The Floquet Theory of the Two-Level System Revisited

2.1 Floquet Theory for SU₂

It appears that the simplest way to explain the general ideas of the Floquet theory for TLSs is by again proving its central claim. In doing so we will emphasise the group-theoretical aspects of the Floquet theory but otherwise will stick closely to [29].

The Schrödinger equation for this system is of the form

Here we have set ℏ = 1 and will assume the Hamiltonian Ĥ(t) to be T-periodic in time,

where throughout in this paper T=2πω>0, (1) gives rise to a matrix equation for the evolution operator U(t, t₀) that reads

with the initial condition

We will assume that U(t, t₀) ∈ SU₂, the Lie group of unitary 2 × 2-matrices with unit determinant. Consequently, the Hamiltonian Ĥ(t) has to be chosen such that 𝗂 Ĥ(t) lies in the corresponding Lie algebra su₂ of anti-Hermitean 2 × 2-matrices with vanishing trace, closed under commutation [ , ]. The relation between (1) and (3) is obvious: If ψ0∈ℂ2 and U(t, t₀) is the unique solution of (3) with initial condition (4), then ψ(t) ≡ U(t, t₀) ψ₀ will be the unique solution of (1) with initial condition ψ(t₀) = ψ₀. Conversely, let ψ₁(t) and ψ₂(t) be the two solutions of (1) with initial conditions ψ1(t0)=(10) and ψ2(t0)=(01), then U(t,t0)≡(ψ1(t),ψ2(t)) will solve (3) and (4).

Further, it follows that any other solution U₁(t, t₀) of (3) with initial condition U₁(t₀, t₀) = V₀ will be of the form

As a special case of (5), we consider

which due to (2), also solves (3) but has the initial condition

Hence (5) implies

ℱ ∈ SU₂ is called the “monodromy matrix”. It can be written as

Now we define

and will show that 𝒫 is T periodic in the first argument:

Summarising this, we have shown that the evolution operator U(t, t₀) can be written as the product of a periodic matrix and an exponential matrix function of time, i.e.

which is essentially the Floquet theorem for TLSs. Equation (15) is also called the “Floquet normal form” of U(t, t₀). For an example where an explicit solution for U(t, t₀) is possible for some limit case, see also subsection 6.2.2.

The derivation of (15) can be easily generalised from SU₂ to any other finite-dimensional matrix Lie group with the property that the exponential map from the Lie algebra to the Lie group is surjective, as this has been implicitly used in (9).

The matrix F is Hermitean and hence has an eigenbasis |n⟩,n=1,2 and real eigenvalues ϵ_n such that

In this eigenbasis, (15) assumes the form

in which the latter functions are called the “Floquet functions” or “Floquet solutions of (1)” and the eigenvalues ϵ_n of F are called “quasienergies”, see [29]. For the TLS, we have exactly two quasienergies ±ϵ such that ϵ ≥ 0 as Tr F = 0. It follows that any solution ψ(t) of (1) with initial condition ψ(t0)=a1|1⟩+a2|2⟩ can be written in the form

with the time-independent coefficients a_n. In this respect u_n(t, t₀), resp. ϵ_n, generalise the eigenvectors, resp. eigenvalues, of a time-independent Hamiltonian Ĥ. The latter is trivially T-periodic for every T > 0 hence also in this case the Floquet theorem (15) must hold. Indeed it does so with 𝒫(t, t₀) = 𝟙 and F = Ĥ.

We remark that the mere analogy between Floquet solutions and eigenvectors can be given a precise meaning by considering the “Floquet Hamiltonian” K defined on the extended Hilbert space L2[0,T]⊗ℂ2, see [29] such that the quasienergies are recovered as the eigenvalues of K. This was already anticipated in [3], [4], but we will not go into the details as the extended Hilbert space will not be used in the present paper.

In this account of Floquet theory we have stressed the dependence of the various definitions of the choice of an arbitrary initial time t₀. It, hence, remains to investigate the effect of changing from t₀ to some other initial time t₁. A straightforward calculation using the semi-group property of the evolution operator

gives the result

It follows that the eigenvalues of F~≡𝒫(t1,t0)F𝒫(t1,t0)−1 and F coincide, hence the change of the initial time will modify the Floquet functions but not the quasienergies. In concrete applications there will often be a natural choice for t₀ and the dependence on t₀ may be suppressed without danger of confusion.

We will add a few remarks on the uniqueness of the quasienergies ϵ_n. It is often argued that the quasienergies are only unique up to integer multiples of ω, see, e.g. [29]. It seems at first glance that in our approach uniqueness is guaranteed by the requirement 𝗂 F ∈ su₂. For example, the replacement ϵ_n ↦ ϵ_n + ω in (16) would result in F ↦ F + ω 𝟙 and violate the condition Tr F = 0. But this uniqueness is achieved by using a complex arg-function with a discontinuous cut. Consider, for example, a smooth 1-parameter family of monodromy matrices ℱ(ω) and the corresponding family ϵ₁(ω) of quasienergies. It may happen that exp(−𝗂Tϵ1(ω)) crosses the cut and hence ϵ₁(ω) changes discontinuously. But this discontinuity is not a physical effect and only due to the choice of the arg-function. It could be compensated by, say, passing from ϵ₁(ω) to ϵ₁(ω) + ω. In this case it would be more appropriate to consider, say, ϵ₁(ω) and ϵ₁(ω) + ω as physically equivalent quasienergies. Generally speaking, the issue of continuity is an argument in favour of considering the quasienergies modulo ω.

2.2 Lift to Higher Spins

A possible physical realisation of the TLS with Hilbert space ℂ² is a single spin with spin quantum number s = 12. Even if the TLS is realised in a different way it will be convenient to adopt the language of spin systems. For example, the Lie algebra su₂ is spanned by the Pauli matrices σ_i, i = 1, 2, 3 (times 𝗂) or equivalently, by the three spin operators s^i≡12σi,i=1,2,3 (times 𝗂). Consequently, the Hamiltonian H^(t) can always be construed as a Zeeman term with a time-dependent dimensionless magnetic field h(t) namely,

where, following the usual convention, we have omitted a minus sign. We will outline the procedure of lifting a solution of the Schrödinger equation for a spin with s=12 in a time-dependent magnetic field to a solution of the corresponding Schrödinger equation for general s. For this lift the T-periodicity of h(t) is not necessary, but it will later be used to draw conclusions about the Floquet state of the system with general s. The lift procedure is less known but has been already applied in 1987 to the problem of an N-level system in a periodic laser field [30]. Also see [31] for a more recent application of the lift procedure to the problems of Landau-Zener transitions in a noisy environment. For the Bloch-Siegert shift in s = 1 systems see also [32].

Let, as in Section 2.1, t ↦ U(t, t₀) be a smooth curve in SU₂, such that U(t₀, t₀) = 𝟙 . It follows that

Hence the columns ψ1(t),ψ2(t) of U(t, t₀) are linearly independent solutions of the Schrödinger equation (1). Next we consider the well-known irreducible Lie algebra representation (shortly called “irrep”), see, e.g. [33] Chapter 10–4,

of su₂ parametrised by the spin quantum number s such that 2s∈ℕ and the corresponding irreducible group representation (also called “irrep”)

It follows that

where the s^i is defined above as the three s=12 spin operators and the S^i denotes the corresponding spin operators for general s.

It follows from (29) that

Hence

will be a matrix solution of the lifted time evolution equation

Note that U(t, t₀)^(s) is a unitary matrix and hence its columns span the general (2s + 1)-dimensional solution space of the lifted Schrödinger equation.

Next we will use that the Hamiltonian is a T-periodic function of time, i.e. that (2) holds. Consequently, we can apply the irrep R^(s) to the Floquet normal form (15) of U(t, t₀) and obtain

where

Recall that the eigenvalues of F are of the form ±ϵ where ϵ ≥ 0 is the quasienergy of the TLS. Moreover, as 𝗂 F ∈ su₂, F can be written in the form

such that ∑i=13|fi|2=1. and hence s^′=∑i=13fis^i is the s=12 spin operator into the direction (f1,f2,f3)⊤. From this it follows that

As S^′=∑i=13fiS^i is the general s spin operator into the direction (f1,f2,f3)⊤ with eigenvalues m = −s, …, s it further follows that the eigenvalues of F^(s) and hence the quasienergies of the lifted Schrödinger equation are of the form

Also the Floquet functions for the lifted problem can be obtained from those of the TLS, and hence the general solution of the lifted Schrödinger equation can be reduced to the general solution of (1).

2.3 Lift to SO₃

We will consider the lift of the two-level problem to the three-level problem with spin s = 1 with more details. Till the end we will not directly use the irrep R⁽¹⁾ but some other well-known representation R that is, however, unitarily equivalent to R⁽¹⁾. It is defined by

and can be restricted to an irrep R:SU2→SO3. The corresponding Lie algebra irrep r:su2→so3 maps 𝗂s^i,i=1,2,3 onto three anti-symmetric real matrices that span so₃. Let

then the lifted evolution equation can be written as

where, as usual, R(t, t₀) ∈ SO₃ and R(t₀, t₀) = 𝟙 . The underlying “Schrödinger equation” has the form

with X(t)∈ℝ3 and can simultaneously be considered as the classical limit of the lifted Schrödinger equation for s → ∞. A more direct derivation of (42) can be given in the following way.

Let ψ(t) be a solution of the Schrödinger equation (1) and

be the corresponding 1-dimensional projector. It can also be written in the form

We will write X(t)=(X1(t),X2(t),X3(t))⊤ and note that 0=detP(t)=14(∥X(t)∥2−1). Hence the 1-parameter family P(t) corresponds to a curve X(t) on the “Bloch sphere”, defined by ∥X(t)∥2=1.

The projector P(t) satisfies the von-Neumann equation

Using the commutation relations of the spin operators ŝ_i one easily derives from (45) the differential equation (42) for X(t). In the special case where ψ(t)=ψ1(t)=u1(t,t0)e−𝗂(t−t0)ϵ1 is a Floquet solution of (1) the curve X(t) is a T-periodic solution of (42) and can be visualised as a closed trajectory on the Bloch sphere. The second Floquet solution ψ2(t)=u2(t,t0)e−𝗂(t−t0)ϵ2 is orthogonal to ψ₂(t) for all t and hence must correspond to the “antipode” periodic solution − X(t) of (42).

It will be instructive to check the consistency of our representation by directly applying the Floquet theory to (41). The corresponding monodromy matrix ℱ(1)=R(t0+T,t0)=exp(−𝗂TF(1)) has as any matrix in SO₃, three eigenvalues of the form 1,e𝗂ρ,e−𝗂ρ. Consequently, F⁽¹⁾ has the eigenvalues 0,+ρ/T,−ρ/T which is in accordance with (38) if 2ϵ = ρ/T. We note in passing that these considerations suggest a simple numerical procedure to determine the quasienergy ϵ: solve (41) numerically over one period T and find the eigenvalues of the corresponding monodromy matrix ℱ⁽¹⁾.

As a byproduct of the Floquet theory for SO₃ we will prove the existence of periodic classical solutions of (42). Let X₀ be the eigenvector of the monodromy matrix ℱ(1)=R(t0+T,t0) corresponding to the eigenvalue 1. If we use X₀ as the initial value X(t₀) of a solution of (42) we conclude

This means that this special solution X(t) will be T-periodic. We may hence ask whether it can be obtained as the lift of a Floquet solution of the Schrödinger equation, and, if so, how this Floquet solution can be reconstructed from X(t).

To this end we will start with a given T-periodic solution X(t) of (42) and want to construct a corresponding Floquet solution of (1). It is not necessary to assume the condition ∥X(t)∥2=1 from the outset. We may rather use the fact that (42) admits the constant of motion

and hence the solutions of (42) are trajectories on the Bloch sphere of radius R. Then the T-periodic 1-parameter family P(t) of one-dimensional projectors defined by

satisfies the von-Neumann equation (45) that is equivalent to (42). As P(t) is a projector and hence satisfies P(t)² = P(t), each non-vanishing column of P(t) will be an eigenvector of P(t) corresponding to the eigenvalue 1. After normalising we thus obtain the T-periodic one-parameter family of vectors

such that |φ(t)⟩⟨φ(t)|=P(t). We note in passing that (54) is undefined at the south pole of the Bloch sphere where X₃(t) = −R. We cannot expect that φ(t) is already a solution of the Schrödinger equation (1) but only that φ(t) differs from a solution ψ(t) of (1) by a time-dependent phase factor e𝗂α(t). After some calculations using (42) we obtain

where the infinite sum in (56) represents the Fourier series of the T-periodic function t↦χ(X(t)). By integrating this Fourier series over t we obtain

(neglecting an additional integration constant that would only yield a constant phase factor) and further

where

According to (61) and (58), ψ(t) is indeed a Floquet solution of (1) with quasienergy ϵ = a₀ modulo ω since u(t) is T-periodic. The quasienergy ϵ is the time average of χ( X(t)) denoted by an overbar:

Thus we have proven the following:

2.4 The RPC Example I

We will check the results of Assertion 1 for the exactly solvable case of the circularly polarised Rabi problem (RPC) where

We obtain T=2πω-periodic solutions X(t) of (42) by the following argument. Obviously,

We set X(t)=h(t)−𝝎 and conclude

and analogously for X(t)=𝝎−h(t). Hence one finds two T-periodic solution of (42) of the form

In this case the function χ( X(t)), see (56), turns out to be time-independent which directly yields the quasienergies

where Ω is the Rabi frequency

Moreover, the corresponding two Floquet solutions ψ_±(t) of the Schrödinger equation (1) can be obtained by (60) with the result:

in accordance with the well-known result, see, e.g. [34].

3.1 Geometry of the Fibre Bundle π:C|1|2⟶𝓢2

The map of wave functions to projectors ψ↦|ψ⟩⟨ψ| can be viewed as a surjective map π from the unit sphere of ℂ|1|2 of ℂ² to the unit sphere of ℝ³, i.e. as π:ℂ|1|2⟶𝒮2. Since ℂ|1|2≅𝒮3 this is essentially the Hopf fibration [35]. The fiber π⁻¹( X) consists of the set of all phase factors e^i⁢α and can hence be identified with the one-dimensional unitary group U₁. Then the map σ:X↦φ(t) according to (54) can be viewed as a local section of the principal fiber bundle π:ℂ|1|2⟶𝒮2 with structure group U₁. The above remark that σ is undefined at the south pole of the Bloch sphere means that σ cannot be extended to a global section due to topological obstacles. The group SU₂ operates in a natural fashion on ℂ|1|2 as well as on 𝒮² via rotations R(u) defined in (39)

Following [27] we may split the function

see (56), into a “dynamical” and a “geometrical part”. The dynamical part is defined as

and represents the expectation value of the energy. Its time average yields the dynamical part of the quasienergy:

The geometrical part of χ( X(t)) is defined as

Using spherical coordinates θ, ϕ for the Bloch sphere with radius 1, we may write the differential χ_g dt in the form

This yields a differential 1-form α on the Bloch sphere and the time average of χg(X(t)) is, up to the factor 1T=ω2π, the integral of α over the closed curve 𝒞 parametrised by X(t). By applying Stoke’s theorem we obtain

where 𝒜 denotes the oriented area enclosed by 𝒞 and |𝒜| the correspondingly signed solid angle. Here we have used that the 2-form

equals 12 times the surface element on the unit sphere.

Thus we obtain the following interpretation of the quasienergy ϵ = ϵ_d + ϵ_g as composed of two parts: the dynamical part ϵ_d is the time average of the energy (76) and the geometrical part ϵ_g is ω4π times the solid angle enclosed by the T-periodic solution of (42).

We refer to [27] for the differential-geometric background of this scenario. Obviously, 1-form α is the retract of the canonical connection 1-form γ of the principal fiber bundle π:ℂ|1|2⟶𝒮2 w. r. t. the local section σ:𝒮2⟶ℂ|1|2 described above. Hence the non-vanishing of ϵ_g is due to the curvature of this principal fiber bundle and generalises Berry’s phase. Recall further that, by definition, γ has values in the Lie algebra of U₁. This Lie algebra is isomorphic to ℝ and an analogous identification has been made by considering above α as a usual real-valued 1-form. However, ϵ_g as the integral over α should properly have values in U₁ and not in ℝ and this subtle difference is in turn in accordance with the general claim that quasienergies are only defined modulo ω.

3.2 The RPC Example II

We will illustrate the results of the preceding subsection by the explicitly solvable case of RPC. The calculation is essentially identical with that in [27]. It follows from (66) and (69) that the dynamical part of the quasienergy assumes the value (note that the involved functions are constant and taking the time average is superfluous)

On the other hand, the vector X₊(t) prescribes a circle on the Bloch sphere with constant Z = ω₀ − ω, see (69). Consider first the case of Z > 0. The corresponding spherical segment has an area of 2πR(R − Z) corresponding to a solid angle of |𝒜|=2π(R−Z)R, where R=Ω=F2+(ω0−ω)2 is the radius of the Bloch sphere. Hence

and

in accordance with (70). For the second periodic solution X₋(t) it follows that both terms ϵ_d and ϵ_g acquires a minus sign, the latter as the spherical segment encircled by X₋(t) has a negative orientation. Hence ϵ−=−ϵ+modω in accordance with (70).

In the case of Z < 0 the dynamical part ϵ_d remains unchanged whereas the solid angle of the spherical segment encircled by X+(t) assumes the form |𝒜|=−2π(R+Z)R. This yields ϵ+=12(−ω+R) which differs from (88) only by −ω and is thus equivalent. Analogous arguments hold for ϵ₋.

3.3 Another Solvable Example

The idea of a “reverse engineering of the control” h(t), see [18], [19], [20], can also be applied to the classical Floquet problem. Given a normalised T-periodic function X(t) we may choose h(t)≡X(t)×X˙(t) such that (42) is satisfied. This special choice of h entails h⋅X=0 and hence the dynamical part χ_d of χ vanishes according to (76). The geometrical part χ_g = χ will be calculated according to (79) for the following example:

f being a real parameter, which leads to

and

see Figure 1 visualising the example f = 1, ω = 1. The Fourier series of χ(t) can be explicitly calculated:

where

Here the J_n(…) denote the Bessel functions of the first kind and integer order. The constant term of (92) is the quasienergy ϵ according to (93) that only depends linearly on ω. The Fourier series of χ(t) could be utilised to explicitly determine the Floquet solutions u±(t)e∓𝗂ϵt of the corresponding Schrödinger equation according to Assertion 1.

Figure 1:

A periodic solution X(t) of the classical equation of motion (42) according to (89) with f = 1, ω = 1 represented by a closed trajectory on the Bloch sphere (blue curve). The corresponding magnetic field h(t) according to (90) is visualised by the red curve. At the time t = π the three vectors X(π), h(π) and X˙(π)=h(π)×X(π) are indicated by the coloured arrows.

Note further that Assertion 3 of Section 4.2 can be sharpened to ∂ϵ∂ω=ϵω as ϵd=0 in this example.

3.4 Classical Mechanics of the Two-Level System

First we will introduce some concepts of classical mechanics suited for the present case. Let z and φ be coordinates of the unit Bloch sphere defined by

Further we define the classical Hamiltonian

and rewrite the differential equation (42) in terms of the two functions z(t) and φ(t):

Note that due to (76), H is twice the expectation value of the Hamiltonian Ĥ. Obviously, (100)–(101) can be viewed as Hamiltonian equations of motions in a two-dimensional phase space isomorphic to 𝒮² with canonical coordinates

We will hence forward often use p and q instead of z and φ. Following [36] we consider the extended phase space 𝒫=𝒮2×ℝ with coordinates p, q, t and the Poincaré-Cartan form

defined on 𝒫 (not to be confounded with the 1-form α defined in Section 3.4). The Hamiltonian equations (100), (101) can be geometrically construed as the direction field on 𝒫 that is given by the unique null direction field of the exterior derivative of the Poincaré-Cartan form

see Section 44 of [36] for the details.

Periodic solutions of (100), (101) correspond to curves γ in 𝒫 that are not closed as after one period T the coordinate t has changed from t = 0 to t = T. This can be repaired by defining another extended phase space 𝒫^′ via identifying points with t-coordinates that differ by an integer multiple of T, or, more formally,

Of course, ℝ/Tℤ is isomorphic to 𝒮¹. Locally the manifolds 𝒫 and 𝒫^′ are isomorphic, but globally they are not. The differential forms 𝜶 and 𝝎 can be transferred to 𝒫^′ as all functions involved in 𝜶 and 𝝎 are T-periodic. Now periodic solutions of (100), (101) correspond to closed curves γ in 𝒫^′.

Next we will rewrite the expression (62) for the quasienergy ϵ. By (76) and (81) we obtain

Hence

where the n in the last term denotes the winding number of γ around the z-axis. This result is in close analogy to equation (2.35) of [26] that represents the semi-classical limit of the quasienergy for integrable Floquet systems. It thus seems that for the TLS, similar as in the case of the driven harmonic oscillator, the semi-classical limit of the quasienergy and the exact quantum-theoretical expression coincide. However, it has not yet been shown that the quantisation procedure adopted in [26] yields the quantum TLS when starting from its classical limit.