Main

We begin by describing the Hamiltonian of each component in the qubit–oscillator circuit, which comprises a superconducting flux qubit and an LC oscillator inductively coupled to each other by sharing a tunable inductance Lc, as shown in the circuit diagram in Fig. 1a.

Figure 1: Superconducting qubit–oscillator circuit.
figure 1

a, Circuit diagram. A superconducting flux qubit (red and black) and a superconducting LC oscillator (blue and black) are inductively coupled to each other by sharing a tunable inductance (black). b, Laser microscope image of the lumped-element LC oscillator inductively coupled to a coplanar transmission line. c, Scanning electron microscope image of the qubit and the coupler junctions located at the red rectangle in image b. The coupler, consisting of four parallel Josephson junctions, is tunable via the magnetic flux bias through its loops (see Supplementary Sections 1 and 2, and Supplementary Fig. 1).

The Hamiltonian of the flux qubit can be written in the basis of two states with persistent currents flowing in opposite directions around the qubit loop18, |L〉q and |R〉q, as , where Δ and ɛ = 2Ipϕ0(nφqnφq0) are the tunnel splitting and the energy bias between |L〉q and |R〉q, Ip is the maximum persistent current, and σx, z are Pauli matrices. Here, nφq is the normalized flux bias through the qubit loop in units of the superconducting flux quantum, ϕ0 = h/2e, and nφq0 = 0.5 + kq, where kq is the integer that minimizes |nφqnφq0|. The macroscopic nature of the persistent-current states enables strong coupling to other circuit elements. Another important feature of the flux qubit is its strong anharmonicity: the two lowest energy levels are well isolated from the higher levels.

The Hamiltonian of the LC oscillator can be written as , where is the resonance frequency, L0 is the inductance of the superconducting lead, Lqc(Lc) is the inductance across the qubit and coupler (see Supplementary Section 2), C is the capacitance, and is the oscillator’s annihilation (creation) operator. Figure 1b shows a laser microscope image of the lumped-element LC oscillator, where L0 is designed to be as small as possible to maximize the zero-point fluctuations in the current and hence achieve strong coupling to the flux qubit, while C is adjusted so as to achieve a desired value of ωo. The freedom of choosing L0 for large Izpf is one of the advantages of lumped-element LC oscillators over coplanar-waveguide resonators for our experiment. Another advantage is that a lumped-element LC oscillator has only one resonant mode. Together with the strong anharmonicity of the flux qubit, we can expect that our circuit will realize the Rabi model19,20,21,22, which is one of the simplest possible quantum models of qubit–oscillator systems, with no additional energy levels in the range of interest.

The coupling Hamiltonian can be written as9 , where g = MIpIzpf is the coupling energy and M(Lc) is the mutual inductance between the qubit and the LC oscillator. Importantly, a Josephson-junction circuit is used as a large inductive coupler23 (Fig. 1c), which together with the large Ip and Izpf pushes the device into the regime where g is comparable to or larger than both Δ and ωo. This regime is sometimes referred to as deep strong coupling24.

The total Hamiltonian of the circuit is then given by

Nonlinearities in the coupler circuit lead to higher-order terms in . The leading-order term can be written as and is known as the A2 term15 in atomic physics. Since this A2 term can be eliminated from by a variable transformation (see Methods), we do not explicitly keep it and instead use equation (1) for our data analysis.

Spectroscopy was performed by measuring the transmission spectrum through a coplanar transmission line that is inductively coupled to the LC oscillator (see Supplementary Section 3). For a systematic study of the g dependence, five flux bias points in three circuits were used. Circuit II is designed to have larger values of g than the other two, and circuits I and II are designed to have smaller values of Δ than circuit III. Figure 2a–d shows normalized amplitudes of the transmission spectra |S21(ωp)|/|S21(ωp)|max from circuits I and II as functions of the flux bias ɛ and probe frequency ωp (see also Supplementary Fig. 5a–d). Characteristic patterns resembling masquerade masks can be seen around ɛ = 0. At each value of ɛ, the spectroscopy data were fitted with Lorentzians to obtain the frequencies ωij of the transitions |i〉 → |j〉, where the indices i and j label the energy eigenstates according to their order in the energy-level ladder, with the index 0 denoting the ground state. Theoretical fits to ωij were obtained by diagonalizing , treating Δ, ωo and g as fitting parameters. The obtained parameters are shown in Table 1. The calculated transition frequencies ωijcal are superimposed on the measured transmission spectra. As g increases, the anticrossing gap between the qubit and the oscillator frequencies at ɛ ±ωo becomes smaller and the signal from the |1〉 → |3〉 transition gradually transforms from a W shape to a Λ shape in the range |ɛ| ωo. These features are seen in both the experimental data and the theoretical calculations, with good agreement between the data and the calculations. Note that ωo depends on the qubit state and ɛ via Lqc, which results in the broad V shape seen in the spectra (see Supplementary Section 2).

Figure 2: Transmission spectra for circuits I and II.
figure 2

Calculated transition frequencies ωijcal are superimposed on the experimental results. As summarized in Table 1, a shows data from circuit I at nφq = −0.5, b shows data from circuit I at nφq = −1.5, c shows data from circuit I at nφq = 2.5, and d shows data from circuit II at nφq = −0.5. The values of g/2π are written in the panels. The red, green, blue, and cyan lines indicate the transitions |0〉 → |1〉, |0〉 → |2〉, |1〉 → |3〉, and |2〉 → |4〉, respectively.

Table 1 Set of parameters obtained from fitting spectroscopy measurements.

To capture signals from more transitions, the transmission spectra in a wider ωp range and a smaller ɛ range were measured, as shown in Fig. 3a for circuit I at nφq = −1.5. As we approach the symmetry point ɛ = 0, the signals from the |0〉 → |2〉 and |1〉 → |3〉 transitions disappear while the signals from the |0〉 → |3〉 and |1〉 → |2〉 transitions appear near ω03cal and ω12cal. The appearance and disappearance of the signals are well explained by the transition matrix elements shown in Fig. 3b: when ɛ → 0, |T02| = |T13| → 0 (forbidden transitions), while |T03| and |T12| are maximum (allowed transitions). As can be seen from the expression for Tij, these features are directly related to the form of the energy eigenstates and can therefore serve as indicators of the symmetry properties of the energy eigenstates, similarly to how atomic forbidden transitions are related to the symmetry of atomic wavefunctions. The weakness of the signals from the |0〉 → |3〉 and |1〉 → |2〉 transitions is probably due to dephasing caused by flux fluctuations. No signals from the |0〉 → |3〉 and |1〉 → |2〉 transitions were observed in circuit I at nφq = 2.5 and in circuit II. The broad dips at ωp/2π = 6.2, 6.38 and 6.45 GHz are the result of a background frequency dependence of the transmission line’s transmission amplitude, and these features can be ignored here. The feature at 6.2 GHz also contains a narrow signal from another qubit–oscillator circuit that is coupled to the transmission line (see Supplementary Section 3).

Figure 3: Selection rules and transmission spectrum around the symmetry point.
figure 3

a, Transmission spectrum for circuit I at nφq = −1.5 plotted as a function of flux bias ɛ. The transition frequencies ωijcal are superimposed on the experimental result in a. b, Matrix elements |Tij| calculated using the parameters shown in Table 1, that is, Δ/2π = 0.430 GHz, ωo/2π = 6.306 GHz, and g/2π = 4.92 GHz. c,d, Calculated transition frequencies around ωo (c) and from the ground state (d) are plotted as functions of g at ɛ = 0. The red, green, black, magenta, and blue lines in all four panels indicate the transitions |0〉 → |1〉, |0〉 → |2〉, |0〉 → |3〉, |1〉 → |2〉, and |1〉 → |3〉, respectively. Solid (dashed) lines in c and d indicate that the corresponding matrix elements Tij are nonzero (zero). Allowed and forbidden transitions cross at g/2π ωo/4π = 3.15 GHz (ref. 25), where there is an energy-level crossing and the energy eigenstates |2〉 and |3〉 exchange their physical states. The black dotted line is at the coupling strength in circuit I at nφq = −1.5, g/2π = 4.92 GHz.

To conclude this analysis of the observed transmission spectra, the fact that the frequencies of the spectral lines and the points where they become forbidden follow, respectively, ωijcal and |Tij| lends strong support to the conclusion that accurately describes our circuits. Importantly, in circuits II and III, g is larger than both ωo and Δ, emphasizing that the circuits are in the deep strong coupling regime [] (ref. 25). The fact that at ɛ = 0 the two forbidden transitions are located between the two allowed transitions is a further sign that g > ωo/2 (see Fig. 3c). In contrast, the highest coupling strengths achieved in previous experiments12,13 give g/ωo = 0.12 and 0.1, respectively. From the spectrum in Fig. 3a, we find that ω01(ɛ = 0)/Δ = 0.13 GHz/0.43 GHz = 0.30, meaning that the Lamb shift26 is 70% of the bare qubit frequency. The same value (0.30) is obtained from theoretical calculations for g/ωo = 0.78.

Using our experimental results, we can make a statement regarding the A2 term and the superradiance no-go theorem15 in our set-up. A direct consequence of the no-go theorem is that, provided that the condition of the theorem (CA2 > g/Δ) is satisfied, the system parameters will be renormalized such that the experimentally measured parameters will satisfy the inequality (see Methods). However, in all five cases in our experiment, we find that , with the ratio on the left-hand side ranging from 2.4 to 9.6 (see Table 1). These results demonstrate that the A2 term in our set-up does not satisfy the condition of the no-go theorem and therefore does not preclude a superradiant state. In fact, we expect that CA2 1 as shown in Methods.

The energy eigenstates of the qubit–oscillator system can be understood in the following way. In the absence of coupling, the energy eigenstates are product states where the oscillator is described by a Fock state |no with n plasmons. Because of the coupling to the qubit, the state of the oscillator is displaced in one of two opposite directions depending on the persistent-current state of the qubit25: and . Here, is the displacement operator, and α is the displacement. The amount of the displacement is approximately ±g/ωo. As the energy eigenstates of an isolated qubit at ɛ = 0 are superpositions of the persistent-current states, and , the energy eigenstates of the qubit–oscillator system at ɛ = 0 are well described by Schrödinger-cat-like entangled states between persistent-current states of the qubit and displaced Fock states of the oscillator , as shown in Table 2. Note that the displaced vacuum state is the coherent state . Although the above picture works best when ωo Δ, theoretical calculations show that it also gives a rather accurate description for circuit III (with ωo/Δ = 1.44) (see Methods). The vanishing of the spectral lines corresponding to the |0〉 → |2〉 and |1〉 → |3〉 transitions at ɛ = 0 is a consequence of the symmetric form of the energy eigenstates. This symmetry is expected from the current-inversion symmetry in the Hamiltonian , and it supports the theoretical prediction that the energy eigenstates at that point are qubit–oscillator entangled states.

Table 2 The energy eigenstates of the qubit–oscillator system.

Using and the parameters shown in Table 1, we can calculate the qubit–oscillator ground-state entanglement (see Supplementary Section 5). In all cases, , and for circuit II in particular . In comparison, the ground-state entanglement for the parameters of refs 12 and 13 is 6% and 4%, respectively. It should be noted here that in all five cases in our experiment there will be a significant population in the state |1〉 in thermal equilibrium, and the thermal-equilibrium qubit–oscillator entanglement will be reduced to below 8% for circuits I and II, and 25% for circuit III (see Supplementary Table 1).

In conclusion, we have experimentally achieved deep strong coupling between a superconducting flux qubit and an LC oscillator. Our results are consistent with the theoretical prediction that the energy eigenstates are Schrödinger-cat-like entangled states between persistent-current states of the qubit and displaced Fock states of the oscillator. We have also observed a huge Lamb shift, 70% of the bare qubit frequency. The tiny Lamb shift in natural atoms, which arises from weak vacuum fluctuations, was one of the earliest phenomena to stimulate the study of quantum electrodynamics. Now we can design artificial systems with light–matter interaction so strong that instead of speaking of vacuum fluctuations we speak of a strongly correlated light–matter ground state, defining a new state of matter and opening prospects for applications in quantum technologies.

Note added in proof: After acceptance of our paper, we became aware of a related manuscript27 taking a different approach to the same theme.

Methods

Laser microscope image.

The laser microscope image in Fig. 1b was obtained by a Keyence VK-9710 Color 3D Laser Scanning Microscope. The magnification of the objective lens is 10. The application ‘VK Viewer’ was used for image acquisition.

Scanning electron microscope image.

The scanning electron microscope image in Fig. 1c was obtained by a JEOL JIB-4601F. The acceleration voltage was 10 kV, the magnification was 6,500, and the working distance was 8.7 mm.

Nonlinearity of M and the A2 term of the total Hamiltonian.

We now consider the nonlinearity of the mutual inductance M between the flux qubit and the LC oscillator. As discussed in the Supplementary Information, M is almost the same as Lc in Fig. 1a, which depends on the current flowing through the Josephson junction Ib as , where acIcIcM is the critical current of the Josephson junction. We thus assume that M can similarly be written as

The nonlinearity of M(Ib) up to second order in δIb can be written as

The coupling Hamiltonian can be written as , where is the persistent-current operator of the qubit, is the current operator of the oscillator, and the current flows through the mutual inductance. Typically, Ip Izpf. Taking into account the nonlinearity of , the coupling Hamiltonian is written as

where

and

Here, we considered terms up to second order in Izpf/Ip. We find that 1 CA2 CA3 considering the following relation, IcM(= acIc) > Ip(a3Ic) Izpf(Ic), where ac 1, 0.4 a3 0.8, Ic is several hundred nanoamperes, and Izpf is several tens of nanoamperes (see Supplementary Section 2). Since the term CA3 is very small, we ignore the third term in equation (4).

The total Hamiltonian of the circuit considering the nonlinearity of M up to first order in Izpf/Ip is given by

where the first term is the Hamiltonian of the flux qubit, the second term is the Hamiltonian of the LC oscillator, and the third term is the coupling Hamiltonian. The fourth term proportional to is known as the A2 term in atomic physics. This term can be eliminated by a variable transformation as

where

and the new field operators,

and

are used. The form of the Hamiltonian in equation (9) is exactly the same as the one where the coupling term is linear in , which is given by

Note that the transformation described by equations (12) and (13) is a Hopfield–Bogoliubov transformation28. It guarantees that . In other words, both the operators and the operators obey the harmonic oscillator commutation relations. The two sets of operators are related to each other by quadrature squeezing operations. The most natural choice among these two and all other quadrature-squeezed variants is the one that leads to the standard form of the harmonic oscillator Hamiltonian, usually expressed as . As such, the operators are the most natural oscillator operators for our circuits. The operators were defined based on an incomplete description of the circuit, considering the properties of the LC circuit and ignoring the qubit and coupler parts of the circuit. In particular, the A2 term in our circuits describes an additional contribution to the inductive energy of the oscillator that arises in the presence of the qubit and coupler circuits. Similarly, the expression given in the main text for the current zero-point fluctuations must be modified in order to correctly describe the fluctuations in the full circuit.

Condition for superradiant phase transition.

In cases where one expects a sharp transition from a normal to a superradiant state, for example, when Δ ωo or when the single qubit is replaced by a large ensemble of N qubits (and g is defined to include the ensemble enhancement factor ), the phase transition condition (without the A2 term) is:

After taking into account the renormalization of ωo and g caused by the A2 term as described above, the condition for the phase transition becomes

or in other words

If the parameters are constrained to satisfy the relation CA2 > g/Δ, the right-hand side increases whenever we increase the left-hand side, and no matter how large g becomes it will never be strong enough to satisfy the phase transition condition. This can indeed be the case with atomic qubits, and it leads to the no-go theorem in those systems15.

Fidelities of qubit–oscillator entangled states for circuit III.

The fidelity between two pure states |φ〉 and |ψ〉 is given by F(|φ〉, |ψ〉) = |〈φ|ψ〉|2. For circuit III, the fidelities between the four lowest energy eigenstates given in Table 2 |iTII〉 and the corresponding exact energy eigenstates of |iexact〉 (i = 0, 1, 2, 3) are calculated to be F(|0TII〉, |0exact〉) = 0.981, F(|1TII〉, |1exact〉) = 0.985, F(|2TII〉, |2exact〉) = 0.975, and F(|3TII〉, |3exact〉) = 0.967. All the other data sets give significantly higher fidelities. In particular, for circuit II F(|0TII〉, |0exact〉) = 0.99994.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon request.