Light-matter Interaction and Zeta Functions

2.1 The prime number theorem

The prime number theorem, a result describing the distribution of prime numbers, was conjectured by Carl Friedrich Gauss and Adrien-Marie Legendre in the 18th century and proved independently by Charles de la Vallé Poussin and Jacques Hadamard in 1896 using the ideas introduced by Bernhard Riemann in his seminal work in number theory.

If 𝜋(x) is the function describing the number of primes less than x(> 0), then the prime number theorem is precisely stated as

Here, ƒ(x)∼g(x) means that the limit ƒ(x)/g(x) → 1 holds as x → ∞. At the heart of the proof is Riemann’s idea that ζ(s) ≠ 0 for ℜ(s) = 1.

The revolutionary contribution of Riemann of recognizing that the seemingly random distribution of prime numbers is intimately related to the analytical properties of ζ(s) may even be a greater achievement that a future proof of the Riemann hypothesis itself.

2.2 Functional equation of the Riemann zeta function

One of the main features of ζ(s) is the functional equation. Let Γ(s) be the gamma function, and set ζ̃(s) := 𝜋^–s/2Γ(s/2)ζ(s), then the functional equation is

The essential idea of the functional equation was discovered by Euler in its computations aimed to assign values to divergent series, including

The computational results obtained by Euler are correct even though the concept of analytical continuation (or even complex function theory) did not exist at the time.

Let us give an outline of the proof of the functional equation to introduce some of the ideas used later for the partition function. To avoid technical complications, we assume that all series and integrals converge and behave in a reasonable manner.

The Mellin transform ℳƒ of a function ƒ is defined by

A fundamental example is given by ƒ(t) = e^–nt with t > 0 with Mellin transform ℳƒ(s) = n^–sΓ(s), verified directly from the definition of Γ(s). Similarly, for a series , the Mellin transform of h(t) = g(e^–t) is easily verified to be ℳh(s). The main point to note here is that the Mellin transform relates the series of exponential with the Dirichlet series.

Let us define the series θ(z) in the upper-half complex plane ℍ := {z ∈ ℂ 𝔍(s) > 0} as

From the foregoing discussion, we verify that by setting , we obtain ζ̃(s) = ℳƒ(s/2). The function θ(t) is an automorphic form called the Jacobi theta function.

Automorphic functions (resp. forms) are functions that are invariant (resp. almost invariant) under certain actions of non-commutative groups. Trigonometric functions are well-known to be invariant under translations (i.e. are periodic functions), in other words, they are invariant under the action of the abelian group ℤ. Thus, in this sense automorphic functions may be thought as non-commutative versions of trigonometric functions.

For z ∈ ℍ, in addition to the translation invariance θ(z + 2) = θ(z), the theta function satisfies the relation . If ƒ̂ is the Fourier transform of a function ƒ, then by the Poisson summation formula

and replacing by the rapidly decreasing function ƒ_t(x) = e^–𝜋tx2, we obtain the desired relation. Finally, for z = it (t > 0), we obtain and applying the Mellin transform to both sides we obtain the functional equation for ζ̃(s).

Let us give an interpretation of the Poisson summation used above. Similar to the idea of the hands on a clock^*3, we consider two real numbers to be equivalent if they have the same fractional part and write this set as ℤ∖ℝ. With this in mind, the right side of the Poisson summation formula is the sum over the lengths of a circumference (i.e., number of turns), and the left side is the sum over the irreducible representations that appear in the Fourier transform, that is, representations x ↦ e^2𝜋iyx of the abelian group ℝ that are trivial on ℤ. In other words, we might think of it as a sum over all y = m ∈ ℤ. The left side may also be interpreted as the sum over the eigenvalues of the Laplacian Δ = – of ℝ. The extension of this idea for non-commutative groups is the celebrated Selberg trace formula^*4.

2.3 Modular and automorphic forms

Let SL₂(ℤ) (respectively SL₂(ℝ)), be the group of 2 × 2 matrices with integer (respectively real) entries and determinant 1. The group SL₂(ℤ) is generated^*5 by matrices

Now, acts on the upper half-plane by the linear fractional transformation z ↦ g. z := for z ∈ ℍ. This action preserves the Poincaré (hyperbolic) metric on ℍ induced by y^–2 dxdy. Since the matrices ±1 (1 is the identity matrix) define the same action, we consider the (projective) modular group Γ = PSL₂(ℤ) := SL₂(ℤ)/±1 and the space Γ∖ℍ of points of ℍ that are not equivalent under the action of Γ.

This is the type of stage where automorphic forms (modular forms) reside. Note that in this case we do not make a distinction between ±1. The theta function θ(z) above, is almost invariant under the action of the subgroup Γ(2) (≅ Γ₀(4)) of SL₂(ℤ) generated by S and T², and is therefore called a Γ(2)-automorphic form.

3.1 QRM

In quantum optics, the QRM is the most fundamental model to describe light-matter interaction. Its Hamiltonian is given by

Here, are the creation and annihilation operators for the quantum harmonic oscillator (bosonic mode, photon or “light”) with angular frequency ω(= 1), the matrices

are the Pauli matrices for a two-level system (particle, qubit or “matter”), g > 0 is the coupling strength between the two-level system and the photon, and 2Δ > 0 is the energy difference between the levels of the two-level system. We may assume that the Hamiltonian H_Rabi acts on the Hilbert space L²(ℝ) ⊗ ℂ² of integrable two-dimensional vector-valued functions.

The addition of a bias term to the Hamiltonian H_Rabi results in the model , called the asymmetric quantum Rabi model (AQRM). The AQRM appears naturally in the experimental realization of deep strong coupling accomplished using the theory of cavity quantum electrodynamics [3].

3.2 Non-commutative harmonic oscillator

The non-commutative harmonic oscillator (NCHO) [4, 5] is defined as a system of ordinary differential equations having a Hamiltonian

acting on L²(ℝ) ⊗ ℂ². When the parameters α, β > 0 satisfy αβ > 1, Q is a self-adjoint operator having only the discrete spectrum (0 <)λ₀ ≤ λ₁ ≤ ⋯ ≤ λ_n ≤ ⋯ ↑ ∞ with multiplicity of at most 2.

The spectral zeta function ζ_Q(s) of the NCHO is given by

Note that when α = β, Q is unitarily equivalent to a couple of harmonic oscillators; thus, ζ_Q(s) = ζ(s). The spectral zeta function ζ_Q(s) has many interesting and fascinating properties. For instance, ζ_Q(s) can be analytically continued to the complex plane with a unique simple pole at s = 1 and similarly to ζ(s), ζ_Q(s) has “trivial zeros” at negative even integers [6].

It is also possible to define analogs of the Apéry numbers from the special values ζ_Q(2), ζ_Q(3), ζ_Q(4) that unveil a rich mathematical structure. For instance, for ζ_Q(2), there are explicit relations with automorphic forms and elliptic curves (Table 1). For ζ_Q(4), one has to venture beyond the usual modular forms and consider natural extensions of Eichler forms (given by generalized Abel integrals) [7] associated with a new cohomology [8, 9]. In the explicit description of the Apéry-like numbers one also encounters integrals of generalized Eisenstein series [10], deeply related to the research started by Shimura in 1982 on holomorphic modular forms for one variable [11].

Table 1. Number-theoretical properties of the spectral zeta function.

3.3 Covering models

One of the motivations for the introduction of the NCHO was to slightly weaken the symmetries imposed on the quantum harmonic oscillator to rise above the Gauss hypergeometric functions appearing in classic representation theory and to consider the resulting spectral zeta function as an extension of ζ(s).

In practice, by using representation theory, we can see that the eigenvalue problem of the NCHO does goes beyond the Gaussian hypergeometric differential equations^*6 and corresponds to the existence of holomorphic solutions of a Heun ordinary differential equation (ODE), with four singular points {0, 1, αβ, ∞}, in a region containing {0, 1} but not αβ [12, 13]. By joining the singular point αβ and ∞ via a confluence process, we obtain a confluent Heun ODE, which corresponds directly to the eigenvalue problem of the QRM.

We may thus say that the NCHO is a covering model of the QRM by looking at the corresponding Heun ODE pictures [13]. The same covering relation holds for the η-NCHO, a shifted version of the NCHO and AQRM [14]. It was discovered [15] that the eigenvalue problem of the NCHO gives rise to a long-established physical model called the two-photon quantum Rabi model (tpQRM) [16]. This shows that the interaction between one photon and a two-level atom (QRM) can be obtained from that between two photons and a two-level atom (tpQRM) via the covering relation. It would be interesting to confirm the mathematical concept of covering between these physical models through actual experiments. It is also worth remarking that in a previous study [15], using representation theory, the covering relation takes a simpler and clearer form. Further exploring the physical and number theoretical implications of the covering relations is one of the promising research directions in this area.

3.4 Partition function and spectral zeta functions

In this section, we consider a quantum system with self-adjoint Hamiltonian H. As mentioned in the introduction, we are interested in knowing the action of the unitary operator exp(–itH) (propagator/heat kernel) and its trace, the partition function Z_H(β) of H. The partition function is defined as the sum of the Boltzmann factors exp(–βE(μ)), where E(μ) is the energy (eigenvalue) of the state μ, that is, it is given by

where Ω is the set of all possible eigenstates of H. The partition function is one of the fundamental tools of statistical mechanics for the study of entropy and other properties of a system in thermodynamical equilibrium.

On the other hand, the spectral zeta function ζ_H(s) is defined as the Dirichlet series determined by the eigenvalue sequence E(μ). We assume for simplicity^*7 that E(μ) ≠ 0. Concretely, ζ_H(s) is given by

Therefore, by the definition of Γ(s), the two functions are connected via the Mellin transform

It is important to mention that the long awaited explicit formulas of the heat kernel and partition function of the QRM and AQRM were finally obtained [17–19]. The technique for the computation is based on the Trotter–Kato product formula, regarded as the mathematical formulation of the Feynman path integral, multivariate Gaussian integrals and the Fourier transform in interpreted as a Weyl representation of SL₂(𝔽₂), where 𝔽₂ denotes the field with 2 elements. The series expression for the heat kernel corresponds to the sum of irreducible representations of the decomposition of the action of the infinite symmetric group 𝔖_∞ over , and each summand is an orbital integral of 𝔖_∞. This is a surprising discovery that gives further hints on the computation of the heat kernel for more general models, and in general to the structure of interaction models.

The partition function also provides a short proof of the analytic continuation of the corresponding spectral zeta function using a path integral expression going from infinity to the origin, then circling the origin and back to infinity^*8, and extensions of Bernoulli numbers for the spectral zeta function (e.g. Rabi-Bernoulli polynomials for the QRM). Note that the partition function can be recovered from the Rabi-Bernoulli polynomials since the Laurent expansion at the origin of the generating functions of the Rabi-Bernoulli polynomials is equal to the partition function. Although integral expressions for the positive integer points for the spectral zeta function of the NCHO are known, an explicit expression for the partition function has not been obtained. The values at negative integers may also be regarded as a generalization of Bernoulli numbers, called NC-Bernoulli numbers. Therefore, we might expect that the partition function is given by the Laurent series at the origin of the generating function of the NC-Bernoulli numbers. Unfortunately, at present, even with strong supporting evidence, it remains as a conjecture [20].

Nevertheless, the research towards conjecture has illuminated several aspects of the theory. For instance, to work with the formal expressions of the special values of the spectral zeta functions arising from the partition functions, it is useful to consider Borel-summation and non-Archimedian methods to deal with certain divergent series. In particular, certain expressions of special values of zeta functions that are divergent in ℝ may be interpreted as special values of zeta (Hurwitz type) functions [21] defined in the p-adic fields ℚ_p [20].