each junction couples independently to the cavity. If junctions are non-identical, there are still Rabi oscillations (frequency proportional to square root of junction number) if junctions slightly different. For larger differences, each junction couples independently to the cavity. In “classical limit” (many photons), dynamical equations predict “self- induced resonant steps” in the IV characteristics (Almaas and Stroud, 2002) in agreement with expt (Barbara et al, PRL, 1999). In “classical limit” (many photons), dynamical equations predict “self- induced resonant steps” in the IV characteristics (Almaas and Stroud, 2002) in agreement with expt (Barbara et al, PRL, 1999).

26. Long Josephson “ring” with crossed static and cavity magnetic fields H Cavity magnetic field H’ Vortex in ring behaves like Cooper pair in small junction Vortex in ring is like a magnetic moment interacting with magnetic fields v I I I

27. Qubit from vortex coupled to cavity magnetic field Magnetic moment of vortex couples to applied static magnetic field +magnetic field of cavity. Magnetic moment of vortex couples to applied static magnetic field +magnetic field of cavity. When current is applied across the ring, vortex is driven around the ring. Applied static magnetic field produces cosine potential.* When current is applied across the ring, vortex is driven around the ring. Applied static magnetic field produces cosine potential.* With magnetic field of cavity mode, quantum Hamiltonian of system is formally identical (for small vortex velocities) to that of small junction With magnetic field of cavity mode, quantum Hamiltonian of system is formally identical (for small vortex velocities) to that of small junction Hence, should be able to make qubit analogous to that produced by small junction coupled to a cavity mode (but via magnetic, not electric, fields. Hence, should be able to make qubit analogous to that produced by small junction coupled to a cavity mode (but via magnetic, not electric, fields. Quantum mechanics in case of static field already seen in experiments, and calculated by Wallraff et al (Nature vol.425, p. 6954 2003).

28. Hamiltonian for vortex-cavity system Hamiltonian is (for slow vortex) Hamiltonian is (for slow vortex) Form of Hamiltonian is identical to that of small junction coupled to electric field of cavity H=H=

29. Summary Have computed quantum states of states of Josephson junctions coupled to single-mode cavities. Have computed quantum states of states of Josephson junctions coupled to single-mode cavities. Junctions may behave like two-level systems which can be strongly entangled with cavity states. Junctions may behave like two-level systems which can be strongly entangled with cavity states. Several junctions in a cavity, can behave like groups of controllable two-level systems which can be entangled via the cavity. Several junctions in a cavity, can behave like groups of controllable two-level systems which can be entangled via the cavity. Work in progress: (i) Other geometries for two level systems; (ii) coupling via magnetic fields of cavity modes; (iii) calculation of dissipation. Work in progress: (i) Other geometries for two level systems; (ii) coupling via magnetic fields of cavity modes; (iii) calculation of dissipation.