Let us focus our attention on our switch for both quantum circuits beginning with the qubit–qubit system.

Qubit–qubit system configuration

We begin by fixing the the qubit frequency of Q 2 at ω 1 = 4.9174 GHz, denoted by the horizontal solid line in Fig. 2a. We then tune the qubit frequency of Q 1 by its z bias line and probe its microwave excitation response. Figure 2a shows the spectroscopic measurement with an anticrossing due to the qubit–qubit coupling clearly observed at 4.9174 GHz. Fitting the spectrum lines (dashed blue curves in Fig. 2a) establishes the coupling strength as g/2π = 0.61 MHz.

Fig. 2 Spectrum of the qubit–qubit and the qubit–resonator systems. In a the spectrum of qubit Q 1 in the qubit–qubit system is displayed. The qubit is driven by a microwave pulse with varied frequency f, which when f matches the qubit frequency ω 1 excites the qubit to its exited state. This increases the excited state occupation probability P. The qubit frequency can then be tuned by a z bias line. With the frequency ω 2 /2π of the second qubit Q 2 fixed at 4.9174 GHz (indicated by the horizontal white dashed line) an anticrossing gap Δ g = 2g/2π = 1.22 MHz of the spectrum lines due to the coupling is clearly visible. Also show as a diagonal white line are the qubit frequencies of Q 1 with zero coupling strength. Further the blue solid curves are fit of the spectrum lines. Next under the longitudinal drive with frequency ω z /2π = 20 MHz the anticrossing gap Δ g (λ z ) variation is depicted in b with the amplitude λ z of the control pulse. Δ g decreases when increasing λ z from zero, and reaches an invisible minimum value at the switch-off point λ z = λ zoff ≈ 1.2ω z , indicated by the vertical dashed line. In c the spectrum of the flux qubit Q 1 in the qubit–resonator system is displayed as a function of ε. The qubit is driven by a microwave pulse with variable frequency f, and when f matches the qubit frequency ω qb the qubit is excited to its exited state. The frequency ω r /2π of the resonator is 2.417 GHz, indicated by the horizontal white dashed line, while the other white dashed line shows the qubit frequencies at zero coupling strength. The blue dashed curves are fit of the spectrum lines. Clearly shown is an anticrossing due to the coupling with gap Δ g = 2g/2π = 18.28 MHz. In d the anticrossing gap Δ g (λ z ) is shown in varying amplitude λ z of the control pulse with frequency ω z /2π = 150 MHz at the qubit optimal point. Δ g decreases when increasing λ z from zero, and reaches a minimum value at the switch-off point λ z = λ zoff ≈ 1.2ω z indicated by the vertical dashed line, and then reopened again to about 0.4g. The color bars in (a, b) is the occupation probability of the qubit excited state and in (c, d) is the readout signal normalized to the range of [0, 1] Full size image

To begin our investigation of switching on/off the qubit–qubit coupling, we apply a control field with frequency ω z /2π = 20 MHz to Q 1 by the z bias line. This results in a longitudinal interaction Hamiltonian H L = ħλ z cos(ω z t)σ z,1 , where λ z is determined by the amplitude of the applied rf driving current I α in the z bias line. By performing an unitary transform U = exp[−i2λ z sin(ω z t)σ z /ω z ], the total Hamiltonian H = H JC + H L of the system reduces to the effective Hamiltonian of the form48,49

$$H_{{\mathrm{eff}}} = \frac{\hbar }{2}\omega _1\sigma _{z,1} + \frac{\hbar }{2}\omega _2\sigma _{z,2} + \hbar g_{{\mathrm{eff}}}\left( {\sigma _{ + ,1}\sigma _{ - ,2} + \sigma _{ - ,1}\sigma _{ + ,2}} \right),$$ (3)

where we have neglected the small fast-oscillating terms assuming \(\omega _z \gg g\). In this case g eff = gJ 0 (2λ z /ω z ) is the effective coupling strength under longitudinal control, where J 0 (x) is the zeroth-order Bessel function of the first kind. Our effective Hamiltonian clearly shows that g eff vanishes when 2λ z /ω z is a zero point of the Bessel function J 0 (2λ z /ω z ), that is when λ z ≈ 1.2ω z . At this point the qubit–qubit coupling is switched off. Moreover, the coupling strength g eff can be continuously tuned between two values with opposite signs by changing the ratio 2λ z /ω z . We can define the switch on/off ratio R as the ratio between vacuum Rabi frequencies with and without the control field. Henceforth, we will call λ zoff the switch-off point and a longitudinal control pulse with such an amplitude a switch-off pulse. Here, we have neglected the fast-oscillating terms to give a simplified description to illustrate the idea of this scheme, in methods we have done numerical simulations with the full Hamiltonian (see Methods). In fact, experiment56 and theory49 have shown that transparency to the transverse classical field can be induced with a longitudinal control pulse. Here, we replace the transverse classical field with a quantum element (qubit or resonator), which becomes decoupled from all transverse interactions. This is the governing principle of our controlled coupling scheme.

For the qubit–qubit case, in view of the qubits small anharmonicities around 250 MHz, we included one extra level in our numerical simulations. Results are consistent with that of the analytical two-level case described by Eq. (3). The driving amplitude used for the qubit-qubit system in this work is well below the qubit anharmonicity and leakage to higher levels is well below 10−4. Note that if the amplitude approaches the anharmonicity, leakage will become nonnegligible.

Next, to test our longitudinal, control field-based switch we perform spectroscopic measurements on the qubit–resonator system under different amplitudes λ z of the longitudinal control fields. These are shown in Fig. 2b where it can be seen that the amplitude of the anticrossing gap Δ g = 2g eff decreases to zero as λ z → 1.2ω z , and then opens up again as λ z further increases. At the switch-off point λ zoff ≈ 24 MHz, the amplitude of the anticrossing Δ g becomes undetectable. This is definite evidence that the coupling can be tuned and switched off by the longitudinal control field.

Qubit–resonator system configuration

Here, we tune the qubit energy gap Δ/2π to be equal to the resonator frequency ω r /2π = 2.417 GHz by applying a long dc bias in the α bias line (see Fig. 1c). Spectroscopic measurements are then performed. Figure 2c shows the measurement results with an anticrossing due to the qubit–resonator coupling clearly observed at 2.417 GHz. Fitting the spectrum lines (dashed blue curves) allows us to determine the coupling strength as g/2π = 9.14 MHz.

Given we now know g we can now test our switching protocol. Applying a longitudinal control field (similar to the qubit–qubit case) with frequency ω z and amplitude λ z , we observed in Fig. 2d that the amplitude of the anticrossing gap Δ g = 2g eff decreases to zero as λ z → 1.2ω z . The bright spot seen at Δ/2π = 2.417 GHz and near λ z /2π ~ 180 MHz is the resonator’s resonance signature. When we drive the flux qubit, the resonator can be excited due to coupling with the qubit microwave driving control line. This excitation can be detected because there is coupling between the resonator and the qubit readout SQUID, which results in the bright spot at the resonator frequency in Fig. 2d.

Performance

Our exploration of the switch using spectroscopic measurements has qualitatively shown its operation in both the qubit–qubit and qubit–resonator configurations, yet it is difficult to quantify exactly how well it is operating from those measurements alone. However, time-domain vacuum Rabi oscillation measurements with the switch turned on and off separately will allow us to measure the time scale of the relaxation decay. Figure 3a for the qubit-qubit system and (c) for the qubit–resonator system shows the vacuum Rabi oscillations when the amplitude λ z of the longitudinal control field is increased from zero. The data clearly shows that the oscillation frequency ω c = 2g eff decreases with increasing λ z . When λ z reaches the switch-off point λ zoff , the oscillation frequency reaches a minimum. Further, Fig. 3b for the qubit–qubit system and (d) for the qubit–resonator system shows the comparison of the dynamics without a control pulse (blue) and with switch-off pulse (orange). At the switch-off point the dynamic behavior is an exponential decay—the oscillation frequency is too small to be observed on the experimental data, indicating a very small effective coupling g eff . The rate of exponential decay due to energy relation in the switched-off state is very close to the decay rate without the switching pulse (shown explicitly in Fig. 3b for the qubit–qubit system). This indicates that the switching pulse does not influence the qubits relaxation time. More quantitatively, the characteristic times for the decays in Fig. 3b are

1. Single-qubit energy relaxation rate: T = 15.58 ± 0.28 μs 2. No switching pulse with maximal coupling: T = 9.57 ± 0.35 μs 3. Switching pulse tuned to turn off the coupling: T = 15.25 ± 0.24 μs

Fig. 3 Switching on/off the coherent oscillation between the qubit–qubit and the qubit–resonator systems. Vacuum Rabi oscillations between the qubit and the resonator under different amplitudes λ z of longitudinal control pulse for the qubit–qubit a and qubit–resonator c configurations. When λ z is increased, the oscillation frequency decreases and reaches a minimum at the switch-off point λ z = λ zoff , indicated by a red dashed line. The color indicates the occupation probability P of the qubit excited state. This is consistent with the numerical simulation in Fig. 5c. In b for the qubit–qubit and d for qubit–resonator situations we display a comparison between the vacuum Rabi oscillation without (blue) and with (orange) longitudinal control for λ z = λ zoff . When the longitudinal control with the amplitude λ zoff is applied to the qubit, the vacuum Rabi oscillation between the qubit and the resonator vanishes, indicating that the qubit–resonator coupling is switched off. Dots are experimental data, solid curves are both exponential decay fit and exponential decay oscillation fit. As a comparison in the qubit–qubit situation, the qubit decay without switching pulse is also plotted (green dots). Here we observe that the qubit decays at the same rate in both cases. We have also compared \(T_2^ \ast\) with and without longitudinal driving and found no noticeable degradation of \(T_2^ \ast\) Full size image

We observed that the difference between the “No switching pulse with maximal coupling” and “Switching pulse tuned to turn off the coupling” situations is 9.57 compared with 15.25 for the “2” and “3” situations, this is because the relaxation time of Q 2 is 7.0 μs, shorter than that of Q 1 (15.58 μs). Further, it is quite interesting that the decay behavior between the “1” and “3” situations are almost the same, even if we attribute all the slight difference to a weak, residual coupling, it should be less than 2.7 kHz (see Methods). The lower limit of the on/off ratio is g/g r = 0.61 MHz/2.7 kHz ~230, where g is the coupling strength between the two qubits. Further, the on/off ratio could be significantly enhanced by improving the qubit coherent properties and the precision of the control pulse, since the effective coupling g eff = gJ 0 (2λ z /ω z ) can be tuned from positive to negative crossing the zero point. Also, as can be seen in the Methods section (Fig. 6) the numerical simulation shows an on–off ratio higher than 105, which hightlights the potential of the longitudinal control field as a high-efficiency way to turn off the coupling.

While we have shown that our longitudinal pulse enables an effective switching operation, it is important to establish its effect on the quantum coherence of the systems. To this end we will now demonstrate the dynamical switching on and off of the coupling. In this case a target qubit is initially prepared in its excited state and brought into coherent resonance with its corresponding quantum element, after a delay, a switch-off pulse is applied over a period of time. In Fig. 4a, d, the coupling is switched off when the target qubit is in the excited state. In Fig. 4b, e, the coupling is switched off when the system is in an entangled state. In Fig. 4c, f, the coupling is switched off when the target qubit is on the ground state. We found that regardless of what state the system is in, when the switch-off pulse is applied, the coherent oscillation is paused (except for free evolution and decay during the switch-off time interval), and when the switch-off pulse is removed, the coherent oscillation resumes.