^{1,a)}, Yang Yu

^{2,b)}and Dong Lan

^{2}

### Abstract

Interaction between quantum two-level systems (qubits) and electromagnetic fields can provide additional coupling channels to qubit states. In particular, the interwell relaxation or Rabi oscillations, resulting, respectively, from the multi- or single-mode interaction, can produce effective crossovers, leading to electromagnetically induced interference in microwave driven qubits. The environment is modeled by a multimode thermal bath, generating the interwell relaxation. Relaxation induced interference, independent of the tunnel coupling, provides deeper understanding to the interaction between the qubits and their environment. It also supplies a useful tool to characterize the relaxation strength as well as the characteristic frequency of the bath. In addition, we demonstrate the relaxation can generate population inversion in a strongly driving two-level system. On the other hand, different from Rabi oscillations, Rabi-oscillation-induced interference involves more complicated and modulated photon exchange thus offers an alternative means to manipulate the qubit, with more controllable parameters including the strength and position of the tunnel coupling. It also provides a testing ground for exploring nonlinear quantum phenomena and quantum state manipulation in qubits either with or without crossover structure.

Thanks to Xueda Wen for useful discussions. This work was supported in part by the State Key Program for Basic Researches of China (2011CB922104, 2011CBA00205), the NSFC (91021003, 11274156), the Natural Science Foundation of Jiangsu Province (BK2010012) and PAPD.

I. Introduction

II. Relaxation-induced interference

III. Competition of two interferences

A. Population dynamics

B. Phenomenological relaxation theory and discussions

IV. Rabi-oscillation-induced interference

V. Superconductingqubits

VI. Conclusion

### Key Topics

- Qubits
- 69.0
- Microwaves
- 15.0
- Electromagnetic interactions
- 10.0
- Electromagnetic interference
- 9.0
- Photons
- 8.0

## Figures

(a) Schematic energy diagram of a driven two-level system. The dotted curve represents the strong driving field A cos ωt. The field through the tunnel coupling Δ forms a LZS interference, exchanging photons with the qubit. (b) Quantum tunnel coupling exists between states and . The interaction between a qubit and an electromagnetic system (such as the environment bath or a single-mode electromagnetic field) would form new couplings between the two states.

(a) Schematic energy diagram of a driven two-level system. The dotted curve represents the strong driving field A cos ωt. The field through the tunnel coupling Δ forms a LZS interference, exchanging photons with the qubit. (b) Quantum tunnel coupling exists between states and . The interaction between a qubit and an electromagnetic system (such as the environment bath or a single-mode electromagnetic field) would form new couplings between the two states.

Schematic energy diagram of relaxation induced interference: (a) refers to the transition from state to through A; (b) refers to the transition from state to through B. The dashed red upward arrows mark the multiphoton absorption from the driving microwave field while the dashed green downward arrows mark the multiphoton release to the microwave. The dotted arrows describe the energy released or absorbed by the bath. Effectively, the resonant conditions are nω = ε + ω c (from state to ) and nω = ε – ω c (from state to ). The blue region I describes effective phase difference contributing to the interference at A. The yellow region II expresses the phase eliminated by the bath. The green region III expresses the effective phase difference contributing to the interference at B.

Schematic energy diagram of relaxation induced interference: (a) refers to the transition from state to through A; (b) refers to the transition from state to through B. The dashed red upward arrows mark the multiphoton absorption from the driving microwave field while the dashed green downward arrows mark the multiphoton release to the microwave. The dotted arrows describe the energy released or absorbed by the bath. Effectively, the resonant conditions are nω = ε + ω c (from state to ) and nω = ε – ω c (from state to ). The blue region I describes effective phase difference contributing to the interference at A. The yellow region II expresses the phase eliminated by the bath. The green region III expresses the effective phase difference contributing to the interference at B.

The stationary population of relaxation induced interference. The pattern is obtained from Eq. (15) . (a) The characteristic frequency ω c /2π = 0.05 GHz with the temperature 20 mK. Features of population inversion and periodical modulation are notable. (b) The characteristic frequency ω c /2π = 6 GHz with the temperature 20 mK. (c) The characteristic frequency ω c /2π = 0.05 GHz with the temperature 2·10–5 mK. The driving frequency ω/2π = 0.6 GHz.

The stationary population of relaxation induced interference. The pattern is obtained from Eq. (15) . (a) The characteristic frequency ω c /2π = 0.05 GHz with the temperature 20 mK. Features of population inversion and periodical modulation are notable. (b) The characteristic frequency ω c /2π = 6 GHz with the temperature 20 mK. (c) The characteristic frequency ω c /2π = 0.05 GHz with the temperature 2·10–5 mK. The driving frequency ω/2π = 0.6 GHz.

Calculated final qubit population versus energy detuning and microwave amplitude. (a) The stationary interference pattern in the weak relaxation situation. The parameters we used are the driving frequency ω/2π = 0.6 GHz, the dephasing rate Г2/2π = 0.06 GHz, the couple tunneling Δ/2π = 0.013 GHz, αϕ2 = 0.0002, the temperature is 20 mK, and the characteristic frequency ω c /2π = 0.05 GHz. The periodical patterns of RII can be seen, although not clear. (b) The stationary interference pattern in the strong relaxation situation with αϕ2 = 0.02 and ω c /2π = 0.05 GHz. Since the relaxation strength is stronger, the periodical interference patterns are more notable. (c) The stationary interference pattern in the weak relaxation situation with αϕ2 = 0.000002 and ω c /2π = 6 GHz. (d) The stationary interference pattern in the strong relaxation situation with αϕ2 = 0.0002 and ω c /2π = 6 GHz. (e) The unsaturated interference pattern in the weak relaxation situation. The system dynamics time t = 0.5 μs. The characteristic frequency ω c /2π = 0.05 GHz, αϕ2 = 0.0002. (f) The unsaturated interference pattern in the weak relaxation situation. The system dynamics time t = 0.5 μs. The characteristic frequency ω c /2π = 6 GHz, αϕ2 = 0.000002. (g) The unsaturated interference pattern in the strong relaxation situation. The system dynamics time t = 0.5 μs. The characteristic frequency ω c /2π = 0.05 GHz, α ϕ2 = 0.02. (h) The unsaturated interference pattern in the strong relaxation situation. The dynamics time t = 0.5 μs. The characteristic frequency ω c /2π = 6 GHz, αϕ2 = 0.0002. The other parameters used in these figures are the same with those in Fig. 4a .

Calculated final qubit population versus energy detuning and microwave amplitude. (a) The stationary interference pattern in the weak relaxation situation. The parameters we used are the driving frequency ω/2π = 0.6 GHz, the dephasing rate Г2/2π = 0.06 GHz, the couple tunneling Δ/2π = 0.013 GHz, αϕ2 = 0.0002, the temperature is 20 mK, and the characteristic frequency ω c /2π = 0.05 GHz. The periodical patterns of RII can be seen, although not clear. (b) The stationary interference pattern in the strong relaxation situation with αϕ2 = 0.02 and ω c /2π = 0.05 GHz. Since the relaxation strength is stronger, the periodical interference patterns are more notable. (c) The stationary interference pattern in the weak relaxation situation with αϕ2 = 0.000002 and ω c /2π = 6 GHz. (d) The stationary interference pattern in the strong relaxation situation with αϕ2 = 0.0002 and ω c /2π = 6 GHz. (e) The unsaturated interference pattern in the weak relaxation situation. The system dynamics time t = 0.5 μs. The characteristic frequency ω c /2π = 0.05 GHz, αϕ2 = 0.0002. (f) The unsaturated interference pattern in the weak relaxation situation. The system dynamics time t = 0.5 μs. The characteristic frequency ω c /2π = 6 GHz, αϕ2 = 0.000002. (g) The unsaturated interference pattern in the strong relaxation situation. The system dynamics time t = 0.5 μs. The characteristic frequency ω c /2π = 0.05 GHz, α ϕ2 = 0.02. (h) The unsaturated interference pattern in the strong relaxation situation. The dynamics time t = 0.5 μs. The characteristic frequency ω c /2π = 6 GHz, αϕ2 = 0.0002. The other parameters used in these figures are the same with those in Fig. 4a .

(a) The unsaturated interference pattern obtained by phenomenological relaxation theory. The system dynamics time is 0.5 μs, Г01/2π= 0.000008 GHz. (b) The stationary interference pattern obtained by the phenomenological relaxation theory. Г01/2π = 0.000008 GHz. (c) Comparison of the results of two theories. The blue dashed line expresses the population in with the characteristic frequency ω c /2π = 6 GHz, the microwave amplitude is fixed at 8 GHz, αϕ2 = 0.000002 and the system dynamics time t = 13 μs, the red dotted line expresses the population with the characteristic frequency ω c /2π = 0.05 GHz, the microwave amplitude is fixed at 8 GHz, αϕ2 = 0.0002, and the system dynamics time t = 16 μs. The black line uses the stationary result with the phenomenological relaxation theory with Г01/2π = 0.000008 GHz, and the microwave amplitude is fixed at 8 GHz. (d) The unsaturated interference pattern obtained by phenomenological relaxation theory. The evolution time is 0.5 μs, Г01/2π = 0.001 GHz. (e) The stationary interference pattern obtained by phenomenological relaxation theory. Г01/2π = 0.001 GHz. Other parameters are identical with those of Fig. 4a .

(a) The unsaturated interference pattern obtained by phenomenological relaxation theory. The system dynamics time is 0.5 μs, Г01/2π= 0.000008 GHz. (b) The stationary interference pattern obtained by the phenomenological relaxation theory. Г01/2π = 0.000008 GHz. (c) Comparison of the results of two theories. The blue dashed line expresses the population in with the characteristic frequency ω c /2π = 6 GHz, the microwave amplitude is fixed at 8 GHz, αϕ2 = 0.000002 and the system dynamics time t = 13 μs, the red dotted line expresses the population with the characteristic frequency ω c /2π = 0.05 GHz, the microwave amplitude is fixed at 8 GHz, αϕ2 = 0.0002, and the system dynamics time t = 16 μs. The black line uses the stationary result with the phenomenological relaxation theory with Г01/2π = 0.000008 GHz, and the microwave amplitude is fixed at 8 GHz. (d) The unsaturated interference pattern obtained by phenomenological relaxation theory. The evolution time is 0.5 μs, Г01/2π = 0.001 GHz. (e) The stationary interference pattern obtained by phenomenological relaxation theory. Г01/2π = 0.001 GHz. Other parameters are identical with those of Fig. 4a .

Schematic energy diagram of a strongly driven two-level system interacting with a weak single-mode field. The green solid curve represents the weak field, forming effective coupling between states and .

Schematic energy diagram of a strongly driven two-level system interacting with a weak single-mode field. The green solid curve represents the weak field, forming effective coupling between states and .

(a) and (b) Schematic energy diagram of Rabi-oscillation-induced interference: (a) describes the transition from state to ; (b) describes the transition from state to . (c), (d), and (e) The interference pattern of population in state obtained from Eqs. (26) , (27) , and (28) , respectively. The parameters used here are ω/2π = 2 GHz, = 0.9, Г01/2π = 0.000008 GHz and the temperature is 20 mK. Other parameters of the qubit are identical with Fig. 4a .

(a) and (b) Schematic energy diagram of Rabi-oscillation-induced interference: (a) describes the transition from state to ; (b) describes the transition from state to . (c), (d), and (e) The interference pattern of population in state obtained from Eqs. (26) , (27) , and (28) , respectively. The parameters used here are ω/2π = 2 GHz, = 0.9, Г01/2π = 0.000008 GHz and the temperature is 20 mK. Other parameters of the qubit are identical with Fig. 4a .

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