Rabi oscillations and Floquet states are likely the most familiar concepts associated with a periodically time-varying Hamiltonian. Here, we present an exactly solvable model of a two-level system coupled to both a continuum and a classical field that varies sinusoidally with time, which sheds light on the relationship between the two problems. For a field of the rotating-wave-approximation form, results show that the dynamics of the two-level system can be mapped exactly onto that for a static field, if one shifts the energy separation between the two levels by an amount equal to ℏ ω⁠, where ω is the frequency of the field and ℏ is Planck's constant. This correspondence allows one to view Rabi oscillations and Floquet states from the simpler perspective of their time-independent-problem equivalents. The comparison between the rigorous results and those from perturbation theory helps clarify some of the difficulties underlying textbook proofs of Fermi's golden rule, and the discussions on quantum decay and linear response theory.

Rabi oscillations arise when a few-level quantum system is subjected to a periodically time-varying field. These oscillations are ubiquitous in many areas of physics, from atomic1,2 and condensed-matter physics to quantum information science, having been reported for Rydberg atoms,3 nuclear spin states of different exchange symmetry,4 cavity polaritons,5–7 impurity states in insulators,8 excitons,9 and spin states in quantum dots10,11 as well as Josephson junctions.12 

Floquet states are stationary solutions of a quantum system in the presence of a periodic time-dependent perturbation. Such states have attracted much attention recently, arising primarily from the possibility that new phases with intriguing topological properties may emerge under high-intensity excitation.13–18 

Past the simplest derivations, students interested in these topics (and researchers from other disciplines) may fail to grasp the connection between Rabi oscillations and Floquet states beyond the obvious fact that they both involve a time-varying field. Equally important, they may struggle to find pedagogical accounts in the literature discussing the effects of dissipation19 and spontaneous decay, which are central to the interpretation of many physical phenomena. They may also ponder about the question as to how Rabi oscillations and Floquet states relate to results of time-dependent perturbation theory, and particularly to linear response theory and Fermi's golden rule.

Some of the underlying difficulties in establishing these relationships stem from the fact that time-dependent two-level models that are exactly solvable are very rare,20 and also because many studies are highly specialized and often rely on the density matrix formalism or Bloch equations, with which students (and researchers) may not be very familiar. Here, we present an exactly solvable two-level model that allows one to understand the role of dissipation and explore the connections between all these matters in a unified manner, including the somewhat hidden relationship that exists between effects due to resonant time-varying excitations and static fields.

We consider the two-level system depicted in Fig. 1, with the ground state | g ⟩ and the excited state | e ⟩ of energies Eg and Ee, respectively. The higher-energy state couples to a continuum of non-degenerate states | χ E ⟩⁠, of energy E, and its energy lies within this continuum. A classical field, of amplitude F 0⁠, which varies periodically in time with frequency ω, can induce transitions between the two discrete levels. The full Hamiltonian is
(1)
with ⟨ g | g ⟩ = ⟨ e | e ⟩ = 1⁠, ⟨ g | e ⟩ = 0⁠, ⟨ χ E | χ E ′ ⟩ = δ ( E − E ′ )⁠, and ⟨ χ E | e ⟩ = ⟨ χ E | g ⟩ = 0⁠; V e g is the matrix element for the interaction of the two-level system with the field and ξ E is the coupling coefficient between | e ⟩ and | χ E ⟩⁠; and asterisks denote the complex conjugate. The classical field is treated here in the rotating-wave approximation (RWA) form,21 which provides a very accurate account of near-resonant situations, that is, ℏ ω ≈ E e − E g (for a brief introduction to the RWA and a numerical example, see the supplementary material).22 Although the above Hamiltonian is self-adjoint, as required, the interaction does not describe coupling to a physical field such as the electric field, which must be expressed in terms of real as opposed to complex quantities (that is, cos   ω t instead of e ± i ω t⁠). The crucial advantage of the RWA form is that it makes the problem exactly solvable.
Fig. 1.
Illustration of the model. A harmonically time-varying classical field of amplitude | F 0 | couples the ground state | g ⟩ to the excited state | e ⟩ of a two-level system, which itself is coupled to a continuum of states | χ E ⟩.
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Illustration of the model. A harmonically time-varying classical field of amplitude | F 0 | couples the ground state | g ⟩ to the excited state | e ⟩ of a two-level system, which itself is coupled to a continuum of states | χ E ⟩⁠.

Fig. 1.
Illustration of the model. A harmonically time-varying classical field of amplitude | F 0 | couples the ground state | g ⟩ to the excited state | e ⟩ of a two-level system, which itself is coupled to a continuum of states | χ E ⟩.
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Illustration of the model. A harmonically time-varying classical field of amplitude | F 0 | couples the ground state | g ⟩ to the excited state | e ⟩ of a two-level system, which itself is coupled to a continuum of states | χ E ⟩⁠.

Close modal

The above Hamiltonian combines two well-known models. For ξ E = 0⁠, H describes, within the RWA, the effect of a time-varying electromagnetic field on a pair of levels, leading to the celebrated Rabi oscillations,23 briefly reviewed here in Sec. III. The case F 0 = 0 reduces to the Friedrichs model,24 which provides a simple and physically well-grounded account of quantum decay. This model is discussed in Sec. IV. Among many other examples, the Friedrichs Hamiltonian has been utilized to explain molecular predissociation25 and radiative decay of an atomic state into a lower-energy level (in which case the continuum is that of one-photon states),26 as well as in a phenomenological analysis of nucleon-antinucleon annihilation,27 and studies of quantum instability in general.28 

For ξ E = 0⁠, the problem reduces to that of solving Schrödinger equation ( H R − i ∂ / ∂ t ) | Ψ ( t ) ⟩ = 0 (we take ℏ = 1 throughout the paper)29 for the Rabi Hamiltonian
(2)
For the benefit of the reader, and because we will use repeatedly the same approach in later derivations, we reproduce here the procedure for solving the problem, following Rabi's original work23 and as presented in many textbooks.30–34 We search for solutions of the form
(3)
leading to
(4)
where Θ = E e − E g − ω⁠. The replacement C κ = a κ e i α κ t⁠, with κ = { e , g }⁠, gives α e − α g = Θ and α g α e = | F 0 V e g | 2⁠. Thus, we obtain two sets of solutions: α g = ( − Θ / 2 ± Ω ) and   α e = ( Θ / 2 ± Ω )⁠, where Ω = Θ 2 / 4 + | F 0 V e g | 2 so that the two normalized, independent wavefunctions are
(5)
Suppose now that the system is in the ground state at t = 0⁠. Using the proper linear combination of the two solutions, we get
(6)
Hence, the probability of finding the system in the excited state is
(7)
The probability oscillations, of period π / Ω⁠, are known as Rabi oscillations.
It is interesting to point out that Eqs. (6) and (7) are identical to those for a time-independent perturbation with unperturbed energy difference E e − E g − ω⁠. This means that the time-varying problem reduces to the static one if one simply changes the gap between the two levels from its actual value, E e − E g⁠, to | E e − E g − ω |⁠. The same result can be obtained directly from the integro-differential equation
(8)
which follows from Eq. (4). Clearly, the time dependence of C g⁠, as it concerns Θ, does not distinguish between the two cases. We will see later that the time-dependent to time-independent transformation also applies to the full Hamiltonian. The equivalence between the two problems is closely related to the transformation from a stationary to a rotating frame used in nuclear magnetic resonance and atomic physics problems.21 Although a result derived from the RWA, we emphasize the fact that this correspondence is applicable to actual physical situations since the RWA is an excellent approximation near resonant conditions, that is, Θ ≈ 0⁠.22 
For F 0 = 0⁠, we obtain the Friedrichs Hamiltonian24,
(9)
which describes the coupling of a discrete state, in this case | e ⟩⁠, to a continuum. Under certain conditions (more on this later), and provided E e lies within the continuum, the spectrum of eigenvalues of H F is a continuum identical to that of ∫ U | χ U ⟩ ⟨ χ U | d U⁠.35 Following closely the notation of Ref. 26, we write the eigenvectors as
(10)
H F | Φ E ⟩ = E | Φ E ⟩ gives
(11)
(12)
If U ≠ E⁠, we get from Eq. (11) B U ( E ) = ξ U A ( E ) / ( E − U )⁠. Moreover, if ξ U = 0⁠, the solutions that belong to the continuum are | Φ E ⟩ ≡ | χ E ⟩ and, therefore, B U ( E ) = δ ( U − E )⁠. Thus, we write
(13)
and get from Eq. (12)
(14)
with (the principal value is understood)
(15)
We obtain A ( E ) from the normalization condition ⟨ Φ E | Φ E ′ ⟩ = δ ( E − E ′ )⁠. A simple calculation gives
(16)
For E ≠ E ′⁠, the integrand can be decomposed into partial fractions to exactly cancel the second term inside the brackets. To account for the double pole, we multiply by e i ε U⁠, as a regulator, and add a small imaginary term to the energy so that the singular term becomes26 
(17)
Finally,
(18)
Since ⟨ e | Φ ( E ) ⟩ = A ( E )⁠, see Eq. (10), and, thus, | e ⟩ = ∫ A * ( E ) | Φ ( E ) ⟩ d E⁠, this Lorentzian-type expression describes the broadening and energy shift of the discrete state resulting from its coupling to the continuum, given, respectively, by | ξ E | 2 and F(E). Coupling renders the discrete state unstable in that, if | Ψ ⟩ = | e ⟩ at t = 0⁠, then
(19)
Hence, the probability of finding the system in the state | e ⟩ at later times is
(20)
The time behavior of the probability depends not only on the strength of the coupling but also on the bandwidth of the continuum spectrum, Γ. The results in Fig. 2 are representative of the case ξ E 2 ≪ Γ and, more broadly, of the generic behavior of unstable quantum systems.36 Here, we assume that the continuum spectrum is bounded from below, with E > 0, and, moreover, that ξ E 2 , F ( E ) ≪ E e⁠, so that the energy-dependence of the coupling and shift can be ignored. The probability exhibits three different regimes. As shown in the inset of Fig. 2, at small times, P depends quadratically on time. This is consistent with the requirement that the mean value of the energy be finite, which dictates that d P / d t | t = 0 = 0⁠.37 Following this so-called Zeno regime,38 the probability exhibits exponential decay, the most common decay behavior found in nature, with the decay time τ = 1 / 2 π ξ 0 2⁠, where ξ 0 = ξ E 0 and E 0 = E e + F ( E e )⁠. Finally, at much longer times, P shows an algebraic dependence, in agreement with general theory.39 The oscillations at the crossover time t ≈ τ   log   ( E 0 τ ) reflect interference between the exponential and algebraic expressions. Since
(21)
it is apparent that the short and long term behaviors of the probability are contained in the integral of | A ( E ) | 2 for the interval [ − ∞ , 0 ]⁠.
Fig. 2.
Friedrichs model ( F 0 = 0) and decay. Probability P of the system remaining in the discrete state | e ⟩ (log scale) as a function of time, τ is the decay time. The main figure shows the transition from exponential to algebraic decay for E 0 = 200 and ξ 0 = 2 (arbitrary units). Dashed curves are asymptotes. The oscillations result from the interference between the expressions for the two regimes. Inset: Zeno region showing deviation from exponential behavior at early times; E 0 = 5000 and ξ 0 = 1 (arbitrary units).
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Friedrichs model (⁠ F 0 = 0⁠) and decay. Probability P of the system remaining in the discrete state | e ⟩ (log scale) as a function of time, τ is the decay time. The main figure shows the transition from exponential to algebraic decay for E 0 = 200 and ξ 0 = 2 (arbitrary units). Dashed curves are asymptotes. The oscillations result from the interference between the expressions for the two regimes. Inset: Zeno region showing deviation from exponential behavior at early times; E 0 = 5000 and ξ 0 = 1 (arbitrary units).

Fig. 2.
Friedrichs model ( F 0 = 0) and decay. Probability P of the system remaining in the discrete state | e ⟩ (log scale) as a function of time, τ is the decay time. The main figure shows the transition from exponential to algebraic decay for E 0 = 200 and ξ 0 = 2 (arbitrary units). Dashed curves are asymptotes. The oscillations result from the interference between the expressions for the two regimes. Inset: Zeno region showing deviation from exponential behavior at early times; E 0 = 5000 and ξ 0 = 1 (arbitrary units).
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Friedrichs model (⁠ F 0 = 0⁠) and decay. Probability P of the system remaining in the discrete state | e ⟩ (log scale) as a function of time, τ is the decay time. The main figure shows the transition from exponential to algebraic decay for E 0 = 200 and ξ 0 = 2 (arbitrary units). Dashed curves are asymptotes. The oscillations result from the interference between the expressions for the two regimes. Inset: Zeno region showing deviation from exponential behavior at early times; E 0 = 5000 and ξ 0 = 1 (arbitrary units).

Close modal
As mentioned earlier, if E e is inside the continuum, the eigenfunctions of the Friedrichs model are usually extended, in that they satisfy the normalization condition ⟨ Φ E | Φ E ′ ⟩ = δ ( E − E ′ )⁠. This is always the case for ξ E → 0⁠; the spectral decomposition and the completeness relation are not analytic functions of the coupling constant, as an infinitesimal interaction leads to the disappearance of the discrete state.24 If the coupling is large enough and E e is not far from the bottom of the continuum, however, a new discrete (or localized) state | d ( − ) ⟩⁠, with ⟨ d ( − ) | d ( − ) ⟩ = 1⁠, may emerge and position itself below the continuum.25,40 Similarly, a state | d ( + ) ⟩ with ⟨ d ( + ) | d ( + ) ⟩ = 1⁠, may appear above a continuum with a high-energy cutoff. From Eqs. (11) and (12), we find that the condition for the occurrence of discrete states of energy E d is
(22)
and, therefore,
(23)
It is thus apparent that the new states, should they exist, will have a significant content of the original state | e ⟩ so that, in Eq. (20), we would need to add terms of the form | A ( E d ) | 2 δ ( E − E d ) to the Lorentzian-like, continuous contribution, Eq. (18). Note that, if the distance from the new states to the edges of the continuum is large enough
(24)
and, therefore,
(25)
which approaches ½ for | E d | ≫ E e (all along, we have assumed that E e is inside the continuum; if it were outside, the coupling will push the state further out because of level repulsion).

Figures 3 and 4 show examples of energy-dependent couplings leading to effective bandwidths that are sufficiently small to allow for the development of extra peaks at large values of the interaction. Results for ξ E = Λ e − ( E − E e ) p / Γ p with p = 2 (Gaussian) and p = 8 (nearly flat) are shown, respectively, in Figs. 3 and 4. For small coupling, | A ( E ) | 2 exhibits a single peak, which turns into a doublet for Λ 2 ≳ Γ / 2 π (also note the three-peak structure at Λ = 0.17 in Fig. 4, which resembles the Mollow triplet).41 Unlike the discrete states of the previous discussion, the peaks in Figs. 3 and 4 are, strictly speaking, resonant states, given that their energies overlap with the continuum. The occurrence of discrete or resonant doublets, containing a large fraction of | e ⟩⁠, profoundly alters the time-dependence of the probability of Eq. (20), since such states necessarily lead to periodic probability oscillations. The correspondence between the static and time-varying problems discussed in Sec. III indicates that the doublet-induced oscillations are the static counterpart to Rabi oscillations (see Sec. V).

Fig. 3.
Friedrichs model ( F 0 = 0). Gaussian energy-dependent coupling ξ E = Λ e − ( E − E e ) 2 / Γ 2 for Γ = 0.1 and E e = 0. The top panel shows A 2 ( E ) vs E for various values of Λ; see Eq. (18). Curves have been shifted vertically for clarity. ξ E is shown in the bottom panel.
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Friedrichs model (⁠ F 0 = 0⁠). Gaussian energy-dependent coupling ξ E = Λ e − ( E − E e ) 2 / Γ 2 for Γ = 0.1 and E e = 0⁠. The top panel shows A 2 ( E ) vs E for various values of Λ; see Eq. (18). Curves have been shifted vertically for clarity. ξ E is shown in the bottom panel.

Fig. 3.
Friedrichs model ( F 0 = 0). Gaussian energy-dependent coupling ξ E = Λ e − ( E − E e ) 2 / Γ 2 for Γ = 0.1 and E e = 0. The top panel shows A 2 ( E ) vs E for various values of Λ; see Eq. (18). Curves have been shifted vertically for clarity. ξ E is shown in the bottom panel.
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Friedrichs model (⁠ F 0 = 0⁠). Gaussian energy-dependent coupling ξ E = Λ e − ( E − E e ) 2 / Γ 2 for Γ = 0.1 and E e = 0⁠. The top panel shows A 2 ( E ) vs E for various values of Λ; see Eq. (18). Curves have been shifted vertically for clarity. ξ E is shown in the bottom panel.

Close modal
Fig. 4.
Same as Fig. 3 but for the nearly flat, energy-dependent coupling ξ E = Λ e − ( E − E e ) 8 / Γ 8 with Γ = 0.15 and E e = 0.
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Same as Fig. 3 but for the nearly flat, energy-dependent coupling ξ E = Λ e − ( E − E e ) 8 / Γ 8 with Γ = 0.15 and E e = 0⁠.

Fig. 4.
Same as Fig. 3 but for the nearly flat, energy-dependent coupling ξ E = Λ e − ( E − E e ) 8 / Γ 8 with Γ = 0.15 and E e = 0.
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Same as Fig. 3 but for the nearly flat, energy-dependent coupling ξ E = Λ e − ( E − E e ) 8 / Γ 8 with Γ = 0.15 and E e = 0⁠.

Close modal
A somehow more direct way27 to obtain | A ( E ) | 2 is to consider, instead of | Φ E ⟩⁠, Eq. (10),
(26)
H F | Ψ ( t ) ⟩ = i ( ∂ / ∂ t ) | Ψ ( t ) ⟩ gives
(27)
If a ( t ) = 1 at t = 0⁠,
(28)
and we get the integro-differential equation
(29)
which is the Friedrichs-model equivalent to Eq. (8). If we write a ( t ) = ∫ a ( E ) e − iEt d E⁠, we get
(30)
Comparing this expression with the identity ∫ | A 2 ( E ) | ( E − U ) − 1   d E + z ( U ) | A 2 ( U ) | = 0⁠, which follows from the closure relation ∫ A ( E ) B U ( E ) d E = 0⁠, it is apparent that a ( E ) = | A 2 ( E ) | and, thus, that, a ( t ) = ∫ | A 2 ( E ) | e − iEt d E⁠, as it should be. Equation (29) can be solved numerically, offering an alternative to the Fourier transform approach of Eq. (20) to calculate the probability. As discussed in Sec. V, the time-varying potential leads to an expression formally identical to Eq. (29), reflecting the already mentioned correspondence between the static and time-varying cases.
For completeness, we conclude this section by describing the time dependence of the continuum as | e ⟩ decays. To that end, we consider massless particles with energy E = c q (c is a velocity and q is a momentum) in s-like extended states ⟨ r | χ q ⟩ = 2 / π   sin   ( q r ) / r⁠, normalized according to
(31)
The continuum component of | Φ E ⟩ is
(32)
Since lim r → ∞ ∫ 0 ∞   sin   ( q r ) ( q − k ) − 1 d q = π   cos   k r⁠, we get at large r
(33)
where Δ q = − arctan [ π / z ( q ) ] is a phase shift reflecting the effect of the interaction with the discrete state.26 Assuming that the coupling is weak enough so that F ( E ) ≈ 0⁠, we obtain at large distances
(34)
After a lengthy but straightforward calculation, we find
(35)
where A 2 = ξ 0 2 / 2 c and θ is the Heaviside step function. This represents an outgoing wave with a sharp edge and maximum at r = c t⁠.42 Note that 4 π ∫ | ψ | 2 r 2 d r = 1 − e − t / τ⁠, as required to conserve probability and that, unlike | ψ E ⟩⁠, this superposition state is localized.
We now have all the ingredients necessary to tackle the full Hamiltonian. In the following, we assume that the spectrum of H F does not contain discrete states. Using the closure relation
(36)
and | e ⟩ = ∫ A * ( E ) | Φ E ⟩ d E⁠, we get
(37)
This Hamiltonian describes the field-mediated coupling of the ground state to the | Φ E ⟩-continuum through a modified coupling constant V e g A * ( E )⁠. As before, we search for solutions of the form
(38)
H | Ψ ( t ) ⟩ = i ∂ | Ψ ( t ) ⟩ gives
(39)
If C g = 1 at t = 0⁠, then C E ( t ) = − i F 0 V e g A * ( E ) × ∫ 0 t C g ( t ′ ) e − i ( ω + E g − E ) t ′ d t ′ and we get
(40)
Based on the discussion of Sec. IV, the comparison of the above expression with Eq. (29) reaffirms our assertion that the time-varying problem is equivalent to a static one with energy difference | E e − E g − ω |⁠, and, moreover, it tells us immediately that the solution is
(41)
where
(42)
The results in Fig. 5 are for the (resonant) case when ω = E e − E g⁠. For small values of the field, that is, in the weak-field limit, | Ξ ( E ) | 2 shows a single peak at ≈ E e and, therefore, C g ( t ) ≈   exp   ( − | F 0 V e g | 2 t / π ξ E e 2 ) (as for | e ⟩ in Sec. IV, if | Ψ ( 0 ) ⟩ = | g ⟩⁠, the probability that the system remains in the ground state goes to zero for t → ∞⁠). The results for F 0 = 0.017 fall in this regime. It is interesting to note that the decay constant decreases with increasing coupling of | e ⟩ to the | χ E ⟩-continuum, a fact that reflects the lower effective density of states, as defined by | A ( E ) | 2⁠, at larger coupling.
Fig. 5.
From decay to Rabi oscillations. The main figure shows the time-dependence of the probability of the system to remain in the ground state under resonant conditions for F 0 = 0.017 (gray) and F 0 = 0.158 (black). Coupling of | e ⟩ to the continuum is energy-independent with ξ E = ξ 0 = 0.1. Inset: | Ξ 2 ( E ) | for the corresponding values of the field and E e = 0; see Eq. (42). The two curves have been shifted vertically for clarity and the units are arbitrary.
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From decay to Rabi oscillations. The main figure shows the time-dependence of the probability of the system to remain in the ground state under resonant conditions for F 0 = 0.017 (gray) and F 0 = 0.158 (black). Coupling of | e ⟩ to the continuum is energy-independent with ξ E = ξ 0 = 0.1⁠. Inset: | Ξ 2 ( E ) | for the corresponding values of the field and E e = 0⁠; see Eq. (42). The two curves have been shifted vertically for clarity and the units are arbitrary.

Fig. 5.
From decay to Rabi oscillations. The main figure shows the time-dependence of the probability of the system to remain in the ground state under resonant conditions for F 0 = 0.017 (gray) and F 0 = 0.158 (black). Coupling of | e ⟩ to the continuum is energy-independent with ξ E = ξ 0 = 0.1. Inset: | Ξ 2 ( E ) | for the corresponding values of the field and E e = 0; see Eq. (42). The two curves have been shifted vertically for clarity and the units are arbitrary.
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From decay to Rabi oscillations. The main figure shows the time-dependence of the probability of the system to remain in the ground state under resonant conditions for F 0 = 0.017 (gray) and F 0 = 0.158 (black). Coupling of | e ⟩ to the continuum is energy-independent with ξ E = ξ 0 = 0.1⁠. Inset: | Ξ 2 ( E ) | for the corresponding values of the field and E e = 0⁠; see Eq. (42). The two curves have been shifted vertically for clarity and the units are arbitrary.

Close modal

The calculations for F 0 = 0.158 show the characteristic splitting in the strong-field limit (referred to as the Autler–Townes or AC Stark effect in atomic physics),43 which leads to the Rabi oscillations shown in Fig. 5. Once again, we emphasize the close relationship that exists between the doublet in Fig. 5 and those of Figs. 3 and 4 (static case), as well as among the single peaks these figures show, all of which lead to a time-dependence of the probability as in Fig. 2.

In this section, we compare the rigorous results with the approximate ones of perturbation theory, limiting the discussion to the decay (as opposed to the oscillatory) aspects of the problem. In Eq. (39), we approximate C g ≈ 1 and get, to first order in F 0⁠,
(43)
Note that | C E ( t ) | 2 ∝ t 2 for t → 0 and that the lowest order correction to | C g | 2 is ∝ | F 0 | 2⁠, which are both consistent with results for the so-called Zeno regime (see the inset of Fig. 2). The first-order correction to the wavefunction is
(44)
For perturbation theory to be valid, one requires that the probability of not being in the state | g ⟩ be small compared with unity, that is,
(45)
The term multiplying | A ( E ) | 2 exhibits a peak at E = ω + E g⁠, the width of which is inversely proportional to t. Provided this width is small compared with the energy scale under which | A ( E ) | 2 varies substantially, we can replace | A ( E ) | 2 with | A ( ω + E g ) | 2 and take it out of the integral. Alternatively, and as presented in most textbooks,44 we can use the approximation
(46)
Either way, the probability of not being in the ground state increases with time as
(47)
This key result is consistent with the weak-field limit result (gray curve in Fig. 5) in that, beyond the Zeno regime, the probability of being in the ground state is
(48)
Identical considerations apply to the Friedrichs problem, that is, to the decay of | e ⟩⁠, where the perturbation does not vary in time. Using the same procedure as above, we obtain from Eq. (26) to first order in ξ E
(49)
Thus, the probability of this state belonging to the continuum, given by
(50)
increases linearly with time, at the rate 2 π | ξ E e | 2⁠, beyond a characteristic time defined by the energy scale of | ξ E | 2⁠. This is Fermi's second golden rule (the first one involves second-order perturbation theory),45 first derived by Dirac.46 
Although the mathematics is the same, a fact that reflects the equivalence between the two problems, it is important to emphasize the differences between the static (Sec. IV) and time-varying (Sec. V) cases. The golden-rule result—that the probability of not being in the discrete state increases at a constant rate—is valid for times that are both sufficiently large so that either | A ( E ) | 2 or | ξ U | 2 can be taken out of the respective integrals and sufficiently small for perturbation theory to apply. Concerning | A ( E ) | 2⁠, Eq. (45), the upper limit to the relevant energy scale is given by | ξ E | 2 and, therefore, the golden rule in the time-varying case applies at least for
(51)
On the other hand, in the static case, Eq. (50), we have
(52)
where Δ ξ denotes the energy scale for which the coupling of | e ⟩ to the continuum changes appreciably. Notice that the right-hand side in the above two equations is the decay time, which generally sets the upper bound for the validity of the golden rule, and that the left-hand side determines the time beyond which the Zeno regime no longer applies.
It is apparent from the previous discussion that Fermi's golden rule applies solely to the early time behavior of the decay and that, as such, it does not explain the exponential law. (This point is not well emphasized in some textbooks.) The earliest theory to account for exponential decay was developed by Weiskopf and Wigner.47 Their model relies in part on arguments similar to those in the derivation of the golden rule, but the physical assumptions are quite different. Focusing on the static case, exponential decay follows from Eq. (29) by assuming that the system has no memory of the past so that a ( t ′ ) can be taken outside the integral, leading to
(53)
Again, at times that are large compared with Δ ξ⁠, we can also take | ξ U | 2 outside the integral arriving at a ̇ ( t ) ≈ − a ( t ) | ξ E e | 2⁠, which gives exponential decay.
Even though the golden-rule by itself does not lead to exponential decay, it does so if the decaying system is constantly monitored at small, regular, or random intervals. This somehow surprising result follows from the identity:
(54)
which is valid for arbitrary (random or otherwise) 0 < δ i ≪ 1⁠, and the assumption that a perfect measurement that fails to observe the decay products collapses the system back to its undecayed state.48 If we take δ i = Δ t i / τ⁠, where Δ t i ≪ τ is the time interval between measurements and τ is the decay time, the above product becomes exp   ( − ∑ i = 1 , N Δ t i / τ )⁠. According to the golden rule, this represents the probability of finding the unstable system in its undecayed state at t = ∑ i = 1 , N Δ t i⁠.
Returning to the driven two-level system, and using the first-order perturbative expression for the wavefunction, Eq. (44), the expectation value of an arbitrary operator O that connects the states of the two-level system is (c.c. denotes complex conjugation)
(55)
where we used ⟨ g | O | Φ E ⟩ = A * ( E ) ⟨ g | O | e ⟩⁠. Assuming that | ξ E | 2 ≪ E e and extending the integration limits to the interval {−∞,+∞}, we get after a contour integration
(56)
After the transient, that is, for t ≫ 1 / π | ξ E e 2 |⁠, one recovers the result of linear response theory49 
(57)
with the response function
(58)
If the operator is the same one associated with the field, R becomes the complex susceptibility
(59)
We recall that the imaginary component of the response function is associated with dissipation, which is another manifestation of the golden rule. Also note that, like the golden rule, Eq. (56) and, more generally, the results of linear response theory are only valid for times that are small compared with the decay time, in our case t ≪ π ξ E e 2 / 2 | F 0 V e g | 2⁠, and that the response, decreasing exponentially with time, vanishes for t → ∞⁠.
For linear differential equations and, in particular, for a Hamiltonian that depends periodically on time, Floquet's theorem tells us that there exists a complete set of solutions of the form50,51
(60)
where E is referred to as the quasienergy and | Φ ( t ) ⟩ is a periodic function with the same period as the Hamiltonian obeying
(61)
The two solutions to Rabi's problem we found in Sec. III, see Eq. (5), are clearly of the Floquet form, with
(62)
Solutions to Eq. (61) can be expressed in terms of the complete set of time-independent eigenstates of H for ω = 0⁠, together with the infinite set of functions of the form e ± i n ω⁠. Given this, one would expect that the number of solutions at a given ω would be infinitely large. The reason why there are only two Rabi eigenfunctions is that the only non-zero matrix elements of the time-dependent perturbation in our case are (p,m are integers)
(63)
where V = ( F 0 e − i ω t V e g | e ⟩ ⟨ g | + F 0 * V e g * e i ω t | g ⟩ ⟨ e | ) and m, p denote, respectively, the functions e im ω t and e ip ω t⁠. This argument also applies to the full Hamiltonian, Eq. (1), and is the primary reason why it can be solved exactly. Had we used cos   ω t in Eq. (1) instead of the RWA form, e ± i ω t⁠, we would have had an extra term in the right-hand side of Eq. (63), namely, F 0 δ m + 2 , p⁠, which would have made the problem analytically intractable.22 
To obtain the Floquet solutions to the full Hamiltonian, replace C g ( t ) = a g e − i α g t and C U ( t ) = a U e − i α U t in Eq. (39). This gives − α E + α g = E − ω − E g and
(64)
If α U ≠ 0⁠, a U = F 0 A * ( U ) a g / α U⁠. Thus, we replace
(65)
to get (principal value is understood)
(66)
Hence, the Floquet solutions are (⁠ a E is a normalization constant)
(67)
with E = E g + α g⁠. Reflecting the correspondence between the two problems, the comparison with Eq. (10) shows that these states are the time-varying counterparts to the eigenstates of the Friedrichs Hamiltonian.

The above results combined with those of Sec. V indicate that for almost any initial condition, specifically for states of the form ∫ C E | Ψ E ⟩ d E , where C E is a continuous function (this excludes the Floquet states themselves), the probability of finding the system in the ground state vanishes at infinite time. This means that, in some sense and regardless of the strength of the applied field, the driven two-level system heats up to ever-larger temperatures as time increases,52 mimicking the thermalization behavior of a generic many-body system.53 

In this work, we obtained exact solutions for the time-dependent wavefunctions of the system depicted in Fig. 1, using the RWA form for the classical field.22 The comparison between the rigorous results and those from perturbation theory helps bring into light some of the subterfuges underlying the proofs of Fermi golden rule,54 quantum decay, and linear response theory. The main takeaway from our results is that the behavior of a two-level system in the presence of a sinusoidally time-varying field can be mapped onto the problem of a static field, if one changes the separation of the two levels from its actual value to | E e − E g − ω |⁠. This correspondence allows one to relate Rabi oscillations and Floquet states to their equivalent time-independent problems.

The author is grateful to P. R. Berman, G. W. Ford, P. W. Milonni, and M. A. Sentef for their comments and suggestions.

1.
B. W.
Shore
 and
P. L.
Knight
, “
The Jaynes-Cummings model
,”
J. Mod. Opt.
40
,
1195
–
1238
(
1993
).
https://doi.org/10.1080/09500349314551321
Google Scholar
Crossref
Search ADS
 
2.
P. L.
Knight
 and
P. W.
Milonni
, “
The Rabi frequency in optical spectra
,”
Phys. Rep.
66
,
21
–
107
(
1980
).
https://doi.org/10.1016/0370-1573(80)90119-2
Google Scholar
Crossref
Search ADS
 
3.
Y. O.
Dudin
,
L.
Li
,
F.
Bariani
, and
A.
Kuzmich
, “
Observation of coherent many-body Rabi oscillations
,”
Nat. Phys.
12
,
790
–
794
(
2012
).
https://doi.org/10.1038/nphys2413
Google Scholar
Crossref
Search ADS
 
4.
G.
Pileio
,
M.
Carravetta
, and
M. H.
Levitt
, “
Extremely low-frequency spectroscopy in low-field nuclear magnetic resonance
,”
Phys. Rev. Lett.
103
,
083002
(
2009
).
https://doi.org/10.1103/PhysRevLett.103.083002
Google Scholar
Crossref
Search ADS
PubMed
 
5.
R.
Houdré
,
R. P.
Stanley
,
U.
Oesterle
,
M.
Ilegems
, and
C.
Weisbuch
, “
Room-temperature cavity polaritons in a semiconductor microcavity
,”
Phys. Rev. B
49
,
16761
–
16764
(
1994
).
https://doi.org/10.1103/PhysRevB.49.16761
Google Scholar
Crossref
Search ADS
 
6.
H.
Deng
,
H.
Haug
, and
Y.
Yamamoto
, “
Exciton-polariton Bose-Einstein condensation
,”
Rev. Mod. Phys.
82
,
1489
–
1537
(
2010
).
https://doi.org/10.1103/RevModPhys.82.1489
Google Scholar
Crossref
Search ADS
 
7.
A.
Amo
,
J.
Lefrère
,
S.
Pigeon
,
C.
Adrados
,
C.
Ciuti
,
I.
Carusotto
,
R.
Houdré
,
E.
Giacobino
, and
A.
Bramati
, “
Superfluidity of polaritons in semiconductor microcavities
,”
Nat. Phys.
5
,
805
–
810
(
2009
).
https://doi.org/10.1038/nphys1364
Google Scholar
Crossref
Search ADS
 
8.
A.
Batalov
,
C.
Zierl
,
T.
Gaebel
,
P.
Neumann
,
I.-Y.
Chan
,
G.
Balasubramanian
,
P. R.
Hemmer
,
F.
Jelezko
, and
J.
Wrachtrup
, “
Temporal coherence of photons emitted by single nitrogen-vacancy defect centers in diamond using optical Rabi-oscillations
,”
Phys. Rev. Lett.
100
,
077401
(
2008
).
https://doi.org/10.1103/PhysRevLett.100.077401
Google Scholar
Crossref
Search ADS
PubMed
 
9.
T. H.
Stievater
,
Xiaoqin
Li
,
D. G.
Steel
,
D.
Gammon
,
D. S.
Katzer
,
D.
Park
,
C.
Piermarocchi
, and
L. J.
Sham
, “
Rabi oscillations of excitons in single quantum dots
,”
Phys. Rev. Lett.
87
,
133603
(
2001
).
https://doi.org/10.1103/PhysRevLett.87.133603
Google Scholar
Crossref
Search ADS
PubMed
 
10.
J. R.
Petta
,
A. C.
Johnson
,
J. M.
Taylor
,
E. A.
Laird
,
A.
Yacoby
,
M. D.
Lukin
,
C. M.
Marcus
,
M. P.
Hanson
, and
A. C.
Gossard
, “
Coherent manipulation of coupled electron spins in semiconductor quantum dots
,”
Science
309
,
2180
–
2184
(
2005
).
https://doi.org/10.1126/science.1116955
Google Scholar
Crossref
Search ADS
PubMed
 
11.
F. H. L.
Koppens
,
C.
Buizert
,
K. J.
Tielrooij
,
I. T.
Vink
,
K. C.
Nowack
,
T.
Meunier
,
L. P.
Kouwenhoven
, and
L. M. K.
Vandersypen
, “
Driven coherent oscillations of a single electron spin in a quantum dot
,”
Nature
442
,
766
–
771
(
2006
).
https://doi.org/10.1038/nature05065
Google Scholar
Crossref
Search ADS
PubMed
 
12.
J. M.
Martinis
,
S.
Nam
,
J.
Aumentado
, and
C.
Urbina
, “
Rabi oscillations in a large Josephson-junction qubit
,”
Phys. Rev. Lett.
89
,
117901
(
2002
).
https://doi.org/10.1103/PhysRevLett.89.117901
Google Scholar
Crossref
Search ADS
PubMed
 
13.
Y. H.
Wang
,
H.
Steinberg
,
P.
Jarillo-Herrero
, and
N.
Gedik
, “
Observation of Floquet-Bloch states on the surface of a topological insulator
,”
Science
342
,
453
–
457
(
2013
).
https://doi.org/10.1126/science.1239834
Google Scholar
Crossref
Search ADS
PubMed
 
14.
T.
Oka
 and
H.
Aoki
, “
Photovoltaic Hall effect in graphene
,”
Phys. Rev. B
79
,
081406(R)
(
2009
).
https://doi.org/10.1103/PhysRevB.79.081406
Google Scholar
Crossref
Search ADS
 
15.
T.
Morimoto
 and
N.
Nagaosa
, “
Topological nature of nonlinear optical effects in solids
,”
Sci. Adv.
2
,
1501524
(
2016
).
https://doi.org/10.1126/sciadv.1501524
Google Scholar
Crossref
Search ADS
 
16.
M. C.
Rechtsman
,
J. M.
Zeuner
,
Y.
Plotnik
,
Y.
Lumer
,
D.
Podolsky
,
F.
Dreisow
,
S.
Nolte
,
M.
Segev
, and
A.
Szameit
, “
Photonic Floquet topological insulators
,”
Nature
496
,
196
–
200
(
2013
).
https://doi.org/10.1038/nature12066
Google Scholar
Crossref
Search ADS
PubMed
 
17.
N. H.
Lindner
,
H.
Netanel
,
G.
Refael
, and
V.
Galitski
, “
Floquet topological insulator in semiconductor quantum wells
,”
Nat. Phys.
7
,
490
–
495
(
2011
).
https://doi.org/10.1038/nphys1926
Google Scholar
Crossref
Search ADS
 
18.
M. A.
Sentef
,
M.
Claassen
,
A. F.
Kemper
,
B.
Moritz
,
T.
Oka
,
J. K.
Freericks
, and
T. P.
Devereaux
, “
Theory of Floquet band formation and local pseudospin textures in pump-probe photoemission of graphene
,”
Nat. Commun.
6
,
7047
(
2015
).
https://doi.org/10.1038/ncomms8047
Google Scholar
Crossref
Search ADS
PubMed
 
19.
S. A.
Sato
,
J. W.
McIver
,
M.
Nuske
,
P.
Tang
,
G.
Jotzu
,
B.
Schulte
,
H.
Hübener
,
U.
De Giovannini
,
L.
Mathey
,
M. A.
Sentef
,
A.
Cavalleri
, and
A.
Rubio
, “
Microscopic theory for the light-induced anomalous Hall effect in graphene
,”
Phys. Rev. B
99
,
214302
(
2019
).
https://doi.org/10.1103/PhysRevB.99.214302
Google Scholar
Crossref
Search ADS
 
20.
See, e.g.,
E.
Barnes
 and
S.
Das Sarma
, “
Analytically solvable driven time-dependent two-level quantum systems
,”
Phys. Rev. Lett.
109
,
060401
(
2012
).
https://doi.org/10.1103/PhysRevLett.109.060401
Google Scholar
Crossref
Search ADS
PubMed
 
21.
I. I.
Rabi
,
N. F.
Ramsey
, and
J.
Schwinger
, “
Use of rotating coordinates in magnetic resonance problems
,”
Rev. Mod. Phys.
26
,
167
–
171
(
1954
).
https://doi.org/10.1103/RevModPhys.26.167
Google Scholar
Crossref
Search ADS
 
22.
See supplementary material at https://www.scitation.org/doi/suppl/10.1119/10.0001897 for an introduction to the rotating-wave approximation.
23.
I. I.
Rabi
, “
Space quantization in a gyrating magnetic field
,”
Phys. Rev.
51
,
652
–
654
(
1937
).
https://doi.org/10.1103/PhysRev.51.652
Google Scholar
Crossref
Search ADS
 
24.
K. O.
Friedrichs
, “
On the perturbation of continuous spectra
,”
Comm. Pur. Appl. Math.
1
,
361
–
406
(
1948
).
https://doi.org/10.1002/cpa.3160010404
Google Scholar
Crossref
Search ADS
 
25.
I.
Reiss
, “
Predissociation
,”
J. Chem. Phys.
52
,
871
–
880
(
1970
).
https://doi.org/10.1063/1.1673067
Google Scholar
Crossref
Search ADS
 
26.
U.
Fano
, “
Effects of configuration interaction on intensities and phase shifts
,”
Phys. Rev.
124
,
1866
–
1878
(
1961
).
https://doi.org/10.1103/PhysRev.124.1866
Google Scholar
Crossref
Search ADS
 
27.
D. S.
Onley
 and
A.
Kumar
, “
Time dependence in quantum mechanics—Study of a simple decaying system
,”
Am. J. Phys.
60
,
432
–
438
(
1992
).
https://doi.org/10.1119/1.16897
Google Scholar
Crossref
Search ADS
 
28.
G.
Ordonez
,
T.
Petrosky
, and
I.
Prigogine
, “
Quantum transitions and dressed unstable states
,”
Phys. Rev. A
63
,
052106
(
2001
).
https://doi.org/10.1103/PhysRevA.63.052106
Google Scholar
Crossref
Search ADS
 
29.
R. H.
Romer
, “
Units—SI-only, or multicultural diversity?
,”
Am. J. Phys.
67
,
13
–
16
(
1999
).
https://doi.org/10.1119/1.19185
Google Scholar
Crossref
Search ADS
 
30.
R.
Loudon
,
The Quantum Theory of Light
, 3rd ed. (
Oxford Science
,
Oxford
,
2000
).
Google Scholar
Crossref
Search ADS
 
31.
C.
Cohen-Tannoudji
,
J.
Dupont-Roc
, and
G.
Grynberg
,
Atom-Photon Interactions: Basic Processes and Applications
(
Wiley
,
New York
,
1992
).
Google Scholar
 
32.
M. O.
Scully
 and
M. S.
Zubairy
,
Quantum Optics
(
Cambridge U. P
.,
Cambridge
,
1997
).
Google Scholar
Crossref
Search ADS
 
33.
D. F.
Walls
 and
G. J.
Milburn
,
Quantum Optics
(
Springer
,
Berlin
,
1994
), Chap. 10.
Google Scholar
Crossref
Search ADS
 
34.
P. R.
Berman
 and
V. S.
Malinovsky
,
Principles of Laser Spectroscopy and Quantum Optics
(
Princeton U. P
.,
Princeton, NJ
,
2011
).
Google Scholar
Crossref
Search ADS
 
35.
R.
Passante
,
F.
Persico
, and
L.
Rizzuto
, “
The bound state in the spectrum of the Lee–Friedrichs Hamiltonian
,”
Phys. Lett. A
274
,
10
–
17
(
2000
).
https://doi.org/10.1016/S0375-9601(00)00540-5
Google Scholar
Crossref
Search ADS
 
36.
P. R.
Berman
 and
G. W.
Ford
, “
Spontaneous decay, unitarity, and the Weisskopf-Wigner approximation
,” in
Advances in Atomic, Molecular, and Optical Physics
, edited by
E.
Arimondo
,
P. R.
Berman
, and
C. C.
Lin
 (
Elsevier Academic Press
,
New York
,
2010
), Vol.
59
, Chap. 5.
Google Scholar
Crossref
Search ADS
 
37.
G. N.
Fleming
, “
A unitarity bound on the evolution of nonstationary states
,”
Nuovo Cimento A
16
,
232
–
240
(
1973
).
https://doi.org/10.1007/BF02819419
Google Scholar
Crossref
Search ADS
 
38.
G. C.
Ghiradi
,
C.
Omero
,
T.
Weber
, and
A.
Rimini
, “
Small-time behaviour of quantum nondecay probability and Zeno's paradox in quantum mechanics
,”
Nuovo Cimento A
52
,
421
–
442
(
1979
).
https://doi.org/10.1007/BF02770851
Google Scholar
Crossref
Search ADS
 
39.
See, e.g.,
P. L.
Knight
 and
P. W.
Milonni
, “
Long-time deviations from exponential decay in atomic spontaneous emission theory
,”
Phys. Lett. A
56
,
275
–
278
(
1976
).
https://doi.org/10.1016/0375-9601(76)90306-6
Google Scholar
Crossref
Search ADS
 
40.
M.
Miyamoto
, “
Bound-state eigenenergy outside and inside the continuum for unstable multilevel systems
,”
Phys. Rev. A
72
,
063405
(
2005
).
https://doi.org/10.1103/PhysRevA.72.063405
Google Scholar
Crossref
Search ADS
 
41.
B. R.
Mollow
, “
Power spectrum of light scattered by two-level systems
,”
Phys. Rev.
188
,
1969
–
1975
(
1969
).
https://doi.org/10.1103/PhysRev.188.1969
Google Scholar
Crossref
Search ADS
 
42.
For the corresponding source-field expression for spontaneous emission, see
P. W.
Milonni
,
The Quantum Vacuum
(
Academic Press
,
San Diego
,
1994
), Chap. 7.
Google Scholar
Crossref
Search ADS
 
43.
S. H.
Autler
 and
C. H.
Townes
, “
Stark effect in rapidly varying fields
,”
Phys. Rev.
100
,
703
–
722
(
1955
).
https://doi.org/10.1103/PhysRev.100.703
Google Scholar
Crossref
Search ADS
 
44.
See, e.g.,
L.
Landau
 and
E.
Lifshitz
,
Quantum Mechanics: Non-Relativistic Theory
(
Pergamon Press
,
Oxford
,
1977
).
Google Scholar
 
45.
E.
Fermi
,
Nuclear Physics
(
University of Chicago
,
Chicago
,
1950
), p.
142
.
Google Scholar
 
46.
P. A. M.
Dirac
, “
The quantum theory of emission and absorption of radiation
,”
Proc. R. Soc. Lond. A
114
,
243
–
265
(
1927
).
https://doi.org/10.1098/rspa.1927.0039
Google Scholar
Crossref
Search ADS
 
47.
V.
Weisskopf
 and
E.
Wigner
, “
Berechnung der natürlichen Linienbreite auf grund der diracschen lichttheorie
,”
Z. Phys.
63
,
54
–
73
(
1930
).
https://doi.org/10.1007/BF01336768
Google Scholar
Crossref
Search ADS
 
48.
R. H.
Dicke
, “
Interaction‐free quantum measurements: A paradox?
,”
Am. J. Phys.
49
,
925
–
930
(
1981
).
https://doi.org/10.1119/1.12592
Google Scholar
Crossref
Search ADS
 
49.
R.
Kubo
, “
The fluctuation-dissipation theorem
,”
Rep. Prog. Phys.
29
,
255
–
284
(
1966
).
https://doi.org/10.1088/0034-4885/29/1/306
Google Scholar
Crossref
Search ADS
 
50.
G.
Floquet
, “
Sur les équations différentielles linéaires à coefficients périodiques
,”
Ann. Sci. l'École Norm. Supérieure
12
,
47
–
88
(
1883
).
https://doi.org/10.24033/asens.220
Google Scholar
Crossref
Search ADS
 
51.
J. H.
Shirley
, “
Solution of the Schrodinger equation with a Hamiltonian periodic in time
,”
Phys. Rev.
138
,
B879
–
B987
(
1965
).
https://doi.org/10.1103/PhysRev.138.B979
Google Scholar
Crossref
Search ADS
 
52.
Implicit in this statement is the requirement that the continuum states be confined to a box (otherwise the probability of finding the two-level system in the ground state would be zero at any temperature). For a box, the probability of finding the thermally equilibrated system in the ground state is ε / ( ε + ∫ e − E / k B T d E )⁠, which becomes vanishingly small as T → ∞⁠; see
S. J.
Strickler
, “
Electronic partition function paradox
,”
J. Chem. Educ.
43
,
365
–
366
(
1966
).
https://doi.org/10.1021/ed043p364
Here, k B is the Boltzmann constant, T is the temperature, ε is the characteristic separation between levels introduced by the box, and the integral is over all the states that fit in the box. Also, note that infinite temperature does not translate into infinite energy and that, as the ground state decays, the field pumps less and less energy into the system.
Google Scholar
Crossref
Search ADS
 
53.
See, e.g.,
D. A.
Abanin
,
W.
De Roeck
,
W. W.
Ho
, and
F.
Huveneers
, “
Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems
,”
Phys. Rev. B
95
,
014112
(
2017
).
https://doi.org/10.1103/PhysRevB.95.014112
Google Scholar
Crossref
Search ADS
 
54.
G. E.
Stedman
, “
Fermi's golden rule—An exercise in quantum field theory
,”
Am. J. Phys.
39
,
205
–
214
(
1971
).
https://doi.org/10.1119/1.1986093
Google Scholar
Crossref
Search ADS
 
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