1. Dressed state amplification by a superconducting qubit E. Il‘ichev, Outline Introduction: Qubit-resonator system Parametric amplification Quantum amplifier Lasing (charge qubit) Dressed state lasing (flux qubit) Conclusion

2. Superconducting Ring + one or more Josephson junctions Contact sizes + process parameters Operation: external magnetic flux near half flux quantum  Persistent current Ip  Energy of external field Flux qubit

3. Qubits energy- Eigenbasis Tunnel-coupling produces splitting ->  – energy gap Eigenstates of the qubit |g> and |e> Splitting adjustable via the external field Hamiltonian in eigenbasis Bias Energy

4. coplanar waveguide resonator 50 ohm characteristic impedance –Length determines the resonance frequency l / 2 fundamental mode 2.5 GHz High Q ~ 10 6 Thermal population at 20 mK Hamiltonian Relaxation with rate  Superconductor CcCc CcCc Resonator

5. In flux basis (g= M Ip Ir) Or in eigenbasis of Qubits Simulation: Due to tapering M is 4 times lager Coupling between resonator and qubit

6. Parametric amplification M. Rehak, Appl. Phys. Lett., 104, 162604, (2014).

7. f f pump signal pump signalidler INOUT

8. Superconducting coplanar waveguide resonator with a pair of flux qubits Resonator design with its nonlinear element – pair of flux qubits

9. Gain

10. Quantum amplifier O. Astafiev, et. al., Phys. Rev. Lett 104, 183603 (2010).

11. 11 Spontaneous emission Noise level of the 4K amplifier is 10 -22 W/Hz! 33 22 11  32  21   13  /2  = 40 MHz  31 /2  (MHz)  31 /2  = 24 MHz  31 /2  (MHz) S (10 -25 W/Hz)  /2  (MHz) Noise spectral density (weak driving limit)  31

12. 12  21 /2  (MHz)  31 /2  (MHz) |t| Arg t Amplification Stimulated emission f 13  21 /2  (MHz) |t| Arg t Amplification

13. 13 Transmission at resonance without pure dephasing Maximum transmission  32  21 33 22 11  31  32 >>  21  21 /2  = 11 MHz  32 /2  = 35 MHz Transmission amplitude Probing amplitude  21 /2  (MHz) Linear amplification regime Quantum amplifier gain

14. 14 Single qubit lasing O. Astafiev, et. al., Nature 449, 588-590 (2007)

15. 3 2 1 D    eff Lasing principle

16. 16 Josephson quasiparticle current 00 22 22 00 2e2e e+e Far away from degeneracy,  0  state is decoupled from  2  11 11 JQP cycle:  2    1    0  22 00 11 I JQP Population inversion V b > 22 e

17. 17 The three level atom in the resonator  island Josephson junctions gate probe electrode to resonator

18. 18 S (10 -21 W/Hz) amplifier noise Emission spectrum  f (MHz) N photon > 2P  p  = 30

19. 19 Laser is OFF Laser is ON Amplification 0.5 1.0 1.5

20. 20 Dress-state lasing

21. 21 2 1  eff Coupling ΦiΦi VLTLT L CTCT IbIb M

22. Tank-qubit arrangement ΦiΦi VLTLT L CTCT IbIb M M 2 =k 2 LL T  > kT > h  Phenomenological approach We found quantum-mechanical correction, but at low temperature kT<<  it is negligible: Ya. S. Greenberg and E. Il’ichev PRB 77, 094513 (2008) Ya. S. Greenberg et al., PRB 66, 214525, 2002 M. Grajcar et al., PRB 69, 060501, 2004

23. Tank cooling M. Grajcar et al., Nature Phys., 4, 612, (2008)

24. Spectroscopy with oscillator as a detector 

25. Rabi resonances

26. Atom + photon field Energy states split on Allowed transitions by dipoles matrix element fluorescence triplet C. Coen-Tannoudji, J. Dupont-Rock, and G. Grynberg, Atom-Photon Interactions. Basic Principles and Applications (JohnWiley, New York, 1998 ) Dressed systems in quantum optics

27. Population depends on detuning Use additional signal with a tunable frequency -> Gain or attenuation Dressed state laser C. Coen-Tannoudji, J. Dupont-Rock, and G. Grynberg, Atom-Photon Interactions. Basic Principles and Applications (JohnWiley, New York, 1998) F. Y. Wu, S. Ezekiel,M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38 1077, (1977) Dressed systems in quantum optics

28. Splitting of the levels in the resonance point is proportional to Hamiltonian Neglecting constant offset the energy is proportional to N For small g and large N a variable: |20> |10> g0 g1 g2e1 e0 Energy detuning Dressed levels

29. Reasonable N=10 5 and g=1MHz Therefore effective two-level system: quasi equilibrium levels |1> and |2> N+1 N N-1 |2> |1>  Detuning changes the role of relaxation  Effective inversion of population    Dressed levels

30. |2> |1>  0 Inversion population

31. System resonator – qubit Qubit:  = 3.6 GHz, Ip = 12 nA, g~0.8 MHz + Gold resistance Fundamental mode below the qubit gap: resonant interaction is absent Additional microwave field generates an effective two-level system Good qubit-resonator coupling High photon numbers in the resonator possible |21> |20> |10> Lasing Damping Lasing: experimental realization

32. Input signal 2.5 GHz Lasing: experimental realization

33. Fitting parameter  = 60 MHz and   = 20MHz Lasing: fitting

34. Emmision G.Oelsner et. al. Phys. Rev. Lett. 110, 053602 (2013)P. Neilingeret. al. Phys. Rev. B 91, 104516 (2015).

35. Conclusion Level inversion of a driven qubits is used to produce lasing at the Rabi frequency The qubit is adjusted for stable resonance conditions and rapid relaxation. Harmonics of the resonator determine the driving field for good coupling and high photon number The experimental results are described by a full quantum theory - on the base of the dressed states.