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

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

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

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

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

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

f f pump signal pump signalidler INOUT

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

Gain

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

11 Spontaneous emission Noise level of the 4K amplifier is 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 ( W/Hz)  /2  (MHz) Noise spectral density (weak driving limit)  31

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

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 Single qubit lasing O. Astafiev, et. al., Nature 449, (2007)

3 2 1 D    eff Lasing principle

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

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

19 Laser is OFF Laser is ON Amplification

20 Dress-state lasing

 eff Coupling ΦiΦi VLTLT L CTCT IbIb M

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, (2008) Ya. S. Greenberg et al., PRB 66, , 2002 M. Grajcar et al., PRB 69, , 2004

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

Spectroscopy with oscillator as a detector 

Rabi resonances

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

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 , (1977) Dressed systems in quantum optics

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

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

|2> |1>  0 Inversion population

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

Input signal 2.5 GHz Lasing: experimental realization

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

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

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.