## Abstract

The Thomas–Reiche–Kuhn (TRK) sum rule is a fundamental consequence of the position–momentum commutation relation for an atomic electron, and it provides an important constraint on the transition matrix elements for an atom. Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light–matter interactions and/or optical nonlinearities. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves expectation values of field operators calculated between general eigenstates of the interacting light–matter system. This sum rule provides constraints and guidance for the analysis of strongly interacting light–matter systems and can be used to test the validity of approximate effective Hamiltonians often used in quantum optics.

## 1 Introduction

### 1.1 A brief history of sum rules in quantum mechanics

Since the beginning of quantum mechanics, sum rules have proved to be very useful for understanding the general features of difficult problems. These relations, obtained by adding (sum) unknown terms, power tool for the study of physical processes [1]. Historically, the first important sum rule is found in atomic physics and concerns the interaction of electromagnetism with atoms: the Thomas–Reiche–Kuhn (TRK) sum rule [2], [3], [4]. It states that the sum of the squares of the dipole matrix moments from any energy level, weighted by the corresponding energy differences, is a constant. The TRK and analogous sum rules, like the Bethe sum rule [5], play an especially important role in the interaction between light and matter. They have widely been applied to the problems of electron excitations in atoms, molecules, and solids [6].

For an atomic electron, the TRK sum rule is a direct consequence, of the canonical commutation relation between position and momentum. It is possible to view it as a necessary condition in order not to violate this commutation relation [7]. Among the many consequences of this sum rule, it constrains the cross sections for absorption and stimulated emission [8]. It has also been shown that useful sum rules can be obtained for nonlinear optical susceptibilities [9], [10], [11]. A modified TRK sum rule for the motion of the atomic center of mass and a generalized TRK sum rule to include ions have been also obtained [12]. Extensions of the TRK sum rule to the relativistic case have been studied (see, e.g., [13], [, 14]). Important sum rules have also been developed in quantum chromodynamics (see, e.g., [15]).

Such sum rules also play a relevant role in the analysis of interacting electron systems [16], [, 17]. Since they are a direct consequence of particle conservation in the system, their satisfaction is necessary to guarantee a gauge-invariant theory [16], [, 17] (see, e.g., [18], [, 19] as two recent examples). In interacting electron systems, the longitudinal version of the TRK sum rule (known as *f*-sum rule) provides a very useful check on the consistency of any approximate theory and can permit a direct calculation of collective mode frequencies in the long wavelength limit [16]. A striking example of the relevance of sum rules in interacting electron systems is constituted by the apparent gauge invariance difficulty in superconductors (Meissner effect), originating by the violation of the *f*-sum rule of approximate models [20].

Almost all the developed sum rules have been derived for the degrees of freedom of particles. One exception is in the study by Barnett and Loudon [21], where optical sum rules have been derived for polaritons propagating through a linear medium.

### 1.2 Summary of our main results

Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light–matter interactions and/or optical nonlinearities [22], [, 23]. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves the expectation values of the field coordinates or momenta calculated between general eigenstates of the interacting light–matter system (dressed light–matter states) and the corresponding eigenenergies of the interacting system.

In this work, we also present a generalized atomic TRK sum rule for atoms strongly interacting with the electromagnetic field. This sum rule has the same form of the standard TRK sum rule but involves the energy eigenstates and eigenvalues of the interacting system.

The sum rules for interacting light–matter systems proposed here can be useful to analyze general quantum nonlinear optical effects (see, e.g., [24], [25], [26], [27]) and many-body physics in photonic systems [28], like analogous sum rules for interacting electron systems, which played a fundamental role for understanding the many-body physics of electron liquids [16], [17], [20]. The proposed sum rules become particularly interesting in the nonperturbative regimes of light–matter interactions.

In the last years, several methods to control the strength of the light–matter interaction have been developed, and the ultrastrong coupling (USC) between light and matter has transitioned from theoretical proposals to experimental reality [22], [, 23]. In this new regime of quantum light–matter interaction, beyond weak and strong coupling, the coupling strength becomes comparable to the transition frequencies in the system or even higher (deep strong coupling [DSC]) [29], [30], [31], [32]. In the USC and DSC regimes, approximations widely employed in quantum optics break down [33], allowing processes that do not conserve the number of excitations in the system (see, e.g., [27], [34], [35], [36], [37]). The nonconservation of the excitation number gives rise to a wide variety of novel and unexpected physical phenomena in different hybrid quantum systems [35], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58]. As a consequence, all the system eigenstates, dressed by the interaction, contain different numbers of excitations. Much research on these systems has dealt with understanding whether these excitations are real or virtual, how they can be probed or extracted, how they make possible higher order processes even at very low excitation densities, and how they affect the description of input and output for the system [22], [, 23].

The eigenstates of these systems, including the ground state, can display a complex structure involving superposition of several eigenstates of the noninteracting subsystems [22], [23], [59] and can be difficult to calculate. As a consequence, a number of approximation methods have been developed [60], [, 61]. Moreover, the output field correlation functions, connected to measurements, depend on these eigenstates (see, e.g., [48], [, 62]). Hence, sum rules providing general guidance and constraints can be very useful to test the validity of the approximations. The general sum rule proposed in this article can also be used to test the validity of effective Hamiltonians often used in quantum optics and cavity optomechanics [58], [63], [64]. In addition, this generalized TRK sum rule applies to the broad emerging field of nonperturbative light–matter interactions, including several settings and subfields, as cavity and circuit quantum electrodynamics (QED) [22], collective excitations in solids [65], optomechanics [63], photochemistry and QED chemistry [59], [, 66].

## 2 Sum rule for interacting photons

A key property used for the derivation of the TRK sum rule is that the commutator between the electron coordinate and the electronic Hamiltonian does not depend on the electronic potential, which is a function of the coordinate only, and hence, it is universal. Considering for simplicity, a single electron 1D system, if *x* is the electron coordinate and

In the Coulomb gauge, the (transverse) vector potential **A** represents the field coordinate, while its conjugate momentum Π is proportional to the transverse electric field:

A general feature of the light–matter interaction Hamiltonians derived from the *minimal coupling replacement* (as for the Coulomb gauge) is that the momenta of the matter system are coupled only to the field coordinate. We can express the total light–matter quantum Hamiltonian as

where the second equality follows from *D* electromagnetic resonator of length *L* so that the expression for the electric field operator can be simplified to *x* is the coordinate along the cavity axis, and *y* is a coordinate along an axis orthogonal to the cavity axis. The vector potential (as well as the electric field operator) can be expanded in terms of photon creation and destruction operators as

and

where

and

Here *SL* is the resonator volume, the subscript *m* labels a generic mode index with frequency *ω*_{m}, and

where

Let us now consider the matrix elements of the operators in (2) between two generic eigenstates

where *j* = 0 indicates the system ground state, and the energy levels are ordered according to their energy: *x*. By defining

and

we obtain the corresponding relation for the individual modes:

It is worth noticing that, in the limit when the light–matter interaction vanishes,

Let us now consider the commutator between the mode coordinate and its conjugate momentum:

where we used

Developing the double commutator, considering its matrix elements between two generic eigenstates of the total Hamiltonian

which reduces (choosing *j* = *i*) to the TRK sum rule for interacting fields:

By using (4), (7) can be also expressed in terms of the momenta matrix elements:

Formally, it coincides with the TRK sum rule for atoms; however, in (7) the matrix elements of the field-mode coordinate replace the atomic *electric dipole* matrix elements. An important difference is that the atomic TRK sum rule [67] considers atomic energy eigenstates, calculated in the absence of interaction with the field. On the contrary, this sum rule is very general since it holds in the presence of interactions with *arbitrary* matter systems every time the interaction occurs via the field coordinate (e.g., Coulomb gauge). We also observe that (7) describes a collection of sum rules, one for each field mode *m*. Actually, following the same reasoning which led us to (7), a generalized atomic TRK sum rule for atoms strongly interacting with the electromagnetic field [analogous to (7)] can be easily obtained, as shown in Section 4.

## 3 Applications

### 3.1 Quantum Rabi model

The quantum Rabi Hamiltonian describes the dipolar coupling between a two-level atom and a single mode of the quantized electromagnetic field. Recently, it has been shown [68] that the correct (satisfying the gauge principle) quantum Rabi Hamiltonian in the Coulomb gauge

strongly differs from the standard model (see also the studies by De Bernardis et al [69], Stokes et al [70], and Settineri et al [71] for gauge issues in the USC regime). Here, *ω*_{c} is the resonance frequency of the cavity mode, *ω*_{0} is the transition frequency of a two-level atom,

(*A*_{0} is the zero-point fluctuation amplitude of the field potential and *d* is the atomic dipole moment) in (A) describes the normalized light–matter coupling strength. When the normalized coupling strength is small (*η* ≪ 1), considering only first-order contributions in *η*, the standard interaction term *η*, calculated after the numerical diagonalization of (8). The two quantities are equal only at negligible coupling. When the coupling strength increases, the two quantities provide very different results. However, in agreement with (4), the numerically calculated

violates (4) providing coupling-independent values

These findings show that, using the wrong quadrature (

In order to understand how the sum rule in (7) applies to the quantum Rabi model, we calculate partial sums with an increasing number of states. Specifically, we calculate

Here and in the following, the eigenstates of the total Hamiltonian, obtained for a given coupling strength *η*, are labeled so that *i* >* j* for *ω*_{i} > *ω*_{j}. Differently from the JC model, the quantum Rabi model does not conserve the excitation number. Therefore, expectation values like *j* > 2. Figure 1(b) displays such partial sums as a function of the number of levels included, obtained for different values of *η*. For small values *η*, where the system ground state *g*) or excited (*e*)], then

### 3.2 Nonlinear electromagnetic resonator

As a further test, we analyze a single-mode nonlinear optical system described by the following effective Hamiltonian:

Here *ω*_{k,0} as a function of *η*. Figure (2) displays the partial sums *η*. Increasing the anharmonicity coefficient *η*, the number of contributions in the sum increases at the expense of the contribution

where the radiative decay rates are

and we assumed an ohmic coupling with the external modes *ω* = *ω*_{c} [dashed curve in Figure 2(b)]. When *η*.

### 3.3 Frequency conversion in ultrastrong cavity QED

The relations in (4) and (7) are very general. So far, we applied them to single-mode fields; however, they are also valid in the presence of (even interacting) multimode fields (see, e.g., [74], [, 75]). Here, we analyze the TRK sum rule for interacting photons in a three-component system constituted by two single-mode resonators ultrastrongly coupled to a single superconducting flux qubit. This coupling can induce an effective interaction between the fields of the two resonators. Using suitable parameters for the three components, the system provides a method for frequency conversion of photons which is both versatile and deterministic. It has been shown that it can be used to realize both single and multiphoton frequency conversion processes [52]. The system Hamiltonian is given as follows:

where *θ* encodes the qubit flux offset which determines parity symmetry breaking. A zero flux offset implies *θ* = 0. Figure 3(a) displays the lowest normalized energy levels *a* and *b*, respectively, while the third entry indicates the qubit state. The second excited level is a horizontal line corresponding to the eigenstate

The mixing is maximum when the level splitting is minimum (at

It has been shown [52] that this effective coupling can be used to transfer a quantum state constituted by an arbitrary superposition of zero and one photon in one resonator (e.g., *a*) to a quantum state corresponding to the same superposition in the resonator at frequency *ω*_{b}.

This system represents an interesting example of two interacting optical modes (with the interaction mediated by a qubit). In order to understand how the sum rule in (7) applies to such a system, we investigate its convergence, calculating partial sum rules for the two modes. Figure 4 shows *N*. The black line describes the zero detuning case, while the dashed blue line, the case

since

are provided in (C), it is easy to obtain

in agreement with the results in Figure 4(a). Notice that for *δ* = 0, it results in *θ* = π/4, and hence,

## 4 TRK sum rule for atoms interacting with photons

The standard atomic TRK sum rule [67] considers atomic energy eigenstates, calculated in the absence of interaction with the transverse electromagnetic field. A recent interesting example of descriptions including the electron–electron interaction can be found in the study by Andolina et al. [18].

Following the same reasoning which led us to (7), a generalized atomic TRK sum rule for atoms strongly interacting with the electromagnetic field [analogous to (7)] can be easily obtained, starting from the dipole gauge. In this gauge (see, e.g., [68]), the light–matter interaction term does not depend on the particle momentum, and the same steps used to obtain (7) can thus be followed. The resulting atomic generalized TRK sum rule formally coincides with the standard one, with the only difference that all the expectation values are calculated using the eigenstates of the *total* light–matter system. For example, we consider a system described by a single effective particle with mass *m* and charge *q* displaying a dipolar interaction with a single mode resonator:

where *A*_{0} is the zero-point fluctuation amplitude of the field potential. The following commutation relation holds:

where *f*-sum rule [16] (the longitudinal analog of the TRK sum rule) for an electron system strongly interacting with a quantized electromagnetic field can be obtained. These sum rules can find useful applications in the study of correlated electron systems strongly interacting with photons (see, e.g., [76]).

## 5 Discussion

The TRK sum rule for interacting photons proposed here can be useful for investigating general quantum nonlinear optical effects and many-body physics in photonic systems (see, e.g., [24], [25], [26], [27], [28]), like the corresponding sum rules for interacting electron systems, which played a fundamental role for understanding the many-body physics of interacting electron systems [16], [17], [20].

We provided a few examples showing how the light–matter interaction can change significantly the number of excited photonic states exhausting the sum rule. Using the sum rule, one can prove without explicit calculations that other excited states have negligible oscillator strength.

The relations in (4) and (7) are very general. They are also valid in systems including several dipoles (see, e.g., [77], [, 78]) and modes (see, e.g., [75]). These relations provide a very useful check on the consistency of approximate models in quantum optics. Approximate Hamiltonians and effective models can violate one of them. Such a violation indicates that the model may miss some relevant physics [16]. For example, we have shown that the JC model, a widespread description for the dipolar coupling between a two-level atom and a quantized electromagnetic field, violates the relation (4). An additional example of a model violating this relation is provided by the well-known and widely employed cavity optomechanical interaction Hamiltonian

An interesting feature of the relations proposed here is that they hold in the presence of light–matter interactions of arbitrary strength. Moreover, the obtained sum rule can be useful for the analysis of strongly interacting light–matter systems, especially when exact eigenstates are not available. These relations in (4) and (7) can provide constraints and a guidance in the development of effective Hamiltonians in quantum optics and cavity optomechanics.

Following the same reasoning leading to (7), we also proposed a generalized TRK sum rule for the matter component involving transitions between the *total* light–matter energy eigenstates [(14)], describing particle conservation in the presence of arbitrary light–matter interactions.

**Funding source: **Army Research Office

**Award Identifier / Grant number: **W911NF-18-1-0358

**Award Identifier / Grant number: **W911NF1910065

**Funding source: **Japan Science and Technology Agency

**Award Identifier / Grant number: **JPMJCR1676

**Funding source: **Japan Society for the Promotion of Science

**Award Identifier / Grant number: **JPJSBP120194828

**Award Identifier / Grant number: **JP20H00134

**Funding source: **Foundational Questions Institute Fund (FQXi)

**Award Identifier / Grant number: **FQXi-IAF19-06

**Funding source: **Silicon Valley Community Foundation

**Funding source: **NTT Research

**Funding source: **Asian Office of Aerospace Research and Development (AOARD)

## Acknowledgments

F.N. is supported in part by the NTT Research, Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technology Agency (JST) (via the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (via the KAKENHI Grant No. JP20H00134, and the grant JSPS-RFBR Grant No. JPJSBP120194828) and the Grant No. FQXi-IAF19-06 from the Foundational Questions Institute Fund (FQXi), a donor advised fund of the Silicon Valley Community Foundation. S.S. acknowledges the Army Research Office (ARO) (Grant No. W911NF1910065).

**Author contribution:**All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.**Research funding:**F.N. is supported in part by the NTT Research, Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technology Agency (JST) (via the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (via the KAKENHI Grant No. JP20H00134, and the grant JSPS-RFBR Grant No. JPJSBP120194828), the Asian Office of Aerospace Research and Development (AOARD), and the Grant No. FQXi-IAF19-06 from the Foundational Questions Institute Fund (FQXi), a donor advised fund of the Silicon Valley Community Foundation. S.S. acknowledges the Army Research Office (ARO) (Grant No. W911NF1910065).**Conflict of interest statement:**The authors declare no conflicts of interest regarding this article.

## Appendix: A Linear response theory and transmission of a nonlinear optical system

This section provides a derivation of the transmission coefficient of a nonlinear optical system based on the dressed master equation approach [80], [, 81].

The dressed master equation in the Schrödinger picture can be written as follows [80], [, 81]:

where

is the system Hamiltonian expressed in the dressed basis, constituted by the energy eigenstates of the nonlinear system. Dissipation is described by the Lindbladian superoperator defined by

This equation includes the thermal populations

and the damping rates

Here, *g*(*ω*) is the system reservoir coupling strength,

At *T* = 0, being

We also consider a coherent drive entering from the left port, described by the following interaction Hamiltonian:

where

is a c-number corresponding to the mean value of the external (left) field operators, assumed to be in a coherent state. We will also assume

where

We assume that the light field from the left port is coherent with driving frequency *ω*:

Retaining only the terms depending linearly from the input field and using Eqs. (A1), (A7), (A8), assuming

(i.e., oscillating resonantly with the driving field), and using the rotating wave approximation, we obtain (to first order in the field)

where, being *T* = 0, only the ground state is populated in the absence of interaction (*LC*-oscillator coupled to a transmission line and use the input–output relations [71] for the positive frequency component of the output (input) vector potential operator defined as follows:

where, for the sake of simplicity, we disregarded the spatial dependence, and *Z*_{0} the impedance of the in-out transmission line(s). In addition, we consider two distinct ports for the input (*L*) and the output (*R*) [for simplicity we assume *g*_{L} (*ω*) = *g*_{R} (*ω*) = *g* (*ω*)], and we have for the output voltage operator [71]

which can be expressed as follows:

where

Assuming *K* and

Using (A.11), we have for the mean value of the input sent through the port (*L*)

where we assumed a coherent drive input at frequency *ω*:

Considering the linear response only, the projection operator oscillates at the frequency *ω* of the drive,

using Eqs. (A13) and (A14), the mean value for the output is given as follows:

where

Using Eqs. (A15)–(A17), we can calculate the transmission coefficient *T*(*ω*) due to the signal detected from the port (*R*) when a driving field is sent through the port (*L*) as follows:

where

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**Received:**2020-07-29

**Accepted:**2020-09-27

**Published Online:**2020-11-18

© 2020 Salvatore Savasta et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.