Fig. 1: Device.<\/b> a<\/b> Optical microscopy image of the measured sample. The sample holder has a coil to bias a uniform magnetic field from the back surface of the chip. Qubit 1 has a local bias line that changes the magnetic flux of the qubit loop. The spectrum is measured using a vector network analyzer (VNA) for probing and reading from the transmission line shown below the circuit. b<\/b> False-color SEM images of qubits 1 and 2. The design parameters of both qubit junctions are the same. Different colors represent different layers of aluminum deposited via double-angle shadow evaporation. c<\/b> Equivalent circuit diagram of the sample. The symbols \u03b1i, \u03b2i, ai, and bi (i \u2208 {1, 2}) label each Josephson junction, while \u03c6 denotes the phase difference across a circuit component.<\/p>\n The Hamiltonian of the entire system is37<\/a>,38<\/a>,39<\/a><\/p>\n $${\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{tot}}}}}={\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{q1}}}}}+{\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{q2}}}}}+{\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{r}}}}}+{\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{int}}}}}\\,,$$<\/p>\n \n (1)\n <\/p>\n where \\({\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{q}}}}k}\\) (k\u2009=\u20091,\u00a02), \\({\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{r}}}}}\\), and \\({\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{int}}}}}\\) represent the qubits, resonator, and atom\u2013resonator plus atom\u2013atom couplings, respectively. The Hamiltonian of the resonator is \\({\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{r}}}}}=\\hslash {\\omega }_{{{{\\rm{r}}}}}({\\hat{a}}^{{{\\dagger}} }\\hat{a}+1\/2)\\), where \\({\\omega }_{{{{\\rm{r}}}}}\\equiv 1\/\\sqrt{{L}_{{{{\\rm{r}}}}}{C}_{{{{\\rm{r}}}}}}\\) is the resonance frequency, \\(\\hat{a}\\equiv ({\\hat{\\phi }}_{{{{\\rm{r}}}}}-i{Z}_{{{{\\rm{r}}}}}{\\hat{q}}_{{{{\\rm{r}}}}})\/\\sqrt{2\\hslash {Z}_{{{{\\rm{r}}}}}}\\) is the annihilation operator, \\({Z}_{{{{\\rm{r}}}}}=\\sqrt{{L}_{{{{\\rm{r}}}}}\/{C}_{{{{\\rm{r}}}}}}\\) is the characteristic impedance of the LC resonator, and \\({\\hat{q}}_{{{{\\rm{r}}}}}\\) is the conjugate variable of \\({\\hat{\\phi }}_{{{{\\rm{r}}}}}={\\Phi }_{0}{\\hat{\\varphi }}_{{{{\\rm{r}}}}}\\). Here, \u03a60 is the flux quantum and the flux \u03c6r is defined in Fig.\u00a01<\/a>c<\/a>. The Hamiltonian of the k-th artificial atom is<\/p>\n $${\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{q}}}}k}\\equiv 4{E}_{{{{\\rm{c}}}}k}{\\hat{{{{\\bf{q}}}}}}_{k}^{{{{\\rm{T}}}}}{{{{\\bf{M}}}}}_{k}^{-1}{\\hat{{{{\\bf{q}}}}}}_{k}+{E}_{{{{\\rm{Lr}}}}}{\\hat{\\varphi }}_{\\beta k}^{2}+{\\hat{{{{\\mathcal{U}}}}}}_{{{{\\rm{J}}}}k}\\,,$$<\/p>\n \n (2)\n <\/p>\n where Eck is the charging energy of the Josephson junction ak (see Fig.\u00a01<\/a>b and 1<\/a>c), \u03c6\u03b2k represents the phase differences in each \u03b2-junction of qubits, M<\/b>k is the normalized mass matrix, \\({E}_{{{{\\rm{Lr}}}}}={\\Phi }_{0}^{2}\/(2{L}_{{{{\\rm{r}}}}})\\), and \\({\\hat{{{{\\mathcal{U}}}}}}_{{{{\\rm{J}}}}k}\\) is the qubit potential energy of Josephson junctions:<\/p>\n $$\\begin{array}{rc} \\,\\,{\\hat{{{{\\mathcal{U}}}}}}_{{{{\\rm{J}}}}k}({\\hat{\\varphi }}_{{{{\\rm{e}}}}k})=& -\\,{E}_{{{{\\rm{J}}}}k}\\,\\left[\\,{\\beta }_{k}\\cos ({\\hat{\\varphi }}_{\\beta k})\\,+\\,\\cos ({\\hat{\\varphi }}_{ak})\\,+\\,\\cos ({\\hat{\\varphi }}_{bk})\\right.\\\\ &\\left.+{\\alpha }_{k}\\cos ({\\varphi }_{{{{\\rm{e}}}}k}-{\\hat{\\varphi }}_{ak}-{\\hat{\\varphi }}_{bk}-{\\hat{\\varphi }}_{\\beta k})\\right]\\,.\\end{array}$$<\/p>\n \n (3)\n <\/p>\n Here, EJk is the current energy of the Josephson junction ak, \u03c6ik (i\u2009=\u2009\u03b1,\u00a0a,\u00a0b) are the phase differences in each junction \u03b1k,\u00a0ak, and bk, and \u03c6ek represents the external flux for the loop of each atom. The interaction Hamiltonian<\/p>\n $${\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{int}}}}}=-{E}_{{{{\\rm{Lr}}}}}({\\hat{\\varphi }}_{\\beta 1}{\\hat{\\varphi }}_{{{{\\rm{r}}}}}-{\\hat{\\varphi }}_{\\beta 2}{\\hat{\\varphi }}_{{{{\\rm{r}}}}}+{\\hat{\\varphi }}_{\\beta 1}{\\hat{\\varphi }}_{\\beta 2})$$<\/p>\n \n (4)\n <\/p>\n is obtained from the boundary condition (Kirchhoff\u2019s voltage law) of the loop forming the resonator with the elements Lr and Cr.<\/p>\n By approximating each atom as a two-level system40<\/a> on the basis of persistent currents of the superconducting loop, we obtain the total Hamiltonian in Eq. (1<\/a>) as<\/p>\n $${\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{tot}}}}}\/\\hslash \\simeq \t \\,\\,{\\omega }_{{{{\\rm{r}}}}}{\\hat{a}}^{{{\\dagger}} }\\hat{a}\\,+\\,\\frac{{\\varepsilon }_{1}}{2}{\\hat{\\sigma }}_{z1}\\,+\\,\\frac{{\\Delta }_{1}}{2}{\\hat{\\sigma }}_{x1}\\,+\\,\\frac{{\\varepsilon }_{2}}{2}{\\hat{\\sigma }}_{z2}\\,+\\,\\frac{{\\Delta }_{2}}{2}{\\hat{\\sigma }}_{x2}\\\\ \t -({g}_{1}{\\hat{\\sigma }}_{z1}-{g}_{2}{\\hat{\\sigma }}_{z2})({\\hat{a}}^{{{\\dagger}} }+\\hat{a})-\\frac{2{g}_{1}{g}_{2}}{{\\omega }_{{{{\\rm{r}}}}}}{\\hat{\\sigma }}_{z1}{\\hat{\\sigma }}_{z2}\\,,$$<\/p>\n \n (5)\n <\/p>\n where \u03b5k is the persistent current energy of each qubit, \u0394k is the qubit energy gap when \u03b5k\u2009=\u20090, while \\({\\hat{\\sigma }}_{zk}\\) and \\({\\hat{\\sigma }}_{xk}\\) are the Pauli matrices for the k-th qubit. We define \u03b5k\u2009>\u20090 when the qubit current flows anticlockwise and vice versa.<\/p>\n After a unitary transformation that diagonalizes the atomic Hamiltonians \\({\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{qk}}}}}\\), we obtain a generalized Dicke Hamiltonian41<\/a> with spin\u2013spin interaction:<\/p>\n $${\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{tot}}}}}\/\\hslash \\simeq \t \\,\\,{\\omega }_{{{{\\rm{r}}}}}{\\hat{a}}^{{{\\dagger}} }\\hat{a}+\\frac{{\\omega }_{{{{\\rm{q1}}}}}}{2}{\\hat{\\sigma }}_{z1}+\\frac{{\\omega }_{{{{\\rm{q2}}}}}}{2}{\\hat{\\sigma }}_{z2}\\\\ \t -({g}_{1}{\\hat{{{\\Lambda }}}}_{1}-{g}_{2}{\\hat{{{\\Lambda }}}}_{2})({\\hat{a}}^{{{\\dagger}} }+\\hat{a})-\\frac{2{g}_{1}{g}_{2}}{{\\omega }_{{{{\\rm{r}}}}}}{\\hat{{{\\Lambda }}}}_{1}{\\hat{{{\\Lambda }}}}_{2}\\,,$$<\/p>\n \n (6)\n <\/p>\n where \\({\\omega }_{{{{\\rm{q}}}}k}={{{\\rm{sgn}}}}({\\varepsilon }_{k}){({\\varepsilon }_{k}^{2}+{\\Delta }_{k}^{2})}^{1\/2}\\) is the qubit frequency and \\({\\hat{{{\\Lambda }}}}_{k}=(\\cos {\\theta }_{k}\\,{\\hat{\\sigma }}_{xk}+\\sin {\\theta }_{k}\\,{\\hat{\\sigma }}_{zk})\\) gives the direction of the interaction, with \\({\\theta }_{k}\\simeq -\\arctan ({\\varepsilon }_{k}\/{\\Delta }_{k})\\) (see Methods for more details). For \u03b8k\u2009=\u20090\u2009(\u03b5k\u2009=\u20090), the interaction is purely transverse. When \u03b8k\u00a0\u2260\u00a00, the interaction has a longitudinal component and the one\u2013photon\u2013exciting\u2013two\u2013atoms effect is allowed.<\/p>\n Energy spectrum<\/p>\n Figure\u00a02<\/a>a<\/a> shows the raw data of the measured spectrum as a function of the persistent current energy \u03b51 of qubit 1, which are obtained after fixing the value of \u03b52\/2\u03c0 at \u00a0\u22123.22 GHz when \u03b51\u2009=\u20090. In Fig.\u00a02<\/a>b<\/a>, the spectrum is fitted with the numerically calculated transition frequencies \u03c9ij between the i-th and j-th eigenstates of the total Hamiltonian \\({\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{tot}}}}}\\). The persistent current energy for qubit 2 and the resonator frequency are affected by the external magnetic flux applied to qubit 18<\/a>. Thus, to derive the transition frequencies \u03c9ij, we substitute \u03b52\u2009\u00a0\u2192\u00a0\u2009\u03b52\u2009+\u2009A\u03b51 and \u03c9r\u2009\u00a0\u2192\u00a0\u2009\u03c9r(1\u2009+\u2009B\u00b1\u03b51) in Eq. (5<\/a>), where A and B\u00b1 are small fitting parameters listed in Table\u00a01<\/a>. We use two different values for B\u00b1 because the spectrum is asymmetric with respect to the sign of \u03b51, i.e., B+ is used when \u03b51\u22650 and vice versa. Including A and B\u00b1, we use 11 parameters in total for the fit. These also include the bias current offset Ib0, when \u03b51\u2009=\u20090, and the persistent current coefficient \\({\\tilde{\\varepsilon }}_{0}\\) to derive \\(\\hslash {\\varepsilon }_{1}={I}_{{{{\\rm{p}}}}}{\\Phi }_{0}({\\varphi }_{{{{\\rm{e1}}}}}-0.5)=\\hslash {\\tilde{\\varepsilon }}_{0}({I}_{{{{\\rm{b}}}}}-{I}_{b0})\\), where Ip is the persistent current of qubit 1, and Ib is the bias current from the room-temperature current source. We use a photo-processing technique to obtain peak points from the spectrum37<\/a>,42<\/a>,43<\/a> and the quantum toolbox in Python for numerical calculations44<\/a>,45<\/a>.<\/p>\n Fig. 2: Transmission spectra.<\/b> Pump frequency \u03c9p from the vector network analyzer versus the persistent current energy \u03b51 of qubit 1. a<\/b> Raw data of the observed single-tone spectrum of the sample shown in Fig.\u00a01<\/a>. b<\/b> Observed single-tone spectrum with fitted curves corresponding to the state transition frequencies \u03c9ij between the i-th and j-th eigenstates of Hamiltonian (6<\/a>). The fit parameters are g1\/2\u03c0\u2009=\u20093.33, g2\/2\u03c0\u2009=\u20093.45, \u03941\/2\u03c0\u2009=\u20091.31, \u03942\/2\u03c0\u2009=\u20091.27, \u03c9r\/2\u03c0\u2009=\u20095.15, and \u03b52\/2\u03c0\u2009=\u2009\u22123.22 GHz. At around \u03c9p\/2\u03c0\u2009=\u20095.09 GHz and 5.57 GHz, parasitic modes can be seen, which originate from, for example, sample ground planes and\/or the measurement environment, which includes the sample holder and microwave components coupled to the system.<\/p>\n Table 1 List of the fitting parameters used in Figs.\u00a02<\/a> and 3<\/a><\/b><\/p>\n Flux qubits 1 and 2 are almost identical except for the loop size; consequently, they have similar fitting parameters; i.e., \u0394q1 \u2243 \u0394q2 \u2243 0.25\u2009\u03c9r. We find atom-resonator coupling rates of g1\/\u03c9r\u2009=\u20090.67 and g2\/\u03c9r\u2009=\u20090.69, indicating that the artificial atoms are ultrastrongly coupled with the resonator.<\/p>\n One photon simultaneously excites two atoms<\/p>\n We indicate with \\(\\left\\vert {\\psi }_{i}\\right\\rangle\\) the eigenstate of the system Hamiltonian \\({\\hat{{{{\\mathcal{H}}}}}}_{{{{\\rm{tot}}}}}\\) with eigenenergies \u210f\u03c9i0. The \\({\\omega }_{{{{\\rm{q}}}}i}{\\hat{\\sigma }}_{zi}\/2\\) terms in Eq. (6<\/a>) define the ground \\(\\left\\vert g\\right\\rangle\\) and excited \\(\\left\\vert e\\right\\rangle\\) atomic bare states.<\/p>\n In Fig.\u00a03<\/a>a, which is an enlarged view of the red dashed rectangle in Fig.\u00a02<\/a>b<\/a>, the black arrow indicates the anticrossing between the eigenstates \\(\\left\\vert {\\psi }_{3}\\right\\rangle\\) and \\(\\left\\vert {\\psi }_{4}\\right\\rangle\\) (see Fig.\u00a03<\/a>b<\/a>), with eigenfrequencies \u03c930 and \u03c940. In agreement with this anticrossing, Fig.\u00a03<\/a>c<\/a> shows the numerically calculated projection \\({P}_{j}^{(i)}\\equiv {| \\left\\langle {\\psi }_{i}| j\\right\\rangle | }^{2}\\) of the third and fourth eigenstates \\(\\left\\vert {\\psi }_{i}\\right\\rangle \\,(i=3,4)\\) on the bare states \\(\\left\\vert j\\right\\rangle=\\{\\left\\vert gg1\\right\\rangle,\\left\\vert ee0\\right\\rangle \\}\\) as a function of \u03b51. Here, it is possible to see that the third and fourth eigenstates are the approximate symmetric and antisymmetric superpositions of \\(\\left\\vert gg1\\right\\rangle\\) and \\(\\left\\vert ee0\\right\\rangle\\), respectively. Considering also that the sum of the dressed qubit frequencies is nearly equal to the dressed resonator frequency, the anticrossing is the signature of the one\u2013photon\u2013exciting\u2013two\u2013atoms effect (see Methods for more details). Half of the minimum split between \u03c930 and \u03c940 in the spectrum gives the effective coupling between \\(\\left\\vert gg1\\right\\rangle\\) and \\(\\left\\vert ee0\\right\\rangle\\), that is 22.8 MHz (see\u00a0Supplementary Information<\/a>).<\/p>\n Fig. 3: Anticross between \\(\\left\\vert gg1\\right\\rangle\\) and \\(\\left\\vert ee0\\right\\rangle\\).<\/b> a<\/b> Enlarged view of the central part of the spectrum in Fig.\u00a02<\/a>b with fitting curve. The fitting reproduces the spectrum well. b<\/b> Enlarged image of the anticrossing between \u03c930 and \u03c940. The white lines represent the eigenmodes of \\(\\left\\vert gg1\\right\\rangle\\) and \\(\\left\\vert ee0\\right\\rangle\\) in the non-interacting Hamiltonian (see Methods for more details). c<\/b> Projection of the third and fourth eigenstates calculated using Hamiltonian in Eq. (6<\/a>) with the fitting parameters to the bare states \\(\\left\\vert gg1\\right\\rangle\\) and \\(\\left\\vert ee0\\right\\rangle\\).<\/p>\n With respect to the theoretical prediction in ref. 28<\/a> (g\/\u03c9r \u2243 0.1\u20130.2), our system has a much larger coupling (g\/\u03c9r \u2243 0.7). This implies that the system eigenstates should have a strong dressing, and in principle we could not observe a clean \u201cone\u2013photon\u2013exciting\u2013two\u2013atoms\u201d effect. On the contrary, Fig. 3<\/a>c<\/a> shows that the dressing is low for those states, and, as shown in Fig.\u00a05<\/a>, our system can still be considered formed by two separated two-level atoms and one cavity mode with shifted eigenfrequencies. This behavior is heuristically justified by the fact that spin\u2013spin and spins\u2013light couplings are competing interactions and that the longitudinal interaction \u201cdecouples\u201d for specific values of the signs of \u03b51 and \u03b52. The signature of this \u201clongitudinal decoupling\u201d is given by the asymmetry in the spectrum with respect to the sign of \u03b51. Assuming that there are only longitudinal couplings, in the interaction part of Eq. (6<\/a>), the operator \\(({\\hat{a}}^{{{\\dagger}} }+\\hat{a})\\) should generate coherent states of light in the ultrastrong coupling regime46<\/a>. However, considering \u03b51\u2009\u03b52\u2009M\u2009=\u2009m1\u00a0\u2212\u00a0m2\u2009=\u20090, where mk\u2009=\u2009\u00b1\u20091 is the eigenstate of \\({\\hat{\\sigma }}_{zk}\\,(k=1,2)\\) (see Methods for more details). As a result, the ground \\(\\left\\vert gg\\right\\rangle\\) and excited \\(\\left\\vert ee\\right\\rangle\\) states, which have M\u00a0=\u00a00, have no coherent states associated with them. Nevertheless, the transverse interactions still affect our system, generating a small dressing that reduces the projections \\({P}_{gg1}^{(4)}\\) and \\({P}_{ee0}^{(3)}\\) to almost 0.8 at \u03b51\/2\u03c0\u2009=\u2009\u22122.4 GHz.<\/p>\n","protected":false},"excerpt":{"rendered":"Device Figure\u00a01a shows an optical microscopy image of the artificial-atom\u2013resonator circuit. 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