† Corresponding author. E-mail:
We experimentally demonstrate the observation of macroscopic resonant tunneling (MRT) phenomenon of the macroscopic distinct flux states in a radio frequency superconducting quantum interference device (rf-SQUID) under a single-cycle sinusoidal driving. The population of the qubit exhibits interference patterns corresponding to resonant tunneling peaks between states in the adjacent potential wells. The dynamics of the qubit depends significantly on the amplitude, frequency, and initial phase of the driving signal. We do the numerical simulations considering the intra-well and inter-well relaxation mechanism, which agree well with the experimental results. This approach provides an effective way to manipulate the qubit population by adjusting the parameters of the external driving field.
Macroscopic resonant tunneling (MRT) has attracted much more research interest in recent years. It has been shown that MRT is one of the most representative macroscopic quantum behaviors based on superconducting Josephson junctions and rf-SQUID system. Such MRT has been demonstrated in current-biased or flux-biased Josephson junctions.[1–7] Also resonant tunneling between energy levels in two adjacent potential wells in a superconducting quantum interference device (SQUID) has been obtained.[8–18] It has also been shown that Landau–Zener transition can enhance the resonant tunneling rate.[19–21] MRT has been used to prove that the low-frequency flux noise in both Josephson junctions and rf-SQUID system is the crucial source of quantum noise.[22–26]
In this article, we experimentally demonstrate the observation of MRT between macroscopic distinct states in an rf-SQUID system, which is driven by an external single-cycle sinusoidal driving signal. The symmetric double-well potential would be detuned under an external driving force. Resonant tunneling between two macroscopic fluxoid wells of the rf-SQUID has been observed when the energy levels in the adjacent wells are aligned. MRT depends significantly on the amplitude, frequency, and initial phase of the external driving field, which agrees with our numerical calculation. The paper is organized as follows. In section
Our rf-SQUID sample is made of an
The probability
The Hamiltonian of the system is
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In order to explore the dynamics of MRT in the double-well of rf-SQUID flux qubit with external driving, we start from a two-level approach developed by Amin and Averin.[22] It has been proved that this two-level model can also be applied to resonant tunneling problem in other double-well systems. In the presence of low-frequency flux noise, the resonant tunneling rate from the initial well to the target well will have a Gaussian line shape, which is given by
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In our experiment, the low-frequency Gaussian-white noise has a bandwidth of about 9 MHz. The Landau–Zener transition rate between quantum states
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In order to calculate the population distribution of the flux qubit, we adopt two different decoherence effects. The first one
Figures
When the initial phase of the signal is 0, in the first half cycle of the sinusoidal waveform, the left well is lifted and the right well is lowered. The particle tunnels mostly from the left well to the right one. It is noted that when the energy levels in the two adjacent wells are aligned, resonant tunneling, MRT, will occur. In the second half cycle, the right well is lifted and the left well is lowered. The dynamics is similar as that in the first half cycle. When the amplitude of the driving field is too small, the potential well is not tilted heavily and MRT is forbidden by the potential barrier. On the other hand, when the amplitude of the driving field is too large, in the second half cycle, the particle is totally kicked to the left well. Because the particle is initialized in the left well, the total effect manifests itself as peaks as shown in Fig.
Due to the different tunneling rate when the ground state in one well is aligned to different excited states in the other well, the peaks are only visible in the corresponding range of frequency and amplitude of the driving field. The spacing of the peaks is equal to that of the anti-crossing in the spectrum (Fig.
When the initial phase of the signal is π, because the particle is initiated in the left well, so the first half cycle of the driving field has no effect on the population. In the second half cycle, the same process happens as that in the first half cycle when the initial phase is 0. The difference is when the amplitude is large, the particle will be totally kicked to the right well, thus the peaks disappear as shown in Fig.
Because the dynamics of
To summarize, using an rf-SQUID system, we have investigated the macroscopic resonant tunneling phenomenon of the macroscopic distinct flux states in a double-well potential when an external single-cycle sinusoidal driving is applied. Coherent resonant peaks due to the MRT have been observed and the dependence of MRT on the amplitude, frequency, and initial phase of the external driving signal is demonstrated. The significant dependence of the population distribution of the qubit on the parameters of the driving signal provides us with a new way to manipulate the quantum states.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] |