1. Quantum Floquet Systems: Topology, steady states, and more
It is a pleasure to speak here tdy. When Shivaji invited to speak, he requested that I talk about Quantum Floquet systems. Having obsessed about those for a while, I was glad to oblige. Floquet systems, are just regyular system driven in a periodic way.
Quantum Floquet Systems:
Topology, steady states, and more
Funding:
Darpa (FENA), NSF,
Packard foundation

2. Hamiltonian construction
quantum J
theorists V U
Thestarting point to a lot fo work, is constructing a hamiltonian. You can already see that I’m a theorist.. You put together some hopping, some interactions.
(taken from: Good night (quantum) construction site)

3. Hamiltonian construction
And out comes this nice looking hmailtonian. But very often, something is missing…

4. Hamiltonian construction
The floqyuet philosophy, is to take your system, and shake it. For some reason or another, that may give you the missing component for your perfect quantum system.

5. Floquet topological phases?
The ‘shaking’ could come from an intese light beam, wich will turn the system into a topological device with electronic edge states.

6. Floquet topological phases
If you’re not into architecture, here is the band structure representation, where a small touch of lightning turns a simple semicondutor into a topological spectrum.

7. Floquet topological phases
Netanel Lindner
Technion
Victor Galitski
Maryland
With all these preliminaries it is time to talk about making a floquet topological insulator out of a trivial semiconductor. I’ll tell you how to do this with a direct resonance, but out work was preceeded by Aoki and Oka, who showed how to obtain a Haldane model with high-frequency light light.
Aoki, Oka (2009)
Lindner, Galitski, Refael (2010)
Kitagawa, Berg, Rudner, Demler (2010) Jiang, Kitagawa,Akhmerov,Alicea,Refael, Pekker, Demler, Cirac, Zoller,Lukin (2011)
Grushin, Gomez-Leon, Neupert (2014)

8. Topological Polaritons
Karzig, Bardyn, Lindner, GR (2014)
So far we talked about using coherent radiation to induce a topological phase in a 2d quantum well. Whenever we use intense radiation in a solid state device, though, people would tell you you are looking for trouble. Can we have similar effects without the intense radiation? How about instead of driving many excitons, as the FTI does, could we have a joint exciton and a photon state that lives only on the edge?
An exciton and a photon together make a polariton. If it is topological, how about calling it a topolariton?
Torsten Karzig
Station Q
Charles Bardyn
Caltech
Netanel Lindner
Technion

9. Floquet-inspired new phenomena
Topolaritons:
Anomalous Floquet Topological Anderson Insulator:
I will show how this is done, and then mov on to new physics inspired by floquet phases. Topological polaritons, as well as anomalous chiral phases – this si a bit of a tongue breaker – but by permuting the words enoguh – you can just call it the faati 

10. Driving uncontrolled heating?
Steady states?
If I were in the audience, or even more so, if I were in this building, I would be really concerned. I’m suggesting shakign it, shining intense lasers on it. It amy just go up in flames – what ever quantum system we have, may just heat up due to the drive.
Driving uncontrolled heating?

11. Steady states?
PRX, 2016
Karthik Seetharam
Caltech
Charles Bardyn
Caltech
Netanel Lindner
Technion
Dehghani, Oka, Mitra (2014) Iadecola, Neupert, Chamon(2015)
Genske, Rosch (2015)
Bilitewski, Cooper (2014)
Chandran, Sondhi (2015)
Mark Rudner
Copenhagen

12. Environment Engineering
Energy filtered lead
Environment E
System V
Towards th end, OI will show how these heating effects ould be controlled with an appropriate environment, both of phonons and of a particle bath.
Phonon bath

13. Mathematical interlude: Floquet Hamiltonians
Periodically varying Hamiltonian:
with
Hamiltonian
Floquet operator:
This was really simple fvloquet. In fact, we should think about talkng a hamiltonain and adding a periodic term to it. Now we need to diagonalize the evolution operator over a period – not the hailtonian. And instead of eigenvalues of the energy, we get roots of unity- which give a set of quasi energies. These quais energies are lke the energy, but are limited ot some segment 0 to omega of energies. Which segment? That’s like choosing the branch of log.
Energy eigenstates
Quasi-energy states: 13

14. Band structure folding and the Floquet zoneP
Let’s apply this to a semiconductor, and see how these quasi energy spectrum is obtained. The band structure has a conduction and valence bands, and driving by omega gives a resonance. 14

15. Band structure folding and the Floquet zoneP
The first resonance we often deal with using the rotating wave approximation. It shifts the relative energy between the bands, to eliminate the time dependence in the drive, which mixes the bands.
(1) 1st resonance: Rotating wave approximation. 15

16. Band structure folding and the Floquet zone
(2) Higher order resonances b b P
Then we get avoided crossings at the intersections. But additional resonances can occur two. This would be a two photon resonance – and we can not brush it off by shifting the bands. To see it better we can make copies of the irginal bnads at hbar omega energy shifts to account for the possibility of absorin a photon.
(1) 1st resonance: Rotating wave approximation. 16

17. Band structure folding and the Floquet zone
3-photon resonance
(2) Higher order resonances
Floquet
Zone b b P b b
2-photon resonance
The copied bands show additional resonances, which get resolved in avoided crossings. This can repeat to third order, and higher resonances. By resolveing all of those, we obtain the quasi energy spectrum in the Floquet zone. This is just like in crystals, when we construc the energy-momentum band structure in the first Briloun zone by contantly folding in the hig momentum statess. Here we do the same, but with energy. Floqeut phases are therefore characterized by sch a band structure in their Floquet zone.
2-photon resonance
(1) 1st resonance: Rotating wave approximation. 17

18. 2d Floquet and topological energy pumping
Floquet Double Drive:
2d Floquet and topological energy pumping
Ivar Martin (Argonne NL)
Frederik Nathan (Copenhagen)
Bertrand Halperin (Harvard)
Gil Refael (Caltech)
So I ill make the following promise: Here I will try to outdo that act – and make a 2d topological insulator using these two drives from a single spin. The work I’ll show you was done in close collaboration with Ivar Mrtin Bert Halprin as well as Frederik Nathan – who Is a student of Mark Rner at copenhagen.

19. And frequency conversion
Overview
Floquet intro (Not again…)
topological phases
0-d topological drive:
Quantized pumping
And frequency conversion
Topological
Cavity amplifier 19

20. Floquet wave functions
Simple drive:
Time evolving wave function:
Time dependent
Schroedinger equation:
That was too fast. I know it. Here it is again. Let’s do something simple – cosine drive. We can put the Floquet ansatz in this hamiltonian, and collect pieces with the same harmonic – and what we get, is a time-dependent SE on a tight binding lattice. With a force – that is this terms which tells you how many photons the system takes away from the external drive. But this is a synthetic dimension. Infinite one!
Synthetic dimension – like a tight binding model
See also: Nathan Goldman; Shanhui Fan; Iacoppo Carussoto.... 20

21. Photon space space vs. real space
Add phase to the drive:
Photon number operator?
How far does this analogy go? Quite far. Let’s cconsider adding a phase to the drive. This phase would get added to each of the harmonics which is just like imprinting momentum on the lattice. So the photon number operator – is just the derivative with respect to this phase. You can see it just by operating with the derivative here - or by analogy with the x operator, which is conjugate to the momentum operator. 21

22. Photon space space vs. real space
Add phase to the drive:
Photon number operator?
By the same token, we can see that the photon absorption rate – by whatever is driven, and fromm the drive, is the commutator of the derivative with the hamiltonian.
Photon absorption rate? 22

23. Phase vs. Momentum Why not identify momentum with drive-phase?
Iff that is so, we might as well write the argument of the cosine – the omega t + phi as k of t. For each k, there is an eigenstate of the tight binding hamiltonian, and the frequency seems to contrl a force on the lattice. The intantaneous photon absorption rate, is then just depsilon dk. Looks like an honest to god 1d system under the infulaence of a fore. 23

24. Can’t access all momentum states.
Caveats
Drive implies constant momentum change:
Can’t access all momentum states.
Dimension of Hilbert space = dim(H0)
Before we go on, I want to emphasize the caveats in thsis analogy. Which I suspect some of you are already muttering to yourselves in secret. Well – certainly we can’t access all momentum states. Momentum just keeps cycling. 24

25. Can’t access all momentum states.
Caveats
Drive implies constant momentum change:
Can’t access all momentum states.
Dimension of Hilbert space = dim(H0)=2
So if the hamiltaoian is for a spin, the hilbert space is just 2d. The wave function is just a 2-tuple spinor. And the force term – which makes this a floquet problem – makes k cycle through the BZ. | -π | π k 25

26. Can’t access all momentum states.
Caveats
Drive implies constant momentum change:
Can’t access all momentum states.
Dimension of Hilbert space = dim(H0)=2
And I can even do the animation for the bloch oscillations simultaneously in real and k spaces! | -π | π k 26

27. Can’t access all momentum states.
Caveats
Drive implies constant momentum change:
Can’t access all momentum states.
Dimension of Hilbert space = dim(H0)=2 z
Adiabatic evolution - if
These bloh oscillations are the periodic floquet states that we are familiar with. The analogy, however, suggests a useful limit. The adiabatic limit where the drive frequency is smaller than the pieces of the hamiltonain. In this case, as k chnges, and as t proggresses, the spinor’s floquet state can follow the effective field pretty well, and at any given time, it is almost the eigen state of the instanteneous hamiltonian. y x 27

28. Higher dimensional analogy?
2d and higher dimensional systems also possible:
And tight-binding effective Hamiltonian:
A bit of a long intro. Apologies. But it accommodate all the people who came late 
But now let’s step on the gas, an move on to 2d floquet. For a 2d problem, we can have two drives. They’s result in a hamiltonian that looks like it has a 2d - BZ, with a forcing term. Why? The wavefunction can now be expended in the produt off harmnics of the two frequences involved. And this gives rise to a nice effective hamiltonian – 2d tight binding model. 28

29. Higher dimensional analogy?
2d and higher dimensional systems also possible:
And here it is – the synthetc diensions, are the number of photons taken from each of the drives. There is a forcing term, which is omega 1 in the n1 dirction, and omega 2 in the n2. These two dimensions are what I need for the topological game Id like to play. 29

30. Commensurate vs incommensurate
There is a caveat here two – an I was expecting you to be whispering it around alrady. In order to make the expansion I did, the two frequencies should be ncommensurate. Their ratio should be irrational, for the space to be truly inependent. What happens if the omega ratio is rational? Say 2/3? Well if you go two sites up or three to the right – you would get the same frequency for n1 omega1 plus n2 omega 2. Sp not all the points in the 2d synthetic space are independent. Only a strip along the omega vector is. And this is then like flattening a nanotube. If we look at how sch a system moves through its dual BZ – phase one vs phase 2, and I’m true to my intention to wrk with k instead of phis, then as time progresses, thepath of the phases crosses the edges of the BZ’s, but after to least common denomeartor of periods, it comes back and repeats the path. To contrast that, if the frequency ratio is irrational – say by the golden mean, which is close to 1.5 – it is 1.618, the path through the BZ is similar atfirst, but then it fails to find its origin, and keeps shifitng. Clearly - this wil cover the entire BZ. There is no translation symmetry in time here. Many of us are out to find a time crystal! This is a time quasi crystal! 
Time
Quasi-crystal! 30

31. Topological synthetic Floquet phases
Use BHZ band structure:
BHZ magic:
I ehxhausted floquet, time to talk 2d topology. I would like to use the 2d structure to do some good. And the BHZ model was always there for me, so why not use it again? Let’s think of spin half electrons, in a 2d band, with spin orbit coupling. The hailtonain is the spin dotted with a3d vector which is a function of momentum, and m and b parameters control to properties of the band. The phase would be topological of a as a function of momenta the direcions of this vector d, would wrap around the unit sphere. In fact - the number of times this vvector wraps abot the sphere, is the chern number, which gives the hall conductance. 31

32. Topological synthetic Floquet phases
Use BHZ band structure:
BHZ magic:
Topo
This object here is the berry curvature. For m=2, this is the phase diagram, with the b’s having to be in a particular range in order to make the phase topological.MOKAE 1d diagram
trivial
Berry curvature 32

33. Topological synthetic Floquet phases
Use BHZ band structure:
BHZ magic:
Topo
This object here is the berry curvature. For m=2, this is the phase diagram, with the b’s having to be in a particular range in order to make the phase topological.MOKAE 1d diagram
trivial
C=-1
C=1
m/2b -1 1 33

34. BHZ Berry curvature and motion:
Semiclassical motion
Berry curvature
BHZ Berry curvature and motion:
Now, we are not going to have a fully ocupied band, we will have to consider a single spin being pushed around, whih is like asingle electron in this band. The band has a berry curvvature, which is the curl of the berry connection, and f a force is applied, then the curvature will be a dual maggnetic field, and will make the particle move normal to the force. I made some maps of the Berry curvTURE. Hre it is in the trivial phase – the b’s being 1, and m about 4. and you can see that there are negative regions and positive ones. The curvature averages to 0. Here is atopological phase: m=1.5, and b=1. If we apply a force in the x direction, we will get a motion in the y direction after a full repeat. You an see that this motion depends on the y momentum. This cconcludes the topological intro. 34

35. Topological synthetic Floquet phases
2f Floquet BHZ:
And no it is time to floquetify it. Make the momentum wind with two frequencies. This makes a double floquet problem.
Where a single spin-1/2, is driven by not one- but 2 elliptically polarized light beams. 35

36. Semiclassical motion:
Synthetic 2f BHZ
2f Floquet BHZ:
Semiclassical motion:
We can now employ the semiclassical equations of motion to determine the motion of the zpin in n-space. Instead of a force - we have omega. Looking at the average motion in n-space, we would find that the average velocity is just the 90 deg rotated omga vector times the average bery curvature, which is the chern number over 2pi. Since multiplng the photon number by omega gives the energy taken from the drives, we get that the wrk the drives do is quantized within semiclassical dynamics. Omega1 oega2 times chern number over 2pi. That’s the effect I wanted to get to.
Quantized energy pumping
Quantized 36

37. Quantized energy pumping
If we take a look at the n space, it looks like this, the system moves from high n2 to low n2 and to high n1. So energy id pumped from laser 2 inro laser 1! This is independet of their frequencis. Why is it good? It can be used to amplify lasers of arbitrary frequency.
Energy pumped
between lasers: Topological
frequency conversion
Quantization - if all BZ is covered… 37

38. Simulations Need to calculate: Parameter regime? Strong driving:
Adiabatic
evolution:
So we need to calcualte this operatore expectation as a function of time. What prameter regime? The best one is when the frequency is lower than the smallest gap in the band structure. Essentally smaller than b and m. This would allow us to have adiabatic evolution, where the spinor is almost the instant eigenstate of the hailtonian. This also lets us know hhow to initialize – we start with the instanteneous eigenstate of the initial hamiltoinan. I hould add that choosing the + woul result in the opposite pumping direction.
Initializing 38

39. Numerics I: Incommensurate Frequencies
Let’s start with the incommenusrate frequencies. We expect that here we will have topolgical drive. 39

40. Numerics I: Incommensurate Frequencies [strong coupling]
Fidelity:
Work:
Power: 40

41. Numerics I: Incommensurate Frequencies [weak copling]
m/2b 41

42. Numerics II: Commensurate Frequencies
Periodic Momentum paths
Initiate w/
Floquet eigenstates 42

43. Numerics II: Commensurate Frequencies [m dependence]43

44. Numerics II: Commensurate Frequencies [phase dependence]44

45. Quantum – Classical hybrid
Replace one of the modes with a cavity:
Much larger Hilbert space… 45

46. Quantum – Classical hybrid
Phase diagram:
(photons in cavity)
Topo
Topo
Trivial 46

47. Quantum – Classical hybrid
Pumping an occupied mode: 47

48. Quantum – Classical hybrid
Pumping from zero: 48

49. Period tripling
Unexpected stable orbit:
Time-crystal??  49

50. Summary and conclusions
Multiple drives could induce topological effects.
Topological frequency conversion, quantized energy pumping, optical amplifier.
Experimental realizations? Spins, qubits, cavities with rare Earth garnets?
Open questions: Temporal disorder effects Using dissipation to increase fidelity? Two cavity systems?

51. Outline Floquet review Double-drive Floquet
Synthetic dimensions and Space-energy duality.
Double-drive topological spin
I will start with a floqet review. And you should pretend like your listening even though it will be the 5th time or so you hear it here. I will then extend to doubly driven sstems, as well show a neat duality between space and eergy- or photon number, which is our synthetic dimension. I will then change gears and do what I romised – which is to turn a single spin into a topological machine. I ill then extend this magic to a driven spin inside a cavty. In both this cases we will have topological fequency conversion alongside quantized energy transfer.
Topologically driven cavity

52. Energy pumping measurement
How to measure n?
Rate of work done:
We should confirm that it actually works. How should we measure n dot? It is the expectation value of the hamiltonaiian derivative vs the phase, or k The rate of work done is just omeag times n dot – and to measure then, we can just measure the expectation value – for the instanteneous wave function, of the time derivative of the relevan t pieve o fthe drive.
Measurement: 52