Periodically driven quantum systems

you okay okay then so let's get back to business after lunch right so my aim with this is basically to give you guys some going on in the field right so just to begin you know what's the motivation so why you I would type up your egg noodle so the few reasons why paraphrasing is an interesting subject what is that you are writing a tool it can be used for probing quantum systems so in the school say spectroscopy like that some system Natalie needs an expanded magnetic field and we apply some radiation and can look for resonances when the frequency matches energy splitting some of the spectrum of the system it's a so that's the waited during driving long been used as a tool for probing places additionally barely driving to be a tool for controlling quantum systems not just for finding out what's there but for getting some new behavior to come out of the system that it wouldn't have done naturally maybe so here's sort of just one particular direction that comes from my own personal interest control could you say take some material you're the best hundreds of years ago with alchemists they had some materials that they weren't so happy with leg and hope that somehow they could get gold and we know that didn't work that well or at least some chemistry so now today we might say well we have some material we like it but maybe there's some even more exotic material that had some different properties can be some folks this simple material to displaying this more interesting quantum behavior by driving in sacred places it's that's one application of this idea so so really in this talk I think my theme will be more aimed towards this idea of controlling both existence because now it's in 21st century I think that's really where the field is going so we want to understand driving from the point of view how do you understand the dynamics induced by driving it so so here's the basic plan in the beginning you know what if the syrup starts starts off on the same foot and so physical pictures and some intuition basically under two-level system with 200 system between the simplest non-trivial quantum system that there is but despite the fact that it's so simple we can learn a lot from it and develop a lot of intuition which can be important to more complicated system so so spend some time reviewing two-level systems and applying certain ideas there that we can then use then in the second part of the talk give a little bit more introduction to some technical machinery this is quite useful in the end for studying beyond the simplest possible and then finally talk about some applications so so far what let me begin by recalling to mind first of all and why my system systems are all over the place so there are many different types of physical devices or systems where similar behavior could be achieved similar from the point of view of the theoretical description in the use ok so so here are some different situations we may describe as a two-level system example superconducting circuits from qubit here you may have clockwise and counter clockwise circulating currents that correspond to different quantum states or say spin in the quantum dot that has two states also have a particle in well potential since the double quantum target is an electron that can tunnel back and forth all of these different physical systems some way basically you know we focus on low energy states have to quantum states to think in the end of the form a basis for the Dilbert space and therefore by studying two-level systems in general actually learning something about all these differences in the center right and so actually there's one sort of unifying picture that I like to use and think about this which is if there's one particular type of two-level system then probably we learn about first in our studies which is the spin 1/2 speak right that is two two states in a similar space and so actually any of these different physical systems that I want to describe once they restrict to the two lowest levels I can think of by mapping it into some kind of spin picture and that's useful because spins are very geometric code if you picture them we can imagine how they rotate and that helps us develop intuition but so just an example suppose have a two-level atom with some kind of ground and excited state and there's some perturbations coming down so this Hamiltonian is very abstract it's just a bunch of symbols on the paper get some bras but if we now define all the operators in terms of these states so Sigma Z would be difference of projectors I'm excited and proud States and so forth then this Hamiltonian which didn't have any particular geometric meaning suddenly who can be expressed essentially as some kind of spin in an effective magnetic field and now there's some geometry can you visualize let me say a couple things I should've said the beginning first of all please stop me in 10 minutes questions much better we stay together the whole time they wait to the end to do anything and second this is warning today they realize just before the talk there are some inconsistencies with with minus signs that they will appear some convention they switch to the middle so if your father just asked so but the basic idea here right so we start with this song had no equipment it's not kind of effective speed in a magnetic field and so one thing that this games is now in this picture you'll never ever again diagonalize the permission to went to matrix by solving a characteristic equation because he'll just take this 2 by 2 matrix and you'll see that well that corresponds to Sun magnetic field which points in a particular direction in three-dimensional space and they remember that the eigenstates of a spin in a magnetic field point along and against that field and such states are the compared States the speed that's Davis beam has maximum projection along the direction of some unit vector a and there's this simple formula all you have to do is remember that theta and Phi are the pole or a 0 so angles in spherical coordinates remember this you already have your eigenstates you're done we used to solve some triangle yeah so we know that your as a vector is defined that one of the angles that correspond to this field in spherical coordinates and I'm done and the energy the eigenvalue this thing is the length of it good okay so so now in terms of dynamics also understand how spins evolved in the presence of a static field so as the equation of motion for the spin is just the last equation the process which describes a lot more precession so the spin is not initializing that they've on the field but at some angle to it it will success with this projection along the field constant isn't so so throughout the top will be a few points for a into this word exercise here which is basically to tell you if this isn't something that's familiar to you or you don't remember how to drive it you should go home and do this take a picture of it or write it down in your face these are all things that you should know we don't dance with the practice I'm to mission something in this equations of because like this now B is fun per second B is one per second are you taking units yeah public units of H bars and yes good so in your exercise right okay so so this is about speed now I say to the men ago did this effective spin language was useful for learning something about all kinds of different physical systems so now if we come back into this thing about tightening say the particle between two quantum dots potential represents and now I want to understand what's the dynamics like suppose I initialize an electron and left dots and have some energy tuning of the two sides and the tunnel coupling J what's going to happen you can understand this by thinking back to this possession second exercise to think about would be you know try to sketch for the probability look like time and you can do it okay so so this is very simple basics now the one new weather control which is something was the goal at the beginning that's not very interesting so this point of view this procession we discuss this well there's a projection of the spin along the field vector field it just conserve as a function of time so if example I wanted to make this spin go somewhere else with that projection would change I won't be able to do it raising static you clearly need some kind of more complicated time dependent fields and want to exhibit some kind of non-trivial evolution of my quantum stick so that will force us to dependence not just and what object this plays a key role in all this is the time evolution operator which is just time ordered exponential integral so this way you get normal age okay so the simplest case that we can imagine that gives us in fact full quantum control over two level system and by full control I mean that any initial state can eventually be brought to any final state is if I just applied to different pendant um Ian's say h1 and h2 I call them statically applied for some amount of times one and the other one for time T 2 and this deposit evolution can get me from any place to any other useful sorry on the previous slide the first equation this is a equation for potential we must find is this integral equation because right okay so everything called capital okay so here's evolution so suppose we take this example this units h1 always x 21 this we some angle theta 1 and then there's some direction unit vector N 1 describes Tony and a different one where and two is not parallel to in watt so this gives me two different rotation axes that you think about then this time evolution will be the product of those two rotations now there's actually two super useful formulas and these will be useful so first one is how to evaluate exponential poly matrices and the second is what happens when I multiply poly matrices like this I didn't say it was an exercise but actually this formula isn't very should also private it just follows straight from conversation relations having these two things then actually we can simplify to this evolution it's still complicated of - you know matrix exponential that's multiply them together but now using these formulas I can combine them into one rotation about some axis T and some angle theta T which you can find you can find that angle and that unit vector using those formulas right so now I think this composite evolution in that single rotation receiving the same fine state that I would get by these two rotations so that's so the simplest version some time dependent Hamiltonian and this one I mentioned soul exactly so it's kind of useful for that point of view and actually is the common common device that's used theoretically if I wanna talk about driven system is to apply sort of stroboscopic with different constant Hamiltonians because other that's something that can be solved where's the generic continuously variable Tanja there's not much we can do exponential other case may be exactly one of the cases it will talk about now which can be solved exactly and that's the case of a rotating okay so now this is this is the first case of periodic convention so here tell me I have some energy splitting which fits on the national and now the off-diagonal some complex exponential which indicates that translate again back to the spin language that I have an effective magnetic field to the static part proceed and then a transverse part which is rotating in the XY plane is feminine right so I mean as you can kind of see from the geometry this pro since everything is rotating it might be useful to jump into a rotating frame and then see if it blows air that's exactly what we do now technically the way to do it just again it's very useful trick which happens over and over again so fingertips is make unitary transformation complexity so sight are means the quantum state in the rotating frame and that's related to the state in the lab frame by this time dependent unitary transformation and remember that the spin operators are generators of rotations right so here I have Sigma Z which is 1/2 Sigma Z SC so this is an operator which rotates spin around the z axis by angle Omega T and that way is undo this rotation now when I was with the short of an equation for the state in the rotating frame so to get that I should just take the time derivative of sine R and you shameful flex up a new chain request I get two times first to take the derivative here it pulls down this Omega Sigma Z and the second piece time derivative X inside and that's just each side that's original equation so this thing if I multiplied by exponential dependence inverse I can express the whole equation is of this type r and now I have whatever this new Hamiltonian is the X in the rotating right just exist these two pieces transform Hamiltonian original Hamiltonian and one piece which transformation okay so the key feature is that for this problem in the rotating field in the rotating frame this Hamiltonian H artists depend on time anymore then it becomes a simple problem so first of all what are these through the first five proportional Sigma Z it essentially modifies diagonal and original secluding Epsilon now it's going to be epsilon minus Omega and then over here it's just a little bit of algebra what happens these poly matrices but you can just multiply the balanced equation and you see that when I sandwich say Sigma plus by the transformation I get e to the I Omega T Sigma plus come back here Sigma plus it was already either the minus I Omega T or cancel so I just get constant V here is my rotate and in the geometrical picture this whole thing that just corresponds to some effective field which is now in the XZ plane and the Z component instead of being epsilon the original one is this reduce splitting stick tuning and then the driving strength V is that okay so so now we can understand the phenomenon of Rabi oscillations familiar in this picture so the last rank remember and a big splitting now if I Drive through the frequency which is not very close to that one then in the rotating frame I still have a big screaming and my effective field mostly points to the z axis and so what this looks like here so here I have this sphere so I would see the if I start say in the excited state and the system evolve in the rotating frame it'll just process around this field but it'll stay near the don't go all the time because this so here I guess these oscillations the rapid but it's smaller now when I Drive very close to the resonance what happens is my detuning Delta Epsilon is supposed to 0 would be 0 and then the field that I see in the rotating frame is actually dominated body by the driving part of beans but mostly plants experinces and the splitting in the rotating frame is in fact dominated exactly the splitting that I see is length that vector which includes both the tuning parts and the small epsilon these things of course they just being feet now the other important feature is that since this field points close to the x axis if I start in the North Pole this lot more possession rotating frame take me almost exactly to the South second residence so so this kind of just precession in the rotating frame when I go back to the lab frame that's some very complicated evolution but I simply okay so that was the end of the reminder of basics so before I go on any questions on that parts yes say what are you allowing us to play just the time I've seen so the plank was yeah if I if I take take is endearing very very precisely so suppose I have two different to have two different unit vectors N 1 and n 2 let's let those be fixed so I whatever it is my system I have just it's fixed it allowed me to control I can apply Ethernet 1 1 2 and then I could choose how long I apply H 1 and on their page - that's enough so if N 1 and n 2 are given to me and they are not parallel even it was a tiny angle between them and probably I may need to alternate these many times did I just do it once the point of view of dynamics coherent States I think the answers yes now when it comes to measurements of course not because spin 1/2 if I measure something I get projection and only two possible answers yeah so so in terms of my mystical here are snakes there's no difference wherever says if you make say single-shot measurements then spin 1/2 is very different from a classical spin if you only look at averages I think then you won't see the difference and if you go beyond spin 1/2 of course they're squeezing in other things yeah you think yep yeah I agree yes but there is a difference when you measure if you have a single shot measurements then you can tell the difference between one to my class averages yes it's the same but individual measurement outcomes will be different evidence today we discuss after okay so in this thing about this school so all right so second ago we talked about this rotating field and there was some magic which was that by going to a rotating frame actually this driven problem actually just became equivalent to a steady but in almost all cases that's not possible so generically there was no simple unitary transformation that can make such did when I'm done I'll have a new Hamiltonian so even the most almost most actually think I let me just apply instead of this weird complex exponential city here right for example if I have this was done on time speed and magnetic field in its apply AC field along one direction let's close this and try going to a rotating frame this okay so so now I want to use this to sort of more general discussion driven systems so what are we talking about here first of all we've already been doing it but talking about systems with a Hamiltonian depends on time now if you want to do some kind of quantum control the principle we may allow any time dependence we want could change parameters in any crazy way but it's not very much you can do on the theoretical level to describe anything general about that fix so restrict our attention to Peoria driving where the hamiltonian repeats itself is some theory which hopefully will be called this right okay so so we don't thought about how to study systems with Hamiltonian and don't depend on time we diagonalizable toniest look at ground States look at the spectrum etc but for a time-dependent Hamiltonian there's no such thing as a ground state there isn't really even such thing as I statement I could instantaneously diagonalize H at every moment of time and look at some kind of instantaneous eigenstates but unless I'm driving very slowly so the system is adiabatic those eigenstates are meaningless they're not useful for understanding the dynamics so just trying to analyze H of T at HT doesn't help me I need to do something else unfortunately when we talk about periodic paper in Hamiltonian there is something we do we can actually recover some aspect of the familiar concept of the spectrum and and basically it's related to the fact that you know we have a conservation law of something or we have a symmetry so usually if age is constant you have this continuous time translation symmetry and that's where energy conservation council in this case we have a discrete time translation symmetry and because of that gives something some kind of conservation its conservation quasi energy which I'll explain now so the analogy I want to make because this is something that's probably more familiar is first about particle moving in a crystal so so recall that if a particle moving in a constant potential have continuous translation symmetry in the space and because of that I have a conservation law of momentum now if you think about electric moving in a crystal if I didn't know anything about quantum mechanics I would say there's no way this particle would ever make it through it's crazy it's going over all kinds of rough potential fluctuations and there's no sense in which its momentum should be conserved in fact those one of the early problems physicists are thinking about before quantum mechanics was developed why is there such thing as a metal electrode should be scattering all the time but in fact once we learned the electrons are waves we can understand it in fact the discrete translation symmetry of the lattice means that there is still something concerned it's not the original momentum but this crystal advancement so so here this this case I can say there's operator addition operator which commutes with the Hamiltonian potential write it simultaneously diagonalize H and this translation operator is unitary you know it's confidence eigenvalues are unit modulus complex numbers which dramatizes like this and it's been suggested this okay it looks like momentum translates by distance thing usually it's almost the same as momentum but it's not quite and the reason is that actually the eigenvalue of this finite discrete translation is this unit modulus complex number and so whatever this eigen value eigen value parameter case i could have picked a different value great change came by two pi over eight times an integer and i'll get the same value same eigenvalue this t so so that's actually the reason why Brillouin zone in a crystal so have a discrete translation symmetry so I have this complex Exponential's when I in value and therefore this K parameter plus living circle not just about realignment indeed okay so this is how I get from having a real number for momentum in the system of continuous symmetry to grab your pain which and having screen now essentially the same idea applies in the time domain I have a perfectly good system so here now I have this evolution operator this is the time ordered exponential is right earlier which probably needs the system forward in time to one complete period of the driving so the Hamiltonian will be the same in this first time zero to T and the next period t the to T and I can think about diagonalizing yes right so so this basically what is it this U is a discrete map of my evolution but have a sustained at zero I can probably get forward to time T but I find this unitary operator and if I want to get to time to TF by the same operator again and again so really in discrete time it acts the same way as the constant Hamiltonian does in the system which is not being driven it's the same operator which operates in each time step and you know why do we spend so much time diagonalizing Hamiltonian supports that existence the point is that if I expand the evolution in the basis of its eigenvectors evolutions pretty simple I just get some faces so the same thing will apply here so expand evolution and eigen basis of this you it'll be very simple each eigen State will just pick up some face right so so I can look for such heightened states of you and again there I can value something you didn't watch in this complex numbers and I can parameterize them in a very suggestive weight because you know if if I didn't if it's Hamiltonian didn't depend on time then I would just pick up a face e to the minus five energy ten spot but that's the way and I can say in space they're not gonna system so here I've written the eigen value of the time evolution operator on its eigenstate e to the minus I epsilon D that defines what it is excellent and so it looks very similar to energy but just like cane which appears for crystalline second ago looked like momentum it wasn't recommends ax this quantity here which looks like energy isn't quite energy called quasi energy and you see they're both at two PI over T continued integer I don't think the same eigen value them so the point is that it appeared the billing system this quantity which is kind of like energies while the energy it has a Brillouin zone itself lives in a circle and you know it's only well-defined up to integer multiples of this 2 PI over T a to gravity is just a magnetism angular frequency and there's a physical meaning to this right I mean it sounds kind of serious listening for energy to be defined mod mega it means that the system can absorb an immense quantity H bar Omega from the driving feel a source of energy it can absorb one full time two photons it could emit one and all these different energies are coupled to each other to be driving but only in integer numbers of photons again so the physical meaning of this condition right so now - yeah right it's an interesting question and and later also I want to make an analogy between cusco driving and and service quantized field picture so from the classical field point of view as you will see how it comes out there are not a good explanation for why the system should know that there's such a thing as quantization of photons in the driving field but so from Fourier analysis in the Schrodinger equation will see that somehow frequencies can only shift by integer multiples of this frequency but also you can there's an alternative viewpoint from you quantize fields where obviously those h-bar Omega if there's a deeper connection they make a deeper connection but I actually don't have a simple answer any other question right okay which shows up all over physics and different guises but since probably everybody basically here okay theorem in space Spearman Bloch's theorem in science okay so - the connection that you should notice it what is this will keep them safe it says I take my time dependent Schrodinger equation with this time going on Sonia and there will be a complete set of solutions in the form of orthonormal basis of the Hilbert space which take the form of some kind of phase which goes linearly in time select the plane waves part of their block state and crystal and then a periodic Park liquor wave function which repeats itself along with you okay so now we need to find one of these instrument month sets or wave function we need to find out whether the commons by this fine and useful thing one useful thing is that since this fly is periodic I know that I can decompose it in a discreet Summa Fourier components with all integer multiples of the vitamin C Omega and I don't need any infraction come on it's because they normally take from on exhibition reproduce this period I can express it with in some particular places but the good thing is now I have some sort of answers for the wave function and if I just know what is this list of Fourier coefficients which sits here they don't have a complete description of my state so that's our goal now is sort of what equation do these Fourier harmonics satisfied so that I can solve it okay so we just plug in to the Schrodinger equation you take take this state with this Fourier decomposition what you do starting your equation well the dynasty time derivative right now it's the left-hand side when you take time derivative I either act it on e to the I epsilon T by M Omega T which gives me this time derivative the right-hand side of this wave is just H times sorry so first thing I can do is this back to front trick question remind all pi alpha is just some time independent basis which you go slow my name is yeah the equations that slice we should go through it so please just ask everything that doesn't make sense so so now the other step is basing this to project this onto a given stage to be given for a moment and then we'll find out with successful project first thing you take the broad of this other state that everybody's coming from the left on both sides I'm equating this line this one and x sub harmonic than what the M prime component come out so I should just multiply by e to the right and find negative integrate my sister I'm doing this for a to enjoy doing and what I get this is something magical so I get to this now five but this final is a shorthand for the list of all that just take all these Fourier coefficients that I have to just by the state and just stack them up into a big vector and amazing we have this giant vector of Fourier coefficients satisfy something that looks like a time independent schrodinger equation so this is just an eigenvalue equation with some hermitian matrix sitting here and then this really giant vector of fourier harmonics there and that's okay Tony inverse it's there it's basically the Fourier transform tell me every Sun extra piece sitting here so it almost sounds too good to be true where we started we had a time dependent problem we don't know how to solve quantum evolution Tony and now I say well this is plugging this particular listen suddenly we're back to solving a time independent equation which presumably be doodling so we must have paid a price and the price to be paid is that you know originally we met some builders base I didn't tell you how big it was but let's say that this alpha you know some phases there any those it was two level system there are two different value systems but now we have a list of Fourier coefficients and for each harmonic M I have n of those but then the same actually runs for - engagement rings I've traded essentially finite dimensional time dependent problem I did not have solve it have an infinite dimensional matrix but it turns out that from a practical point of view we've actually gained a lot first of all this there's some witnessin instructions let's take a look at this matrix and then I'll show you the effect we never have to do in many cases in practice you don't have to worry about the fact that it's in two-dimensional we can truncate it and use some finite dimensional crystal structures thank you it's particularly simple if I have a driving which contains a single frequency any kind of so first of all stand matrix and put a header block structure and on the diagonal have blocks which are essentially the DC parts zero frequency just copied over and over again shifted by integer multiples of magnet 1600 plus omega2 omega3 on that guy's a part this is bad luck and then the off diagonal blocks are just related to the company so this term here with frequency in it you can see we think about the Fourier transform that we did a second ago any term attendee to the vibes I can teach and change my structure right so there's another question which comes up should come up yes I told you we started with a system whose nth dimension now we have this infinite dimensional matrix and analyze it and we get infinitely many item values and eigen vectors my problem is still living in an n-dimensional Hilbert space so we're all these solutions that are where's the meat why don't have some a solution well the fact is that they're not all independent of each other in fact I had an n-dimensional problem when it started I'll only find n distinct physical solutions when I go back to writing the actual stable system so so to show you here here's this okay onsets wave function with quasi energy epsilon in this expansion they see I can just grab an harmonics here so it's just like was the energy by some integer times Omega I can compensate by also shifting up by harmonics here by the same number and and this I didn't do anything this is nothing the same things that I start of it but it has a different eigen value it's probably epsilon plus so so the point is that there's a huge sort of degeneracy of solution we could solve this dimensional matrix but there's only n solutions we need to find and then there's many other ones if you just copy the original it's basically like you know there's no momentum like the Sun extended some scheme where I have KX over the neck so but but actually somehow I'm just getting copies so that's the important point now what's really useful about this matrix is this block structure that I mentioned actually has some intuitive picture so so block tridiagonal matrix basically describes the nearest neighbors like binding right so if I think of each of these blocks the site then it's some kind of engineering fictitious lettuce my mind then this Delta is a hopping which takes me from one lettuce tight to the next so so I can relate this basically this strange eigenvalue problem is going to give me the solution to the system some kind of hopping on an effective one dimensional lattice and this slide is a little bit funny because the lattice with a linear potential I see that the energy that energy of each of these active sites changes by a mega each time I take a step forward so so this okay diagnose a shoe problem is closely related to put the bunny a stork problem finding what are the eigenstates of the particle and vice in the linear potential field and here this there's some useful picture to keep in mind which which tells you that in fact the solutions are localized in space in space at this price and this goes into different names one is this one you start problem the others Bluff consolation you know if we have an electron in a band and I apply an electric field that one way to view it is that the momentum of this particle will shift in your lead-time and they'll just move around the mystical cart and velocity which is the slope will spend the same amount of time being negative as positive in the particle of oscillates space miserable situation and now if one asks how far can this part of the region we can explore basically this banded has so if I would ignore the linear potential right it's part of the topic so there's a bandwidth to Delta that's the maximum amount of kinetic energy I can give or take its particle so I started up in some location it can explore the region of the slightest changing its potential energy until it has no more kinetic energy so that means there's some kind of bicycle or semi classical turning points the particle can't go beyond so the reason I'm going through all this is to tell you that in terms of this matrix we actually from this picture understand hopefully that the eigenstates are localized in this M direction I have some state which is localized starting around M equals 0 it will explore some region of M supposed to 0 but at some point it will stop some exponential tails and that's what allows me to truncate this matrix I just need to find solutions you know somehow centered around here but go out far enough past these turning points and they just chop off the matrix I won't really mess up my answer pretty much that's what from practical point of view this so one exercise is a bit more challenging probably answer itself is what interests me I do this for some problems research and it just so happened that in class I was teaching with an explanation and so you can do some type analysis right so so now let's try to use this to understand some example okay so we should use this for something so we already talked about this rotating field problem we know the answer we understand it very well that's a great place to start checking if you understand this is normal is here you should understand how is that map so if I take the rotating field example that we already discussed and map it into this placate language but it fine is there are disconnected two by two blocks in this giant matrix which allow me just simply diagonalize so even in this case even though I have this infinite dimensional matrix second so that's basically related but now that we have this formalism and our disposal we can go beyond the rotating fields case say what happens if I add so counter-rotating terms the terms I would get from a cosine writing they will appear you know some other positions in the space and couple different plots and because of that it's no longer available from the salt but now understanding this structure it gives me some way either perturbatively over in America we ever have to release understand what happens and also this is sort of a new physical phenomenon that appear so so for just the rotating field all I have this so the signal looks like resonates I can only expect the system if my driving field has the same frequency as it energy splitting but once I have other terms which couple here I kind of multi floats on resonances suppose I Drive at half the frequency of the splitting the magnitude is big enough I can still excite the system and you'll find that if you go and play with this matrix you find that that's possible by including these effects you won't get a 2,000 residents this can get three so you have an exercise to understand which types of multi put some processes would be allowed by this step we're driving and there'll be some big debates so what would you have to change about the driving in order to get the rest okay so in the last two minutes I'll just tell you some sort of modern application of this suppose now instead of return the level two little sisters so if I was a system with two bands at a particular lattice to pass for each K basically two little system so this is basically an extension of what we've been talking about so now suppose I have this system which discuss from manipulating bad structure called you bands and so forth in such systems through time-dependent writing and the basic idea just like you know we start the very beginning as I went through those simple examples was you know suppose they go here imagine this rotating field suppose that okay the rotating frame essentially I can take in a one branch of states and shift it up and give these resonant points the driving will cover the two beds you can open new gaps change and it's a one application also which has been very popular and we're doing some experiments trying to probe it is this can be used to open gaps at the direct points of either graphene or the services later certainly quite simple limits take with the frequencies let's get there's an even simpler version there less exercise you can use understand it either you can solve this problem to use fear not actually if you want to understand here here the Hamiltonian involves only Sigma X and Sigma Y open the gap again Sigma Z what weight is shining circularly polarized light give me a signal see the simplest way that I know of to convince yourself little work is take this circularly polarized field and just break it into four steps interfere in sequence again heading back to we discussed before one two three four and these will of Sigma X & Y is indexing Y and when you look at the commutator but a quick temperature reading statistics okay so that's that's an excellent question so so the point is that in empirically driven system for many reasons the there's no simple statistical mechanics or thermodynamics like we usually have the energy coming in on the system a one one particular issue that should be mind is that I mentioned this energy like variable is causing energy actually lives in a circle therefore letting them clear which state is above it which states below it shifts my definition and suddenly in the ground state with the clinic sites so so what happens in practice may I assume you when you ask that question what happens in temperature I think what you mean somehow is suppose I connect to a bath and let it thermal eyes and they go with the distribution be safe on these locations first question is is there a distribution just in terms of 4k States or you know is this a good basis even for describing the same states sometimes yes sometimes no but in general whereas in usual step back the form of the bath doesn't matter the way that it's covered with the system doesn't matter as long as it's coming in as long as it's a large reservoir here everything matters so the specific form of the coupling between the system in the bath is actually important there's no Universal answer that's a question in relation to this it'll be the most painful aspects or business anticipation I understand they're 12 in depending on the situation but instead strategy as clever inside some special cases of classes of cases when this the patient will be somehow mitigated yeah innocently by the ball it's a question yeah so so first of all you know here I was mostly focusing on few level system anybody system environment thus essentially three approaches system so so one is okay banana bath and then you know we you know the people have worked understanding what type of coupling to the bath second is okay let's say on your side I'll say since is the third is that there is something called three thermalization so there are regimes such as very high sweetly subscribing but the driving frequency let's see that the energy of a single photon the driving feel is much bigger than any local engines the other systems so I have to make many excitation strips or one photon then my or fragment right so love resident for any kind of dissipation system and it's sort of a different person thanks what simplified needs so if you would really take it's a one way to think about taking idiomatic living here would be just throw away Tobago under agonal I can do that and then in fact this sort of fake type binding problem the dimensions the claim for a transform it it actually gives me back the instantaneous Hamiltonian and tells me the flow case takes or instantaneous Monday and can this approach be used to solve systems peace non-radiative pathways for radiation or excitation so do you mean that I'm driving it and there's some additional speciality of the way to lose energy right thickness related to to job loss question about the sufficient so this strictly speaking what I described what it depends what you include in your handle tummy that you would include everything then it's there if it'll be useful in that case trying to do this say for the system and the whole environment so that each one of these blocks actually is a holding anybody our space problem is very messy not too much you can see what about conclusion something like a photon subsidy for non subsystem right so if it's weakly coupled then you can develop some kind of flock a master equation and try to solve it that the various approaches to solving sir open system plus spirit writing probably this with you questions the inventions that now when you have this matrix you then have to restrict a number of solutions is there a way when you look in the simple case of two-level system to see how this restriction has to work out and stupid maybe a little bit more so there's two two versions of restricting so it's a white person illustrate things that we have too many solutions so in fact all we need to do is pick two of them so so here I mean you can see was this I'm gonna clear why I have to make practice them right so in practice if breakfast all you need to do is is basically defined one real one zone of energy quasi energy so this fix it integral from zero to H bar Omega and you actually guarantee that they'll be exactly two solutions which fall that interval and those are the year two independent solutions and you can pick those and ignore the rest so basically the solutions are repeated every song is for Omega and within each domain have exactly the number of solutions which the second part was I mentioned this truncation related to the fact that the solutions are localized Banach space do that safely you can solve the problem first or you use this kind of argument to understand how many harmonisation keep before I truncate and then you should only take solutions in the very center of this band that you kept the question is is related to and I guess what I should say is this in general these steady state of the system when it's coupled to a thermal bath will not be described by Gibbs factors at all so there's no need of the minus beta quasi energy which shows up only at very very specific conditions interesting like otherwise distribution rights to this different visit there's a truncation of Hilbert space so if we start here we're describing the Sun thank you this aids which is here saying I start with finite dimensional matrix meaning I've already truncated my Hilbert space problems some you know n levels and I don't want any observation out of that subspace of my full Hilbert space in the universe now if I have guaranteed for myself say because I have you know bunch of levels and then a huge gap before any other ones but guarantee not only stay within those levels then this whole description is ok but I can't tell you anything general about occupation among those levels which are down there and rather that becomes very sensitive to the coupling there would be oxy occupation here nothing in excited states of the hilbert space but within this low-energy original subspace distribution can be very in this industry oh I in energy yes also to make this a well well defined problem it should say yeah defined some original system and in any real physical system actually is two-dimensional Hilbert spaces universe i somehow yeah there's some real energy scale that i start with which is large and I say that I never populate any levels above just real energy scale now I respected myself to some finite number of levels and everything that I said this talk was about what's happening in this number of levels given that an ever excite could you please return to mattress various strengths of oscillator hidden so what determines at what speed the system will do this transition except from the frequency difference and the strength external field yes there is a sort of amazing there should be the one which would determine the properties of the material so for example different materials would rack differently to the same intensity and the same frequency even if they all do this this matrix element V here is actually you know it's matrix element specific to this system so I have you know if I measure my intensity of applied electric field and you know in volts per meter I have to convert that through some coupling constants to get this so it's already inside the speed but it's from transformation of external driving force right yes my you know some AC electric field and then that turns into some matrix element between states that's all right either a contribution from the form of original potential for example if we have an atom's we have potential from other electrons which don't interact in this example this external field and determine likely it's a level right so those I mean this external potential is what determines the United States a subspace are talking about and this B is just a matrix element between those states sorry I answer to my own question it goes into H 0 coefficient which is not present on this slide but it was on previous so it goes into other constant contributions so it goes with constant external field yeah it should be both in the DC part and question so do you need to have some weak coupling between the bends you know initially or it's your yes so initially bet bends are already vacuum eyes so there shouldn't be any coupling between them field it's actually distracted the driving that provides the company eventually to really the driving strength to get the gap will be proportional to space crazy question with others it's not driving we're just working the imaginary time everything's free electron you know zero to pee this is not driving too strong some techniques some have pejorative concept of this okay here so it's a principle that the folk a theory which describes is not a weak company doesn't depend on the country so actually you can solve this problem you'd for a strong country it's more complicated to do it but again you'll get some effective bands which come out there can be additional sort of phase transitions to face with conditions where gaps open and close as a function of driving strength my just Requa see this is interesting to explore one last pressing questions to family that doesn't mean what that it was seized and it was minus 1000 times 10 plus it to the people who actually to get the physical observables which do not change after this discussion by superiors so the reason for that is the observable I had to ask this box and see how my observable such words so in other words like having a high-frequency signal which is what reportable people do because they do the magnetics passionately there's more complicated before anybody

Channel: SuperEd International

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Id: ZW2LQ8iVet0

Length: 81min 45sec (4905 seconds)

Published: Sat Oct 08 2016