Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics

TL;DR

This paper presents a first-principles multimode circuit QED model showing convergence without cutoff, explaining how higher modes naturally decouple and enabling analysis of ultra-strong coupling regimes.

Contribution

It introduces a novel multimode circuit QED model that demonstrates convergence without cutoff and clarifies the physical mechanisms behind mode decoupling.

Findings

Quantities like the Lamb shift do not diverge without a cutoff.

Higher modes naturally decouple due to circuit-based rescaling.

Framework for analyzing ultra-strong coupling in multimode circuit QED.

Abstract

Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number $n$ scales as $\sqrt{n}$ for $n$ up to $\infty$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a non-zero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultra-strong coupling regime of multimode circuit QED.

Equations

$$\begin{array}{rl}{\hat{H}}^{(M)}& =\sum _i\hslash {ϵ}_i^{(M)}{|i⟩}^{(M)}{⟨i|}^{(M)}+\sum _{m=0}^{m<M}\hslash {\omega }_m{\hat{a}}_m^{†}{\hat{a}}_m\\ & +\sum _{i,j}\sum _{m=0}^{m<M}\hslash g_{m,i,j}^{(M)}{|i⟩}^{(M)}{⟨j|}^{(M)}({\hat{a}}_m+{\hat{a}}_m^{†}).\end{array}$$

$${\hat{H}}_{\text{AA}}^{(M)}=4E_C^{(M)}{\hat{N}}_J-E_J\cos (\hat{\delta }).$$

$$E_C^{(M)}=\frac{e^2}{2}\frac{C_0+M{C}_c}{C_0{C}_c},$$

$${\tilde{E}}_C^{(M)}=E_C^{(M+1)}-\hslash ({\bar{g}}_M^{(M+1)}{)}^2/4{\omega }_M,$$

$${\tilde{E}}_C^{(L)}=\underset{M\to \infty }{\lim }{E}_C^{(M)}-\sum _{m\ge L}^M\hslash ({\bar{g}}_m^{(M)}{)}^2/4{\omega }_m.$$

$$E_C^{(M)}=\frac{e^2}{2}\frac{C_0+M{C}_c}{M{C}_c{C}_J+C_0(C_c+C_J)},$$

Full text

Quantum electrodynamics (QED) explores one of the most fundamental interactions in nature, that of light and matter. In a typical cavity QED scenario, an individual atom interacts through its dipole moment with the electric field of cavity modes, as described by the Rabi model Rabi (1936) and depicted in Fig. 1(a). One consequence of this interaction is the Lamb shift of the atomic transition frequencies Lamb and Retherford (1947); Heinzen (1988). Early attempts at calculating this shift led to the first shortcomings of QED theory, mainly that the transition energies of the atom diverge as the infinite number of electromagnetic modes are considered. Efforts to address these issues gave birth to renormalization theory Bethe (1947). Akin to cavity QED is the field of circuit QED Wallraff et al. (2004), where artificial atoms such as anharmonic superconducting LC circuits couple to the modes of a waveguide resonator or an open transmission line. Such systems allow the study of a wealth of quantum effects Schuster et al. (2007); Bishop et al. (2009), and are one of the most promising platforms for the realization of quantum processors Takita et al. (2016); Kelly et al. (2015); Ristè et al. (2015). Despite experimental successes, “ad hoc” multimode extensions of the Rabi model suffer from divergences when considering the limit of infinite modes in a waveguide resonator Houck et al. (2008); Filipp et al. (2011).

Aware of this problem, Nigg et al. Nigg et al. (2012); Girvin (2014) and Solgun et al. Solgun et al. (2014) developed the method of black-box quantization to obtain effective Hamiltonians with higher predictive power. While a practical tool for weakly anharmonic systems, this method was not designed for systems with strong anharmonicity, such as a Cooper pair box Koch et al. (2007). Furthermore, in applying the black-box procedure, the form of the quantum Rabi Hamiltonian is not preserved, and while it gives the correct energy spectrum, it is not clear how to identify and connect it to a coupling rate between the two bipartite systems. Doing so, it is no longer possible to directly identify which parts of the Rabi interaction lead to certain energy shifts of bare quantities, such as the Bloch-Siegert shift, highly relevant for studying the physics of ultra-strongly coupled (USC) systems Forn-Díaz et al. (2010); Bosman et al. (2017).

In this paper, we derive a first-principles Hamiltonian model addressing these issues. This Hamiltonian is expressed in the basis of the uncoupled resonator modes and the atom, is valid for arbitrary atomic anharmonicities, and allows us to understand why previous attempts at extending the Rabi Hamiltonian have failed. The presence or not of a Josephson capacitance $C_{J}$ in our study leads to two important results. First, in the limit $C_{J}\rightarrow 0$, the coupling rates follow a square root increase up to an infinite number of modes. Without introducing a cutoff in the number of coupled modes, we show that a first-principles analysis of the quantum circuit leads to convergence of the energy spectrum. The $C_{J}=0$ limit also highlights a natural renormalization of Hamiltonian parameters, arising from the circuit analysis, which is essential to understanding how to reach correct multimode extensions of the Rabi model. Secondly, we study the experimentally relevant case $C_{J}>0$, which introduces a cutoff that suppresses the coupling to higher modes De Liberato (2014); García-Ripoll et al. (2015); Malekakhlagh and Türeci (2016). In particular, we provide an analysis of this regime and discuss the physics of this cutoff in the context of a lumped element circuit model. This results in a useful tool for studying multimode circuit QED in the framework of the Rabi Hamiltonian, or for studying strongly anharmonic regimes, out of reach of the black-box quantization method. This model was indispensable in extracting the Bloch-Siegert shift in the experiment of Ref. Bosman et al. (2017), where a naive extension of the Rabi model would predict a Lamb shift of more than three times the atomic frequency due to 35 participating modes before any physically-motivated cutoff, such as the qubit’s physical size or the Junction capacitance, becomes relevant.

The circuit QED system studied in this work is an artificial atom (AA) formed from an anharmonic LC oscillator Koch et al. (2007), capacitively coupled to a quarter-wave ($\lambda/4$) transmission line resonator Pozar (2009) as depicted in Fig. 1(b). The AA is a superconducting island connected to ground through a Josephson junction characterized by its Josephson energy $E_{J}$. It has a capacitance to ground $C_{J}$ and is coupled to the voltage anti-node of the resonator, with characteristic impedance $Z_{0}$ and fundamental mode frequency $\omega_{0}/2\pi$, through a capacitance $C_{c}$. The Josephson junction acts as a non-linear inductor, providing a source of single-photon anharmonicity in the oscillations of current flowing through it. In order to clearly illustrate the most novel aspect of our model, the renormalization of the charging energy, we first consider the case $C_{J}=0$. Despite the absence of a cutoff in the number of coupled modes in this case, we find that the energy spectrum still converges. The case of $C_{J}>0$ is discussed at the end of this article and in detail in the Supplementary Material SI .

Circuit Hamiltonian — We consider a Hamiltonian in which each uncoupled harmonic mode of the resonator, with resonance frequency $\omega_{m}$ and annihilation operator $\hat{a}_{m}$, is coupled to the transition between the bare atomic states $\ket{i}$, $\ket{j}$ with energies $\hbar\epsilon_{i}$, $\hbar\epsilon_{j}$ through a coupling strength $\hbar g_{m,i,j}$ Zhu et al. (2013); Sundaresan et al. (2015). We derive such a Hamiltonian by constructing a lumped element equivalent circuit, or Foster decomposition, of the transmission line resonator as represented in Fig. 1(c). The input impedance of a shorted transmission line, at a distance $\lambda/4$ from the short, $Z(\omega)=iZ_{0}\tan(\pi\omega/(2\omega_{0}))$, is equal to that of an infinite number of parallel LC resonators with capacitances $C_{0}=\pi/(4\omega_{0}Z_{0})$ and inductances $L_{m}=4Z_{0}/((2m+1)^{2}\pi\omega_{0})$. In order to consider a finite number of modes $M$ in the model, one replaces the $m\geq M$ LC circuits in Fig. 1(c) by a short circuit to ground. This removes the $m\geq M$ resonances in the resonator input impedance $Z(\omega)$ with little effect on $Z(\omega)$ for $\omega\ll\omega_{M}$. The focus of this paper is on the evolution of the Hamiltonian parameters as a function of this system size $M$, and the consequences on the energy spectrum. Using the tools of circuit quantization Devoret (1997), we obtain as the Hamiltonian of the system SI

[TABLE]

The eigenfrequencies of the higher resonator modes $\omega_{m}$ are related to that of the fundamental mode through $\omega_{m}=(2m+1)\omega_{0}$. The coupling strength $\hbar g^{(M)}_{m,i,j}=2eV_{\text{zpf},m}\bra{i}^{(M)}\hat{N}_{J}\ket{j}^{(M)}$ scales with the square root of the mode number $m$ through the zero-point voltage fluctuations of the $m$-th mode $V_{\text{zpf},m}=\sqrt{2m+1}\sqrt{\hbar\omega_{0}/2C_{0}}$. Since we will concentrate on the frequency and coupling of the first atomic transition $\ket{g}\rightarrow\ket{e}$, we use the shorthand $\omega_{a}^{(M)}=\epsilon_{e}-\epsilon_{g}$ and $g_{m}^{(M)}=g_{m,g,e}^{(M)}$ throughout this paper.

The (bare) AA eigenstates $\ket{i}^{(M)}$ and energies $\hbar\epsilon_{i}^{(M)}$ in Eq. (1) are those that diagonalize the Hamiltonian

[TABLE]

Here $\hat{N}_{J}$ is the quantum number of Cooper-pairs on the island conjugate to $\hat{\delta}$ the superconducting phase difference across the junction, and $E_{C}^{(M)}$ is the charging energy of the island. This choice of the decomposition of the Hamiltonian is one in which the bare atom corresponds to purely anharmonic degrees of freedom (currents flowing only through the junction) and the bare cavity to purely harmonic degrees of freedom (currents flowing only through the linear cavity inductors).

The crucial consequence of quantizing our model is a renormalization of the parameters of the Hamiltonian as modes are added. In particular, the charging energy $E^{(M)}_{C}$ of the (bare) AA depends explicitly on the number of modes included in the equivalent circuit

[TABLE]

as reported previously in the case of multiple atoms coupled to a single mode Jaako et al. (2016), or in the context of metallic dots coupled to a resonator Bergenfeldt and Samuelsson (2012). For the case of $M\rightarrow\infty$ with $C_{J}=0$, the charging energy of the bare atom diverges. This divergence arises from the definition of the bare atom as current oscillations flowing only through the junction. As $M\rightarrow\infty$, the impedance path through only the series capacitors of the resonator equivalent circuit diverges. Charge from currents through the junction can no longer oscillate on $C_{c}$ and $\omega_{a}^{(M)}$ diverges. For the case of $M=1$ and $C_{c}\ll C_{0}$, Eq. (3) simplifies to the standard definition of the charging energy $E_{C}=e^{2}/2C_{c}$ Koch et al. (2007). With $M>1$, we will see that a more complex picture emerges.

Renormalization of the atomic parameters — In Fig. 2, we explore the renormalization of the parameters of our model as the number of modes $M$ is increased. Through the change in charging energy, both the eigenstates $\ket{i}^{(M)}$ and coupling strengths $g_{m}^{(M)}$ depend on $M$. For a fixed number of modes $M$, the coupling $g_{m}^{(M)}$ of the atom to mode $m$ scales with the square root of the mode number $m$: $g_{m}^{(M)}=g_{0}^{(M)}\sqrt{2m+1}$. From this coupling, each mode will induce a Lamb shift of the atomic energy \chi_{m}\simeq-2\big{(}g_{m}^{(M)}\big{)}^{2}/\omega_{m}=-2\big{(}g_{0}^{(M)}\big{)}^{2}/\omega_{0}, a formula valid in the transmon regime when $\omega_{m}$ is much larger than the bare atomic frequency. With the typical assumption of coupling and bare atomic frequency independent of $M$, summing the Lamb shifts of every mode would lead to diverging values of the dressed atomic frequencies. This leads to the divergences found in typical multimode extensions of the quantum Rabi model.

In the model presented here, however, we find that the full quantization of the lumped-element circuit leads to a Hamiltonian in which both the bare atomic frequency $\omega_{a}$ and the couplings to the modes $g_{m}$ are explicitly dependent on the number of modes $M$ included in the model. As the number of modes $M$ in the model increases, the bare atomic couplings $g_{m}^{(M)}$ are suppressed (Fig. 2(a,b)), and the bare atomic frequency $\omega_{a}^{(M)}$ increases (Fig. 2(c)), diverging for an infinite number of modes. As we will see, however, convergence is obtained in the dressed transition energy of the atom when including the Lamb-shift from higher modes of the resonator.

As an illustration of how renormalization in our model leads to convergence of the spectrum, let us consider the case shown in Fig. 2(d–e) in which the fundamental mode is resonant with the atomic frequency $\omega_{a}^{(1)}$ when $M=1$ (Fig. 2(d)). Including an additional mode with frequency $\omega_{1}$ will lead to an upwards shift of the bare atomic transition $\omega_{a}^{(1)}\rightarrow\omega_{a}^{(2)}>\omega_{a}^{(1)}$ and a change of the coupling $g^{(1)}_{0}\rightarrow g^{(2)}_{0}<g^{(1)}_{0}$ through the renormalization of the charging energy (Fig. 2(e)). Diagonalizing the subsystem of the atom and mode 1 in our model, the transition energy of the atom is shifted down again near resonance with the fundamental mode $\omega_{a}^{(2)}\rightarrow\tilde{\omega}_{a}^{(2)}\sim\omega_{a}^{(1)}$ by the dispersive shift, and the coupling of the atomic transition to the fundamental mode is increased $g_{0}^{(2)}\rightarrow\tilde{g}_{0}^{(2)}\sim g^{(1)}_{0}$ (Fig. 2(f)). In this way, the resulting vacuum Rabi splitting of the fundamental mode is found to be similar to that of the $M=1$ model, despite the decrease in the bare coupling rates $g_{0}^{(2)}$.

Note that in our model, the value of $E_{J}/E_{C}^{(M)}$ of the bare atom, which determines its anharmonicity Koch et al. (2007), is also a function of $M$. It would seem that in the limit $M\rightarrow\infty$, the bare atom would be deep in the Cooper pair box limit. However, including the hybridization with the cavity, the low energy sector of $\hat{H}^{(M+1)}$ is well approximated by a model with $M$ modes where the charging energy is not $E_{C}^{(M)}$ but

[TABLE]

where $\bar{g}^{(M)}_{m}$ is the coupling constant without the dipole moment $\bra{i}^{M}\hat{N}_{J}\ket{j}^{M}$. For this to hold, $\hbar\omega_{M}$ must be larger than the characteristic energy of the low energy sector of $\hat{H}^{(M)}$. In this case, the vacuum of the $(M+1)$-th mode, shifted by $\left(\bar{g}^{(M+1)}_{M}/\omega_{M}\right)\hat{N}_{J}$, is a good variational choice for the low lying energy sector of $\hat{H}^{(M+1)}$. In this subspace, the effective Hamiltonian is of the same form as $\hat{H}^{(M)}$, but with charging energy $\tilde{E}_{C}^{(M)}$ SI . This result matches with the zero-th order of a Schrieffer-Wolff approximation Schrieffer and Wolff (1966); Bravyi et al. (2011). We can iterate this procedure to a mode $L$. For $M\rightarrow\infty$, an effective Hamiltonian with $L$ modes will have a finite charging energy

[TABLE]

The interaction with higher modes therefore modifies the charging energy of the dressed atom, leading to a convergence of the atomic anharmonicity as well. This formula applies for all values of $C_{J}$, but for $C_{J}=0$, we have $\tilde{E}_{C}^{(M)}=E_{C}^{(M)}$, i.e., the dressing from higher modes exactly compensates the renormalization of the charging energy.

In order to illustrate the effectiveness of this renormalization, in Fig. 3 we compare a diagonalization of Hamiltonian (1) to a non-renormalized multimode extension of the quantum Rabi model, implemented by removing the $M$-dependence of the charging energy $E_{C}^{(M)}\rightarrow e^{2}/(2C_{c})$. The dashed line indicates the result of the black-box quantization (BB) method Nigg et al. (2012) as a point of reference. The calculations are performed using the same physical parameters as in Fig. 2. Compared to the non-renormalized model, which diverges linearly, a diagonalization of the first-principle Hamiltonian (1) converges towards the value expected from BB.

It is also interesting to note that the corrections from our model are non-perturbative: perturbation theory fails to give a value for the Lamb shift resulting from including an extra mode. Using a circuit analysis of coupled LC oscillators (see SI ), in the transmon regime, $E_{J}\gg E_{C}^{(M)}$, we find an estimate of the shift in the dressed AA energy when including an additional mode in the model given by $\chi_{m}\simeq-2(g_{m}^{2}/\omega_{m})(\tilde{\omega}_{a}^{2}/\omega_{m}^{2})$. This formula can be used to estimate the number of relevant modes to include in a simulation, and can be though of as a replacement of the usual expression for the Lamb shift $\chi_{m}^{\text{Lamb}}\simeq-2g_{m}^{2}/\omega_{m}$.

Consequence of introducing a high-frequency cutoff — In a realistic system, higher modes will tend to decouple from the atom due to several coexisting physical mechanisms Filipp et al. (2011). One such mechanism is the capacitance of the Josephson junction $C_{J}$. In particular, the capacitive loading of the cavity from the AA illustrated in the inset of Fig. 4 leads to a decreasing impedance to ground $Z_{c}(\omega)\simeq i(C_{J}+C_{c})/\omega C_{J}C_{c}$ at the end of the resonator when $\omega\gg\omega_{a}$. When the mode frequencies become such that this impedance is lower that the characteristic impedance of the resonator $|Z_{c}(\omega)|\ll|Z_{0}|$, this voltage anti-node of the resonator, to which the AA couples, becomes a voltage node, and the coupling vanishes. Additionally, the eigen frequencies will span from those of a $\lambda/4$ resonator for the lower modes to those of a $\lambda/2$ resonator $\omega_{m}\rightarrow 2m\omega_{0}$ for the higher lying modes.

This effect can be captured with the same quantization procedure applied to the circuit in Fig. 1(c) with $C_{J}\neq 0$ and is detailed in the supplementary material SI . Mathematically, the cutoff in the coupling is due to a mode-mode coupling term of the form $\sum_{m=0}^{m<M}\sum_{m^{\prime}=m+1}^{m<M}G^{(M)}_{m,m^{\prime}}(\hat{a}_{m}+\hat{a}_{m}^{\dagger})(a_{m^{\prime}}+a_{m^{\prime}}^{\dagger})$, which arises naturally from the circuit quantization. This is the equivalent of the $A^{2}$ term discussed in Refs. De Liberato (2014); García-Ripoll et al. (2015); Malekakhlagh and Türeci (2016). Diagonalizing the Hamiltonian of coupled resonator modes leads to decreasing zero-point voltage fluctuations of the modes at the coupling node. As shown in Fig. 4, with a capacitance to ground $C_{J}=5$ fF close to the experimental parameters of Ref. Bosman et al. (2017), the expected cutoff occurs when $|Z_{c}(\omega_{m_{c}})|\simeq|Z_{0}|$, or equivalently at the mode number $m_{c}\simeq(C_{J}+C_{c})/2\omega_{0}Z_{0}C_{J}C_{c}$. This mechanism is accompanied by the appearance of an upper bound in the renormalized charging energy, such that Eq. (3) becomes

[TABLE]

and $E^{(M)}_{C}\rightarrow e^{2}/2C_{J}$ for $M\rightarrow\infty$. We emphasize however, that this cutoff is not a necessary condition for the convergence of the energy spectrum: the model described above with $C_{\textrm{J}}=0$ converges even in the absence of such a cutoff. This is to be contrasted with typical models of (natural) atoms coupled to cavity modes where high frequency cutoffs must be imposed to obtain finite predictions Mandel and Wolf (1995). It would be interesting to study if the ideas developed in this work apply to such systems.

Conclusion — We have developed a first-principles multimode quantum Rabi model of circuit QED from a compact lumped element equivalent circuit. Using this formulation, we derived the convergence of quantities such as the Lamb shift in the absence of any high frequency cutoff, arising from a natural renormalization of the Hamiltonian parameters as modes are added. We also study the implications of a finite junction capacitance, which introduces a cutoff in the coupling to high frequency modes, but does not change the renormalization that occurs when additional modes are included in the circuit. For both cases with and without a junction capacitance, we show that when constructing a quantum Rabi model from this compact lumped element equivalent circuit, it is crucial to include this renormalization to get correct Hamiltonian parameters from the values of the circuit elements. This work provides a useful framework for an intuitive understanding and modeling of experiments in the multimode ultra-strong coupling regime. This formulation of the multimode quantum Rabi model in the context of circuits hints at an intuitive picture on how this renormalization can arise physically, and it suggests the study of how this proposed physical picture could be applied to other problems in quantum field theory.

Acknowledgments A. P-R and E. S. thank Enrique Rico and Íñigo Egusquiza for useful discussions. M. G. and G. S. thank Yuli V. Nazarov for useful discussions. The authors acknowledge funding from UPV/EHU UFI 11/55, Spanish MINECO/FEDER FIS2015-69983-P, Basque Government IT986-16 and PhD grant PRE-2016-1-0284, the Netherlands Organization for Scientific Research (NWO), the Dutch Foundation for Fundamental Research on Matter (FOM), and the European Research Council (ERC).

Note After we finished this manuscript, we became aware of Ref. Malekakhlagh et al. (2017), which arrives at a similar conclusions as the last section of this paper through a different approach.