Strong Coupling, Energy Splitting, and Level Crossings: a Classical Perspective

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Strong coupling, energy splitting, and level crossings: A classical perspective ͒ Lukas Novotnya Institute of Optics, University of Rochester, Rochester, New York 14627 ͑Received 20 January 2010; accepted 6 July 2010͒ Strongly interacting quantum systems are at the heart of many physical processes, ranging from photosynthesis to quantum information. We demonstrate that many characteristic features of strongly coupled systems can be derived classically for a simple coupled harmonic oscillator. This model system is used to derive frequency splittings and to analyze adiabatic and diabatic transitions between the coupled states. A classical analog of the Landau–Zener formula is derived. The classical analysis is intuitive and well suited for introducing students to the basic concepts of strongly interacting systems. © 2010 American Association of Physics Teachers. ͓DOI: 10.1119/1.3471177͔

I. INTRODUCTION tions exist only if det͓MJ ͔=0. The resulting characteristic The interaction between quantum systems is commonly equation yields studied in the weak coupling and the strong coupling limits. 2 1 2 2 ͱ 2 2 2 2 ␻Ϯ = ͓␻ + ␻ Ϯ ͑␻ − ␻ ͒ +4⌫ ␻ ␻ ͔, ͑2͒ Weak coupling can be treated within perturbation theory, and 2 A B A B A B various classical analogs have been developed.1–3 Examples ␻ ͱ͑ ␬͒/ ␻ ͱ͑ ␬͒/ of the effects of weak coupling are changes in atomic decay where A = kA + mA, B = kB + mB, and rates4 ͑Purcell effect͒ and Förster energy transfer5 between a ͱ␬/ ͱ␬/ donor and acceptor atom or molecule. Förster energy transfer mA mB ⌫ = . ͑3͒ assumes that the transfer rate from donor to acceptor is ͱ␻ ␻ A B smaller than the relaxation rate of the acceptor. This assump- 10 tion ensures that once the energy is transferred to the accep- In analogy to the dressed atom picture, the eigenfrequen- tor, there is little chance of a back transfer to the donor. Once cies ␻Ϯ can be associated with dressed states, that is, the the interaction energy becomes sufficiently large, and a back oscillator frequencies of systems A and B in the presence of transfer to the donor becomes possible, the system is in the mutual coupling. ͑ ͒ strong coupling regime. In this limit, it is no longer possible To illustrate the solutions given by Eq. 2 we set kA =ko, ⌬ ͑ ͒ to distinguish between donor and acceptor. Instead, the exci- kB =ko + k, and mA =mB =mo. Figure 2 a shows the frequen- tation becomes delocalized, and we must view the pair as cies of the two oscillators in the absence of coupling ͑␬ ͒ ⌬ one system. A characteristic feature of the strong coupling =0 .As k is increased from −ko to ko, the frequency of ͱ ␻ regime is energy level splitting, a property that can be well oscillator B increases from zero to 2 o, while the frequency understood from a classical perspective. of oscillator A stays constant. The two curves intersect at Coupled harmonic oscillators are an intuitive and popular ⌬k=0. Once coupling is introduced, the two curves no model for many phenomena, including electromagnetically longer intersect. Instead, as shown in Fig. 2͑b͒, there is a 6 7 induced transparency, level repulsion, nonadiabatic characteristic anticrossing with a frequency splitting of processes,8 and rapid adiabatic passage.9 In this article, we ͓␻ ␻ ͔ ⌫ ͑ ͒ use a coupled oscillator model as a canonical example for + − − = . 4 strong coupling and derive frequency splittings and condi- Anticrossing is a characteristic fingerprint of strong cou- tions for adiabatic and diabatic transitions. pling. Because ⌫ϰ␬, the splitting increases with coupling strength. Note that we have ignored damping in the analysis of the II. COUPLED OSCILLATORS coupled oscillators. Damping can be readily introduced by ␥ ˙ ␥ ˙ ͑ ͒ ͑␬ ͒ adding the frictional terms AxA and BxB to Eqs. 1 . The In the absence of coupling =0 , the two oscillators introduction of damping gives rise to complex frequency ei- ␻0 ͱ / ␻0 shown in Fig. 1 have eigenfrequencies A = kA mA and B ͱ / ͑␬ ͒ = kB mB, respectively. In the presence of coupling 0 , the equations of motion become System A System B ␬͑ ͒ ͑ ͒ mAxÄ + kAxA + xA − xB =0 1a mA mB ␬͑ ͒ ͑ ͒ mBx¨B + kBxB − xA − xB =0. 1b k κ k ͑ ͒ 0 ͓ ␻ ͔ A B We seek solutions of the form xi t =xi exp −i Ϯt , where x (t) x (t) ␻Ϯ are the new eigenfrequencies. We insert this ansatz into A B ͑ ͒ 0 Eqs. 1 and obtain two coupled linear equations for xA and 0 0 0 T Fig. 1. Strong coupling regime illustrated by mechanical oscillators. The x , which can be written in matrix form as MJ ͓x ,x ͔ =0. ␬ ͑ ͒ B A B coupling of the two oscillators mass mi and spring constant ki leads to a Nontrivial solutions for this homogeneous system of equa- shift of the eigenfrequencies and a characteristic frequency splitting.

1.2 We now investigate what happens if one of the oscillator o ω+ / ωo ωA / ωo parameters changes as a function of time. For example, to 1 ⌬

o determine the curves shown in Fig. 2, we need to tune k

ω Γ / ωo ⌬ / 0.8 from an initial value of −ko to a ﬁnal value ko. Thus, k

ω becomes a function of time, a fact that we have ignored in 0.6 the analysis in Sec. II. We assumed that ⌬k is tuned so 0.4 o slowly that for every measurement window ⌬t the system ωB / ωo ω- / ωo 0.2 parameters can be regarded as constant. Thus, if we initially ⌬ ␻ 0 have k=−ko, the coupled system oscillates at frequency − -1-0.5 0 0.5 1-1-0.50 0.5 1 ͑bottom curve in Fig. 2͑b͒͒, and the system will follow the Δk/k Δk/k o o same curve as we slowly increase ⌬k. We can fine tune the oscillation frequency by adjusting ⌬k. The same applies if Fig. 2. ͑a͒ Eigenfrequencies of two uncoupled oscillators ͑␬=0͒ with equal ␻ ͓ ͑ ͔͒ ⌬ ͑ ͒ we initially start with frequency + top curve in Fig. 2 b . mass and spring constants ko and ko + k. b Frequency anticrossing due to ␬ ͓␻ ␻ ͔ In both cases the anticrossing region is passed by staying on coupling of strength =0.08ko. The frequency splitting + − − scales linearly with the coupling strength ␬. the same branch. This scenario is referred to as an adiabatic transition and is illustrated in Fig. 3͑a͒. In the adiabatic limit, it is possible to transfer the energy from one oscillator to the other by slowly tuning the coupled genvalues, the imaginary part of which represents the line- system through resonance. To see this effect, we introduce ͑ ͒ widths. The latter gives rise to a “smearing out” of the curves normal coordinates x+ ,x− defined by shown in Fig. 2, and for very strong damping, it is no longer ͑ ͒ ͑ ͒ ␤ ͑ ͒ ␤ ͑ ͒ ͓␻ ␻ ͔ xA t = x+ t sin + x− t cos , 6a possible to discern the frequency splitting + − − . There- fore, to observe strong coupling, the frequency splitting x ͑t͒ = x ͑t͒cos ␤ − x ͑t͒sin ␤, ͑6b͒ needs to be larger than the sum of the linewidths, B + − ␤ ␤ ͑␻2 ␻2͒/͑␬/ ͒ ͑␻2 where is determined by tan = B − + mB =− A ␻2͒/͑␬/ ͒ ⌫ − mB . We substitute these expressions for xA and xB Ͼ 1. ͑5͒ − ␥ / ␥ / into Eqs. ͑1͒ and obtain A mA + B mB ͑ ͒ ␻2 ͑ ͒ ͑ ͒ x¨+ t + +x+ t =0, 7a In other words, the dissipation in each system needs to be smaller than the coupling strength. ͑ ͒ ␻2 ͑ ͒ ͑ ͒ x¨− t + −x− t =0, 7b Coupled mechanical oscillators are a generic model system for many physical systems, including atoms in external which represent two independent harmonic oscillators oscil- 11 12 fields, coupled quantum dots, and cavity lating at the eigenfrequencies ␻Ϯ defined by Eq. ͑2͒. In other 13 optomechanics. Although our analysis is purely classical, a words, there are two sets of coordinates, x+ and x−, which quantum mechanical analysis yields the same result for the oscillate independently from each other. frequency splitting shown in Eq. ͑2͒.14 The coupling between Now, imagine that ⌬k is slowly tuned in time from the ⌬ ⌬ energy states gives rise to avoided level crossings. The initial value k=−ko through resonance to a value of k ͑␻ coupled oscillator picture can be readily extended by exter- =ko. According to Fig. 2, at the initial time we have A ͑ ͒ ͑ ͒ ␻ ͒ӷ⌫ ␤ϳ ␲/ nal forces FA t and FB t acting on the masses mA and mB to − − and therefore − 2. If we use this value in account for externally driven systems, as in the case of elec- Eqs. ͑6͒ and assume that initially only oscillator A is active, tromagnetically induced transparency.6 we find that all the energy is associated with normal mode

1.6 a b 1.4

Padiab 1.2 o

ω 1 / Pdiab ω 0.8

0.6

0.4 -1-0.5 0 0.5 1 -1-0.50 0.5 1

Δk/ko Δk/ko

Fig. 3. Adiabatic and diabatic transitions caused by a time-varying ⌬k. ͑a͒ In the adiabatic case, the dynamics of the system is not affected by the time dependence of ⌬k, and the system evolves along the eigenmodes ␻Ϯ. ͑b͒ In the diabatic case, the time dependence of ⌬k gives rise to “level crossing,” a transition from one eigenmode to the other.

1200 Am. J. Phys., Vol. 78, No. 11, November 2010 Lukas Novotny 1200 ⌬ x+, that is, x− =0. Once k is tuned past resonance to ko,we temporal behavior of cA but rather in the value that cA ͑␻ ␻ ͒Ӷ⌫ ␤Ϸ ͑ ͒ have A − − and 0. According to Eqs. 6 , the en- assumes after the anticrossing regime has long passed. By ergy of mode x now coincides with oscillator B, and hence using contour integration, we find that the solution for + ͑ →ϱ͒ 15 the energy is transferred from oscillator A to oscillator B as cA t is the system is slowly tuned through resonance. If ⌬k changes ␻ ␻ ␲ in time, then kB, B, and the eigenfrequencies Ϯ become ͑ϱ͒ ͫ ⌫2/␣ͬ ͑ ͒ cA = exp − . 15 time-dependent. If we assume a slowly varying ␻Ϯ͑t͒ in Eqs. 4 ͑ ͒ 7 ,wefind ϰ͉ ͉2 Because the energy of oscillator A is EA cA , the probabil- t ity for level crossing is xϮ͑t͒ = xϮ͑t ͒Reͭexpͫi͵ ␻Ϯ͑tЈ͒dtЈͬͮ, ͑8͒ i ti ␲ P = expͫ− ⌫2/␣ͬ, ͑16͒ ͑ ͒ ͑ ͒ ͓ ͑ ͔͒ diab where we used the ansatz xϮ t =xϮ ti exp if t and 2 ͓ 2 / 2͔Ӷ͓ / ͔2 ͑ ͒ d f dt df dt . Equation 8 describes the adiabatic which is also referred to as a diabatic transition, a transition evolution of the normal modes. involving loss or gain. ⌬ We now analyze what happens if we change k more Equation ͑16͒ is the classical analog of the Landau–Zener rapidly. We use the ansatz formula in quantum mechanics.16,17 As illustrated in ͑ ͒ ͑ ͒ ͓ ␻ ͔ ͑ ͒ ͑ ͒ ͓ ␻ ͔͑͒ Fig. 3͑b͒, Eq. ͑16͒ defines the probability that the energy of xA t = xocA t exp i At , xB t = xocB t exp i At 9 oscillator A remains the same after transitioning through the and assume that initially only oscillator A is active, that is, anticrossing region. P is the probability for passing ͑ ϱ͒ diab cB − =0. The amplitude xo is a normalization constant that through the anticrossing region by switching branches, that ͉ ͉2 ͉ ͉2 ͓ ensures that cA + cB =1. We substitute these expressions is, for jumping from one eigenmode to the other see ͑ ͒ ͑ ͔͒ for xA and xB into Eqs. 1 and obtain the following coupled Fig. 3 b . Consequently, the probability of an adiabatic tran- differential equations for cA and cB: sition is Padiab=1−Pdiab. ␻ ͑␬/ ͒ ͑ ͒ The probability of a diabatic transition depends on the cÄ +2i Ac˙A = mA cB, 10a ⌫ ͓␻ ␻ ͔ ␶ϳ⌫/␣ frequency splitting = + − − and the time that it takes to transition through the anticrossing region. A diabatic ␻ ͓␻2 ͑ ͒ ␻2 ͔ ͑␬/ ͒ ͑ ͒ c¨B +2i Ac˙B + B t − A cB = mB cA, 10b transition is likely for ⌫␶Ӷ1, which corresponds to a rapid where we emphasized the time dependence of ␻ . For weak transition through the anticrossing region. In contrast, for a B ͑⌫␶ӷ ͒ coupling between the two oscillators, the amplitudes c ͑t͒ slow transition 1 , an adiabatic transition is more prob- A ⌫␶ and c ͑t͒ in Eq. ͑9͒ vary much slower in time than the oscil- able. Note that the product has analogies with the time- B energy uncertainty principle. For times ␶Ӷ1/⌫, the energy latory term exp͓i␻ t͔. Therefore, c¨ Ӷi␻ c˙ and c¨ Ӷi␻ c˙ , A A A A B A B uncertainty becomes larger than the level splitting thereby which allows us to drop the second-order derivatives in Eqs. ͑ ͒ “closing up” the anticrossing region and making diabatic 10 to obtain transitions possible. ␻ ͑␬/ ͒ ͑ ͒ ͑ ͒ 2i Ac˙A = mA cB, 11a In our example, we can control which branch eigenmode we end up with by setting the speed at which ⌬k͑t͒ changes. 2 2 ͑ ͒ ͉ ͑ ͉͒2 2i␻ c˙ + ͓␻ ͑t͒ − ␻ ͔c = ͑␬/m ͒c . ͑11b͒ Figure 4 b shows computed results for cA t for two time A B B A B B A ⌬ ⌬ dependences of k. In both cases, k changes from −ko to From Eq. ͑11a͒,wefindc and c˙ ͑by taking a derivative͒ ͑ ͒ B B ko, but the speed of this change is different. Figure 4 a and substitute the results into Eq. ͑11b͒ to find, after some shows the corresponding time dependence of the frequency algebra, shifts. For the computations of cA, we assumed that only ͑ ϱ͒ ␻2 ͑t͒ − ␻2 ␬2/͓m m ͔ oscillator A is active initially, that is, cA − =1 and c¨ − ic˙ ͫ B A ͬ + c A B =0. ͑12͒ c ͑−ϱ͒=0. As time evolves, we observe small oscillations in A A ␻ A ␻2 B 2 A 4 A cA and then an abrupt change in the transition region. This ␻ ͑ ͒ change is followed by a slowly damped oscillation. The two The time dependence of B makes Eq. 12 nonlinear. Dur- ing the time interval of interest, close to the anticrossing limiting values for the diabatic probability Pdiab are indicated ͑ ͒ region, ␻ ͑t͒Ϸ␻ , and hence ͓␻ ͑t͒2 −␻2 ͔/͓2␻ ͔Ϸ͓␻ ͑t͒ in Fig. 4 b . Although one of the curves represents a nearly B A B A A B ͑ ϳ ͒ −␻ ͔. For the same reason, we can set ⌫2 Ϸ␬2 /͓m m ␻2 ͔ adiabatic transition Pdiab 0 , the other represents a nearly A A B A ͑ ϳ ͒ ͓see Eq. ͑3͔͒. Finally, we assume that near the anticrossing diabatic transition Pdiab 1 . Situations in between can be ⌬ ͑ ͒ region, the frequency difference of oscillators A and B selected by adjusting the speed at which k t changes. changes linearly in time, that is, Let us now verify the transition probabilities using the Landau–Zener formula. A linear approximation to the curves ͓␻ ͑ ͒ ␻ ͔ ␣ ͑ ͒ B t − A = t. 13 ͑ ͒ ␣ ␻2 ␣ ␻2 in Fig. 4 a yields the slopes 1 =0.003 A and 2 =0.075 A. ⌫ ͑␬/ ͒/␻ According to Eq. ͑13͒, the anticrossing region is passed at The level splitting of the two oscillators is = mo A, ␻2 ͑ ␬͒/ ␬ time tϷ0, and the frequency difference is negative for t with A = ko + mo.Ifweuse =0.08ko and substitute the ⌫ ␣ ͑ ͒ ͑␣ ͒ Ͻ0 and positive for tϾ0 ͑see Fig. 2͒. With these approxi- expressions for and into Eq. 16 ,wefindPdiab 1 ͑ ͒ ͑␣ ͒ mations, Eq. 12 becomes =0.06 and Pdiab 2 =0.89, in agreement with the computed results in Fig. 4͑b͒. c¨ − ic˙ ␣t + c ⌫2/4=0. ͑14͒ A A A The accuracy of the classical Landau–Zener formula can Despite the approximations, there is no analytical solution of be tested numerically for different time dependences of ⌬k͑t͒ ͑ ͒ ͑ ͒ Eq. 14 for cA t . However, we are not interested in the and for different initial conditions. Students can combine the

1201 Am. J. Phys., Vol. 78, No. 11, November 2010 Lukas Novotny 1201 0.8 a ACKNOWLEDGMENT

A 0.6

ω α2 The author is grateful for the ﬁnancial support by the U.S.

/ 0.4

] ͑ ͒

) Department of Energy Grant No. DE-FG02-01ER15204 . t

( 0.2

B α1 ω

- 0 a͒

A Electronic mail: [email protected] 1 ω -0.2

[ R. R. Chance, A. Prock, and R. Silbey, in Molecular Fluorescence and -0.4 Energy Transfer near Interfaces, Advances in Chemical Physics, Vol. 37, edited by I. Prigogine and S. A. Rice ͑Wiley, New York, 1978͒, pp. 1–65. 1.2 2 ͑ b L. Novotny and B. Hecht, Principles of Nano-Optics Cambridge U. P., 1.0 Cambridge, 2006͒. 3 0.89 Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical 0.8 analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, ͑ ͒

(t) 033807-1–13 2000 . 0.6 4 E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” diab

P ͑ ͒ 0.4 Phys. Rev. 69, 681–681 1946 . 5 Th. Förster, “Energiewanderung und Fluoreszenz,” Naturwiss. 33,166– 0.2 175 ͑1946͒; Th. Förster, “Zwischenmolekulare Energiewanderung und Fluoreszenz,” Ann. Phys. 2, 55–75 ͑1948͒ An English translation of 0 0.06 0400200 600 800 1000 1200 Förster’s original work is provided by R. S. Knox, “Intermolecular en- ω t ergy migration and fluorescence,” in Biological Physics, edited by E. A Mielczarek, R. S. Knox, and E. Greenbaum ͑American Institute of Phys- ics, New York, 1993͒, pp. 148–160. Fig. 4. Diabatic transition probability computed for two different time de- 6 C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical pendencies of ⌬k͑t͒. ͑a͒ Time dependence of the frequency shift. The linear ͑ ͒ ␣ ␻2 ␣ ␻2 ͑ ͒ analog of electromagnetically induced transparency,” Am. J. Phys. 70, approximation Eq. 13 yields 1 =0.003 A and 2 =0.075 A. b The tran- .2002͒͑ 41–37 ء sition probability P ͑t͒=c ͑t͒c ͑t͒ for the two functions in ͑a͒. For a fre- diab A A 7 W. Frank and P. von Brentano, “Classical analogy to quantum-mechanical quency difference that changes slowly ͑curve labeled ␣ ͒, we obtain a nearly 1 level repulsion,” Am. J. Phys. 62, 706–709 ͑1994͒. adiabatic transition with the limiting value of P =0.06, whereas a rapidly diab 8 H. J. Maris and Q. Xiong, “Adiabatic and nondiabatic processes in clas- changing frequency difference ͑curve labeled ␣ ͒ results in a nearly diabatic 2 sical and quantum mechanics,” Am. J. Phys. 56, 1114–1117 ͑1988͒. transition with the limiting value of P =0.89. k =k , k =k +⌬k, m diab A o B o A 9 B. W. Shore, M. V. Gromovyy, L. P. Yatsenko, and V. I. Romanenko, =m =m , and ␬=0.08k . B o o “Simple mechanical analogs of rapid adiabatic passage in atomic physics,” Am. J. Phys. 77, 1183–1194 ͑2009͒. 10 ͑ ͒ C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon In- two coupled equations in Eq. 10 to obtain two decoupled teractions ͑Wiley-VCH Verlag, Weinheim, 2004͒, Chap. VI. fourth-order differential equations and implement a finite dif- 11 M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, “Stark ͑ ͒ ͑ ͒ ference method to follow the evolution of cA t and cB t and structure of the Rydberg states of alkali-metal atoms,” Phys. Rev. A 20, ͑ ͒ 2251–2275 ͑1979͒. obtain curves similar to those shown in Fig. 4 b . To resolve 12 all the temporal details, it is important to keep the differential M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. time step ⌬t sufficiently small. The asymptotic values after Wasilewski, O. Stern, and A. Forchel, “Coupling and entangling of quantum dot states in quantum dot molecules,” Science 291, 451–453 ͑2001͒. transitioning through the anticrossing region can then be 13 18 T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at compared with the Landau–Zener formula. the mesoscale,” Science 321, 1172–1176 ͑2008͒. The Landau–Zener formula is an important result because 14 A. R. Bosco de Magalhães, C. H. d’Avila Fonseca, and M. C. Nemes, it finds application in a wide range of problems. It allows us “Classical and quantum coupled oscillators: Symplectic structure,” Phys. to readily calculate the transitions between eigenmodes of Scr. 74, 472–480 ͑2006͒. 15 two coupled systems. The result in Eq. ͑16͒ can be extended C. Wittig, “The Landau–Zener formula,” J. Phys. Chem. B 109,8428– ͑ ͒ to a quantum mechanical system with eigenstates ͉1͘ and ͉2͘ 8430 2005 . 16 L. Landau, “Zur Theorie der Energieübertragung bei Stössen II,” Phys. Z. ប⌫/ →͉͗ ͉ ˆ ͉ ͉͘ ប␣ by using the substitutions 2 1 Hint 2 and Sowjetunion 2, 46–51 ͑1932͒. 17 → ͓ ͑ ͒ ͑ ͔͒/ ˆ C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. Lon- d E1 t −E2 t dt, where Hint is the interaction Hamil- don, Ser. A 137, 696–702 ͑1932͒. 18 tonian and Ei the energy eigenvalues. These substitutions S. Geltman and N. D. Aragon, “Model study of the Landau–Zener ap- yield the original quantum Landau–Zener formula. proximation,” Am. J. Phys. 73, 1050–1054 ͑2005͒.

1202 Am. J. Phys., Vol. 78, No. 11, November 2010 Lukas Novotny 1202