Thomas–Reiche–Kuhn (TRK) sum rule for interacting photons

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].

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, ω₀ is the transition frequency of a two-level atom, aˆ and aˆ† are the destruction and creation operators for the cavity field, respectively, while the qubit degrees of freedom are described by the Pauli operators σˆi. The parameter

(A₀ 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 ℏω0η(aˆ+aˆ†)σˆy is recovered. If the system is prepared in its first excited state, the photodetection rate for cavity photons is proportional to |P1,0|2 (see [62], [, 71]). Figure 1(a) displays this quantity (black dashed curve) as well as |Q1,0|2 (dotted blue) versus the normalized coupling 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 (ω1,02/ωc2)|Q1,0|2 coincides with |P1,0|2. In contrast, the Jaynes Cummings (JC) model,

violates (4) providing coupling-independent values |Q1,0|2=|P1,0|2 [the horizontal line in Figure 1(a)].

Figure 1:

(a) P−Q relation: calculation of |P1,0|2 (proportional to the photodetection rate for cavity photons) (black dashed) and of |Q1,0|2 (dotted blue) versus the normalized coupling η. (b) Thomas–Reiche–Kuhn (TRK) sum rule for interacting fields: partial sums ∑j=0Nℱ0j as function of the number N of levels included for different normalized coupling rates η. Inset: energy spectrum for the first energy levels ω_k,0 versus the normalized coupling strength.

These findings show that, using the wrong quadrature (Q instead of P) for the calculation of the photodetection rate for systems in the USC regime can result into significantly wrong results. This is a direct consequence of (3).

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 Q0,j (and hence ℱ0,j) can be different from zero also for 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=0.01), only the two lowest excited levels contribute to the sum with approximately equal weights, in good agreement with the JC model. For η=0.2, still only two transitions contribute to the sum rule; however, the second transition provides a larger contribution to the sum. For η=0.5, the contribution of the lowest energy transition becomes smaller, while ℱ02=0 owing to the parity selection rule. Note that, at η=ηcr≃0.44, there is a crossing between the levels 2 and 3 [see inset in Figure 1(b)] so that, for η>ηcr, state |2〉 has the same parity of state |0〉. It is sufficient to include ℱ03 to approximately satisfy the sum rule. For η=1, ℱ0,1 is very small and ℱ0,2=0. In this case, the sum rule is satisfied mainly with the contributions ℱ0,j with 3≤j≤6. Finally, for very high values of the normalized coupling strength (η=1.8), only one contribution (ℱ0,3) becomes relevant. This effect is due to the light–matter decoupling [71] which occurs at very high values of η, where the system ground state |0〉 is well approximated by |g,0〉 [the first entry in the ket labels the photon number, the second labels the qubit state: ground (g) or excited (e)], then |1〉≃|e,0〉, |2〉≃|e,1〉, |3〉≃|g,1〉, and so on: the higher energy levels are of the kind ≃|g(e),n>1〉. This explains why for η=1.8, the only significant contribution to the sum is ℱ0,3. These behaviors of the partial sums and of the terms ℱi,j are closely connected to accessible experimental features, as explicitly shown in the example below.

3.2 Nonlinear electromagnetic resonator

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

Here HˆF=ℏωcaˆ†aˆ, while the nonlinear terms are assumed to arise from the dispersive interaction with some material system [72]. Note that the nonlinear terms in (9) commute with the field coordinate Qˆ=(aˆ+aˆ†)/2; hence, Eqs. (4) and (7) hold. In contrast, the presence of a standard self-Kerr term ∝aˆ†2aˆ2 (see, e.g., [73]) would violate them. The inset in Figure 2 shows the anharmonic energy spectrum ω_k,0 as a function of η. Figure (2) displays the partial sums ∑j=1Nℱ0j as versus the number of included levels, calculated for different values of η. Increasing the anharmonicity coefficient η, the number of contributions in the sum increases at the expense of the contribution ℱ01 of the lowest energy transition. This behavior is closely connected with accessible experimental features which can be observed, e.g., in linear transmission spectra. For a two-port (equally coupled to the external modes) nonlinear resonator, the transmission spectrum (see Appendix A for supporting content) can be written as follows:

where the radiative decay rates are

and we assumed an ohmic coupling with the external modes (g2(ω)∝ω). When the anharmonicity is switched off (η=0), Γk,0∝ℱ0k=0 for k≠1, and the transmission spectrum presents a single peak at ω = ω_c [dashed curve in Figure 2(b)]. When η≠0, Γk,0∝ℱ0k≠0, and the transmission spectrum in Figure 2 evolves accordingly (the blue continuous curve shows the spectrum calculated for η=0.12). By integrating the individual spectral lines in (10), we obtain for each line a contribution ≃πΓk,02/Γk, which is approximately proportional to ℱ0k in the sum (notice that Γk∼kΓ1). The inset in Figure 2 shows the integrated lines for two values of η.

Figure 2:

(a) Thomas–Reiche–Kuhn (TRK) sum rule for a single-mode nonlinear system: partial sums ∑j=0Nℱ0j versus the number (N) of levels included for different normalized coupling strengths η. Inset: anharmonic energy spectrum ω_k,0 versus η. (b) Transmission spectrum T(ω) for a two-port nonlinear resonator for η=0.12. The inset shows the integrated lines for two values of 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 (aˆ,ωa,ga) and (bˆ,ωb,gb) describe the photon operator, the frequency mode, and the coupling with the qubit for the two resonators. The angle 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 (ω−ωg)/ω‾0 (we indicated with ℏωg the ground state energy) versus the qubit frequency ω0/ω‾0 obtained diagonalizing numerically the Hamiltonian in (11). We used the parameters ωa=3ω‾0, ωb=2ω‾0, θ=π/6, ga=gb=0.2ω‾0, where ω‾0 is a reference point for the qubit frequency. Notice that the two resonators are set in order that their resonance frequencies satisfy the relationship ωa=ωb+ω‾0. The first excited level is a line with slope ≃1, corresponding to the approximate eigenstate |ψ1〉≃|0,0,e〉, where the first two entries in the ket indicate the number of photons in resonator a and b, respectively, while the third entry indicates the qubit state. The second excited level is a horizontal line corresponding to the eigenstate |ψ2〉≃|0,1,g〉; the next two lines on the left of the small rectangle in Figure 3(a) (for values of ω0/ω‾0 before the apparent crossing) correspond to the states |ψ3〉≃|0,1,e〉 and |ψ4〉≃|1,0,g〉. The apparent crossing in the rectangle is actually an avoided level crossing, as can be inferred from the enlarged view in Figure 3(b). It arises from the hybridization of the states |0,1,e〉 and |1,0,g〉 induced by the counter-rotating terms in the system Hamiltonian. The resulting eigenstates can be approximately written as follows:

The mixing is maximum when the level splitting is minimum (at ω0/ω‾0≃1.056). In this case, θ=π/4.

Figure 3:

Energy spectum obtained from the numerical diagonalization of (11). (a) Lowest normalized energy levels versus the qubit frequency. (b) Enlarged view of the spectrum inside the rectangle in (a) showing the presence of an avoided level crossing. Parameters are given in the text.

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 ∑j=0Nℱ0ja (a) and ∑j=1Nℱ1jb (b) for different values of N. The black line describes the zero detuning case, while the dashed blue line, the case δ=(ω0−ω‾0)/ω‾0=−6×10−3. The results in Figure 4(a) can be understood observing that

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, ℱ03a≃ℱ04a. A similar analysis can be carried out for the results in Figure 4(b).

Figure 4:

Thomas–Reiche–Kuhn (TRK) sum rule for interacting photons in the three-component system described by the Hamiltonian in (11). (a) Partial sum rules ∑j=1Nℱ0ja relative to the first resonator and (b) ∑j=1Nℱ1jb relative to the second resonator, both for different values of levels N. The black segmented line describes the zero detuning case δ = 0, while the dashed blue segmented lines refer to the case δ=(ω0−ω‾0)/ω‾0=−6×10−3. Parameters are given in the text.