1. S. V. Remizov, A. A. Zhukov, D. S. Shapiro, W. V. Pogosov, Yu. E. Lozovik All-Russia Research Institute of Automatics, Moscow Parametrically driven hybrid qubits- photon systems: dissipation-induced quantum entanglement and photon production from vacuum to be submitted to Phys Rev A

2. Motivation: cavity QED nonstationary effects in the context of quantum information processing Basic idea: entanglement dynamics under the parametric excitation of hybrid qubits-photon systems Theoretical model: parametrically driven Dicke model, energy dissipation, master equation, tools to characterize entanglement Results: energy dissipation in one of the subsystems is able to enhance quantum effects in another subsystem Summary Outline

3. Artificial quantum systems and especially superconducting quantum circuits (thanks to their high flexibility and tunability). Applications: First observation (?) of the dynamical Casimir effect – tuning boundary condition for the electric field via an additional SQUID С. M. Wilson, G. Johansson, A. Pourkabirian, J. R. Johansson, T. Duty, F. Nori, P. Delsing, Nature (2011). Initially suggested as photon production from the “free” space between two moving mirrors due to zero-point fluctuations of a photon field Motivation-1: cavity QED with qubits - Quantum computation (qubits) - A unique platform to study cavity QED nonstationary phenomena

4. In superconducting quantum circuits: dynamically tunable qubit-resonator coupling - In contrast to the optical system with natural atoms, it is possible to dynamically tune also this degree of freedom A lot of potential applications: decoupling qubit and resonator Motivation-2: parametric driving Single qubit and its parametric excitation: Phys. Rev. A 2015, 2016 Dynamical Lamb effect: Lozovik and coworkers 2001 Atom excitation due to the nonadiabatic modulation of Lamb shift

5. Dynamically-tunable coupling: examples of implementation Motivation-3: experimental implementation Two strongly coupled transmon-like qubits with hybridized energy levels

6. Motivation-4: experimental implementation 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)

7. Basic idea: hybrid photon-qubits system Parametric modulation of coupling between qubit and cavity subsystems. What is going to happen?

8. Dicke model beyond the rotating wave approximation (one-mode photon field). Applicable for various physical platforms. Photons (one mode field) two qubits coupling between subsystems Model-1: Dicke model beyond RWA Rotating-wave contribution (Tavis-Cummings). Conserves excitation number (number of photons + excited spins). Counter rotating-wave contribution (Anti-Tavis-Cummings). No number conservation, but parity conservation

9. In the case of a single qubit: Rotating wave approximation (RWA), conserves excitation number. Counterrotating wave term. Responsible for the dynamical Lamb effect Model-2: more on interaction Structure of bare energy levels (resonance)

10. Decoherence effects (open quantum system) Lindblad (master) equation, Markovian approximation, no memory Model-3: decoherence All these parameters as well as relations between them can be very different for different physical realizations of qubits-photon hybrid systems In superconducting circuits, dissipation in a qubit >> dissipation in a cavity In microscopic systems, dissipation in a qubit << dissipation in a cavity

11. Model-4: master equation For illustrative purposes: system of equations in the case of single qubit and only energy dissipation in a qubit. Basis consists of bare states. Resonance:

12. Quantum information tools Model-5: entanglement Entanglement Key resource for quantum computation

13. Model-6: quantum concurrence Simplest example: pure two-qubit state It can be factorized provided Quantum concurrence Мера того, насколько данное состояние далеко от факторизуемого состояния

14. Model-7: quantum concurrence

15. Model-8: entropy von Neumann (quantum) entropy It is zero for pure states and positive for mixed states (less information). It is not larger than classical (Shannon) entropy – quantum states contain larger amount of information Entanglement of formation – quantifies resources needed to create a given entangled state (eigenvalues)

16. Model-9: entropy & concurrence

17. Model-10: quantum mutual information Quantum mutual information (bipartite system) Quantifies quantum correlations between two subsystems In our case, there also photons in addition to two qubits. Therefore, we trace out photon degrees of freedom when evaluating all these quantities.

18. Model-11: back to our system Problem formulation t = 0: qubits are uncoupled from the resonator Interaction constants g start to be modulated periodically in time. Counterrotating terms are essential.

19. Model-12: signal decomposition p – rotating-wave (Tavis-Cummings) channel q – counter-rotating-wave (Anti-Tavis-Cummings) channel + full resonance. Qubit subsystem as a whole.

20. Model-13: averaging out fast oscillations Time-averaging procedure: eluminating fast and low-amplitude oscillations, while keeping slow and large-amplitude Rabi-like oscillations Hamiltonian in the interaction picture …already time independent…

21. Results-1: dynamics of decoherence-free system No decoherence for the moment. Shrodinger equation. The effect is very strong even in the limit of weak interaction!

22. Results-2: analytics vs numerics Unfortunately, even in absence of dissipation explicit solution is generally not attainable. However, there are three special cases, when explicit solution can be obtained.

23. Results-3: details on analytics Zero-order solution: First-order solution:

24. Results-4: details on analytics Quantum concurrence: It is produced by Excellent agreement with numerics

25. Results-5: details on analytics In zero order: In this order, the concurrence is identically zero

26. Results-6: details on analytics In first order

27. Results-7: details on analytics The solution in first order Quantum concurrence Again an excellent agreement with numerics

28. Results-8: details on analytics Quantum concurrence is zero. Again in a full agreement with numerics.

29. Results-9: energy dissipation in cavity The effect of cavity relaxation Pedestrian way: step by step Nothing bad !

30. Results-10: energy dissipation in qubits We increase cavity relaxation AND stabilize entanglement !

31. Results-11: steady state Dissipation is of importance: Qubit relaxation is opposite to the dynamical Lamb effect

32. Results-12: qualitative picture Very delicate balance !

33. Results-13: occupations in steady state Here we see amplification of photon generation from vacuum due to the qubit relaxation. Thus, in both cases dissipation in one of the subsystems enhances quantum effects in another subsystem !

34. Results-14: energy dissipation assisting driving Ladder of bare energy states (single qubit) - Energy dissipation brings the driven system UP - New channel of photon generation with assistance of dissipation - This dynamical regime does not exist in a dissipationless system Two processes:

35. Results-15: energy dissipation assisting driving Ladder of bare energy states (two qubits) -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

36. Results-16: potential applications Dissipative quantum computation – an alternative for error correction codes An engineering of the photonic 'reservoir' might be perspective in the context of quantum information processing. A presence of a additional bosonic subsystem with well defined discrete energy levels allows to activate various nontrivial correlations between the qubits by utilizing special types of parametric drivings. Some speculations…

37. Results-17: more on concurrence Conditional concurrence

38. Dynamics of Dicke model under parametric excitation of coupling constant between the subsystems Entanglement generation in qubit subsystem. The effect is strong! Energy dissipation in one of the subsystems of a hybrid system enhances quantum effects in another subsystem Entanglement in the steady state Cavity QED in context of quantum information processing Summary

39. 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

40. Entropy evaluation

42. Дипольное приближение