 That means that I will try to be a little bit less understandable, okay? So we were talking about Floquer states and I remind you there was some so many things has has been said have been said Between the previous lecture in this lecture. So let me remind you what? Where the energy states are what floquia floquia states are we have a Hamiltonian, which is periodic in time and We look for the wave functions Which when we increment time by the period of the Hamiltonian that they acquire a trace factor which is determined by the Value epsilon and epsilon is called is called the quasi energy or the floquia eigenvalue Now I must say that the term floquia eigenvalue is used in completely different context in the in the theory of dynamical systems and therefore I prefer to call it quasi energy because this is this should not this should not leave it's any confusion and And then what where the steps that we Made to arrive to our effective Hamiltonian of the Duffing oscillator We started with this periodic Hamiltonian which had this form One half p squared plus one half omega naught squared q squared plus one quarter gamma q to the fourth minus qf cosine omega f t We consider the case where omega f was close to the eigenfrequency So that absolute value of delta omega was much less than omega f And we use the rotation wave approximation but we did it in two steps the first made economical transformation and Then we did rescaling the canonical transformation essentially tells us that the Hamiltonian At which we arrived which had this form To the one half you that this Hamiltonian has a Complete set of eigenvalues. This is a Hermitian Hamiltonian, and we have heard these words that Hermitian Hamiltonian has complete has a Has a the eigenvalues and eigenfunctions of the Hermitian Hamiltonian are orthogonal the eigenvalues are real and Because we were making set of canonical transformations we expect that they form a full Set and I will not be Proving this but what is important is the following that if We are talking about a harmonic oscillator we use a Floquence state of a harmonic oscillator and we use floquence state of the harmonic oscillator which has Frequency not omega naught, but omega f. So Like Fox states of Harmonic Oscillator with frequency omega f and These states are defined as eigenstates of the raising and lowering operator and this this set of states we know is Orthogonal and complete Now we have found the eigenstates of the operator g Which have the property that g psi g is equal to Psi g n g n Psi g n There's a eigenstates of the operator g. So in this rotating frame We have a Hamiltonian which is not a sum of the kinetic and potential energy But it's still a Hamiltonian. It's fine Where it has a eigen functions it has eigen values and it is eigen values form Orthogonal eigen functions, excuse me are orthogonal and what I can do I can write this eigen function So I g n Is a sum of Eigen Vector n that is I expand my eigen function of The oscillator in the rotating frame of this Hamiltonian. I Expanded in the eigen functions of the oscillator With frequency omega f This is this is just a formal operation. This is a purely formal operation and this is like you expand you have a vector and You say that this vector is a sum of this basis vectors with certain coefficients and Essentially this folk states that they provide a set of Vectors in the Hilbert space and we expand our Eigen vector of the operator g in these states and there is one-to-one correspondence between these functions and these functions Both sets are complete and orthogonal this must be familiar to you already after lectures today And so far so good This by the way well, I'll come back. I'll come later to this the interesting thing that comes from this Representation emerges once we start talking about relaxation So let's think for a second about the way Operator oscillator decays in Fox state. So this are states and Equal to zero This is state one two Three and so on now I put along with this states of the Eigen states of operator g n and This is psi zero psi one psi two and so on The standard dissipation of the oscillator that we were using when we were describing Decay of the oscillator Where does it come from? It comes from the coupling of the oscillator to a thermal reservoir For example from emission of photons you couple an oscillator to the electromagnetic field and And also later can emit a photon and make a transitions between the levels More generally it can emit phonons it can Excite electron hole pairs. There are many ways an electron can Give energy and exchange energy with a thermal reservoir. This is probably familiar to you although sometimes it is kind of hidden under the rug that relaxation is a microscopic process The other name for it is back-action And probably for many people in the audience the word back-action is more familiar than relaxation But essentially one way or the other what happens is you couple your oscillator to a system This oscillator perturbs the system And the system doesn't like it it hits back and When it hits back it takes energy away from the oscillator or it can give energy to the oscillator What is the result of this energy exchange? Well, my oscillator can emit a photon and go down or it can absorb a photon and Go up I will be talking about absorption and emission of photons because this is probably the most familiar a Mechanism but I can as well as I said any any coupling to it can be pair to two photons or two phonons or Whatever them Specific mechanism depends on a system So in non-mechanical systems if you are talking about non-mechanical modes These modes can decay usually that important mechanism is decay into phonons and then this phonons propagate through the Through the connection of the system to to something to the support it go into support If phonons go into support and disappear So this is one of the mechanisms of decay of non-mechanical modes another mechanism is There is current generated by these Vibrations of a non-mechanical system this current again It's essentially the creation of a lack of whole pair excitation of a lack of whole pairs and these elections and holes. They they go away Just current and resistors its dissipation So there are many mechanisms of Relaxations and essentially identifying these mechanisms is one of the important and Sometimes critical problems of Non-mechanical and micromechanical systems essentially when you are talking about high q systems These are systems where you have managed to suppress these dissipation processes to Make a support such that there is huge mismatch of The resistances and therefore phonons don't propagate into into the support you can isolate them from from Resistors you can make lifetime long, but you cannot make it infinite No matter how hard you try and What happens as a result of this process is so An oscillator can make a transition here can make a transition between these states and maybe up there The transitions go both ways It can emit excitations in the thermal reservoir and it can absorb excitations from the thermal reservoir As a result of these transitions ultimately when you have prepared an oscillator and you don't drive it The oscillator comes to if your bus is in some of the equilibrium the oscillator comes also to some of the equilibrium and The probability to find a system in a state n is Proportional to e to the minus e n over k Bt and the way we are considering an oscillator with frequency omega f e n is h bar omega f n and If you want to put here the partition function to To make sure that your oscillator has come to thermal equilibrium now How does the thermal equilibrium? Form how is it formed on at the microscopic level? the critical thing for systems in thermal equilibrium is at a condition of detailed balance and This is also sometimes Not properly defined, so let me try to define it in this context So my oscillator is in state n and it can go from state n to state n What I drew it's It goes to neighboring states. So I am it's n plus one or n minus one It's not necessarily nearest state. It can go over two levels or over three levels It it depends on the microscopic mechanism of relaxation It can also come from state m To state n So as I draw here The oscillator can go from stage two to state one if it just is coupling to photons it emits a photon But then it can absorb a photon if it isn't state one and go to stage two or it can emit a Paraphotons and go from state one to state three if the coupling allows Processes like that Now we leave the system alone and it comes to thermal equilibrium But first it comes to what we say a stationary state Stationary state means that if I look at a given state There is a total probability to leave this state And if I call the population of state n rho n Then there is a total probability to leave this state And so I can write the equation for the population of this state which will be Minus sum over m W n to m rho There's also a total probability to come back to this state From all other states So this is an inflex term and Of course m is not equal to n and then is not equal to n here The stationary regime means that rho n dot is equal to zero Does it mean that our system is in thermal equilibrium? Does it mean thermal equilibrium yes No, it does not It does not it means that we have a stationary state But it doesn't mean that it is in thermal equilibrium now in systems in thermal equilibrium There is time reversibility of all processes you inverse time and Nothing happens that is if I have a transition from state n to state m I In worse time it will be a transition from state m to state n if I look backward in time And the property of thermal equilibrium is The detailed balance that the total probability of transitions between any pairs of states What I wrote here is that the total probability to leave this states and go anywhere Is equal to the total probability to come to this states from anywhere? detail balance Means that for any pair of states that the probability to go from one to the other is Equal to the probability to go in the opposite direction So if I say that W nm or Rho n is equal to W mn n Rho m that is each transition is balanced and This property is satisfied in thermal equilibrium now what we have from here is that we know that W nm over W mn is Rho n over Rho m and since we know the Boltzmann factor for this No, no, no, no the other way around right W nm over W mn is Rho m over Rho n of course Then we see that the ratio of the transition probability is e to the minus e n minus e m over kb So now what is relaxation in this language? Relaxation is I prepared my system somewhere here and Let's assume that the temperature is very low Well, the oscillator will most likely go down It can go up because there is a probability to go up But if this exponential factor is small most likely it will go further down It will go further down all the way until it will be more or less localized in the ground state This is what relaxation is quantum mechanically Classically we describe relaxation by the friction force quantum mechanically relaxation is a result of of the transitions between the levels and These transitions are mostly in one direction And if can't be she's here the just in one direction and this is what I want to illustrate for you because this is the next point that I want to make So I am erasing this part we have we are done with detailed balance I want to look at the eigen states of my oscillator in the rotating frame and As you remember, I was telling that we have this eigen states and there is a stable state in the system Now I want to understand what the stable state means in terms of the transitions between quasi energy levels I look at this expression and and I understand a very funny thing so again, this is The quasi energy level that corresponds to a stable state you remember we had this this line and There were of amplitude or versus field and I was saying that this is a stable state of the system of a given oscillator Now I have quantized my oscillator. I have this quasi energy levels I want to understand what happens with the distribution over this quasi energy levels in the presence of Now this is a little bit tricky. So please interrupt me if It's incomprehensible. Okay promise Okay, so I have one to one correspondence between my fox states and my oscillator Relaxation comes Not from the driving relaxation comes from the coupling of my oscillator to the thermal reservoir Therefore relaxation processes in this case where I that I consider when the driving is comparatively weak Remember that the driving force and on linearity and the decay we're competing with each other But not with this big frequency omega f So my dream and oscillator Decays in the same way as in the absence of the drive Do we see any signature of that in in our analysis? We did so so tell me where I wrote the equation of motion for the oscillator, right? classical equation of motion It had a friction coefficient Did I say that this friction coefficient changes when I turn on the driving field I? Did not so I say that Relaxation occurs in the same way as without a drive Which means that in terms of Fox states? Relaxation still corresponds. I don't know so later, but what? relaxation it means is that an oscillator makes transitions between Its states and emits photons so temperature is zero it can only emit photons How is the stationary state form if it is just emitting photons? How is this state formed? No, no I'm talking about virtualization because if if the relaxation is strong all the quasi energy levels are Gone and it makes no sense, of course So how is the stationary state formed? With this finite amplitude The dissipation mechanism of course does not change if my relaxation is not weak dissipation relaxation Dispatch and mechanism remain the same until I make a really strong driving field and my whole dynamics changes But what I what I want to talk about now is the case where the relaxation is comparatively weak So again this picture Describe it refers to this case of the relaxation So how how does it happen that my oscillator is not in the ground state? Well, it happens because I have a drive So I put energy back from this field cosine omega f t and this field causes transitions up Essentially and they are balanced by relaxation processes and as a result of this balance that there is form a Stationary distribution a stationary state. This is a stationary state But now I want to look at it from the point of view of quasi energy levels Before before I do it. Let's let's come back to this because I know that this is one of the sticking points So where does relaxation come from? I wrote classical equation of motion plus And so on Right, I have this friction coefficient here Where does it come from? microscopically We have to have something that takes energy from the oscillator Some drain for the energy. Where does this drain come from quantum mechanically? It comes from emitting energy to be specific photons into the environment this is the source of this drain and I can write a simple expression for it for a simple model, but let me let me not do it Just emission of excitations you emit excitations This way you lose energy This energy loss corresponds to friction in this simple model the simple Hamiltonian that describes this process is you couple your Hamiltonian to a thermal reservoir and You consider coupling which is linear in the Operators of the thermal reservoir. So for example, for example Q sum over K Excellent K BK derger plus BK where BK and BK derger are creation and annihilation operators of photons This is a specific example. This is a more general form that can be phonons. It can be it doesn't have to be linear It can be by linear It can be more complicated But this is the microscopic mechanism that underlies the restriction coefficient So the restriction coefficient can be expressed in terms of the correlation function of This phonons. It's essentially integral DT Omega naught T commutator average value Don't want to write it Something like that probably imagine a part Not your part So it's it's proportional You can express the friction coefficient in terms of the coupling But if you look at this and if you write that Q is a plus a derger Square root of H bar over 2 omega Not omega F You see that there are processes a BK derger For example terms like that. What is this process? This is a process which drives your operator your oscillator from state and to n minus one This is the lowering operator, right? This is a creation operator of your photon So this is how relaxation comes into play Relaxation comes into play by emitting excitations into the thermal reservoir This is where the friction coefficient This is what defined the friction coefficient that we were having before so Now this is the picture of relaxation. I emit an excitation from this state I made an excitation from this state I go down down down if there were no drive I would go down to the ground state Because the driver is there, I am brought back into the excited state and ultimately there is a balance between emission of excitations into the thermal reservoir and Getting energy from the driving field and the stationary distribution is formed Are you comfortable with this? More or less It's tricky. You did not think about this In these terms because you were saying okay. Well, there is relaxation. There is drive and And the and the oscillator it's the same for a linear oscillator And then you have this variance and response for the amplitude of force vibrations, right? What what is it in terms of quantum mechanics? How does it work in terms of quantum mechanics? Just think about it If you if you write equation of motion q double dot plus two gamma q dot plus Omega not squared q linear oscillator is equal to f cosine omega f t And we know the answer that q is in the stationary regime is a cosine Omega f t plus some phase classically and you were taught Today I think or yesterday that it's actually true quantum mechanical for coherent state You can you can write the same answer What does it mean in terms of the energy levels? What does it mean in terms of the microscopic processes that underlie this solution? There must be some microscopic picture behind and this is this microscopic picture Now what how does this microscopic picture? translate to the quasi energy states well, I have transitions between and between Fox states But I know that my quasi energy states are linear combinations of Fox states therefore if there occurs a transition between Fox states it means that there is also a transition between quasi energy states So I start with this by the energy state and as a result of a mission of a photon I Make a transition between quasi energy states. I can make a transition here or there or there even in this Model where transitions between the Fox states are transitions between the nearest levels Why because there is a sum here My quasi energy state is a superposition of Fox states and vice versa My Fox state is a superposition of quasi energy states therefore a transitions between a transition between Fox neighboring Fox states corresponds to transitions between several quasi energy states But moreover It's not necessarily transitions down in quasi energy Right This is the tricky part I do it. I do it at a Purely formal level and what I am doing is the following I'm saying that if I have my system prepared in a quasi energy state Now I turn on relaxation If my system is an a quasi energy state it is actually in a superposition of Fox states right At the initial moment at the initial instant of time Right, I prepare my system in a quasi energy state Initially, and I want to see what happens because of relaxation now relaxation is transitions between Fox states and all these transitions go in the one direction down but this transition down even between neighboring states Correspond because my my quasi energy state is a superposition Even though I make here transitions between neighboring states in terms of quasi energy states I make transitions between remote states as well And I don't necessarily make transitions down. I also make transitions up And now we are really confused. I hope we are go ahead I switch between different coherence. I better use Fox states because Fox states Yeah, quasi energy states are not necessarily exactly for coherent states. They they become coherent states only the semi-classical limit So they are not they are not exactly coherent states. So I don't want I don't want to Mislead you too far I'm trying to chip on a little bit So So I have just from this formal picture What I see is that I make transitions down, but I also can make transitions up and then Do I have relaxation? How how is this state formed? What does relaxation mean in these terms if from any Energy state my dissipative process Listed transitions up and down. How does system possibly arrive at a Stationary state at the stable state of force vibrations Well, the answer to this question is that you have to compare the transition probabilities So suppose I have this state I Will use capital letters for this state and minus one and plus one and so on and I have a probability of transition n to n Minus one and I have a probability of transition n to n plus one what happens is that If the probability of transitions from n to n minus one and all the way down and this is up if my Rate down is bigger than the rate up Then the picture is quite clear. I start in this state for example and I Can make a transition down? So suppose I have made a transition down I can make a transition up but more likely I Will go further down Again from this point I can make a transition up with non-zero probability But more likely I will go further down and further down and further down and therefore In terms of quasi energies what happens is that relaxation is the result of the probability of transitions in one direction Being higher than the probability of transitions in the opposite direction And then I made made a step here Most likely I will make a step here again. I can make a step back But then again, I will most likely step here And most like a step step here. I can make a step back make me can make two steps back But it's unlikely most likely I will be I will keep going going going till I fall down. So this is what? What? relaxation means the excision means the transitions in in one direction are more probable than in the opposite direction but at the very fact that I can make a transition in the opposite direction has a very profound consequence It tells me that there is absolutely no way That even where the temperature of my thermal reservoir is zero. I will go down to the lowest Pleasant energy state here. I'm using a jargon. So I Just want to be understood The minimum at the meaning is that I will necessarily have a finite probability to be in this state and To be in this state and to be in this state That is I cannot cool the system down to the ground quasi energy state no matter what the temperature of my thermal reservoir is and This effect is called quantum heating. Yes Yeah, that's the right. This is why this is what I said before when I say when I said that I'm I'm oversimplifying this there is certainly in quasi energy state. There is absolute. There is actually no up and down It's it depends on what you are talking about But if I made a plot of this function G As a function of coordinates this function G in the simplest possible case Would look like that It's actually quite fancy, but the simplest possible case is where this function looks like that This is the case where there is just one stable state on this diagram. It corresponds to this region large compared to the large fields and then my eigenvalues are here and The transition that I am talking about are transitions between this by the energy states This is the stable state Dynamically classically, I will come here to this state want to mechanically I will be in in I would think that I would Occupy quasi energy state, which is closest to this minimum This is what I do in a system in thermal equilibrium But what I am saying is that no no such luck. I will necessarily Have a finite probability to occupy this state and occupy this state and to occupy excited states Even where the temperature of the thermal reservoir is zero So this is a profound effect of a non-equilibrium system being away from equilibrium Prevents you from being really cool down to the ground Flocky eigen state down down to the Down to one quasi energy state And there was an experiment which very nicely Confirmed this prediction people just measured occupation of the of the Lost occupy a lot next to the lowest state here for this oscillator. This experiment was done in Sackley and And now people are talking about quantum heating due to quantum measurements, which of course another another way of talking about About coupling to a thermal reservoir and back-action from thermal reservoir I doubt it I doubt it I don't know how to do it let me put it that way I You the transition that you make When you couple to any real systems are transitions between between four states No matter what you do, there are super positions You can cool it down to the ground state, right, but Doesn't mean anything So this is I think this is just an unavoidable consequence of the pleasure of having Having quasi energy states. You cannot it's very hard to prepare a given quasi energy state It can be done. Let me let me take it back if you do it fast Sufficiently fast so that relaxation doesn't come into play you can't prepare your system in the quasi energy state It will stay there for during the lifetime But it will not be a stationary state the stationary state is necessarily a Superb you you you find your system in excited questions the states And this has a profound consequence in a Yet another context So I have time to 505 right Sure I Can I can I Can and and the equation will have the same form raw and dot is equal It's it will be the same equation and Thank you for the question. It's very much very much to the point the equation will be the same raw and dot is Sam Wmm Raw and with the sign minus I go from a quasi energy state and let me use capital plus some Wmm Raw and and I can look for the stationary solution of this equation Right, this will give me populations of quasi energy states But there is no this condition. There is no detailed balance in this system Therefore, I No matter how hard I try I cannot localize my system in single state So this is the difference between a stationary state and the state of detail with detailed balance now Why I you to you remember that I said the detailed balance is a property of systems and thermal equilibrium Related to time reversibility So where did I break time reversibility in my oscillator? It was emitting phonons anyway What did I change compared to the oscillator which was in some of the equilibrium It's written. It's written in the Hamiltonian the Hamiltonian does this Hamiltonian have have time invariance This Hamiltonian Hamiltonian of the original system. Of course not time has become non-uniform. I Have broken time reversibility My time is now chopped into portions Here force goes up here force goes down here force goes up here force goes down. It's no longer a uniform time And this is this is why I lose detail balance and this is why the solution of this equation is Does not satisfy this condition for detail balance It's stationary this this sum is equal to this sum But not each condition not each transition is balanced Detailed balance thermal equilibrium is a very delicate and very actually restrictive and amazing condition each transition is balanced if I make a step to the right I Know that I will have to make the step to the left with absolutely certain probability in the stationary regime I can go several times right and then several times left or Three three steps right three steps left or five steps right Four steps left and then maybe another step left On average I my population does not change but not each transition is balanced Because my time is My time reversibility is broken No, no, no, of course not of course not think about think about what happens when I Consider this approaching thermal equilibrium I Couple my system to a thermal reservoir right now. I pump energy Into my oscillator This energy is dumped into thermal reservoir There is an energy flux Into thermal reservoir if I inverse time There will be energy flux into my oscillator from thermal reservoir don't think so my shield Breaks time reversibility. I have dissipation of energy from the field the oscillator takes energy from the field and dumps it into thermal reservoir and there is a flux on average and Non-zero energy flux from my oscillator into thermal reservoir Once you have a flux You don't have time reversibility because time reversibility would change the sign of this flux Do you agree with this? Yeah, you're better Well, well these things are very confusing I can tell you These things are very confusing and it requires it requires thinking but the idea of this is very Very straightforward once you have a flux It means that if you reverse time the flux would go in the opposite direction But you have flux only in one direction you have energy dissipated into thermal reservoir no the other way around so let me try Let me try to tell you a more recent story related to this business and related to the driven systems and This story is kind of entertaining so you all know the EPR paradox right and Paradox was a mistake made by a great man Which turned out to be very stimulating? Recently there was another mistake made by by an outstanding physicist Which turned out to be very stimulating and I just want to tell you about this mistake and how it is related to Nanomechanics and things like that So the mistake was called time Crystals and the person who made this mistake is Frank Wilczek And if you know the word combination standard model, you should also know the name Wilczek so Wilczek said that if Atoms can arrange themselves into a lettuce Why wouldn't we have a lettuce in time? Why wouldn't we have the ground state of a quantum system? Periodic in time So this was done Wilczek. I'm I misspell. I'm sorry There is Z something like that no, I Like that. Yeah, so it's Wilczek 2012 and Seals later there was published a paper which explained that no no no This is not happening in systems in thermal equilibrium. And again This is related to what we were just saying that in some of the equilibrium. There is no time direction all time Time is uniform This is the definition of some of the equilibrium. So in some of the equilibrium you may not possibly have a time crystal then There was a parallel development which started actually yearly and a year earlier Which was related to periodically modulated systems the question was The following suppose I have a system which is periodically modulated like this what is The expectation value of my operator in the quasi-energy state So the expectation value is psi epsilon of t Some operator L doesn't matter what it is psi epsilon of t Let me now calculate it at time t plus tf I know this factor of the wave function, right? Calculated in your head. What is the answer? It's the first line, right? Because I have here this is either this is complex conjugate So this is e to the i epsilon tf over h bar Psi epsilon of t L e to the minus i epsilon tf over h bar Psi epsilon of t which is Equal to this right So what it means is that any absorbable quantity in a quasi-energy state oscillates with the period of the drive then there was a paper that said and then the experiment that independently done Kind of observe the phenomenon of period doubling that is You have this an operator L you look at the system and the observer and the expectation value of this operator oscillates with Period doubling with periods 2 tf How does it happen? Well, let's say that I have I Have a state phi of t Which is a 1 psi epsilon 1 of t plus a 2 psi epsilon 2 of t And now I want to calculate The expectation value of my operator L In this state I prepare the coherency preposition Right, so I will have of course the diagonal terms, right? So I will have a 1 squared Psi 1 L of t. Let me write it carefully psi 1 of t L psi 1 of t Plus a 2 squared Psi 1 psi psi 2 psi psi 2 of t L Psi 2 of t and to make it meaningful I will take Phi of t plus t f so this is from the diagonal terms and we know that for diagonal terms There is no time-dependent factor here, right? This is just the matrix element which calculated But then there is something weird because then there is this term a 1 a 2 Either the I Epsilon 1 minus epsilon 2 t f over h bar Phi Psi 1 of t L psi 2 of t Plus complex function. That's luck Nothing special right except for the case where epsilon 1 minus epsilon 2 is Pi h bar over t f That is the difference of the quasi energies is Half the modulation period half the Half the energy h bar omega f if this is Pi h bar over t f for some reason then this factor Becomes e to the i pi and so the expectation value is not periodic But if I calculate this expectation value at time 2 tf If I put 2 tf here, but this will not change here I will have 2 tf and it will be e to the i 2 pi therefore my expectation value in a superposition of two states that Differ in quasi energy by half of the Was energy bandwidth h bar omega f I Can create Expectation values in these states that will be periodic with Period to we'll get to it probably How can it possibly happen that you have this relation applied well the papers are that where one paper was about cold atoms Fermi atoms in Bosonic reservoir or vice-versa Bosonic atoms in Fermi Fermi atom reservoir by Zoller Gil Raphael and and several other authors in 2011 All these references are in this file that I put on on the On the Internet so you can find all these references and I put These references with the titles of the paper so you can see what how the paper was What the title was? right And the states that were found there we Spent quite a bit of time trying to understand what it was but ultimately this were a state where there were two myranna fermions at the at the edges and these two myrannas had Difference of why the energy exactly like that And then there was in 2012 there was an experiment and the first author on this paper is Kitagawa the experiment where essentially time periodic time periodicity was mimicked by a Set of polarizers and the photon was propagating through this set of polarizers and in this system there also emerged a couple of boundary states which have exactly this Condition and they have seen that this period doubling in This In this chain, so this boundary states had periods two So periods two means that it's 2 tf the periodicity of the system is tf However, you don't come back after one period. Do you know any example where you don't come back after one period? spin, right Spin one half is is an example like that. So here is a time analog of spin one half if you like right Well along and behold Last year There was actually the first people was on the archive in 2015 a bison he and his group and This year you could find in all Popular journals from Wall Street Journal to Go down down the line to nature that there is a new state of matter where you drive your system and This system displays period doubling and The theory was done for a spin chain The experiments there were two experimental results done with an interval of About one week and one of the people on the both papers both papers have the same courses Although the systems the systems are completely different one experiment was done with With cold irons in Monroe group at Maryland the other was done with Envy centers in diamond in looking son in looking's group and in both groups they saw that when they drive the system by a Specifically designed sequence of pulses they kept they could see Period doubling and looking even saw period tripling now Do you know of any system that you know that they displaced period doubling? You know such systems trivial What are we doing? What about parametric oscillator? What do you do with the parametric oscillator? Imagine that you are this is this is how I usually describe it you are on swings and You do squats to make them Go Right, whatever apparently ancient Greeks had swings already on some basis there are There are pictures of swings and So you do this squats and you count one forward to backwards C forward for backward five forward six backward all odd forward all all Even backward What is the relation between the period of your vibrations and the period of sports? to squats for vibration period This is called period doubling Now you necessarily have two states because we had all odd forward and even Backward, but you could as well have all odd Backward and even forward so you have two states but you are in one of them and This system is of course Your swings are pretty classical system very very Disciptive system so it is different from the statement of Of the new state of matter which displays period doubling in the quantum coherence regime The question is how to link these two Cases and essentially a nonlinear oscillator provides a very natural way to linking period doubling in a quantum coherence system and see How what happens when you switch to incoherence regime and the question number one is can you have period doubling in a Parametric oscillator in the quantum coherence regime, so no dissipation Can you have your? Parametric oscillator that will be in a state that displays period doubling The answer is yes, but it's a non-trivial. It's a non-trivial statement so If you are prepared to listen for another 15 minutes how to do it, I will I will tell you how to do it If you are that tired, I can stop here. It's a yes or no question. Let's take a vote Oh Don't those who those who are not interested. I just just just leave and or or check your email Whatever Let me let me try to show you what happens here So first of all, what is the Hamiltonian of my German oscillator Parametrically driven oscillator in the real Hamiltonian. It's one half you square Except that Delta omega is now One half omega f minus a mega note that is I drive my oscillator I modulate the frequency look at these terms. This is q squared and this is q squared Now my mega note squared has become a mega note squared plus f cosine omega f t That is I modulate the frequency. Is it related to what I'm doing on swings What do I do when I do a squadron on swings? What do I change? the lengths What is does the frequency of a pendulum depend on the links? It does so when I'm doing Squatting I'm changing the look I'm modulating the frequency of the swings and this is how I excite them So swings are a parametric oscillator So I drive it close to a resonance in this case It's one half omega f is close to a mega note And if I do the system classically exactly as I did before I will find that my Oscillator can have two states with two stationary states a Cosine one half omega f t plus five one state and second state a Cosine one half omega f t plus five plus five They are just shifted in phase by pi as are these These vibrations on swings they are shifted in phase by pi Why even at odd different phase just by pi right? You don't see it here is the vibration One vibration another vibration and here is my modulation frequency. Let me try to see so this is f cosine omega f t and These are two vibrations one this and the other a This is number one and this is number two now in the notes I describe a much more interesting and more sophisticated case of period tripling but It's not something I can describe in in in a fast way. Oh, let's think about what happens with these periods to vibrations These are two states of the oscillator Which are identical? It reminds you a particle in a double well potential in symmetric double well potential Here there are also two states which are identical just shifted in sign have opposite sign So believe me or not I can go through the same calculation that I did here And I will obtain my function G Which will look almost like that It will have a form plus one half one minus mu p squared minus one half one plus mu q squared and we know is Proportional to this delta omega and if you see if you look at this Now Hamiltonian in the rotating frame if I look at it at p equal to zero at p equal to zero It becomes one quarter q for the force minus one half one plus mu q squared right So if I plot this function g of p equals to zero q Like that. This is the potential that I will have Again, I cannot separate kinetic and potential energy. So what I'm saying now is a cartoon So now again, I have eigenvalues of this operator g and eigenfunctions So what do you know about eigenfunctions and eigenvalues of a particle in a symmetric potential well If I change q to the minus q I Have the same Function at the same Hamiltonian. So my eigenfunctions have to be either even or odd Does it ring the bell? Not every bell So I have an even state plus and Antisymmetric states so state minus psi minus of q is minus psi minus of minus q and Psi plus of q is psi plus of minus q Which you can guess from here if I change the sign of q my function g does not change Therefore my change of sign should My eigenfunctions should be an eigenfunction of the operator of China of sign change and this is either even or odd And now if I am in any of this of the corresponding quasi energy state What is my? period for each of these states It's period one It's period one because after a period one I come back to that same state The only option for me to have period doubling Is where these two states coincide Have the same eigenvalues gm and Have you ever heard about? level repulsion anti-crossing any of these words, right? so levels anti-cross they don't cross they They don't like to coincide Well, this is this is our mentality based on on the Hamiltonian which is a sum of the kinetic and potential energy and actually Hilbert's theorem Related was related to the Hamiltonian over the four p squared over two plus potential energy so this anti-crossing is kind of an artifact of Simple Hamiltonian for more complicated Hamiltonians like we have here This symmetric and anti-symmetric state can have at the same Eigenvalue chip and there's no way that I can show you how it works But you can read about it But the amazing part is that yes the eigenvalues can coincide and when they coincide this is exactly that situation of quite the energy levels different by half half the energy H bar omega F so With a parametric oscillator, we can naturally have a quantum coherent regime where also where expectations values are Periods, too, but now we know what happens when we turn on dissipation in this system Because when we turn on dissipation ultimately, we end up on swings, right? And on swings we vibrate with a certain phase which has periods to and So you can carefully see how by turning on dissipation you switch from this period doubling In the coherence regime where these two states coincide To period doubling which is fully incoherent so none of the mechanics provides you with this sweet opportunity of connecting Quantum coherent and quantum incoherent period doubling and it also allows you to study period tripling which is a far more interesting phenomenon with different terminology and different properties and And this is one of the games none mechanics allows you to play. Thank you very much