Time Dependent Perturbation Theory, Lecture notes of Physics

Download Time Dependent Perturbation Theory and more Lecture notes Physics in PDF only on Docsity! Chapter 12 Time-dependent perturbation theory So far, we have focused largely on the quantum mechanics of systems in which the Hamiltonian is time-independent. In such cases, the time depen- dence of a wavepacket can be developed through the time-evolution operator, Û = e!iĤt/! or, when cast in terms of the eigenstates of the Hamiltonian, Ĥ|n! = En|n!, as |!(t)! = e!iĤt/!|!(0)! = ! n e !iEnt/!cn(0)|n!. Although this framework provides access to any closed quantum mechanical system, it does not describe interaction with an external environment such as that im- posed by an external electromagnetic field. In such cases, it is more convenient to describe the induced interactions of a small isolated system, Ĥ0, through a time-dependent interaction V (t). Examples include the problem of magnetic resonance describing the interaction of a quantum mechanical spin with an external time-dependent magnetic field, or the response of an atom to an ex- ternal electromagnetic field. In the following, we will develop a formalism to treat time-dependent perturbations. 12.1 Time-dependent potentials: general formalism Consider then the Hamiltonian Ĥ = Ĥ0 + V (t), where all time-dependence enters through the potential V (t). In the Schrödinger representation, the dynamics of the system are specified by the time-dependent wavefunction, |!(t)!S through the Schrödinger equation i!"t|!(t)!S = Ĥ|!(t)!S. However, in many cases, and in particular with the current application, it is convenient to work in the Interaction representation,1 defined by |!(t)!I = eiĤ0t/!|!(t)!S where |!(0)!I = |!(0)!S. With this definition, one may show that the wave- function obeys the equation of motion (exercise) i!"t|!(t)!I = VI(t)|!(t)!I (12.1) where VI(t) = eiĤ0t/!V e!iĤ0t/!. Then, if we form the eigenfunction expansion, |!(t)!I = ! n cn(t)|n!, and contract the equation of motion with a general state, "n|, we obtain i!ċm(t) = " n Vmn(t)ei!mntcn(t) , (12.2) 1Note how this definition di!ers from that of the Heisenberg representation, |!!H = eiĤt/! |!(t)!S in which all time-dependence is transferred into the operators. Advanced Quantum Physics 12.1. TIME-DEPENDENT POTENTIALS: GENERAL FORMALISM 140 where the matrix elements Vmn(t) = "m|V (t)|m!, and #mn = (Em #En)/! = ##nm. To develop some intuition for the action of a time-dependent potential, it is useful to consider first a periodically-driven two-level system where the dynamical equations can be solved exactly. $ Info. The two-level system plays a special place in the modern development of quantum theory. In particular, it provides a platform to encode the simplest quantum logic gate, the qubit. A classical computer has a memory made up of bits, where each bit holds either a one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can hold a one, a zero, or, crucially, any quantum superposition of these. Moreover, a pair of qubits can be in any quantum superposition of four states, and three qubits in any superposition of eight. In general a quantum computer with n qubits can be in an arbitrary superposition of up to 2n di!erent states simultaneously (this compares to a normal computer that can only be in one of these 2n states at any one time). A quantum computer operates by manipulating those qubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm. An example of an implementation of qubits for a quantum computer could start with the use of particles with two spin states: |$! and |%!, or |0! and |1!). In fact any system possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and su"ciently spaced consecutive eigenval- ues, is a suitable candidate for implementing a qubit. This is true because any such system can be mapped onto an e!ective spin-1/2 system. $ Example: Dynamics of a driven two-level system: Let us consider a two-state system with Ĥ0 = # E1 0 0 E2 $ , V (t) = # 0 %ei!t %e!i!t 0 $ . Specifying the wavefunction by the two-component vector, c(t) = (c1(t) c2(t)), Eq. (12.2) translates to the equation of motion (exercise) i!"tc = % # 0 ei(!!!21)t e!i(!!!21)t 0 $ c(t) , where #21 = (E2 # E1)/!. With the initial condition c1(0) = 1, and c2(0) = 0, this equation has the solution, |c2(t)|2 = %2 %2 + !2(# # #21)2/4 sin2 #t, |c1(t)|2 = 1# |c2(t)|2 , where # = ((%/!)2+(###21)2/4)1/2 is known as the Rabi frequency. The solution, which varies periodically in time, describes the transfer of probability from state 1 to state 2 and back. The maximum probability of occupying state 2 is a Lorentzian with |c2(t)|2max = &2 &2 + !2(# # #21)2/4 , taking the value of unity at resonance, # = #21. $ Exercise. Derive the solution from the equations of motion for c(t). Hint: eliminate c1 from the equations to obtain a second order di!erential equation for c2. $ Info. The dynamics of the driven two-level system finds practical application in the Ammonia maser: The ammonia molecule NH3 has a pryramidal structure with an orientation characterised by the position of the “lone-pair” of electrons sited Advanced Quantum Physics 12.3. “SUDDEN” PERTURBATION 143 12.3 “Sudden” perturbation To further explore the time-dependent perturbation theory, we turn now to consider the action of fast or “sudden” perturbations. Here we define sudden as a perturbation in which the switch from one time-independent Hamiltonian Ĥ0 to another Ĥ "0 takes place over a time much shorter than any natural period of the system. In this case, perturbation theory is irrelevant: if the system is initially in an eigenstate |n! of Ĥ0, its time evolution following the switch will follow that of Ĥ "0, i.e. one simply has to expand the initial state as a sum over the eigenstates of Ĥ "0, |n! = ! n! |n"!"n"|n!. The non-trivial part of the problem lies in establishing that the change is sudden enough. This is achieved by estimating the actual time taken for the Hamiltonian to change, and the periods of motion associated with the state |n! and with its transitions to neighboring states. 12.3.1 Harmonic perturbations: Fermi’s Golden Rule Let us then consider a system prepared in an initial state |i! and perturbed by a periodic harmonic potential V (t) = V e!i!t which is abruptly switched on at time t = 0. This could represent an atom perturbed by an external oscillating electric field, such as an incident light wave. What is the probability that, at some later time t, the system lies in state |f!? From Eq. (12.4), to first order in perturbation theory, we have c(1)f (t) = # i ! % t 0 dt""f|V |i!ei(!fi!!)t! = # i!"f|V |i! ei(!fi!!)t # 1 i(#fi # #) . The probability of e!ecting the transition after a time t is therefore given by Plot of sin2()t)/)2 for t = 1. Note that, as t , ), this func- tion asymptotes to a %-function, (t%()). Pi#f(t) * |c (1) f (t)| 2 = 1 !2 |"f|V |i!| 2 # sin((#fl # #)t/2) (#fl # #)/2 $2 . Setting ) = (#fl # #)/2, the probability takes the form sin2()t)/)2 with a peak at ) = 0, with maximum value t2 and width of order 1/t giving a total weight of order t. The function has more peaks positioned at )t = (n+1/2)(. These are bounded by the denominator at 1/)2. For large t their contribution comes from a range of order 1/t also, and as t ,) the function tends towards a %-function centred at the origin, but multiplied by t, i.e. the likelihood of transition is proportional to time elapsed. We should therefore divide by t to get the transition rate. Finally, with the normalisation, ,$ !$ d)( sin("t) " ) 2 = (t, we may e!ect the replacement, limt#$ 1t ( sin("t) " ) 2 = (%()) = 2(%(2)) leading to the following expression for the transition rate, Enrico Fermi 1901-1954: An Italian physi- cist most noted for his work on the development of the first nuclear reactor, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics. Fermi was awarded the Nobel Prize in Physics in 1938 for his work on induced radioactivity and is today regarded as one of the most influential scientists of the 20th century. He is acknowledged as a unique physicist who was highly accomplished in both theory and experiment. Fermium, a synthetic element created in 1952 is named after him. Ri#f(t) = lim t#$ Pi#f(t) t = 2( !2 |"f|V |i!| 2%(#fl # #) . (12.5) This expression is known as Fermi’s Golden Rule.2 One might worry that, in the long time limit, we found that the probability of transition is in fact 2Curiously, although named after Fermi, most of the work leading to the Golden Rule was undertaken in an earlier work by Dirac, (P. A. M. Dirac, The quantum theory of emission and absorption of radiation. Proc. Roy. Soc. (London) A 114, 243265 (1927)) who formulated an almost identical equation, including the three components of a constant, the matrix element of the perturbation and an energy di!erence. It is given its name due to the fact that, being such a useful relation, Fermi himself called it “Golden Rule No. 2” (E. Fermi, Nuclear Physics, University of Chicago Press, 1950). Advanced Quantum Physics 12.3. “SUDDEN” PERTURBATION 144 diverging — so how can we justify the use of perturbation theory? For a transition with #fl &= #, the “long time” limit is reached when t - 1/(#fl##), a value that can still be very short compared with the mean transition time, which depends on the matrix element. In fact, Fermi’s Rule agrees extremely well with experiment when applied to atomic systems. $ Info. Alternative derivation of the Golden Rule: When light falls on an atom, the full periodic potential is not suddenly imposed on an atomic time scale, but builds up over many cycles (of the atom and of the light). If we assume that V (t) = e$tV e!i!t, with * very small, V is switched on very gradually in the past, and we are looking at times much smaller than 1/*. We can then take the initial time to be #), that is, c(1)f (t) = # i ! % t !# "f|V |i!ei(!fl!!!i%)t ! dt" = #1! ei(!fl!!!i$)t #fl # # # i* "f|V |i! , i.e. |cf(t)|2 = 1!2 e2!t (!fl!!)2+$2 |"f|V |i!| 2. Applied to the transition rate ddt |c (1) f (t)|2, the identity lim$$0 2$(!fl!!)2+$2 , 2(%(#fl # #) leads to the Golden Rule. From the expression for the Golden rule (12.5) we see that, for transitions to occur, and to satisfy energy conservation: (a) the final states must exist over a continuous energy range to match $E = !# for fixed perturbation frequency #, or (b) the perturbation must cover a su"ciently wide spectrum of frequency so that a discrete transition with a fixed $E = !# is possible. For two discrete states, since |Vfi|2 = |Vif |2, we have the semiclassical result Pi#f = Pf#i – a statement of detailed balance. 12.3.2 Info: Harmonic perturbations: second-order transi- tions Although the first order perturbation theory is often su"cient to describe transition probabilities, sometimes first order matrix element, "f|V |i! is iden- tically zero due to symmetry (e.g. under parity, or through some selection rule, etc.), but other matrix elements are non-zero. In such cases, the transition may be accomplished by an indirect route. We can estimate the transition probabilities by turning to the second order of perturbation theory (12.4), c(2)f (t) = # 1 !2 " m % t t0 dt" % t! t0 dt""ei!fmt !+i!mit!!Vfm(t")Vmi(t"") . If, as above, we suppose that a harmonic potential perturbation is gradu- ally switched on, V (t) = e#t V e!i!t, with the initial time t0 , #), we have c(2)f (t) = # 1 !2 " m "f|V |m!"m|V |i! % t !$ dt" % t! !$ dt""ei(!fm!!!i#)t ! ei(!mi!!!i#)t !! . The integrals are straightforward, and yield c(2)n = # 1 !2 e i(!fi!2!)t e 2#t #fi # 2# # 2i* " m "f|V |m!"m|V |i! #m # #i # # # i* . Then, following our discussion above, we obtain the transition rate: d dt |c(2)n |2 = 2( !4 ///// " m "f|V |m!"m|V |i! #m # #i # # # i* ///// 2 %(#fi # 2#). Advanced Quantum Physics 12.3. “SUDDEN” PERTURBATION 145 This is a transition in which the system gains energy 2!# from the harmonic perturbation, i.e. two “photons” are absorbed in the transition, the first taking the system to the intermediate energy #m, which is short-lived and therefore not well defined in energy – indeed there is no energy conservation requirement for the virtual transition into this state, only between initial and final states. Of course, if an atom in an arbitrary state is exposed to monochromatic light, other second order processes in which two photons are emitted, or one is absorbed and one emitted (in either order) are also possible. Advanced Quantum Physics