1. Stabilization of entanglement in qubits- photon systems: Effects of parametric driving and inhomogeneous broadening S. V. Remizov 1,2, A. A. Zhukov 1,3, D. S. Shapiro 1,2, W. V. Pogosov 1,4, and Yu. E. Lozovik 1,3,5 (1) All-Russia Research Institute of Automatics, Moscow (2) V. A. Kotel'nikov Institute of Radio Engineering and Electronics RAS, Moscow (3) National Research Nuclear University (MEPhI), Moscow (4) Institute for Theoretical and Applied Electrodynamics RAS, Moscow (5) Institute of Spectroscopy RAS, Troitsk LPHYS’17, Kazan’, Russia

2. Motivation: Search for robust entanglement. Superconducting quantum circuits – qubits-cavity systems with high tunability. Part I: Parametrically-driven circuits Basic idea: entanglement stabilization under the parametric pumping Theory: parametrically driven Dicke model, energy dissipation, master equation Results: energy dissipation in one of the subsystems is able to enhance quantum effects in another subsystem Summary Outline / Main results

3. Part II: Mesoscopic ensemble of qubits Basic idea: stability of entangled states encoded into the qubit subsystem Theory: mesoscopic regime of inhomogeneous Dicke model & Bethe ansatz Results: entanglement stabilization due to dark-states induced Zeno-like effect and quantum interference Summary Outline / Main results

4. Motivation-1: quantum entanglement Quantum entanglement is a key resource for quantum computation In 2017 Google (J. Martinis group) is planning to create 50-qubits superconducting quantum processor that achieves “quantum supremacy”. Generation of highly entangled state. Storage and manipulation of this state is beyond the capabilities of most powerful modern supercomputers.

5. Motivation-2: robust entanglement Entanglement is a very fragile characteristics Full implementation of error-correcting codes is still challenging despite of the progress of Google, IBM and other teams Search for robust entanglement. Reduced sensitivity to environment. Superconducting systems: tunability, relatively strong qubit-cavity coupling

6. Part I Parametrically-driven circuits

7. Basic idea: parametric pumping Parametric modulation of coupling between qubit and cavity subsystems. - What is going to happen in the qubit subsystem? Circuit QED architecture

8. Dynamically-tunable coupling: examples of implementation Examples of experimental implementation Two strongly coupled transmon-like qubits with hybridized energy levels

9. Three-level system under the coherent drive: both amplitude and phase tuning of g S. Gasparinetti, S. Berger, A. A. Abdumalikov, M. Pechal, S. Filipp, A. J. Wallraff, "Measurement of a Vacuum-Induced Geometric Phase", Sci. Adv. 2, e1501732 (2016). S. Berger, M. Pechal, P. Kurpiers, A.A. Abdumalikov, C. Eichler, J. A. Mlynek, A. Shnirman, Yuval Gefen, A. Wallraff, S. Filipp, "Measurement of geometric dephasing using a superconducting qubit", Nat. Comm. (2015) Examples of experimental implementation

10. Dicke model beyond the rotating wave approximation (one-mode photon field). Photons (one mode field) Qubits coupling between subsystems Theory: Quantum optics language Rotating-wave contribution (Tavis-Cummings). Conserves excitation number (number of photons + excited spins). Counterrotating contribution (Anti-Tavis-Cummings). No number conservation, but parity conservation

11. Master equation Theory: decoherence and master equation In superconducting circuits, dissipation in a qubit >> dissipation in a cavity In microscopic natural systems, dissipation in a qubit << dissipation in a cavity

12. Theory: signal decomposition p – rotating-wave (Tavis-Cummings) channel q – counter-rotating-wave (Anti-Tavis-Cummings) channel + full resonance. A. A. Zhukov, D. S. Shapiro, W. V. P., and Yu. E. Lozovik, Phys. Rev. A 93, 063845 (2016). D. S. Shapiro, A. A. Zhukov, W. V. P., and Yu. E. Lozovik, Phys. Rev. A 91, 063814 (2015).

13. Results-1: dynamics of quantum concurrence of qubits The effect is very strong even in the limit of weak interaction! Cavity decay – no bad effect

14. Results-2: steady state in presence of qubit relaxation Optimal cavity relaxation rate and optimal parametric pumping 2

15. Results-3: qualitative picture - Delicate balance between several processes. - Both rotating wave and counterrotating terms are of importance.

16. Results-4: energy dissipation assisting pumping Ladder of energy states -Some upward paths are cut (fully polarized states). -De-excitation can help to re-excite and to populate levels with odd excitation number! -This effect heavily relies on two-level nature of qubits

17. Entanglement generation in qubit subsystem due to parametric processes. The effect is strong even in the weak coupling limit ! Energy dissipation in one of the subsystems of a hybrid system enhances quantum effects in another subsystem Steady-state entanglement Alternative to error-correcting codes? Entangling gates Summary for Part I

18. Part II Inhomogeneously broadened mesoscopic ensemble of qubits

19. Basic idea: excitations within qubit subsystem -Crossover from few-qubit system to large ensembles; unavoidable splitting in excitation energies -How are collective properties formed? Total spin vs splitting. -What about excitations of qubit subsystem and their relaxation? Relevant for superconducting quantum circuits (mesoscopic and coherent ensembles are under the development) Arrays of transmon qubits, Ustinov group (2017).

20. Inhomogeneous Dicke model within rotating wave approximation This system is integrable and has similarities to Gaudin magnets (nuclear physics and superconductivity) W. V. P., D. S. Shapiro, L. V. Bork, and A. I. Onishchenko "Exact solution for the inhomogeneous Dicke model in the canonical ensemble: thermodynamical limit and finite-size corrections", Nuclear Physics B 919, 218 (2017) Even static picture is very rich (for more detail see our poster presentation) Model-1: inhomogeneous Dicke model Bethe equations:

21. Spectrum within one-excitation sector: location of Hamiltonian eigenstates along the energy axis (Full resonance) Two bright states – strong hybridization with light (standard Jaynes-Cummings model)

22. Spectrum within one-excitation sector: location of Hamiltonian eigenstates along the energy axis (Full resonance) Two bright states and set of dark states -- Dark states are collective; they are formed gradually as the number of qubits increases -- They play a very important role in the dynamics Two bright states – strong hybridization with light (standard Jaynes-Cummings model)

23. Single-qubit excitation along the crossover from few qubit system to large ensembles L =4 L = 6 L =20 Results-1: dark-states induced Zeno-like effect - Relaxation slows down as number of qubits L increases - Freezing of qubit excited state despite of the ‘bath’ of remaining qubits - Dark states affect system dynamics by suppressing qubit-qubit interactions - What about robustness of collective excitations? Dynamics of fidelity (overlap)

24. Results-2: Bell states encoded into the spin subsystem Dynamics of fidelity (overlap) - Antisymmetric two-qubit states are even more stable due to quantum interference effects (dramatic reduction of coupling to the light). - Another example of robust entanglement.

25. Testing ensembles of artificial spins Excite single qubit or affect a couple of qubits via additional waveguides Single qubit or two-qubit tomography Zeno-like effect heavily relies on collective character of spin ensemble (disorder vs total spin) Bell states stability is linked to quantum interference effects Very deep test of nature of an artificial quantum system

26. Dark-states induced Zeno-like effect in mesoscopic ensemble of qubits in circuit QED architecture Entangled Bell states encoded into spin subsystem are very stable (Zeno-like effect together with quantum interference) Tool to deeply probe nature of artificial quantum systems – artificial spins coupled to cavities Summary for Part II

27. Resonator frequency– 10 GHz g – 1-100 MHz Decoherence – 1-30 MHz or smaller in new transmons Quality factor 10^4 Resonator size - centimeter Bifurcation oscillators, Josephson ballistic interferometers, 1 picosecond

29. Дипольное приближение

31. Resonator frequency– 10 GHz g – 1-100 MHz Decoherence – 1-30 MHz or smaller in new transmons Quality factor 10^4 Resonator size - centimeter Bifurcation oscillators, Josephson ballistic interferometers, 1 picosecond

32. Entropy evaluation

34. Дипольное приближение