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 ω, where ω is the frequency of the field and 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 |χE, of energy E, and its energy lies within this continuum. A classical field, of amplitude F0, which varies periodically in time with frequency ω, can induce transitions between the two discrete levels. The full Hamiltonian is

H=Eg|gg|+Ee|ee|+E|χEχE|dE+(F0Vegeiωt|eg|+F0*Veg*eiωt|ge|)+(ξE|χEe|+ξE*|eχE|)dE,
(1)

with g|g=e|e=1, g|e=0, χE|χE=δ(EE), and χE|e=χE|g=0; Veg is the matrix element for the interaction of the two-level system with the field and ξE is the coupling coefficient between |e and |χE; 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, ωEeEg (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, cosωt instead of e±iωt). 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 |F0| couples the ground state |g to the excited state |e of a two-level system, which itself is coupled to a continuum of states |χE.

Fig. 1.

Illustration of the model. A harmonically time-varying classical field of amplitude |F0| couples the ground state |g to the excited state |e of a two-level system, which itself is coupled to a continuum of states |χE.

Close modal

The above Hamiltonian combines two well-known models. For ξE=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 F0=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 ξE=0, the problem reduces to that of solving Schrödinger equation (HRi/t)|Ψ(t)=0 (we take =1 throughout the paper)29 for the Rabi Hamiltonian

HR=Eg|gg|+Ee|ee|+(F0Vegeiωt|eg|+F0*Veg*eiωt|ge|).
(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

|Ψ(t)=eiEgtCg(t)|g+eiEetCe(t)|e,
(3)

leading to

iĊg(t)=F0*VgeeiΘtCe(t),iĊe(t)=F0VegeiΘtCg(t),
(4)

where Θ=EeEgω. The replacement Cκ=aκeiακt, with κ={e,g}, gives αeαg=Θ and αgαe=|F0Veg|2. Thus, we obtain two sets of solutions: αg=(Θ/2±Ω)andαe=(Θ/2±Ω), where Ω=Θ2/4+|F0Veg|2 so that the two normalized, independent wavefunctions are

|Ψ±(t)=e±iΩtei(ωEeEg)t/24|F0Veg|2+(Θ2Ω)2(2F0*Vge|g+(Θ2Ω)eiωt|e).
(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

|Ψ(t)=ei(ωEeEg)t/2[(cosΩt+iΘ2ΩsinΩt)|giF0VegΩeiωtsinΩt|e].
(6)

Hence, the probability of finding the system in the excited state is

|F0VegΩ|2sin2Ωt.
(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 EeEgω. 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, EeEg, to |EeEgω|. The same result can be obtained directly from the integro-differential equation

Ċg(t)=|F0Veg|20teiΘ(wt)Cg(w)dw,
(8)

which follows from Eq. (4). Clearly, the time dependence of Cg, 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, Θ0.22 

For F0=0, we obtain the Friedrichs Hamiltonian24 

HF=Ee|ee|+U|χUχU|dU+(ξU|χUe|+ξU*|eχU|)dU,
(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 Ee lies within the continuum, the spectrum of eigenvalues of HF is a continuum identical to that of U|χUχU|dU.35 Following closely the notation of Ref. 26, we write the eigenvectors as

|ΦE=A(E)|e+BU(E)|χUdU.
(10)

HF|ΦE=E|ΦE gives

A(E)ξU+UBU(E)=EBU(E),
(11)
ξU*BU(E)dU+EeA(E)=EA(E).
(12)

If UE, we get from Eq. (11)BU(E)=ξUA(E)/(EU). Moreover, if ξU=0, the solutions that belong to the continuum are |ΦE|χE and, therefore, BU(E)=δ(UE). Thus, we write

BU(E)=A(E)ξU[1EU+z(E)δ(UE)]
(13)

and get from Eq. (12)

z(E)=EEeF(E)|ξE|2,
(14)

with (the principal value is understood)

F(E)=|ξU|2EUdU.
(15)

We obtain A(E) from the normalization condition ΦE|ΦE=δ(EE). A simple calculation gives

ΦE|ΦE=A*(E)A(E)[|ξE|2z2(E)δ(EE)+F(E)F(E)EE+|ξU|2(EU)(EU)dU].
(16)

For EE, 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 eiεU, as a regulator, and add a small imaginary term to the energy so that the singular term becomes26 

limEEε,δ0|ξU|2eiεUdU(EUiδ)(EU+iδ)=limEEε,δ01EE+2iδ|ξU|2eiεUdU(1EUiδ1EU+iδ)=δ(EE)π2|ξE|2.
(17)

Finally,

|A(E)|2=|ξE|2z2(E)+π2=|ξE|2[EEeF(E)]2+π2|ξE|4.
(18)

Since e|Φ(E)=A(E), see Eq. (10), and, thus, |e=A*(E)|Φ(E)dE, this Lorentzian-type expression describes the broadening and energy shift of the discrete state resulting from its coupling to the continuum, given, respectively, by |ξE|2 and F(E). Coupling renders the discrete state unstable in that, if |Ψ=|e at t=0, then

|Ψ(t)=|ΦEΦE|eeiEtdE=A*(E)|ΦEeiEtdE.
(19)

Hence, the probability of finding the system in the state |e at later times is

P(t)=||A(E)|2eiEtdE|2.
(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 ξE2Γ 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 ξE2,F(E)Ee, 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 dP/dt|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=1/2πξ02, where ξ0=ξE0 and E0=Ee+F(Ee). Finally, at much longer times, P shows an algebraic dependence, in agreement with general theory.39 The oscillations at the crossover time tτlog(E0τ) reflect interference between the exponential and algebraic expressions. Since

ξ02eiEtdE(EE0)2+π2ξ04=eiE0teπξ02t,
(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 (F0=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 E0=200 and ξ0=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; E0=5000 and ξ0=1 (arbitrary units).

Fig. 2.

Friedrichs model (F0=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 E0=200 and ξ0=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; E0=5000 and ξ0=1 (arbitrary units).

Close modal

As mentioned earlier, if Ee is inside the continuum, the eigenfunctions of the Friedrichs model are usually extended, in that they satisfy the normalization condition ΦE|ΦE=δ(EE). This is always the case for ξE0; 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 Ee 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 Ed is

EdEe=|ξU|2EdUdU,
(22)

and, therefore,

|A(Ed)|2=(1+|ξU|2(EdU)2dU)1.
(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(Ed)|2δ(EEd) 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

|ξU|2dU(EdEe)Ed
(24)

and, therefore,

|A(Ed)|2(2Ee/Ed)1,
(25)

which approaches ½ for |Ed|Ee (all along, we have assumed that Ee 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 ξE=Λe(EEe)p/Γp 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 Λ2Γ/2π (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 (F0=0). Gaussian energy-dependent coupling ξE=Λe(EEe)2/Γ2 for Γ=0.1 and Ee=0. The top panel shows A2(E) vs E for various values of Λ; see Eq. (18). Curves have been shifted vertically for clarity. ξE is shown in the bottom panel.

Fig. 3.

Friedrichs model (F0=0). Gaussian energy-dependent coupling ξE=Λe(EEe)2/Γ2 for Γ=0.1 and Ee=0. The top panel shows A2(E) vs E for various values of Λ; see Eq. (18). Curves have been shifted vertically for clarity. ξE is shown in the bottom panel.

Close modal
Fig. 4.

Same as Fig. 3 but for the nearly flat, energy-dependent coupling ξE=Λe(EEe)8/Γ8 with Γ=0.15 and Ee=0.

Fig. 4.

Same as Fig. 3 but for the nearly flat, energy-dependent coupling ξE=Λe(EEe)8/Γ8 with Γ=0.15 and Ee=0.

Close modal

A somehow more direct way27 to obtain |A(E)|2 is to consider, instead of |ΦE, Eq. (10),

|Ψ(t)=a(t)eiEet|e+bU(t)eiUt|χUdU.
(26)

HF|Ψ(t)=i(/t)|Ψ(t) gives

ȧ(t)=ieiEetξU*bU(t)eiUtdU,ḃU(t)=iξUa(t)ei(UEe)t.
(27)

If a(t)=1 at t=0,

bU(t)=iξU0ta(t)ei(UEe)tdt,
(28)

and we get the integro-differential equation

ȧ(t)=0ta(t)dt[|ξU|2ei(UEe)(tt)dU],
(29)

which is the Friedrichs-model equivalent to Eq. (8). If we write a(t)=a(E)eiEtdE, we get

[EEeF(E)]a(E)=|ξE|2a(U)(EU)dU.
(30)

Comparing this expression with the identity |A2(E)|(EU)1dE+z(U)|A2(U)|=0, which follows from the closure relation A(E)BU(E)dE=0, it is apparent that a(E)=|A2(E)| and, thus, that, a(t)=|A2(E)|eiEtdE, 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=cq (c is a velocity and q is a momentum) in s-like extended states r|χq=2/πsin(qr)/r, normalized according to

0r2χq|χqdr=δ(qq).
(31)

The continuum component of |ΦE is

|ψE=A(E)[ξUEU|χUdU+ξEz(E)|χE].
(32)

Since limr0sin(qr)(qk)1dq=πcoskr, we get at large r

r|ψE=cqA(q)ξqr2π[z(q)sinqrπcosqr]2/πsin(qr+Δq)/r,
(33)

where Δq=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

ψ(r,t)=A(q)r|ψE=cqeicqtdq2/πrξ3eicqtc2(qqe)2+π2ξ4dq[c(qqe)ξ2sinqrπcosqr].
(34)

After a lengthy but straightforward calculation, we find

ψ(r,t)=Arθ(ctr)e(ctr)πξ2/ceiqe(ctr),
(35)

where A2=ξ02/2c and θ is the Heaviside step function. This represents an outgoing wave with a sharp edge and maximum at r=ct.42 Note that 4π|ψ|2r2dr=1et/τ, as required to conserve probability and that, unlike |ψE, 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 HF does not contain discrete states. Using the closure relation

HF=E|ΦEΦE|dE
(36)

and |e=A*(E)|ΦEdE, we get

H=Eg|gg|+E|ΦEΦE|dE+[F0VegeiωtA*(E)|ΦEg|dE+F0*Veg*eiωtA(E)|gΦE|dE].
(37)

This Hamiltonian describes the field-mediated coupling of the ground state to the |ΦE-continuum through a modified coupling constant VegA*(E). As before, we search for solutions of the form

|Ψ(t)=Cg(t)eiEgt|g+CE(t)eiEt|ΦEdE.
(38)

H|Ψ(t)=i|Ψ(t) gives

iĊg(t)=F0*Veg*[A(U)CU(t)ei(ω+EgU)tdU],iĊE(t)=F0Vegei(ω+EgE)tA*(E)Cg(t).
(39)

If Cg=1 at t=0, then CE(t)=iF0VegA*(E)×0tCg(t)ei(ω+EgE)tdt and we get

Ċg(t)=0tCg(t)dt[|F0VegA(U)|2ei(UωEg)(tt)dU].
(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 |EeEgω|, and, moreover, it tells us immediately that the solution is

Cg(t)=|Ξ(E)|2ei(EωEg)tdE,
(41)

where

|Ξ(E)|2=|F0VegA(E)|2[EEgω|F0VegA(U)|2EUdU]2+π2|F0A(E)|4.
(42)

The results in Fig. 5 are for the (resonant) case when ω=EeEg. For small values of the field, that is, in the weak-field limit, |Ξ(E)|2 shows a single peak at Ee and, therefore, Cg(t)exp(|F0Veg|2t/πξEe2) (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 F0=0.017 fall in this regime. It is interesting to note that the decay constant decreases with increasing coupling of |e to the |χE-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 F0=0.017 (gray) and F0=0.158 (black). Coupling of |e to the continuum is energy-independent with ξE=ξ0=0.1. Inset: |Ξ2(E)| for the corresponding values of the field and Ee=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 F0=0.017 (gray) and F0=0.158 (black). Coupling of |e to the continuum is energy-independent with ξE=ξ0=0.1. Inset: |Ξ2(E)| for the corresponding values of the field and Ee=0; see Eq. (42). The two curves have been shifted vertically for clarity and the units are arbitrary.

Close modal

The calculations for F0=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 Cg1 and get, to first order in F0,

CE(t)iF0VegA*(E)0tei(ω+EgE)tdt=F0VegA*(E)ei(ω+EgE)t1(ω+EgE).
(43)

Note that |CE(t)|2t2 for t0 and that the lowest order correction to |Cg|2 is |F0|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

|Ψ(t)eiEgt|g+F0VegA*(E)ei(ω+EgE)t1(ω+EgE)eiEt|ΦEdE.
(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,

|F0Veg|2|A(E)|2|ei(ω+EgE)t1(ω+EgE)|2dE=|F0Veg|2|A(E)|24sin2(ω+EgE)t/2(ω+EgE)2dE1.
(45)

The term multiplying |A(E)|2 exhibits a peak at E=ω+Eg, 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+Eg)|2 and take it out of the integral. Alternatively, and as presented in most textbooks,44 we can use the approximation

4sin2(ω+EgE)t/2(ω+EgE)22πtδ(ω+EgE).
(46)

Either way, the probability of not being in the ground state increases with time as

2π|F0VegA(ω+Eg)|2t.
(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

exp[2π|F0VegA(ω+Eg)|2t]12π|F0VegA(ω+Eg)|2t.
(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 ξE

|Ψ(t)eiEet|e+ξUei(EeU)t1(EeU)eiUt|χUdU.
(49)

Thus, the probability of this state belonging to the continuum, given by

|ξU|24sin2(EeU)t/2(EeU)2dU,
(50)

increases linearly with time, at the rate 2π|ξEe|2, beyond a characteristic time defined by the energy scale of |ξE|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 |ξU|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 |ξE|2 and, therefore, the golden rule in the time-varying case applies at least for

1ξEe2tπξEe22|F0Veg|2.
(51)

On the other hand, in the static case, Eq. (50), we have

1Δξt12πξEe2,
(52)

where Δξ 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

ȧ(t)a(t)0tdt[|ξU|2ei(UEe)(tt)dU]=a(t)[|ξU|21ei(UEe)ti(UEe)dU].
(53)

Again, at times that are large compared with Δξ, we can also take |ξU|2 outside the integral arriving at ȧ(t)a(t)|ξEe|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:

limNi=1,N(1δi)=eiδi,
(54)

which is valid for arbitrary (random or otherwise) 0<δi1, 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 δi=Δti/τ, where Δtiτ is the time interval between measurements and τ is the decay time, the above product becomes exp(i=1,NΔti/τ). According to the golden rule, this represents the probability of finding the unstable system in its undecayed state at t=i=1,NΔti.

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)

Ψ(t)|O|Ψ(t)F0Vegg|O|eeiωt|A(E)|21ei(ω+EgE)t(ω+EgE)dE+c.c.,
(55)

where we used g|O|ΦE=A*(E)g|O|e. Assuming that |ξE|2Ee and extending the integration limits to the interval {−∞,+∞}, we get after a contour integration

Ψ(t)|O|Ψ(t)F0Vegg|O|eeiωt[1ei(EeEgω)teπ|ξEe|2tEeEgωiπ|ξEe|2]+c.c..
(56)

After the transient, that is, for t1/π|ξEe2|, one recovers the result of linear response theory49 

Ψ(t)|O|Ψ(t)F0eiωtR(ω)+F0*eiωtR*(ω)
(57)

with the response function

R(ω)=g|O|eVegEeEgωiπ|ξEe|2.
(58)

If the operator is the same one associated with the field, R becomes the complex susceptibility

χF(ω)=|Veg|2EeEgωiπ|ξEe|2.
(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 tπξEe2/2|F0Veg|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

eiEt|Φ(t),
(60)

where E is referred to as the quasienergy and |Φ(t) is a periodic function with the same period as the Hamiltonian obeying

(Hi/t)|Φ=E|Φ.
(61)

The two solutions to Rabi's problem we found in Sec. III, see Eq. (5), are clearly of the Floquet form, with

E=12(ωEeEg)±Ω.
(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±inω. 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)

e,p|V|g,m+1=g,m|V|e,p1*=F0δm,p,
(63)

where V=(F0eiωtVeg|eg|+F0*Veg*eiωt|ge|) and m, p denote, respectively, the functions eimωt and eipωt. This argument also applies to the full Hamiltonian, Eq. (1), and is the primary reason why it can be solved exactly. Had we used cosωt in Eq. (1) instead of the RWA form, e±iωt, we would have had an extra term in the right-hand side of Eq. (63), namely, F0δm+2,p, which would have made the problem analytically intractable.22 

To obtain the Floquet solutions to the full Hamiltonian, replace Cg(t)=ageiαgt and CU(t)=aUeiαUt in Eq. (39). This gives αE+αg=EωEg and

αgag=F0*Veg*A(U)aUdU,αUaU=F0VegA*(U)ag.
(64)

If αU0, aU=F0A*(U)ag/αU. Thus, we replace

aU=F0VegA*(U)ag[1/αU+s(αg)δ(UωEgαg)]
(65)

to get (principal value is understood)

s(αg)=[|F0VegA(U)|2UωEgαgdU+αg]/|F0VegA(ω+Eg+αg)|2.
(66)

Hence, the Floquet solutions are (aE is a normalization constant)

|ΨE(t)=aEeiEt{|g+eiωt[F0VegA*(U)ω+EU|ΦUdU+s(E)F0VegA*(ω+E)|Φω+E]},
(67)

with E=Eg+αg. 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 CE|ΨEdE, where CE 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 |EeEgω|. 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.

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Supplementary Material