Introduction to Floquet

1 Introduction to Floquet - Lecture notes -Giuseppe E. SantoroSISSA, Trieste2 PrefaceThese notes are based on a short course on the topic given at SISSA and at GGI Floquet theory: a classical warmup5 Periodically driven pendulum ..6 Bloch electron in a periodic potential ..10A perturbative approach to the periodic Bloch function ..13 Lyapunov exponents ..15 Classical Floquet -Lyapunov theory ..16 Application to the Mathieu equation ..172 Floquet theory in quantum mechanics19 The Floquet theorem ..19 Proof of the Floquet theorem ..20 Proof of the Floquet theorem for a classical linear system ..22 The periodic moving frame and the extended Hilbert space ..24 Thet tformalism ..27 Magnetic spin resonance and two-level atom ..28 Exact solution for the circularly polarised case ..28 The rotating wave approximation for the linearly polarised case ..31 The Shirley- Floquet approach.

2 32 Increasing the dimensionality of the system ..351 Floquet theory: a classical warmupThe term Floquet is associated to periodicity in time. More specifically, in a classicalcontext the Floquet theory was introduced to describe the behaviour of a set oflineardif-ferential equations with a time-periodic coefficients, which was in turn originating from theproblem of the stability of periodic orbits in classical mechanics. In the quantum world, wherethe linearity of the Schr odinger equation is guaranteed from the start, the Floquet theoryapplies whenever the Hamiltonian governing the system is time-periodic, H(t) = H(t+ T),where T is speaking, one would need distinguish two main regimes of interest:1)a regime ofslow driving, often connected to some form ofadiabatic limit, where the drivingfrequency =2 Tis suitably small, formally 0;2)a regime offast driving, when is larger than the proper frequencies of the system thatis is nowadays a very intense field of research, with many interesting topics, rangingfrom quantum pumping (in the adiabatic regime) to Floquet engineering (tayloring non-trivial topological properties).

3 Among the many interesting phenomena, I will start my discussion with some classicalphysics, related to the following two classes of phenomena:Parametric resonance :the example of the familiarswing(a planar pendulum) illustratethis phenomenon. You remember that a way to increase the amplitude of the swingoscillations is to perform a characteristic up-and-down movement of the center-of-mass of your body as the swing oscillates. You do thattwiceevery period: goingforward, and also backward. This corresponds to a driving period T = T0/2, whereT0is the natural period T0= 2 / 0, hence a frequency 1= 2 0. But when youwere a little boy, your father would push you, usuallyonceevery period, hence with 2= 0. More generally, you could drive the system by pushing only once everynhalf-periods, hence with a driving frequency: n=2 0nwithn= 1,2,3, ( )These are the resonant driving frequencies of the ordinary pendulum, where 0is itsnatural stabilisation:An inverted pendulum is not stable.

4 Nevertheless, as pointed outby Kapitza in 1951, if you oscillate vertically the suspension point at a sufficientlyfast frequency , you can stabilise it. The following video illustrates this. A similar6 Floquet theory: a classical warmup(Notes by Santoro)phenomenon occurs in arotating saddle(see video), and is also used to create radio-frequencyionic will illustrate both phenomena with the example of the periodically driven precisely, a pendulum in which you oscillate the suspension pointvertically. In oneshot, we will be able to capture both phenomena in the same simple , we will move to the quantum world. As an application, probably the simplest one,we will consider the problem of NMR or of a two-level atom under laser irradiation, close toresonance. We will discuss, using this example, the Shirley- Floquet approach that essentiallypromotes the Fourier index to a new extra references on different aspects of the story are: a tutorial review by a leading expertin the field, M.

5 Holthaus [1], a very nice review on the issue of high-frequency expansions bythe group of A. Polkovnikov [2], and two older excellent reviews [3, 4], the latter includingalso some mathematical aspects concerning the nature of the Floquet spectrum and driven pendulumLet us start considering a very familiar one-dimensional system: a planar pendulum madeof a massless rod of lengthlending with a point massm. In the familiar swing, the drivingoccurs in different ways: if you drive the swing yourself, you do it by effectively modifyingthe position of your center-of-mass , hence the effective lengthl(t) of the pendulum . Ifyou are pushed by someone else, then you have a pendulum with a periodic external force .We will drive the pendulum in a third (different) way, by oscillatingverticallyits suspensionpoint, which has the advantage that we can describe in the same framework also the invertedpendulum stabilisation Ifq= denotes the angle formed with the vertical ( = 0 being theFigure :(a) Periodically shaken pendulum and (b) Kapitza pendulum.)

6 Figure taken from Ref. [2]. Periodically driven pendulum(Notes by Santoro)7downward position), andy0(t) denotes the position of its suspension point, we can derive theequations of motion from the Lagrangian formalism. In a short while we will assume thaty0(t) =Acos t, whereAis the amplitude of the driving and the driving frequency, butfor the time being, let us proceed by keepingy0(t) to be general. In a system of referencewith they-axis oriented upwards and thex-axis horizontally, the positionx(t) andy(t) ofthe massmis:{x(t) =lsin (t)y(t) =y0(t) lcos (t) { x(t) =l cos y(t) = y0+l sin ( )The LagrangeanL( , ,t) is therefore given by:L( , ,t) =m2( x2+ y2) mgy=ml2 22+ml y0 sin +mglcos + (m2 y20 mgy0),( )where we drop the last terms, which are simply functions of time which would not enter inthe Euler-Lagrange equations. The associatedmomentumis given by:p = L =ml2 +ml y0sin =p ml2 y0lsin.}}

7 ( )A simple calculation will give us the HamiltonianH( ,p ,t), which we denote byHlabbecauseit is the Hamiltonian in the laboratory reference frame where you observe the suspension pointto oscillate:Hlab( ,p ,t) =p L=p2 2ml2 p y0lsin +m2 y20sin2 mglcos =(p ml y0sin )22ml2 mglcos .( )Observe that the laboratory Hamiltonian contains a non-standard kinetic term. But weshould be able to describe the same phenomenon in a reference frame that moves togetherwith the suspension point: simply imagine that that pendulum, and the suspension point, arelocated into an elevator from which you would not see the outside world. In that non-intertialsystem, the mass would feel anon-inertial forcedue to y0, hence experiencing an effectivelytime-dependent acceleration of gravity:g g(t) =g+ y0=g A 2cos t .( )In essence, by theequivalence principle, you would anticipate a moving-frame Hamiltonianwith a standard kinetic term and a modifiedg(t):Hmov( ,p ,t) =p2 2ml2 m(g+ y0)lcos ( )To make sense of the equivalence of the previous two descriptions, one should perform acanonical transformationof variables in the Hamiltonian formalism.

8 If you are a bit rustyabout canonical transformations in classical Hamiltonian dynamics, here is a nice detour8 Floquet theory: a classical warmup(Notes by Santoro)which performs this transformation in aquantum framework, which I personally find muchmore , let us promote our Hamiltonians to be quantum, writing: Hlab(t) =( p ml y0sin )22ml2 mglcos ( )where p is the canonical momentum, with p = i~ Lz.( )Notice that p is theangular momentumaround thez-axis. The quantum problem is set inthe Hilbert space ofperiodic functions ( ) = ( + 2 ), and the time-dependent Schr odingerequation would read:i~ t ( ,t) = Hlab(t) ( ,t).( )Consider now a time-dependentunitary transformationperformed with an operator Ut= eif( ,t),( )wheref( ,t) is periodic in , and should be suitably chosen so that the transformed kineticenergy is standard. Let us see how the momentum is transformed: p U t p Ut= p +~f ( ,t).

9 ( )wheref = f. The transformation of the Hamiltonian is therefore: Hlab(t) U t Hlab(t) Ut=( p +~f ( ,t) ml y0sin )22ml2 mglcos ,( )where we see that the kinetic term becomes standard provided~f ( ,t) =ml y0sin ~f( ,t) = ml y0cos .( )But you should refrain from thinking that this transformed Hamiltonian, which in essencebecomes that of the standard un-driven pendulum, governs the motion. The unitary trans-formation, being time-dependent, adds an extra term in the Schr odinger dynamics. Thisfact is so general, that we formulate it in a more abstract ket-notation, without referenceto the specific problem at hand. The result is the following. If| (t) = Ut| (t) , then theSchr odinger equation for| (t) reads:i~ t| (t) =[ U 1t Hlab(t) Ut i~ U 1tddt Ut]| (t) H(t)| (t) ,( )where U 1t= U tfor a unitary transformation, and the transformed Hamiltonian governingthe dynamics contains a characteristicextra term: H(t) = U t Hlab(t) Ut i~ U t Ut.

10 ( )In the specific case we are studying, Ut= eif( ,t), the extra term reads: i~ U t Ut=~ tf( ,t) = ml y0cos ,( )1 Sometimes, the quantum route is simpler (in the sense that we got more school training to it) that thecorresponding classical one. Think of Kubo linear-response theory, as another Periodically driven pendulum(Notes by Santoro)9where we used the choice of~f( ,t) = ml y0cos which simplifies the kinetic term. Hence,we get: H(t) = U t Hlab(t) Ut ml y0cos = p2 2ml2 m(g+ y0)lcos Hmov(t).( )So, the transformed Hamiltonian is precisely the moving-frame Hamiltonian we had guessedon the basis of the equivalence us now return to classical mechanics. From now on we derive our Hamilton s equationsfrom the moving-frame Hamiltonian, which we simply denote byH. The Hamilton s equationsread: = H p =p ml2 p = H = m[g A 2cos ( t)]sin .( )Hence, transforming it into a second-order equation:d2 dt2= [ 20 2 Alcos ( t)]sin ,( )where 0= g/lis the frequency of the unperturbed pendulum in the linear regime.