4.4 Microscopic derivation of the Lindblad equation

Transcription

1 This is a really powerful result: any trace-preserving Liouvillian must have a zero eigenvalue. Its right eigenvector will be the steay-state of the equation, whereas the left eigenvector will be the ientity. Of course, a more subtle question is whether this steay-state is unique. That is, whether the eigenvalue is egenerate or not. I woul say quite often the steay-state is unique, but unfortunately this really epens on the problem in question. Let us now return to the general solution (4.9). Using the iagonal ecomposition (4.95) we get (t) = e t x appley vec(()) = c e t x, (4.98) where c = y vec(()) are just coe cients relate to the initial conitions (you may see a similarity here with the usual solution of Schröinger s equation). From this result we also arrive at another important property of Liouvillians: the eigenvalues must always have a non-positive real part. That is to say, either = or Re( ) <. Otherwise, the exponentials woul blow up, which woul be unphysical. As an example, the Liouvillian in Eq. (4.89) has eigenvalues eigs( ˆD) =, 2, 2,. (4.99) In this case they turn out to be real. But if we also a a unitary term, then they will in general be complex. Notwithstaning their real part will always be non-positive. Assume now that the zero eigenstate is unique. Then we can write Eq. (4.98) as (t) = c x + c e t x. (4.1) I really lie this result. First, note that in the first term c = y vec(()) = 1 by normalization. Seconly, in the secon term all eigenvalues have negative real part so that, in the long-time limit, they will relax to zero. Consequently, we see that, lim (t) = x. (4.11) t!1 which, as expecte, is the steay-state (4.97). Thus, we conclue that if the steaystate is unique, no matter where you start, the system will always eventually relax towars the steay-sate. The real part of the eigenvalues therefore tell you about the relaxation rate of the i erent terms. That is, they give you information on the time-scale with which the relaxation will occur. 4.4 Microscopic erivation of the Linbla equation In this section we will begin our iscussion concerning microscopic erivations of quantum master equations. The iea is quite simple: we start with a system S interacting with an environment E through some coupling Hamiltonian V. We then assume E is enormous, chaotic, filthy an ugly, so that we can try to trace it out an 127

2 obtain an equation just for S. Our hope is that espite all the approximations, our final equations will have Linbla s form [Eq. (4.42)]. But reality is not so in, so you may get a bit frustrate as we go along. This is ue to two main reasons. First, we will have to o several approximations which are har to justify. They are har because they involve assumptions about a macroscopically large an highly chaotic bath, for which it is really har to o any calculations. Seconly, these erivations are highly moel epenent. I will try to give you a general recipe, but we will see examples where this recipe is either insanely har to implement or, what is worse, leas to unphysical results. The erivation of microscopic equations for quantum systems is a century ol topic. An for many ecaes this i not avance much. The reason is precisely because these erivations are moel epenent. In classical stochastic processes that is not the case an you can write own quite general results. In fact, you can even write them own without microscopic erivations, using only phenomenological ingreients. For instance, Langevin augmente Newton s equation with a ranom force an then later euce what the properties of this force ha to be, using equilibrium statistical mechanics. Langevin s equation wors great! It escribes a ton of experiments in the most wie variety of situations, from particles in a flui to noise in electrical circuits. In the quantum realm, unfortunately, that is simply not the case. It is impossible to write own general equations using only phenomenology. An everything has to be moel-epenent because operators on t commute, so if a a new ingreient to the Hamiltonian, it will not necessarily commute with what was there before. Setting up the problem O. Sorry about the bla-bla-bla. Let s get own to business. Here is the eal: we have a composite S+E system evolving accoring to a Hamiltonian H = H S + H E + V, (4.12) where H S an H E live on the separate Hilbert spaces of S an E, whereas V connects the two. We now consier their unitary evolution accoring to von Neumann s equation = i[h,], (4.13) where is the total ensity matrix of S+E. What we want is to tae the partial trace of Eq. (4.13) an try to write own an equation involving only S =. Everything is one in the interaction picture with respect to H = H S + H E [see Sec. 3.3]: That is, we efine = e iht e iht. Then will also satisfy a von Neumann equation, but with an e ective Hamiltonian Ṽ(t) = e iht Ve iht. In orer not to clutter the notation I will henceforth rop the tile an write this equation simply as = i[v(t),], V(t) = eiht Ve iht. (4.14) I now this seems sloppy, but you will than me later, as these tile s mae everything so much uglier. Just please remember that this is not the same as the appearing in 128

3 Eq. (4.13). In consiering the evolution of Eq. (4.14), we will also assume that at t = the system an the environment were uncorrelate, so that () = S () E () = S () E (). (4.15) The initial state of S can be anything, whereas the initial state of E () is usually assume to be a thermal state, E () = e HE, = tr(e HE ), (4.16) although other variations may also be use. The result I want to erive oes not epen on the moel we choose for the environment. However, in 99.99% of the cases, one chooses the bath to be compose of a (usually infinite) set of harmonic oscillators. That is, the Hamiltonian H E in Eq. (4.12) is usually chosen to be H E = b b, (4.17) where the b are a set of inepenent bosonic operators 7 an are some frequencies. Usually it is assume that the varies almost continuously with. The logic behin the assumption (4.17) is base on the fact that the two most wiely use baths in practice are the electromagnetic fiel an the phonons in a crystal, both of which are bosonic in nature. As for the system-environment interaction, it is usually assume that this is linear in the bosonic operators b. So V woul loo something lie V = g M b + g A b, (4.18), where M are system operators an g are numbers. The justification for this in of coupling is two-fol. First, it turns out there are many system in the literature where this type of coupling naturally appears; an secon because this is one of the few types of couplings for which we can actually o the calculations! An important property of the a coupling such as (4.18) an a thermal state such as (4.16) is that, since hb i th =, it follows that applev E () =. (4.19) This property will greatly simplify the calculations we shall o next. 8 7 Whenever we say inepenent set we mean they all commute. Thus, the bosonic algebra may be written as [b, b q] =,q, [b, b q ] =. 8 Pro-tip: If you ever encounter a moel where for which (4.19) is not true, reefine the system Hamiltonian an the interaction potential to rea V = V (V E ()), H S = H S + (V E ()). This of course oesn t change the total Hamiltonian. But now (V E ()) =. But please note that this only wors when [ E (), H E ] =. 129

4 The Naajima-wanzig metho We are now reay to introuce the main metho that we will use to trace out the environment. There are many ways to o this. I will use here one introuce by Naajima an wanzig 9 because, even though it is not the easiest one, it is (i) the one with the highest egree of control an (ii) useful in other contexts. The metho is base on a projection superoperator efine as P(t) = S (t) E () = ((t)) E (). (4.11) Yeah, I now: this loos weir. The iea is that P almost loos lie a marginalizator (I just came up with that name!): that is, if in the right-han sie we ha E (t) instea of E (), this P woul be projecting a general (possibly entangle) state (t) into its marginal states S (t) E (t). This is lie projecting onto a uncorrelate subspace of the full Hilbert space. But P oes a bit more than that. It projects onto a state where E in t really move. This is motivate by the iea that the bath is insanely large so that as the system evolves, it practically oesn t change. For the purpose of simplicity, I will henceforth write E () as simply E. So let us then chec that P is inee a projection operator. To o that we project twice: apple P 2 (t) = P S (t) E = ( S (t) E ) E = S (t) E, which is the same as just projecting once. Since this is a projection operator, we are naturally le to efine its complement Q = 1 P, which projects onto the remaining subspace. It then follows that Q 2 = Q an QP = PQ =. To summarize, P an Q are projection superoperators satisfying P 2 = P, Q 2 = Q, QP = PQ =. (4.111) If you ever want to write own a specific formula for this P operator, it is actually a quantum operation efine as P() = p (I S ihq )(I S qih ), (4.112),q where qi is an arbitrary basis of the environment, whereas i is the eigenbasis of E = E () with eigenvalue p [i.e., E i = p i]. I on t thin this formula is very useful, but it is nice to now that you can write own a more concrete expression for it. If we happen to now P, then it is easy to compute S (t) because, ue to Eq. (4.11), S (t) = P(t). (4.113) We will also nee one last silly change of notation. We efine a superoperator V t () = i[v(t),], (4.114) 9 S. Naajima, Progr. Theor. Phys. 2, 984 (1958) an R. wanzig, J. Chem. Phys. 33, 1338 (196). 13

5 so that Eq. (4.14) becomes = V t. (4.115) We are now reay for applying the Naajima-wanzig projection metho. We begin by multiplying Eq. (4.114) by P an then Q on both sies, to obtain P = PV t, Q = QV t. Next we insert a 1 = P + Q on the right-han sie, leaing to P = PV tp + PV t Q, (4.116) Q = QV tp + QV t Q. (4.117) The main point is that now this loos lie a system of equations of the form x = A(t)x + A(t)y, (4.118) y = B(t)x + B(t)y, (4.119) where x = P an y = Q are just vectors, whereas A(t) = PV t an B(t) = QV t are just matrices (superoperators). What we want in the en is x(t), whereas y(t) is the guy we want to eliminate. One way to o that is to formally solve for y treating B(t)x(t) as just some timeepenent function, an then insert the result in the equation for x. The formal solution for y is t y(t) = G(t, )y() + G(t, t )B(t )x(t ), (4.12) where G(t, t ) is the Green s function for the y equation, t t G(t, t ) = T exp sb(s) = T exp t t sqv s (4.121) Here I ha to use the time-orering operator T to write own the solution, exactly as we i when we ha to eal with time-epenent Hamiltonians [c.f. Eq. (3.62)]. One lucy simplification we get is that the first term in Eq. (4.12) turns out to be zero. The reason is that y = Q so y() = Q(). But initially the system an the bath start uncorrelate so P() = (). Consequently, Q() = since Q = 1 P. Inserting Eq. (4.12) into Eq. (4.118) then yiels x t = A(t)x + A(t) 131 G(t, t )B(t )x(t ). (4.122)

6 This is not ba! We have succee in eliminating completely y(t) an write own an equation for x(t) only. The ownsie is that this equation is not local in time, with the erivative of x epening on its entire history from time to time t. Here Eq. (4.19) starts to become useful. To see why, let us rewrite the first term A(t)x: A(t)x = PV t P = PV t S E We further expan this recalling Eq. (4.114): A(t)x = ip[v(t), S E ] = i [V(t), S E ] E. But the guy insie is zero because of Eq. (4.19) an this must be true for any S. Hence, we conclue that V(t) E =! PV t P =. (4.123) This, in turn, implies that A(t)x =, so only the last term in Eq. (4.124) survives: x t = Going bac to our original notation we get t P = A(t)G(t, t )B(t )x(t ). (4.124) PV t G(t, t )QV t P(t ). A final naughty tric is to write Q = 1 PV t P =. Hence we finally get P. Then there will be a term which is again t P = PV t G(t, t )V t P(t ). (4.125) This is the Naajima-wanzig equation. It is a reuce equation for P(t) after integrating out the environment. An, what is coolest, this equation is exact. Thin about it: we in t o a single approximation so far. We i use some assumptions, in particular Eqs. (4.15) an (4.19). But these are not really restrictive. This is what I lie about this Naajima-wanzig metho: it gives us, as a starting point, an exact equation for the reuce ynamics of the system. The fact that this equation is non-local in time is then rather obvious. After all, our environment coul very well be a single qubit. The next step is then to start oing a bunch of approximations on top of it, which will be true when the environment is large an nasty. 132

7 Approximations! Approximations everywhere! Now we have to start oing some approximations in orer to get an equation of the Linbla form. Justifying these approximations will not be easy an, unfortunately, their valiity is usually just verifie a posteriori. But essentially they are usually relate to the fact that the bath is macroscopically large an usually, highly complex. Here is a quic ictionary of all the approximations that are usually one: Born (or wea-coupling) approximation: assume that the system-environment interaction is wea so that the state of the bath is barely a ecte. Marov approximation: assume bath correlation functions ecay quicly. That is, bath-relate stu are fast, whereas system stu are slow. This is similar in spirit to classical Brownian motion, where your big grain of pollen moves slowly through a bunch of rapily moving an highly chaotic molecules (chaos helps the excitations ie out fast). Rotating-wave (secular) approximation: lie when the CIA tries to ill Jason Bourne to clean up the mess they create. We will now go through these approximations step by step, starting with Born/wea coupling. In this case it is useful to rescale the potential by a parameter V! V where is assume to be small (in the en we can reabsorb insie V). In this case the same will be true for the superoperator V t in Eq. (4.114). Hence, Eq. (4.125) is alreay an equation of orer 2. We will then neglect terms of higher orer in. These terms actually appear in the Green s function G(t, t ), which we efine in Eq. (4.121). Since now the exponential is of orer, if we expan it in a Taylor series we will get G(t, t ) = 1 + T t t sqv s + O( ) 2. Thus, if we restrict to an equation of orer 2, it su ces to approximate G(t, t ) ' 1 in Eq. (4.125). This is essentially a statement on the fact that the bath state practically oes not change. We then get t P(t) = PV t V t P(t ), (4.126) where I alreay reabsorbe insie V, since we won t nee it anymore. Next we tal about the Marov approximation. You shoul always associate the name Marov with memory. A Marovian system is one which has a very poor memory (lie fish!). In this case memory is relate to how information about the system is isperse in the environment. The iea is that if the environment is macroscopically large an chaotic, when you shae the system a little bit, the excitations will just i use away through the environment an will never come bac. So the state of the system at a given time will not really influence it bac at some later time by some excitations that bounce bac. 133

8 To impose this on Eq. (4.126) we o two things. First we assume that (t ) in the right-han sie can be replace with (t). This maes the equation time-local in (t). We then get t P(t) = PV t V t P(t). Next, change integration variables to s = t t : t P(t) = s PV t V t s P(t). This guy still epens on information occurring at t =. To eliminate that we set the upper limit of integration to +1 (then then term V t s will sweep all the way from 1 to t). We then get 1 P(t) = s PV t V t s P(t). (4.127) This equation is now the starting point for writing own actual master equations. It is convenient to rewrite it in a more human-frienly format. Expaning the superoperators, we get PV t V t s P(t) = PV t V t s S (t) E = ( i) 2 P[V(t), [V(t s), S (t) E ]] Thus Eq. (4.127) becomes = [V(t), [V(t s), S (t) E ]] E S (t) E = 1 s [V(t), [V(t s), S (t) E ]] E. Taing the trace over E is now trivial an thus gives S = 1 s [V(t), [V(t s), S (t) E ]]. (4.128) This is now starting to loo lie a recipe. All we nee to o is plug in a choice for V(t) an then carry out the commutations, then the E-traces an finally an integral. The result will be an equation for S (t). Once we are one, we just nee to remember that we 134

9 are still in the interaction picture, so we may want to go bac to the Shcröinger picture. For practical purposes it is also convenient to open the commutators an rearrange this as follows: S 1 = s V(t) S (t) E V(t s) V(t s)v(t) S (t) E + h.c., (4.129) where h.c. stans for Hermitian Conjugate an simply means we shoul a the agger of whatever we fin in the first term. This is convenient because then out of the four terms, we only nee to compute two. As we will see below, it turns out that sometimes Eq. (4.129) is still not in Linbla form. In these cases we will have to mae a thir approximation, which is the rotatingwave approximation (RWA) we iscusse in the context of the Rabi moel (Sec. 3.3). That is, every now an then we will have to throw away some rapily oscillatory terms. The reason why this may be necessary is relate to an argument about time-scales an coarse graining. The point worth remembering is that bath-stu are fast an systemstu are slow. This is again similar to classical Brownian motion: if the bath is a bear, trying to score some honey, then the bath is a swarm of bees esperately fighting to save their home. 1 During the time scale over which the bear taes a few steps, the bees have alreay live half their lifetime. Due to this reason, a master equation is only resolve over time-scales much larger than the bath scales. If we ever encounter rapily oscillatory terms, then they mean we are trying to moel something in a time scale which we are not resolve. That is why it is justifie to throw them away. So, in a sense, the RWA here is us trying to fix up the mess we ourselves create. 4.5 Microscopic erivation for the quantum harmonic oscillator Now that we have a general recipe for fining Linbla equations, we nee to learn how to apply it. Let s o that in the context of a quantum harmonic oscillator couple to a bath of harmonic oscillators. The three Hamiltonians in Eq. (4.12) will be taen to be H =!a a + b b + (a b + b a). (4.13) This is the simplest moel possible of an open quantum system. An, it turns out, this moel can actually be solve exactly, as we will in fact o in the next chapter. It also turns out that for this moel the RWA is not necessary (we will see a variation of it in the next section). The first step is to compute the interaction picture potential V(t). Using Eq. (3.32) 1 A. A. Milne, Winnie-the-Pooh. New Yor, NY: Penguin Group,

10 we get V(t) = e iht Ve iht = e i t a b + e i t b a, (4.131) where =! is the etuning between the system oscillator an each bath moe. Now we plug this into the first term in Eq. (4.129). This gives a messy combination of four term: V(t) S E V(t s) = q e i( + q)t e i qs a b S E a b q,q +e i( q)t e i qs a b S E ab q +e i( q)t e i qs ab S E a b q +e i( + q)t e i qs ab S E ab q. We will continue to wor with this guy, which is only one of the terms in Eq. (4.129). However, once we learn how to eal with it, it will be easy to repeat the proceure for the other terms. Next we tae the trace over the environment. This is a nice exercise on how to move things aroun. For instance, a b S E ab q = a S a b E b q = a S ahb qb i. Thus, we get V(t) S E V(t s) =,q q e i( + q)t e i qs a S a hb q b i +e i( q)t e i qs a S ahb qb i +e i( q)t e i qs a S a hb q b i +e i( + q)t e i qs a S ahb qb i. (4.132) We are finally reay to use the initial state of the bath. If we assume that the bath is in the thermal Gibbs state (4.16), then we get hb q b i =, (4.133) hb qb i = q, n( ), (4.134) hb b qi = hb qb i +,q =,q ( n( ) + 1), (4.135) where n(x) = 1 e x 1. (4.136) 136

11 Plugging everything we then get 2 V(t) S E V(t s) = e i(! )s n( ) a S a + e i(! )s [ n( ) + 1] a S a. (4.137) This is now a goo point to introuce an important concept nown as the spectral ensity of the bath. It is efine as J( ) = 2 2 ( ). (4.138) This is efine to be a function of a continuous variable an, in general, this function will loo lie a series of elta-peas whenever equals one of the. However, for a macroscopically large bath, the frequencies will change smoothly with an therefore J( ) will become a smooth function. In terms of Eq. (4.138), Eq. (4.137) can be written as an integral V(t) S E V(t s) = 1 2 J( ) e i(! )s n( )a S a + e i(! )s ( n( ) + 1)a S a, (4.139) where we use the fact that cannot be negative, since it is a frequency. It is really nice to notice that all properties of the system-bath interaction are summarize in the spectral ensity J( ). This means that the tiny etails of o not matter. All that matter is their combine e ect. Finally, still with Eq. (4.129) in min, we compute the integral in s. 1 s V(t) S E V(t s) = 1 1 s 2 J( ) e i(! )s n( )a S a+e i(! )s ( n( )+1)a S a. (4.14) To continue, the best thing to o is to tae the s integral first. An, of course, what we woul love to o is use the elta-function ientity 1 1 s 2 ei(! )s = (! ). (4.141) There is one tiny problem: in Eq. (4.14) the lower limit of integration is instea of 1. The correct ientity is then 1 s 2 ei(! )s = 1 2 (! ) i 2 P 1!, (4.142) where P enotes the Cauchy principal value. It can be shown that this last term only causes a tiny rescaling of the oscillator frequency! (i.e., it leas to a unitary contribution, instea of a issipative one). For this reason, it is usually calle a Lamb shift. 137

12 However, in orer to actually compute the Lamb shift, we nee more etails about the bath an the calculations are quite har. Hence, this is selom one in practice. Usually, we just remember that! has been renormalize a bit. Neglecting the lamb-shift we then get 1 s V(t) S E V(t s) = 1 = J(!) 2 J( ) 2 (! ) n( )a S a + ( n( ) + 1)a S a n(!)a S a + ( n(!) + 1)a S a. (4.143) We i it! We starte all the way with Eq. (4.129) an compute the first term. Now, I now this seems there is still a long way to go, but computing the other terms is actually quite easy. Going bac to Eq. (4.132), we now compute V(t) S E V(t s) = e i( + q)t e i qs a a S hb q b i,q q +e i( q)t e i qs aa S hb qb i +e i( q)t e i qs a a S hb q b i +e i( + q)t e i qs aa S hb qb i. (4.144) The wor now stops here. If we just thin about this for a secon we will notice that this has the exact same structure as (4.132), except that the orer of the operators is exchange. For instance, in the secon line of (4.132) we ha a S a. Now we have aa S. You can see Linbla s form LL 1 2 L L 1 2 L L is starting to appear. If we now repeat the same proceure we will get at the same form as (4.143), but with the operators exchange: V(t s)v(t) S E = J(!) 2 For simplicity, I will now change notations to J(!) :=, n(!) := n = n(!)aa S + ( n(!) + 1)a a S. (4.145) 1 e! 1. (4.146) Then, combining (4.143) an (4.145) into Eq. (4.129) we get S = n a S a aa S + ( n + 1) a S a a a S + h.c We cannot forget the h.c. Plugging it bac, we then finally get S apple = n a S a 1 apple 2 {aa, S } + ( n + 1) a S a 1 2 {a a, S }. (4.147) 138

13 Yay! We i it. We erive the master equation for a harmonic oscillator (is anyone still here?). 139