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Quantum circuit model

Let us consider the multimode circuit QED system in Fig.1a, namely a Josephson junction coupled to a lumped-element transmission line. As detailed in the Methods, the quantum Hamiltonian has been derived from the circuit Lagrangian depending on C and , respectively, the capacitance and inductance matrices. The system degrees of freedom are described by the vector , containing independent flux variables41, indicated in Fig.1a. To get the final form, we have performed a simultaneous diagonalization of C and via the basis change =P. Through a Legendre transform and identifying as the junction phase =kP0kk, we have obtained the classical Hamiltonian (for an analogous procedure see42). Finally, by quantizing the degrees of freedom, we get the quantum Hamiltonian in the charge gauge

$$hat{{{{{{{{mathcal{H}}}}}}}}}={hat{{{{{{{{mathcal{H}}}}}}}}}}_{J}+sumlimits_{k=1}^{{N}_{m}}hslash {omega }_{k}{hat{a}}_{k}^{{{{dagger}}} }{hat{a}}_{k}+ihat{N}sumlimits_{k=1}^{{N}_{m}}{g}_{k}({hat{a}}_{k}^{{{{dagger}}} }-{hat{a}}_{k}),$$

(1)

where ({hat{{{{{{{{mathcal{H}}}}}}}}}}_{J}=4{E}_{C}{hat{N}}^{2}-{E}_{J}cos hat{varphi }) and ({hat{a}}_{k}^{{{{dagger}}} }) is the bosonic creation operator for the kth mode and ({g}_{k}=2sqrt{hslash {omega }_{k}{E}_{C}}{P}_{0k}) its coupling to the junction. Here we used the following sum rule, due to the normalization identity PTCP=I,

$${P}_{00}^{2}+frac{1}{4{E}_{C}}sumlimits_{k=1}^{{N}_{m}}frac{{g}_{k}^{2}}{hslash {omega }_{k}}=1.$$

(2)

a Representation of a circuit consisting of a Josephson junction (red) of Josephson energy EJ and capacitive energy EC coupled to a transmission line resonator. The fluxes used to describe the circuit are indicated with black dots. b Energy bands of the uncoupled junction in terms of the quasi-charge for EJ=0.5EC. c Frequencies of the transmission line modes versus the mode number. d Junction-line couplings in the charge gauge as a function of the mode frequency for different values of the impedance Z. For these last two panels: plasma frequency p=5EC and free spectral range =0.2EC (see definition in the text).

The frequencies and couplings in (1) depend on the capacitive and inductive matrices. Illustrative values are shown in Fig.1c, d. A lumped-element circuit possesses an upper frequency cutoff, known as the plasma frequency p. Overall, the line has three key parameters: the impedance (Z=sqrt{L/C}), the plasma frequency ({omega }_{p}=2/sqrt{LC}), and the free spectral range ({{Delta }}=frac{pi }{sqrt{LC}}frac{1}{{N}_{m}}), where Nm is the number of modes. The phase transition is expected to emerge in the thermodynamic limit Nm, that is equivalent to 0.

It is insightful to consider the basis diagonalizing the last two terms of (1), namely (leftvert N,{{{{{{{bf{n}}}}}}}}rightrangle={leftvert Nrightrangle bigotimes }_{k=1}^{{N}_{m}}leftvert {alpha }_{N,k},{n}_{k}rightrangle), where (leftvert {alpha }_{N,k},{n}_{k}rightrangle={D}_{k}({alpha }_{N,k})leftvert {n}_{k}rightrangle), with ({D}_{k}(alpha )=exp (alpha {hat{a}}_{k}^{{{{dagger}}} }-{alpha }^{*}{hat{a}}_{k})) the displacement operator43 for the kth mode with ({alpha }_{N,k}=iNfrac{{g}_{k}}{hslash {omega }_{k}}), and n is a vector with the number of photons in each mode. The only nontrivial term in (1) is the junction Hamiltonian: the dressing of the junction by the line can be determined by inspecting its matrix elements in such a basis. As detailed in the Methods, the coupling to the line effectively produces a renormalized Josephson energy

$${widetilde{E}}_{J}=exp left(-frac{1}{2}sumlimits_{k=1}^{{N}_{m}}frac{{g}_{k}^{2}}{hslash {omega }_{k}^{2}}right){E}_{J},$$

(3)

independent of the number of photons in the modes. The remaining effect is a renormalized capacitive energy

$${widetilde{E}}_{C}={P}_{00}^{2}{E}_{C} , < ,{E}_{C}.$$

(4)

P00 vanishes in the thermodynamic limit and has been taken to be exactly zero in a recent study of the phase transition23. However, the way it approaches zero in the thermodynamic limit is important and cannot be neglected. Indeed, also the renormalized Josephson energy ({widetilde{E}}_{J}) tends to zero for 0.

The properties of the infinite system will hence be determined by the asymptotic behavior of their ratio. We computed such analytical expression by injecting the numerically evaluated values of gk, k and P00 for different values of the impedance Z and decreasing values of . The results are summarized in Fig.2. One can see that both ({widetilde{E}}_{C}) and ({widetilde{E}}_{J}) decrease for all values of Rq/Z while is decreased (Fig.2b, c). However, while ({widetilde{E}}_{C}) monotonically decreases while increasing Rq/Z, ({widetilde{E}}_{J}) first decreases and then increases. The ratio of the two quantities ({widetilde{E}}_{J}/{widetilde{E}}_{C}) instead displays a striking behavior (Fig.2d): for small enough it sharply decreases from 1 to very small values and then grows again. Remarkably, the curves for different values of all cross for Z=Rq, with the curves for larger sizes (smaller ) tending to smaller values for Rq/Z<1, and to larger values for Rq/Z>1. The inset Fig.2e shows that the slope of the curves at Z=Rq grows logarithmically with 1. We can conclude that in the thermodynamic limit the ratio ({widetilde{E}}_{J}/{widetilde{E}}_{C}) tends to 0 for Rq/Z<1 (apart from Rq/Z=0, for which no renormalization can occur) and to infinity for Rq/Z>1.

a In the thermodynamic limit the full system can be understood as a renormalized junction. b, c Renormalized capacitive and Josephson energies computed from (4) and (3) for different values of . d Ratio of the renormalized Josephson and charging energies. e Slope of the ratio at Z=Rq showing logarithmic growth in the thermodynamic limit. For these plots EJ=EC and p=2EC.

This points at the two expected phases: for Rq/Z<1 the charging energy dominates (so-called insulating behavior), for Rq/Z>1 the charging energy is negligible (so-called superconducting behavior). We emphasize that our theoretical analysis predicts a singular point for Z=Rq, independently of the bare ratio EJ/EC, and that the compact or extended nature of the junction phase does not play a role here.

To complete our investigation and fully characterize the emergence of the two phases we have applied exact diagonalization techniques44,45 to our Hamiltonian (1) for finite-size systems. To extend the reachable sizes, we have taken full advantage of Josephson potential periodicity and of Bloch theorem. Indeed, we can introduce the eigenstates of the junction Hamiltonian (leftvert nu,srightrangle={e}^{inu varphi }{u}_{nu }^{s}(varphi )), where is the quasi-charge and s is the band index. In our charge gauge representation, the full Hamiltonian is block diagonal and for each we need to diagonalize the following matrix:

$${hat{H}}_{nu }= sumlimits_{s}{varepsilon }_{nu }^{s}leftvert nu,srightrangle leftlangle nu,srightvert+sumlimits_{k=1}^{{N}_{m}}hslash {omega }_{k}{hat{a}}_{k}^{{{{dagger}}} }{hat{a}}_{k}\ +isumlimits_{k=1}^{{N}_{m}}{g}_{k}({hat{a}}_{k}^{{{{dagger}}} }-{hat{a}}_{k})sumlimits_{s,r}leftlangle nu,rrightvert hat{N}leftvert nu,srightrangle leftvert nu,rrightrangle leftlangle nu,srightvert,$$

(5)

where ({varepsilon }_{nu }^{s}) are the eigenenergies of the bare junction with [0.5,0.5] (Brillouin zone). Note that the spectrum is even with respect to , so we can restrict to positive values only.

Illustrative energy bands for the bare junction are reported in Fig.1b. An example of the obtained energy bands for the interacting system is shown in Fig.3 versus Z. Different colors correspond to different values of the free spectral range for the same p (i.e., to different numbers of modes), and the energies are rescaled by . For Rq/Z0, the junction and the modes are decoupled: the spectrum is given by the bare junction bands with replicas due to a finite number of bosons in the modes. For finite Rq/Z instead, the interaction manifests in energy anticrossings. To investigate a quantum phase transition, the behavior of the ground state and of the low-lying excited states is crucial. While increasing Rq/Z, the first energy band becomes narrower, corresponding to a reduced effective charging energy (see Fig.2b). However, the curvature of the first band cannot change, since the sum rule (2) implies that ({widetilde{E}}_{C}ge 0). This means that the ground state is always at =0. The first excited band instead changes its shape while varying Z. When the system is not coupled (Rq/Z=0), it is simply a one-photon replica of the first band. Instead, when Rq/Z is increased, the slope changes (see Fig.3f, g). The low-energy spectrum of the full system at low impedances is reminiscent of the bare junction spectrum, although at much smaller energies. This qualitative behavior of the first few bands occurs for different sizes, although at smaller energies for smaller (different colors in Fig.3). Interestingly, the rescaled bands overlap in the best way for Z=Rq, confirming the universality observed in Fig.2d. Away from this point, for both larger and smaller values of Z, the rescaled spectra do not overlap any longer. This is reminiscent of the finite-size scaling of continuous phase transitions.

ag Bands for different impedances Z, as indicated in the subplot titles. The different line colors correspond to different free spectral ranges (see legend). The differences of the energies and the ground state energy EG are plotted and the vertical axis is rescaled by . In these plots EJ=0.5EC and p=2EC.

In the exact diagonalization studies, we considered an extended phase and applied Blochs theorem. Now, since the full Hamiltonian for our finite-size system does not couple different quasi-charges , the results for a compact phase are independent of the phase being fundamentally extended or compact. Indeed, we found that the ground state always occurs for =0, which corresponds to a periodic wavefunction. This independence of the phase transition on the extended/compact nature of the junction phase was also discussed in22. For more general shunts (e.g., inductive) an extended phase is relevant27 and this may be needed to describe transport through the junction.

In Fig.4a we plot the ground state energy versus Rq/Z: surprisingly, it has a weak dependence on the system size, and no precursor of a critical behavior is apparent. Figure4b instead reports the energies of the first few excited states, separated from the ground state by an energy of the order of for all the values of Z. In the thermodynamic limit, the system is expected to be gapless for all values of Z: hence, the phase transition needs to emerge with a different mechanism. Indeed, around Z=Rq the first three excited levels exhibit an anticrossing. This corresponds to the observation we already made for Fig.3 that the first excited band passes from being a one-photon replica to a dressed Josephson band. For a compact phase, this occurs around Z=Rq, with the splitting of the resulting anticrossing (but not the position) depending on EJ, as displayed in Fig.5a. As in Fig.3, while decreasing all the rescaled spectra exhibit a universal behavior at Z=Rq. Hence, when approaching the thermodynamic limit, for a fixed EJ the anticrossing becomes narrower and narrower, meaning that the signature of the phase transition first emerges in the first excited states. At the same time, the spectrum is gapless in the 0 limit so that a singular behavior in the first excited state necessarily reflects on the ground state.

a Ground state energy for different free spectral ranges. b Corresponding dependence of the first few excited state energies for a compact phase (=0). c Junction charge fluctuations in the ground state for different free spectral ranges. d Energy difference between the first two bands at the edge of the Brillouin zone =0.5. In all the panels EJ=0.5EC and p=2EC.

a Dependence on EJ of the anticrossing involving the first three excited states for a compact phase (=0). Here =0.55EC and p=2EC. b Energy difference between the first two bands at the edge of the Brillouin zone (=0.5) showing the universality at Rq/Z=1 for a wide range of EJ. Solid lines =0.55EC, dashed lines =0.41.

While a compact phase is the only relevant quantity for the ground state of the system, the response of the system to external perturbations can depend on the full bands. To better highlight the behavior of the low-lying bands, in Fig.4d we show the behavior of the gap between the first and second bands at the edge of the Brillouin zone. The ratio of the gap and the free spectral range displays a behavior analogous to the renormalized parameters of Fig.2, with the gap closing for larger systems for Rq/Z<1. This means that the low energy spectrum is approaching a capacitive free particle" behavior. The opposite is true for Rq/Z>1. As shown in Fig.5b, this behavior extends to larger values of EJ.

Importantly, however, the ground state does not behave in the same way, as can be seen for example by computing observables for the junction degrees of freedom. As an example, we show in Fig.4c the charge fluctuations ({sigma }_{N}^{2}={{{{{{{rm{tr}}}}}}}}({hat{rho }}_{J}{hat N}^{2})-{{{{{{{rm{tr}}}}}}}}{({hat{rho }}_{J}hat N)}^{2}), with ({hat{rho }}_{J}) the reduced density matrix for the junction (analogous results are obtained, for example, for ({{{{{{{rm{tr}}}}}}}}({hat{rho }}_{J}cos hat{varphi }))). For Rq/Z>1 the charge fluctuations have a strong dependence on the system size. For Rq/Z<1 instead, the size dependence is much smaller, and the fluctuations never fall below the Cooper pair box value that is obtained for Rq/Z0 (horizontal dashed line). This, however, does not mean that the transition does not affect the ground state, as signaled by the different size dependence of the charge fluctuations on the two sides of Rq/Z=1. The emergence of the singularity in the ground state in the thermodynamic limit is, however, slow, and we can only see a precursor.

We believe that this peculiar difference of behavior between the ground state and the excited states is at the origin of much of the recent controversy over this phase transition, specifically over the characterization of the so-called insulating state. In particular, in the thermodynamic limit, the ground state is not approaching a purely capacitive behavior. However, the response of the system to a gate charge (i.e., to a change in the quasi-charge) changes sharply at the transition point.

To understand the fate of the higher frequency modes at the transition point, we focus on the compact-phase Hamiltonian, given by equation (5) for =0. For EJ=0, the dressed excited bands (charge states) energies are ({varepsilon }_{0}^{n}({E}_{J}=0)=4{widetilde{E}}_{C}{n}^{2}), with n an integer. In this limit, the effect of the transition on the photons can be understood by simply plotting the Z-dependent mode energies and the same energies shifted by the first dressed charge state ({varepsilon }_{0}^{1}({E}_{J}=0)). We show these two sets of energy branches in Fig.6. The two lowest solid and dashed curves are respectively the first photon energy 1 and ({varepsilon }_{0}^{1}({E}_{J}=0)). Note that they cross at Z=Rq. With a finite EJ the crossing is replaced by the anticrossing highlighted in Fig.5a. Above these levels, for energies small enough with respect to p, also the kth photonic mode energy k crosses the energy ({varepsilon }_{0}^{1}({E}_{J}=0)+hslash {omega }_{k-1}) at Z=Rq. This degeneracy is again lifted for EJ0. Hence, around the transition point, all the single photon states will hybridize. Note that the crossing of the first two curves,which is essentially determined by the universality highlighted in Fig.2, is robust with respect to the intrinsic ultraviolet cutoff given by the Josephson plasma frequency p. For the excited states, there is eventually an important shift of the crossing point at high energies, that would be absent in the limit p (a non-dispersive line). The same shift is also present for much smaller free spectral ranges, of the order of the ones in the experiment39, as shown by the red line in Fig.6. For modes above the third, also states involving higher dressed bands play a significant role. Moreover, states with more than one photon are also present. For energies small enough with respect to p (equidistant modes) these are also degenerate with the single-photon energies at the transition point, increasing the number of levels that participate the hybridization.

Single photon energies (solid lines) and the same energies shifted by the energy of the first excited renormalized junction band for =0 (dashed lines) with EJ0, =0.5EC and p=5EC. The nth mode energy and the nth level of the shifted branch have the same color. Their intersection point is highlighted by a black dot. The solid red line shows the intersections for a much smaller free spectral range =0.02EC.

Recent experiments have observed a striking signature of the transition from the dispersion of high-frequency modes39. With the results of our exact diagonalization, we can calculate the linear-response photonic spectral function

$$D(E)=sumlimits_{n}frac{{gamma }^{2}}{{gamma }^{2}+{(E-{E}_{n}+{E}_{G})}^{2}}| leftlangle Grightvert sumlimits_{k}({a}_{k}+{a}_{k}^{{{{dagger}}} })leftvert {E}_{n}rightrangle {| }^{2},$$

(6)

where En ((leftvert {E}_{n}rightrangle)) is the nth excited eigenenergy (eigenstate) of the full system, EG ((leftvert Grightrangle)) the ground energy (state) and is a phenomenological broadening. Here, as in the experiment, we have considered a SQUID that is equivalent to a junction with EJ that can be tuned by a magnetic field. Illustrative spectra versus the external magnetic flux are shown in Fig.7. We observe a clear change of sign of the photon frequency dispersion across the transition, as observed in39. This effect was linked to a capacitive/inductive behavior of the junction and here we see that it microscopically originates from the crossings observed in Fig.6. The broadening observed in39 for high energy modes is instead not present because of the moderate size of the system simulated here with exact diagonalization techniques. In fact, it relies on multiple resonances between single and multi-photon states37. In Fig.7, one can see the state belonging to the shifted set of levels acquiring a finite single-photon component near the crossing point for /00.5, that is for ({E}_{J} neq 0). For larger and smaller impedances, this state is dark, i.e., it does not contribute to this spectral function.

Color plot of D(E) around the bare energy of a photonic mode versus normalized flux bias (0=h/2e). The panels correspond to different values of Rq/Z with EJ=0.1EC, =0.5EC and p=5EC.

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Emergent quantum phase transition of a Josephson junction coupled to a high-impedance multimode resonator - Nature.com