Adiabatic transfer of light in a double cavity and the optical Landau-Zener problem
N. Miladinovic, F. Hasan, N. Chisholm, I. E. Linnington, E. A. Hinds, D. H. J. O'Dell
AAdiabatic transfer of light in a double cavity and the optical Landau-Zener problem
N. Miladinovic, F. Hasan, N. Chisholm, I. E. Linnington, E. A. Hinds, and D. H. J. O’Dell Department of Physics and Astronomy, McMaster University,1280 Main St. W., Hamilton, ON, L8S 4M1, Canada School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, 02138, USA Centre for Cold Matter, Imperial College, Prince Consort Road, London, SW7 2AZ, United Kingdom
We analyze the evolution of an electromagnetic field inside a double cavity when the differencein length between the two cavities is changed, e.g. by translating the common mirror. We findthat this allows photons to be moved deterministically from one cavity to the other. We are ableto obtain the conditions for adiabatic transfer by first mapping the Maxwell wave equation for theelectric field onto a Schr¨odinger-like wave equation, and then using the Landau-Zener result for thetransition probability at an avoided crossing. Our analysis reveals that this mapping only rigorouslyholds when the two cavities are weakly coupled (i.e. in the regime of a highly reflective commonmirror), and that, generally speaking, care is required when attempting a hamiltonian descriptionof cavity electrodynamics with time-dependent boundary conditions.
I. INTRODUCTION
One of the outstanding challenges facing quantum in-formation proposals based on cavity quantum electro-dynamics (CQED) is the development of a method fortransferring photons between different cavities [1]. Witha robust photon transfer method, a cavity containing oneor more atoms can act as a single node in a quantum net-work made up of many such cavities connected by opticalfibres [2]. Quantum networks provide a way to solve theproblem of scalability in quantum information process-ing, scalability being one of the Divincenzo criteria forthe realization of a quantum computer [3].A scheme for the efficient transfer of photons betweendistant atoms has been put forward by Cirac et al. [4].They considered what happens when a photon wavepacket is emitted by an atom into a high-Q optical cav-ity which is connected, via an optical fibre, to a secondcavity containing a second atom. A generic wave packetreflects from the highly reflective mirror of the secondcavity, thus leaving the wave packet to slosh around in-side the cavity-fibre system. However, Cirac et al. real-ized that if the emitted wave packet was time-symmetricthen it could be absorbed by the second atom with highefficiency. The coherent exchange of photons betweenan atom and a cavity mode can be controlled by time-dependent laser pulses applied through the side of a cav-ity [5], and this allows the temporal shape of the wavepacket to be controlled.A different approach has been suggested by Pellizari[6, 7]. In analogy with stimulated Raman adiabatic pas-sage (STIRAP) [8, 9], which uses two or more controllasers to transfer an atom between two internal statesvia an intermediate state which is never populated, Pel-lizari proposed that laser pulses applied to the two atomscould be used to control the transfer of photons betweenthem along the fibre, the role of the intermediate statebeing played by the cavity and fibre photon states. Theappeal of an adiabatic passage method lies in the factthat the details of the pulse shape and timing are not
Figure 1. (Color online) Double cavity setup consisting of twoperfectly reflecting mirrors separated by a partially transmis-sive common central mirror. ∆ L ≡ L − L is the differencein length between the two cavities. important. Furthermore, because the cavity and fibremodes are only virtually populated, the scheme is insen-sitive to cavity and fibre losses. Inspired by [6], the theoryof adiabatic passage of photons between two atoms viacavities and optical fibres has been developed by a num-ber of groups, see e.g., [10–12]. We also note that thebasic concept of STIRAP has also been discussed in thecontext of the adiabatic passage of matter waves, suchas atoms and electrons, where it is referred to as coher-ent tunneling adiabatic passage (CTAP). For example,references [13] and [14] consider the adiabatic transfer ofultracold atoms along a chain of traps.In this paper we consider the adiabatic transfer of lightbetween two coupled cavities by changing their lengths.This can be achieved, for example, by translating the cav-ity mirrors using piezo-electric actuators (motors). Al-ternatively, if the cavity takes the form of a dielectricwaveguide the mirror positions can remain fixed but the a r X i v : . [ qu a n t - ph ] N ov optical length of the cavities can be controlled by mod-ulating the refractive index [15]. The specific model sys-tem we choose to illustrate the adiabatic transfer processin either of these experimental scenarios is a double cav-ity in one dimension as depicted in Figure 1. Changingthe lengths of the two cavities changes the structure ofthe electromagnetic modes in such a way that a modewhich is initially localized in one cavity can be smoothlytransferred into the other cavity. Our study of the spa-tial adiabatic passage of light is in the same spirit asPelizzari’s, although we shall not consider the atoms, butfocus instead on the global normal modes of Maxwell’sequations whose frequencies form a net of avoided cross-ings as a function of mirror displacement. It would bestraightforward to introduce atoms into the scheme: forcertain mirror displacements a quasi-resonant global lightmode can be strongly localized in one of the cavities, andso a photon emitted by an atom in that cavity will finditself in that chosen mode. This global mode is thentransferred to the other cavity, where the other atomcan interact with the photon with high efficiency. Onthe other hand, if during the emission the mirror is notat a position which localizes the relevant global modein the first cavity, then the photon will be placed intoa superposition of global modes and the resulting time-dependence of the superposition leads to an oscillationof the photon back and forth between the two cavities,thereby lowering the transfer efficiency to the other atomand/or introducing the necessity for good timing.Because our model is simple, it allows a straightfor-ward analysis of the conditions for adiabatic transfer,and a number of analytic results can be obtained. Fur-thermore, the Yale opto-mechanical group have realizeda double cavity [16, 17] which is close to that consid-ered by us here. Their experimental setup consists ofan optical cavity divided in half by a thin silicon mem-brane. When the cavity is driven through an end mirrorby an external laser, the membrane vibrates due to ra-diation pressure. Indeed, a recent theoretical paper [18],discusses photon shuttling between the two halves of thecavity, in close correspondence to what we will considerhere. Other theoretical studies of this system treat it as aphotonic version of the Josephson junction [19–21]. Theimportant difference between our paper and previous op-tomechanical studies is that we propose to control themotion of mirror from outside rather than allowing it tomove under the action of radiation pressure.The problem of a cavity with moving mirrors is also ofgeneral interest from the point of view of studying CQEDwith time-dependent boundary conditions. This is aninteresting generalization of CQED with many poten-tial applications, including the field of opto-mechanics.Other related applications include the possibility of mod-ifying the decay rate of light from a cavity by modulatingthe position of the mirrors in time [22–24], and if the mir-rors accelerate very rapidly, photon pair production [25–27]. Observations of this dynamical Casimir effect haverecently been reported in microwave cavities [28]. Also on the experimental side, advances in the ultrafast ma-nipulation of the refractive index of silicon microcavities(which is equivalent to changing the cavity’s length) havebeen used to demonstrate transitions between modes [29].When the refractive index is changed more slowly, thelight adiabatically follows the modes, leading to a changein its color [15, 30–32].The programme we follow in this paper is to treat cav-ity electrodynamics with time-dependent boundary con-ditions from first principles, by starting with Maxwell’swave equation. From this we derive an effective ‘hamil-tonian’ which can be used for the purposes of first-ordertime propagation. More precisely, by making an approxi-mation analogous to the paraxial approximation familiarfrom optics, but in time rather than space, we find thatthe Maxwell wave equation reduces to an equation ofmotion for the classical Maxwell field that is first orderin time and has a mathematical resemblance to the time-dependent Schr¨odinger equation [22]. (For a different ap-proach to mapping the Maxwell wave equation onto theSchr¨odinger equation see [33, 34]). Under certain condi-tions we are therefore able to apply Landau-Zener theory[35–37], familiar from non-relativistic quantum mechan-ics, to the calculation of the coupling between classicalMaxwell modes induced by changing the lengths of thecavities. We emphasize that our method is different fromone in which the hamiltonian obtained from the static problem is used for time propagation. In particular, weshow that there are higher order terms which providesignificant corrections to the latter scheme [22].The idea of using the Landau-Zener model to under-stand time-dependent processes in optics is not new. Forexample, the experiment reported in [38] studied the timeevolution of light in cavity where an electo-optic modu-lator was used to couple the two polarization states of asingle longitudinal mode. A second electo-optic modula-tor was used to vary the energy separation between twopolarization states: varying this separation linearly intime whilst maintaining a constant coupling correspondsto the Landau-Zener model. Another system which isuseful for studying the analogies between optical andquantum dynamics is provided by evanescently coupledwaveguides, where the coupling between the waveguideschanges with distance along the waveguide [39]. Thissystem has been studied experimentally by a number ofgroups and various optical analogues to quantum phe-nomena have been observed, including Zener tunneling[41], adiabatic evolution [40], and dynamical localization[42]. There have also been numerous studies of the condi-tions necessary for adiabaticity in CQED when atoms areincluded, see, for example, [43]. In [44], the dynamics of atwo-level system coupled to a cavity field is studied in thesituation where the separation between the two energylevels is varied linearly in time, and in [45] the principleof STIRAP is applied to a two-mode Jaynes-Cummingsmodel with degenerate mode frequencies, with the aim ofadiabatically transferring photons between modes. Weshow in this paper that the application of the Landau-Zener model to the problem of a double cavity withchanging cavity lengths illustrates in a simple fashionsome of the similarities and differences between classicalelectrodynamics obeying Maxwell’s equations and quan-tum mechanics obeying Schr¨odinger’s equation.This paper is organized as follows. In Section II weintroduce a simple model for the spatial dependence ofthe dielectric permittivity function inside a double cavity.This model treats the central mirror as a Dirac δ -functionwhich facilitates analytic calculations later in the paper.In Section III we find the global static solutions (normalmodes) of Maxwell’s wave equation subject to this di-electric function. In Section IV we use these solutions tocompute the transfer ratio, i.e. the degree to which thelight can be localized on one side of the central mirroror the other depending on mirror position. In SectionV we introduce time-dependence by relating the problemof the transfer of light in a double cavity with a mov-ing mirror to the well known Landau-Zener problem inquantum mechanics. In order to apply the results fromthe Landau-Zener problem we need to map the Maxwellwave equation, which is second order in time, onto theSchr¨odinger wave equation, which is first order in time,and this is accomplished in Section VI. The electric fieldis quantized in Section VII, and in Section VIII we ex-amine the regimes of validity of the mapping onto theSchr¨odinger wave equation. The δ -function mirror modelis compared to a more realistic model for a mirror in Sec-tion IX, and in Section X we discuss the experimentalfeasibility. Conclusions are drawn in Section XI.This paper has six appendices: in Appendix A we com-pare and contrast the double cavity problem for lightwith the double well problem in quantum mechanics. InAppendix B we give some results on the transmissionamplitudes for finite and δ -function mirrors, and in Ap-pendix C we explain how the results given in this papercan be applied to the case of two coupled waveguides witha fixed mirror position but two refractive indices, n ( t )and n ( t ), that can be separately varied in time. In Ap-pendix D we define the diabatic basis and explain its con-nection to the adiabatic (global) basis. In Appendix E wediscuss the approximation whereby the time-dependenceof diabatic basis functions is neglected. In Appendix Fwe briefly consider modifications to Maxwell’s wave equa-tion due to moving dielectric mirrors. II. δ -FUNCTION MIRROR MODEL Consider a double cavity formed from two end mirrorsplus a common mirror located between them, as shownFigure 1. We assume that the end mirrors are muchmore reflective than the central mirror and that the lossof light from the double cavity is negligible during thetime taken for the light to be transferred from one sideto the other. The stringent conditions this places on anexperiment will be examined in Section X. As mentionedabove, our proposed setup is motivated by remarkable developments in opto-mechanics [16, 17]. The additionalfeature we require here is that the common mirror beexternally controlled, i.e. can be moved along the cavityaxis.A simple theoretical model describing a double cavityhas been given in a classic paper by Lang, Scully andLamb [46]. For the purposes of solving Maxwell’s waveequation in the double cavity, they treated the end mir-rors as perfect reflectors and the central mirror as a thinslab of dielectric material which is modelled by a Dirac δ -function spatial profile. The double cavity model isthereby encoded in a dielectric permittivity function ofthe form ε ( x ) = (cid:40) ε (1 + αδ ( x )) − L < x < L ∞ elsewhere (1)where x = − L , and x = L are the positions of theend mirrors. α is a parameter which determines the re-flectivity of the common mirror, see the Appendices formore details. The total length of the double cavity is L ≡ L + L , and we also define the difference betweenthe lengths of the two cavities to be ∆ L ≡ L − L , whichis also twice the displacement of the common mirror fromthe center of the whole cavity.Maxwell’s wave equation for the electric field E ( x, t ) inthe double cavity is ∂ E ( x, t ) ∂x − µ ε (1 + αδ ( x )) ∂ E ( x, t ) ∂t = 0 . (2)We use this δ -mirror model for the bulk of this paperbecause its simplicity facilitates analytic results. How-ever, in Section IX we compare the results of the δ -mirror model to the more realistic case of a mirror offinite width.Note that for mathematical convenience we have cho-sen a coordinate system in Eq. (1) where the commonmirror is always located at x = 0. However, we do notintend to attach any physical significance to this choice:physically, one can either take the view that the com-mon mirror has a fixed position and it is the two endmirrors that are displaced, or vice versa. The latter viewcorresponds to an experimental situation, such as thatdescribed in [16] and [17], where a common membrane isdisplaced, whilst the former view is closer in spirit to anexperiment in which the position of the common mirroris held fixed, but the refractive index of the two cavitiesis modulated [15], see Appendix C for more details. InSections V and VI we treat the case of moving mirrors:providing the mirrors are moving at constant velocities, itis physically equivalent to imagine either translating theend mirrors or the common mirror because these two situ-ations simply correspond to two different inertial frames.However, there are some interesting considerations to betaken into account when treating light propagating inmoving dielectrics, as briefly discussed in Appendix F. III. SOLUTIONS OF MAXWELL’S WAVEEQUATION
We write the solutions to the Maxwell wave equation as E m ( x, t ) = U m ( x ) exp( − iω m t ), where ω m = k m / √ ε µ isthe angular frequency and m = 1 , , . . . is an integerlabelling the modes. The dimensionless mode functions U m ( x ) can be chosen to be orthonormal in the Sturm-Liouville sense by ensuring that they obey1 ε (cid:90) L − L ε ( x ) U l ( x ) U m ( x ) dx = δ lm . (3)Inserting the above form for E ( x, t ) into Eq. (2) givesd U m ( x )d x + k m (1 + αδ ( x )) U m ( x ) = 0 . (4)Solutions satisfying the boundary conditions U m ( − L ) = U m ( L ) = 0 are given by U m ( x ) = (cid:40) A m sin [ k m ( x + L )] − L ≤ x ≤ B m sin [ k m ( x − L )] 0 ≤ x ≤ L . (5)One boundary condition is given by the continuity of theelectric field across the δ -mirror, U m (0 + ) = U m (0 − ). Asecond boundary condition is given by integrating Eq. (4)over a vanishingly small interval containing the mirror,leading to U (cid:48) m (0 + ) − U (cid:48) m (0 − ) = − αk m U m (0).Combining the two boundary conditions one is led tothe following equation for the wave numbers k m of theallowed modes [46]tan( k m L ) = tan( k m L ) αk m tan( k m L ) − . (6)This transcendental equation can in general only besolved numerically. It is useful to re-write it ascos( k m ∆ L ) − cos( k m L ) = 2 sin k m Lαk m , (7)so that when αk is large the sinc function on the righthand side is small. The left hand side may then be ex-panded around its roots and this permits approximateanalytic solutions (Eqns (9) and (10) below).Consider first the situation when the common mirror isperfectly reflecting ( α → ∞ ), so that the two sides of thedouble cavity are uncoupled. The modes in this case arespecified by the wave numbers k n = 2 nπ/ ( L ± ∆ L ), whichcome in pairs that cross at ∆ L = 0, and are plotted asthe dashed red lines in Figures 2 and 3. The mode whosewave number decreases with increasing ∆ L correspondsto light trapped on the left hand side of the mirror, andvice versa. When the modes are coupled there are stilltwo for each value of the integer n , whereas each value ofthe integer m introduced previously labels a single mode:from henceforth we shall use the “ n ” convention.We refer to the solutions of the wave equation in theperfectly reflecting case as localized modes. To find the
802 804 806 808 810 812 814 816 0 0.002 0.004 0.006 0.008 0.01 W a v e N u m b e r k ( U n it s o f / L ) Length Difference (cid:54)
L (Units of L)
Global wave numbersLocalized wave numbers
Figure 2. (Color online) The wave numbers k of the modesin a double cavity plotted as a function of ∆ L ≡ L − L which is the difference in length between the two cavities.The wave numbers are obtained by solving the transcendentalequation (7). When the reflectivity of the common mirror isinfinite ( α → ∞ ), we obtain the localized modes (dashedred lines), which are trapped on one side or the other of themirror. When the reflectivity is finite we obtain the globalmodes (solid blue curves) which feature avoided crossings.The total cavity length is set at L = 1 × − m, and theglobal curves have α = 1 × − m, which translates to atransmission probability of the common mirror of about 4%. modes when the coupling between the two sides of thecavity is finite, one must solve Eq. (7) for k , with α ap-propriately chosen to give the desired reflectivity of thecommon mirror. As shown in Appendix B, the reflectionprobability from the δ -mirror is given by R = k α /
41 + k α / . (8)We shall refer to solutions of Eq. (7) for finite α as global solutions, because they extend throughout the doublecavity, and they are plotted as the solid curves in Fig-ures 2 and 3. As can be seen in these figures, the effectof the coupling is to turn crossings into avoided crossings.At ∆ L = 0 the cavity is symmetric about the centralmirror and so the solutions of Maxwell’s wave equationmust have well defined parity: for each n we find a pairof modes, one of which is even and the other odd. Asthe common mirror is moved away from the center of thecavity the global modes lose their well defined parity, butwe shall continue to label them as “even” and “odd” forthe sake of simplicity. By expanding Eq. (7) to secondorder in ∆ L about ∆ L = 0, we obtain the following ex-pressions for the wave numbers of the global modes valid W a v e N u m b e r k ( U n it s o f / L ) Length Difference (cid:54)
L (Units of L)
Global wave numbersLocalized wave numbersEqn (11), (12)
Figure 3. (Color online) A zoom-in of one of the avoidedcrossings shown in Figure 2 corresponding to n = 128. Threepairs of curves are shown: the exact numerical solutions ofEq. (7) are plotted as solid (blue) curves, the approximateanalytical solutions given by Eqns (11) and (12) are plottedas short dashed (black) curves, and the localized solutions areplotted as long dashed (red) curves. The difference betweenthe exact numerical solutions and the approximate analyticalones is hard to discern. Note that in contrast to the doublewell problem in quantum mechanics, the even mode lies at ahigher frequency ω = ck than the odd mode. close to avoided crossings centered at ∆ L = 0 k n,e ≈ πnL + 2 πnL
11 + n π α/L + 2 π n α ∆ L L (9) k n,o ≈ πnL − π n α ∆ L L . (10)These analytic approximations give very good agreementto the exact numerical solutions of Eq. (7) provided one isquite high up in the spectrum (optical regime), i.e. when n (cid:29)
1, but are only valid in the immediate vicinity ofan avoided crossing. In particular, being quadratic theydo not display the asymptotically linear behavior awayfrom the avoided crossing that can be seen in Fig. 3. Thiscan be remedied by fitting the wave numbers close to anavoided crossing to the functions k n,e = 2 πnL + ∆ (cid:126) c + 1 (cid:126) c (cid:112) ∆ + γ ∆ L (11) k n,o = 2 πnL + ∆ (cid:126) c − (cid:126) c (cid:112) ∆ + γ ∆ L (12)which are the forms suggested by the Landau-Zenerhamiltonian (24), as will be explained in Section V. Thegap between the wave numbers of the even and odd globalsolutions at an avoided crossing is given by 2∆ / (cid:126) c , wherefor later convenience we have defined ∆ as an energy.The parameters γ and ∆ can be obtained by matchingthe Taylor expansions of Eqns (11) and (12) to Eqns (9) and (10), respectively. We find γ = 2∆ (cid:126) c π n αL (13)∆ = (cid:126) cL nπ n π α/L . (14)Note that this expression for ∆ is valid up to second or-der in the quantity (cid:15) = kL − πn , i.e. the deviation ofthe solutions from the perfectly localized case. (cid:15) is smallif αk (cid:29)
1, which is the case for reasonably reflective mir-rors and optical wave numbers. A comparison betweenthe values of k found from an exact numerical solutionof Eq. (7), and the approximate analytic solutions givenby Eqns (11) and (12) is included in Fig. 3.Away from the avoided crossings the global modes be-come localized, each member of a pair localizing on adifferent side of the common mirror. The larger α is,the stronger this localization becomes (and the more theavoided crossings close up). The general procedure forthe adiabatic transfer of a light mode from one side ofthe double cavity to the other is now apparent: if thecommon mirror position is initially set to one side of anavoided crossing (the optimal position is roughly halfwayto the next avoided crossing–see Section IV) then thelocalized and global modes approximately coincide andthe mode is localized on one side on the common mirror.If the mirror is now slowly translated to the equivalentpoint on the other side of the avoided crossing, the sys-tem will follow the global curve on which it began. Onthe opposite side of the avoided crossing this global modeapproximately coincides with the localized mode on theother side of the common mirror. This procedure is il-lustrated in Figure 4.It is notable that the wave number of the degeneratelocalized solutions at ∆ L = 0 coincides exactly with thelower branch of the corresponding global solutions at theavoided crossing, rather than lying halfway between thetwo global branches. This is because the lower globalbranch corresponds to an odd solution in the double cav-ity. Odd solutions have a node at the δ -mirror and hencedo not see the mirror at all. Connoisseurs of the relatedproblem of the double well potential in quantum mechan-ics [47] will immediately observe that this is the reverseof the situation found there, where the lower branch isdue to the even global solution. This reversal of the or-dering of the solutions with respect to the double wellproblem in quantum mechanics is due to the fact thatfor light propagating in 1D there can be no evanescentwaves (assuming the refractive index is > O dd m od ea m p lit ud e (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) E v e n m od ea m p lit ud e Position x (cid:72)
Units of Μ m (cid:76) Figure 4. This series of plots shows how each global wavemode swaps from one side of the double cavity to the otheras the common mirror is moved through an avoided crossing.The odd mode U n, odd ( x ) plotted as a function of position x is shown in the left column, and the even mode U n, even ( x )is shown in the right column. Each row corresponds to adifferent value of ∆ L/L : the top row has ∆
L/L = − × − ,the second row has ∆ L/L = − . × − , the middle rowhas ∆ L/L = 0, the fourth row has ∆
L/L = 0 . × − , andthe bottom row has ∆ L/L = 4 × − . Strictly speaking,the modes only have well defined parity at ∆ L = 0, but it isconvenient to retain the labels “even” and “odd” for ∆ L (cid:54) = 0.The parameters used to make this figure were L = 1 × − m, α/L = 0 .
3, and n = 128 so that k ≈ × m − . IV. TRANSFER RATIO
Once the wave numbers have been determined, thefield amplitudes A o,en and B o,en on the two sides of thecommon mirror can be calculated. From the continuitycondition for the field across the mirror we find that A n B n = − sin( k n L )sin( k n L ) = − sin[ k n ( L − ∆ L ) / k n ( L + ∆ L ) /
2] (15)where we have suppressed the even/odd labels. Usingthis ratio one can construct the global field distributioninside the cavity, as shown in Figure 4 for a pair of modes.The mirror only needs to be moved by approximately onequarter of a wavelength to achieve the transfer of lightfrom one cavity to the other. × -3 -4 × -3 × -3 × -3 A m p lit ud e R a ti o Length Difference (cid:54)
L (Units of L) |A/B||B/A|
Figure 5. (Color online) The relative amplitudes A n /B n and B n /A n of the modes as a function of the displacement ∆ L ofthe common mirror. The parameters used to make this plotwere α/L = 0 .
3, and n = 128. The ratio given by Eq. (15) is a key quantity becauseit measures the degree to which the field is localized onone side or the other. In Figure 5 we plot
A/B as afunction of ∆ L , for the case α/L = 0 .
3. The maximaof the ratio
A/B are located at values of the mirror dis-placement which are roughly halfway between pairs ofavoided crossings. We shall denote these values of themirror displacement by ∆ L n(cid:63) . Accordingly, we shall de-note the values of the wave numbers at these points by k n(cid:63) . These wave numbers obey the equationcos( k n(cid:63) L ) + sin( k n(cid:63) L ) k n(cid:63) L (cid:18) α/Lα/L (cid:19) = 0 , (16)which can be obtained by finding the stationary pointsof the ratio (15) with respect to ∆ L , and eliminatingthe quantity d k n(cid:63) / d∆ L , which appears as a result of thedifferentiation, by finding an expression for d k n(cid:63) / d∆ L from Eq. (7).Equation (16) can be used to find an analytic expres-sion for k n(cid:63) . In the regime αk (cid:29)
1, the wave numbers k n(cid:63) are located very close to the points halfway betweenthe avoided crossings, k n(cid:63) ≈ (4 n ± π/ (2 L ), where the ± sign refers to the even/odd mode. Expanding Eq. (16)about these points we find that to first order k n(cid:63) L = (4 n ± π/ α/L πnα/L . (17)The mirror displacement ∆ L n(cid:63) corresponding to wavenumber k n(cid:63) is found by solving Eq. (7) for ∆ L and eval-uating it at k n(cid:63) , i.e.∆ L n(cid:63) = 1 k n(cid:63) arccos (cid:20) k n(cid:63) Lαk n(cid:63) + cos k n(cid:63) L (cid:21) (18) ≈ k n(cid:63) (cid:18) π − αk n(cid:63) + k n(cid:63) L − (4 n ± π (cid:19) (19)where to obtain the second line we expanded the arccosterm under the assumption that αk n(cid:63) (cid:29)
1, and alsomade the approximation sin( k n(cid:63) L ) ≈ k n(cid:63) and ∆ L n(cid:63) , we arenow in a position to obtain analytic estimates for themaximum value of the transfer ratio given in Eq. (15).This ratio can first be simplified by noting that at a max-imum we expect the numerator to be near unity and thedenominator to be near zero. Since it is the inverse ra-tio B/A which is large for ∆ L small and positive, let usconsider that. Expanding the numerator about the pointwhere the phase is equal to an odd integer multiple of π/ (cid:18) k n L + ∆ L n ) (cid:19) ≈ − (cid:18) k n L + ∆ L n ) − π − πn (cid:19) (20)and expanding the denominator about the nearest zerogivessin (cid:18) k n L − ∆ L n ) (cid:19) ≈ k n ( L − ∆ L n ) − πn (21)so that close to a maximum the amplitude ratio can bewritten B n A n (cid:12)(cid:12)(cid:12)(cid:12) max ≈ − ( k n(cid:63) ( L + ∆ L n(cid:63) ) − ( π + 2 nπ )) k n(cid:63) ( L − ∆ L n(cid:63) ) − nπ . (22)Upon inserting Eqns (17) and (19) into this equation westill obtain a rather complicated result. However, if thisexpression is expanded for large values of n and kα itsimplifies to B n A n (cid:12)(cid:12)(cid:12)(cid:12) max ≈ − nπL α . (23)This result is compared against the exact numerical resultin Fig. 6. The agreement is excellent, except when α/L becomes extremely small, in which case αk n(cid:63) ≈ nπα/L is no longer large, and the approximations made in theabove derivation break down.As can be seen in Fig. 6, as α increases (i.e. as thereflectivity of the common mirror increases), so too doesthe maximum possible shift in amplitude. However, thereis a trade-off: higher reflectivity gives smaller gaps at theavoided crossings, which requires slower mirror speeds ifthe transfer is to be adiabatic. This is discussed in Sec-tion V. Furthermore, whilst it is important to know themaximum value A n /B n can take in order to evaluate theoptimal performance of the double cavity as a light trans-fer device, it is also useful to know the ratio near ∆ L = 0.The reason for this is that, depending on the parameters,a sizeable amplitude ratio can already be reached at verysmall displacements, see Figure 5. In practice, therefore,it may not be worthwhile going to the maximum, espe-cially if that involves a longer transfer time. This issueis discussed further in Section X. Using Eq. (9) or (10)in the ratio (15) yields the desired result. This permits a M a x A m p lit ud e R a ti o | B / A | (cid:95) (Units of L) NumericAnalytic
Figure 6. (Color online) The absolute value of the maximumvalue the relative amplitude B n /A n can take as a function ofthe common mirror parameter α . The exact numerical resultis plotted as the solid curve, and the approximate analyticalresult, as given by Eq. (23), is plotted as the dashed curve.Note that the difference between these two curves is hard todiscern. We set n=128 to make this plot. very quick calculation of amplitude ratios near the equi-librium position, which along with the results of SectionV may be used to minimize losses. V. TIME-DEPENDENCE: THELANDAU-ZENER PROBLEM
So far we have found the time-independent solutionsto the double cavity problem: the global modes are thestatic solutions to Maxwell’s wave equation for a fixedmirror displacement. Even if the system begins in a sin-gle (global) mode, a mirror moving at a finite speed willinduce transitions and the light will therefore be excitedinto a superposition of modes. This is a complicatedproblem to treat analytically, but if we can map the pas-sage of our system through an avoided crossing onto therelated Landau-Zener (LZ) problem in quantum mechan-ics, then we can take advantage of existing solutions.The LZ problem considers a quantum system with twostates, the energy separation of which is varied linearlyin time so that at t = 0 they are degenerate. The twostates have a time-independent coupling ∆. Stated inthis way, the two states being coupled are known as the diabatic states, and the time-dependent hamiltonian inthe diabatic basis is H LZ = (cid:20) E ( t ) ∆∆ − E ( t ) (cid:21) , (24)where the diabatic energies ± E ( t ) vary linearly in time: E = ϑt/ . (25)Diagonalizing this hamiltonian at each instant of timegives the energies E ± of the adiabatic states E ± ( t ) = ± (cid:112) ϑ t / (26)which form an avoided crossing. At the point of closestapproach ( t = 0) the magnitude of the gap between thetwo adiabatic energies is 2∆, and at t = ±∞ the diabaticand adiabatic energies coincide.The adiabatic states interchange their character be-tween t = −∞ and t = + ∞ , which is the crucial prop-erty we need in order to transfer modes between the twosides of the double cavity. The exact solution of the LZproblem shows that if at t = −∞ the two state systemis prepared in one of the two states, then the probabilitythat it has made a transition to the other adiabatic stateby t = + ∞ is [35–37] P LZ = exp( − π ∆ / (cid:126) ϑ ) . (27)Of course, the LZ problem assumes that there are onlytwo modes present, but due to the exponential suppres-sion in the adiabatic regime of transitions as a functiontheir energy difference, we need only consider a singleeven and odd pair in the double cavity if we start from asingly occupied mode and move slowly through a singleavoided crossing.It is tempting to apply the result (27) directly to thetime-dependent cavity problem by identifying the globalmodes we found in previous sections as the adiabaticstates E + ≡ (cid:126) c k e (28) E − ≡ (cid:126) c k o (29)and extracting from them the parameters ∆ and ϑ thatare needed to calculate the transition probability P LZ .Note that the appearance of Planck’s constant in theserelations is purely for dimensional purposes (we have notyet quantized the electromagnetic field, see Appendix Afor more discussion). In fact, the parameter ∆ has al-ready been found above and is given in Eq. (14). Fur-thermore, if the common mirror is moved at a constantvelocity v , so that ∆ L = 2 vt , then the key requirement ofthe LZ model that the energy difference between the di-abatic states is varied linearly with time will be fulfilled.From Eqns (11)–(14) we find that ϑ = 4 v √ γ . For large n this gives ϑ ≈ π (cid:126) cnv/L (30)which coincides with the result obtained by assumingthat the wave numbers are those given by localized solu-tions.However, this whole approach of directly applyingthe LZ results to the time-independent solutions of theMaxwell equations is wrong in principle because theMaxwell and Schr¨odinger equations evolve differently intime. Therefore, although the time-independent Maxwelland Schr¨odinger equations are identical (compare Eqns (A4) and (A5)), one should not blindly apply the LZ so-lution to the Maxwell problem. Rather, in Section VIwe justify the application of the LZ result to the movingmirror problem by deriving an approximate version ofthe Maxwell equation which is first order in time and isvalid in the adiabatic regime. In order to set this up, weneed to express our system in the LZ diabatic basis. Al-though they are similar in character, the diabatic statesdo not correspond exactly to the localized modes unless α → ∞ (the LZ diabatic energies must cross halfwaybetween the two adiabatic curves whereas the localizedmodes do not). There is a precise way to obtain the LZdiabatic modes and this “undiagonalization” is presentedin Appendix D.The reason it is advantageous to work in the diabaticbasis is worth mentioning. Consider the time-dependentsolution expressed in the diabatic and adiabatic bases.Denoting the diabatic modes as φ L ( x ) and φ R ( x ), wehave E ( x, t ) = A L ( t ) φ L ( x, t ) + A R ( t ) φ R ( x, t ) (31) E ( x, t ) = B e ( t ) U e ( x, t ) + B o ( t ) U o ( x, t ) . (32)The time-dependence extends not only to the superposi-tion coefficients but also to the basis modes themselvesand this complicates the treatment. As can be seen aposteriori from the LZ transition probability (27), thechanges in the adiabatic superposition coefficients B e ( t )and B o ( t ) are exponentially small during a transfer inthe adiabatic regime. Almost all the change is incorpo-rated in the adiabatic modes. By contrast, the changes inthe diabatic coefficients during a transfer in the adiabaticregime are of order unity because in this basis the systemchanges its state. As pointed out by Zener himself [36],the changes in the diabatic basis in the transition region(close to the avoided crossing) are typically negligible incomparison to the changes in the coefficients, and so onecan take φ L ( x ) and φ R ( x ) to be independent of time.This approximation is analyzed in Appendix E. Thus,by working in the diabatic basis we can focus purely onthe time-dependence of the coefficients A L ( t ) and A R ( t ),which is a significant simplification. VI. FROM MAXWELL TO SCHR ¨ODINGER
Our aim in this section is to derive from Maxwell’swave equation a Schr¨odinger-like (first order in time)equation for the electric field amplitudes. This will allowus to put the use of the LZ result in the time-dependentdouble cavity problem on a firm footing.As discussed above, it is more convenient to solve theLZ problem in the diabatic basis. On the other hand, theexact solutions we have already found by solving Eq. (7)are for the global modes, i.e. for the adiabatic basis. Weshall therefore switch backwards and forwards betweenthe two bases as needed. We begin in the diabatic basisand substitute expression (31) for the electric field intothe Maxwell wave equation ∂ x E = µ ε ( x, t ) ∂ t E , neglect-ing the time-dependence of the mode functions φ ( x, t ).The validity of this approximation is examined in Ap-pendix E. This gives A L ∂ x φ L + A R ∂ x φ R = µ ε ( x, t ) (cid:0) ¨ A L φ L + ¨ A R φ R (cid:1) (33)where the dots indicate time derivatives. In fact, thetime dependence of the dielectric function ε ( x, t ) impliesthat there are also corrections to the above quoted formof Maxwell’s wave equation [48]; these corrections arediscussed in Appendix F.We tackle the left and right hand sides of Eq. (33)separately. To simplify the left hand side we substitutethe expansions (D6) and (D7) of the localized modes interms of the global modes and then make use of the factthat the global solutions, being eigenmodes of the system,satisfy ∂ x U e = − µ ε ( x, t ) ω e U e (34) ∂ x U o = − µ ε ( x, t ) ω o U o (35)to give ∂ x φ L = − µ ε ( x, t ) (cid:0) − sin( θ ) ω e U e + cos( θ ) ω o U o (cid:1) (36) ∂ x φ R = − µ ε ( x, t ) (cid:0) cos( θ ) ω e U e + sin( θ ) ω o U o (cid:1) . (37)Now that we have made use of the exact results for theglobal modes we can go back to the diabatic basis byusing (D8) and (D9) to obtain ∂ x φ L = − µ ε ( x, t ) (cid:110) φ L (cid:2) sin ( θ ) ω e + cos ( θ ) ω o (cid:3) + φ R (cid:2) sin( θ ) cos( θ ) ω o − sin( θ ) cos( θ ) ω e (cid:3)(cid:111) (38) ∂ x φ R = − µ ε ( x, t ) (cid:110) φ R (cid:2) cos ( θ ) ω e + sin ( θ ) ω o (cid:3) + φ L (cid:2) sin( θ ) cos( θ ) ω o − sin( θ ) cos( θ ) ω e (cid:3)(cid:111) . (39)In order to explicitly express the adiabatic transferproblem in LZ form we let ω e = (cid:112) E + ∆ / (cid:126) + ω av (40) ω o = − (cid:112) E + ∆ / (cid:126) + ω av , (41)cf. Eqns (11) and (12). In these equations ± E/ (cid:126) arethe frequencies of the diabatic modes, as discussed inAppendix D. Indeed, Eqns (40) and (41) can be viewedas a way of defining the diabatic modes, the adiabatic(global) modes with frequencies ω e and ω o being alreadyknown from solving the transcendental equation (7). Wehave also introduced ω av which is the average frequencyof the even and odd modes ω av = ω e + ω o E/ (cid:126) = 0, but are in fact upin the optical spectrum. From Eqns (11) and (12) we seethat ω av = 2 πncL + ∆ (cid:126) . (43) The definitions (40) and (41) can be combined with ex-pressions (D4) and (D5) for cos( θ ) and sin( θ ) in terms ofthe diabatic variables to givesin( θ ) cos( θ ) (cid:0) ω o − ω e (cid:1) = 2∆ ω av / (cid:126) (44)which is one of the combinations appearing in (38) and(39). In a similar fashion we find that the other combi-nations are given bysin ( θ ) ω e + cos ( θ ) ω o = ( E/ (cid:126) − ω av ) + ∆ / (cid:126) (45)cos ( θ ) ω e + sin ( θ ) ω o = ( E/ (cid:126) + ω av ) + ∆ / (cid:126) . (46)Putting all this together, and combining with the righthand side of Eq. (33), we arrive at the following expres-sion for the two-mode Maxwell wave equations2(∆ / (cid:126) ) ω av A L φ R + { ( E/ (cid:126) − ω av ) + ∆ / (cid:126) } A L φ L +2(∆ / (cid:126) ) ω av A R φ L + { ( E/ (cid:126) + ω av ) + ∆ / (cid:126) } A R φ R = − (cid:16) ¨ A L φ L + ¨ A R φ R (cid:17) . (47)Multiplying through by ε ( x ) φ L ( x ) or ε ( x ) φ R ( x ) and in-tegrating over space in order to make use of the orthog-onality conditions (D10) and (D11), we can pick out thecoupled equations of motion for the amplitudes − (cid:18) ¨ A L ¨ A R (cid:19) = (cid:32) ( E (cid:126) − ω av ) + ∆ (cid:126) ω av / (cid:126) ω av / (cid:126) ( E (cid:126) + ω av ) + ∆ (cid:126) (cid:33) (cid:18) A L A R (cid:19) (48)The two-mode equations of motion (48) are second or-der in time. They are dominated by the fast evolutiongenerated by the diagonal terms containing ω av which isan optical frequency of order 10 s − . This frequency isfar greater than the other frequencies in the problem andcomes from the stationary solutions. We can remove thediagonal terms by defining A L/R ( t ) ≡ (cid:101) A L/R ( t ) exp (cid:20) − i (cid:90) t β L/R ( t (cid:48) )d t (cid:48) (cid:21) , (49)where β L/R ( t ) = (cid:112) ( E ( t ) / (cid:126) ∓ ω av ) + ∆ / (cid:126) . (50)Time derivatives of A L/R ( t ) then give rise to the terms¨ A L/R = (cid:0) ¨˜ A L/R − iβ L/R ˙˜ A L/R − i ˙ β L/R ˜ A L/R − β L/R ˜ A L/R (cid:1) × exp (cid:20) − i (cid:90) t β L/R ( t (cid:48) )d t (cid:48) (cid:21) . (51)By virtue of the fact that β L and β R are so much greaterthan the other frequencies, we make a ‘paraxial approxi-mation in time’ (or equivalently, a slowly varying ampli-tude approximation) and drop the first and third termson the right hand side of Eq. (51), and thereby obtain anapproximation to Maxwell’s wave equation that is firstorder in time i d ˜ A L/R d t = ω av ∆ (cid:126) β L/R ˜ A R/L exp (cid:20) ± i (cid:90) t ( β L − β R )d t (cid:48) (cid:21) . (52)0These equations can be further simplified by noting that β L/R = ω av (cid:18) ∓ E (cid:126) ω av + 12 ∆ (cid:126) ω + . . . (cid:19) (53)and β R − β L = 2 E (1 −
12 ∆ (cid:126) ω + . . . ) (54)and truncating each of these expansions at their firstterm. In this way we arrive at i dd τ (cid:18) ˜ A L ˜ A R (cid:19) = (cid:32) − i ˜ ϑτ / e i ˜ ϑτ / (cid:33) (cid:18) ˜ A L ˜ A R (cid:19) (55)where τ ≡ ∆ t/ (cid:126) and ˜ ϑ ≡ (cid:126) ϑ/ ∆ are both dimension-less quantities. These coupled equations are mathemati-cally equivalent to the time-dependent Schr¨odinger equa-tion i d ψ/ d τ = H ( τ ) ψ with the LZ hamiltonian (24),as can be seen by letting ˜ A L → ˜ A L exp[ − i ˜ ϑτ /
4] and˜ A R → ˜ A R exp[ i ˜ ϑτ / L = 100 µ m and α/L = 0 . n = 128and Eq. (14) gives ∆ / (cid:126) ≈ × s − . Furthermore, E = ∆ at the avoided crossing and remains of this or-der of magnitude. One therefore finds from expansion(53) that the first correction to the result β L/R = ω av isof magnitude 3 × − ω av and so relatively small. Also,from expansion (54) we see that the first correction tothe result β R − β L = 2 E is of second order and equals5 × − E . The smallness of this latter correction is sig-nificant because the combination β R − β L appears in thephases which are generally more sensitive to approxima-tions than amplitudes.A further very important check is to see if the neglectof second order derivatives in Eq. (51) is consistent withour final result (55). Differentiating Eq. (55), we find thatthe largest term neglected, namely ¨˜ A L/R , has a magni-tude of order (∆ / (cid:126) ) ˜ A L/R , whereas the smallest termretained, namely 2 iβ L/R ˙˜ A L/R , has a magnitude of or-der (∆ / (cid:126) ) ω av ˜ A L/R and so is a factor of 10 bigger. Tocheck this point more systematically, we have numericallysolved the first and second order equations and their so-lutions are compared in Figures 7 and 8. The initialconditions were taken to be A L = 1 and A R = 0 at theinitial time τ = −
25 and the equations were integratedup to the final time τ = 25. Note that in order to solvethe second order equations we also require a second setof boundary conditions, e.g. the values of the first deriva-tives of the amplitudes at the initial time. Because the (cid:45) (cid:45)
10 0 10 200.00.20.40.60.81.0 Τ (cid:200) A L , (cid:200) A R a (cid:45) (cid:45)
10 0 10 200.00.20.40.60.81.0 Τ (cid:200) A L , (cid:200) A R b (cid:45) (cid:45)
10 0 10 200.00.20.40.60.81.0 Τ (cid:200) A L , (cid:200) A R c Figure 7. (Color online) Comparison of second-order-in-time dynamics, as given by the full Maxwell wave equa-tion (48), versus first-order-in-time dynamics, as given bythe Schr¨odinger-like equation (55), as the mirror is sweptthrough an avoided crossing. The initial conditions are thesame in every panel: A L = 1 and A R = 0 at τ = − τ ≡ ∆ t/ (cid:126) is dimensionless time. The first-order-in-time dynamics are also the same in every panel and are setby ˜ ϑ ≡ (cid:126) ϑ/ ∆ = 1, which is the single dimensionless param-eter appearing in the Landau-Zener formula (27) and in Eq.(55). The second-order-in-time dynamics depend additionallyon the value of ∆ / (cid:126) ω av ; panel a has ∆ / (cid:126) ω av = 1 / b has ∆ / (cid:126) ω av = 1 /
500 and panel c has ∆ / (cid:126) ω av = 1 / τ = −
25 both give A L . The black curve (which climbs higher) is calculated fromEq. (48). The red curve is given by Eq. (55). The other twocurves (blue and green) that end on the upper right at τ = 50give A R . The blue curve (which is higher) is calculated fromEq. (48) and the green curve is from Eq. (55). Τ (cid:200) A R Figure 8. (Color online) A zoom-in of Fig. 7 combining theupper right part of all the panels in order to show the time-dependence of A R at the end of the sweep. Like Fig. 7, this fig-ure illustrates how the solutions of the (second order) Maxwellequation reduce to those of the (first order) Schr¨odinger equa-tion as ∆ / ( (cid:126) ω av ) →
0. The solid black curve is calculatedfrom the first order equations (55), and the remaining curvesfrom the second order equations (48); ∆ / ( (cid:126) ω av ) = 1 /
100 (blueshort-dashed), ∆ / ( (cid:126) ω av ) = 1 /
500 (green medium-dashed),and ∆ / ( (cid:126) ω av ) = 1 / values of these derivatives are required only at the initialtime, i.e. before the mirror has started moving, we cantake their values from the known solutions for a staticmirror. We find that the first derivatives of the ampli-tudes for a static mirror are given by i (cid:18) ˙ A L ˙ A R (cid:19) = (cid:18) ω av − E (cid:126) ∆ (cid:126) ∆ (cid:126) ω av + E (cid:126) (cid:19) (cid:18) A L A R (cid:19) . (56)This equation should not be confused with Eq. (55).The first order dynamics generated by the Schr¨odinger-like Eq. (55) depend only upon the value of the combi-nation ˜ ϑ . They do not depend upon the value of ∆ byitself. By contrast, the second order dynamics generatedby Eq. (48) depend upon the separate values of both ˜ ϑ and ∆ (or more precisely ∆ / (cid:126) ω av ) as can be seen if thetime variable in Eq. (48) is scaled by (cid:126) / ∆, like in Eq.(55). Therefore, we expect that as ∆ / (cid:126) ω av →
0, but ˜ ϑ is held constant, the second order dynamics will reduceto the first order dynamics. This is exactly what hap-pens, as illustrated by the successive panels of Figure 7.In these panels the first order dynamics is held constantby fixing ˜ ϑ = 1, but ∆ / (cid:126) ω av takes the successive values1 / , /
500 and 1 / A L and A R give the electromagnetic energy in thedouble cavity H Maxwell = 12 (cid:90) ε ( x ) |E ( x, t ) | d t = ε (cid:0) | B e | + | B o | (cid:1) = ε (cid:0) | A L | + | A R | (cid:1) . (57) This energy will not in general be conserved under time-dependent boundary conditions. In quantum mechanics A L and A R have a radically different interpretation interms of probability amplitudes and do not by themselvesgive the energy. More discussion of this point is given inAppendix A. Of course, the energy will also not in gen-eral be conserved in time-dependent quantum mechan-ics, however the hamiltonian structure of the Schr¨odingerequation means that the quantity | A L | + | A R | = 1 isconserved even for time-dependent problems. This im-plies that by reducing the Maxwell wave equation to firstorder in time we artificially enforce energy conservationin our system.Interestingly, whatever the value of ∆ / (cid:126) ω av , the agree-ment between first and second order in time dynamics al-ways seems to be very good at the end of the evolution,as can be seen in Figure 8. This is probably connectedwith the quite precise vanishing of the energy differencebetween the initial and final states if they are symmet-ric about the avoided crossing, as illustrated in Figure9. This figure shows how energy is initially pumped intothe electromagnetic field from the moving mirror as theavoided crossing is approached and is then removed af-ter it has been passed. Indeed, Figure 9 also helps usinterpret the behaviour of the amplitudes seen in Figure8 whereby | A L | and | A R | as calculated from the sec-ond order equations climb above the values calculatedfrom the first order equations (which strictly conserve | A L | + | A R | = 1). This is a good place to remind thereader that the curves plotted in Figures 2 and 3 showingthe avoided crossings of the wave numbers (or equiva-lently, frequencies) do not correspond to energies as theywould in the quantum mechanical case. Indeed, start-ing with A L = 1 and A R = 0 at τ = −
25 correspondsto starting approximately in the even global mode onthe left of the avoided crossing, and providing the mirrormotion is slow, as here, remaining in that mode. Thewave number of the even global mode is given by the up-per curve in Figure 3 which decreases as it approachesthe avoided crossing and increases once it has passed it.This is opposite to the behaviour of the energy shownin Figure 9, and underscores the difference between thequantum mechanical and classical optical cases.
VII. QUANTIZATION
Everything we have discussed in this paper so far hasconcerned classical electrodynamics. We now wish tobriefly see how the hamiltonian is quantized in our for-malism. However, as emphasized in Section VI, we cau-tion the reader again that this hamiltonian approach onlystrictly applies for the static mirror case because in thedynamic mirror case Maxwell’s wave equation providescorrections to the Landau-Zener hamiltonian.We begin by quantizing in the global basis. The evenand odd modes provide the normal modes of the dou-ble cavity and so the hamiltonian can be written in the2 (cid:45) (cid:45)
10 0 10 200.000.020.040.060.080.10 Τ (cid:200) A L (cid:43) (cid:200) A R (cid:45) Figure 9. (Color online) A plot showing the deviation of | A L | + | A R | from unity as a function of time, as calculatedfrom the full two-mode Maxwell equations (48) which includethe second-order time derivatives. As shown in Eq. (57), thisquantity is proportional to the energy stored in the electro-magnetic field. Each curve in this plot has ˜ ϑ = 1, but theupper curve (blue, short dashed) has ∆ / (cid:126) ω av = 1 / / (cid:126) ω av = 1 / / (cid:126) ω av = 1 / diagonal form H = (cid:126) ω e b † e b e + (cid:126) ω o b † o b o (58)where b e is the annihilation operator for a photon in theeven mode and b o is the annihilation operator for a pho-ton in the odd mode. In contrast to the classical case,the energy now depends upon the frequency of the light.Referring to Appendix D, we see that we can transformto the diabatic basis by letting b e = cos θ a R − sin θ a L (59) b o = sin θ a R + cos θ a L (60)where a R and a L are the right cavity and left cavity an-nihilation operators, respectively. We therefore find that ω e b † e b e + ω o b † o b o = ( ω e cos θ + ω o sin θ ) a † R a R + ( ω e sin θ + ω o cos θ ) a † L a L + cos θ sin θ ( ω o − ω e ) a † R a L + cos θ sin θ ( ω o − ω e ) a † L a R . (61)Using the explicit expressions for cos θ and sin θ given byEqns (D4) and (D5), as well as those for ω e and ω o givenby Eqns (40) and (41), we arrive at the second quantizedhamiltonian in the diabatic basis H = ( (cid:126) ω av + E ) a † R a R +( (cid:126) ω av − E ) a † L a L +∆( a † R a L + a † L a R )(62)where ± E are the diabatic energies and ∆ the gap ap-pearing in the Landau-Zener hamiltonian (24).Writing the hamiltonian in this second quantized formmakes it clear that ∆ is the matrix element of H between the left and right diabatic states, i.e. ∆ = (cid:104) φ L | H | φ R (cid:105) .When ∆ is small, i.e. when the two cavities are weaklycoupled, the transition probability per unit time betweenthe diabatic states should obey Fermi’s golden ruled P d t = 2 π (cid:126) ∆ ρ (63)where ρ = L/ π (cid:126) c is the density of states in either of theuncoupled cavities. This is indeed the case, as can beseen by first noting that in the weak coupling limit wehave, from Eq. (14), that∆ ≈ (cid:126) cnπα ≈ kα (cid:126) cL , (64)where to obtain the second equality we have put k =2 πn/L , which is exact in the uncoupled limit. Second, weobserve from Eq. (B2) that in the weak coupling regime kα is large, and the transmission probability can be writ-ten T ≈ k α = ∆ (cid:18) L (cid:126) c (cid:19) = (2 πρ ∆) , (65)where we have used Eq. (64) to obtain the second equal-ity. Finally, using the fact that d P/ d t = T / ( L/c ), where
L/c is the “return time” of a photon in one of the cav-ities, i.e. the time between collisions with the commonmirror, we recover Fermi’s golden rule Eq. (63).We saw in Section VI that in the weak coupling regime,i.e. when ∆ / (cid:126) ω av →
0, the Maxwell wave equation re-duces to a Schr¨odinger-like equation for the classical elec-tromagnetic field. The above quantization procedureshows us that in this regime the Schr¨odinger-like equa-tion can be promoted to a true Schr¨odinger equation.
VIII. REGIMES OF VALIDITY
According to the Landau-Zener theory, the adiabaticregime occurs when, from Eq. (27),2 π ∆ (cid:126) ϑ (cid:29) . (66)Various physical interpretations can be given to this con-dition, as we now show. The rate of change of the sepa-ration of the diabatic energies, given in Eq. (30), can bewritten ϑ ≈ (cid:126) δω dop L/c = 4 (cid:126) ω av L/c vc (67)where δω dop ≡ kv ≈ ω av v/c is the change in angularfrequency of a localized mode during the photon returntime L/c , or, equivalently, the Doppler shift acquiredwhen a light wave reflects from a mirror moving at speed v . Then, using Fermi’s golden rule Eq. (63) to replace ∆ with d P/ d t , we find that the adiabatic condition becomes2 π ∆ (cid:126) ϑ ≈ πδω dop d P d t (cid:29) . (68)3 AdiabaticSchrödinger-like (cid:45) (cid:45) (cid:45) (cid:45) Ω FSR Ω av T r a n s m i ss i on Figure 10. A plot showing the various regimes discussed inSection VIII as a function of the transmission T of the com-mon mirror and ratio of the free spectral range ω FSR to theaverage optical frequency ω av . The Schr¨odinger-like equation(55) is valid below the dashed line, which is given by Eq. (70),the left hand side of which we have assumed to be equal to10 . This is a very conservative estimate, but is still a factorof 10 less than the value given by assuming the parameters: L = 100 µ m, α = 0 . L , and λ = 780 nm. If the Schr¨odinger-like theory holds, then we can apply the Landau-Zener crite-rion for adiabaticity, given by Eq. (69), and which is plottedas the solid line. To plot this line we assumed adiabaticitysets in when the left hand side is greater than 10 (the expo-nential dependence of P LZ certainly ensures that this is thecase). The regime of adiabatic evolution of the electromag-netic field is above the solid line. The shaded region showsthe regime where the Schr¨odinger-like theory applies and theevolution it predicts is adiabatic. This says that the change in frequency of the diabaticmodes during a photon return time must be much smallerthan the transition rate between the diabatic states(roughly speaking, the rate of photon jumps across thecommon mirror).The adiabatic condition must also have a classical in-terpretation, because Eq. (27) applies to classical electro-magnetic waves. To find this we use Eq. (65) to replace∆ by the transmission probability T to obtain2 π ∆ (cid:126) ϑ ≈ T ω FSR δω dop = T ω FSR ω av cv (cid:29) . (69)This says that the ratio of the shift in frequency dur-ing the photon return time to the free spectral range ω FSR ≡ πc/L (angular frequency difference betweencavity modes) must be much smaller than the fraction ofthe incident intensity which is transmitted by the com-mon mirror.In terms of the same physical parameters as introducedabove, the condition under which the Maxwell wave equa-tion reduces to the Schr¨odinger-like equation can be writ- ten (cid:126) ω av ∆ ≈ π √ T ω av ω FSR (cid:29) . (70)This is a distinctly different condition from Eq. (69) foradiabaticity, and this means that it is possible to be eitheradiabatic or non-adiabatic, but still be in the Schr¨odingerregime where Eq. (55) applies, see Fig. 10.It is perhaps counter-intuitive that Eq. (70) for the va-lidity of the reduction of the Maxwell to the Schr¨odinger-like theory does not depend on the mirror speed. This isbecause in this paper we have chosen to work in the di-abatic basis, which is not the eigenbasis, except far fromthe avoided crossings. When the dynamics of the elec-tromagnetic field is seen from the point of view of thediabatic basis it is, therefore, not generally stationary,even in the case of a stationary mirror. The characteris-tic frequency ∆ / (cid:126) for the intrinsic dynamics in the dia-batic basis is set by the coupling between the two cavities,and this can dominate the extra time-dependence intro-duced by a moving mirror if the mirror is moving slowlyenough. Therefore, for a slowly moving mirror the limit-ing factor that sets the condition (70) for the validity ofthe Schr¨odinger-like theory does not involve v .A direct dependence on the mirror speed only entersthe dynamics of diabatic mode amplitudes in the thirdterm on the right hand side of Eq. (51), which we haveso far ignored. Let us estimate the conditions underwhich we can ignore it. We begin by recalling that con-dition (70) comes from comparing the orders of mag-nitude of the first and second terms on the right handside of Eq. (51) for ¨ A , namely ¨˜ A ≈ O [(∆ / (cid:126) ) ˜ A ] and β ˙˜ A ≈ O [(∆ / (cid:126) ) ω av ˜ A ]. The magnitude of the third termon the right hand side of Eq. (51) is˙ β ˜ A ≈ (cid:126) d E d t ˜ A = ϑ (cid:126) ˜ A = δω dop L/c ˜ A . (71)The condition under which this term can be neglected incomparison to the first order time derivative term, is O [ β ˙˜ A/ ˙ β ˜ A ] = ω av δω dop ∆ (cid:126) Lc ≈ ω av δω dop √ T = 12 √ Tv/c (cid:29) . (72)This condition is obeyed for most practical situations,but does require a certain finite value of the transmission T and/or a non-relativistic mirror. If we assume a mirrorwith α/L = 0 . T = 7 × − , implying that theSchr¨odinger-like equation (55) holds if v/c (cid:28) . IX. FINITE WIDTH MIRROR
We now go beyond the δ -function model introduced inSection II and consider the case of a mirror with a fi-nite width. For many purposes the δ -function model is4 T r a n s m i ss i on Wave Number k (Units of 1/L) T FM T (cid:98) Figure 11. A comparison of the transmission probability T for the finite and δ -function mirrors, as given by Eqns (B1)and (B2), respectively. The resonances of the finite mirroroccur at the peaks of its transmission function. For the pa-rameters used in this plot, which are the same as those givenin the caption of Fig. 12, the two curves intersect near a min-imum of the finite barrier transmission. This means that thetwo transmission probabilities remain in good agreement fora larger range of k than they might otherwise do. perfectly sufficient and has the advantage of being mathe-matically simple, but there are phenomena which it is notcapable of describing such as resonances within the mir-ror. In this section we compare the wave numbers of thenormal modes produced with the δ -function model withthose of a model of a mirror as a uniform slab of dielec-tric material of finite width (we have in mind somethinglike the SiN membrane used in [16], although perhapssomewhat thicker). Our main purpose is to check howwell the δ -function model reproduces the predictions ofthe finite width model. Of course, the finite width modelconsidered here does not accurately capture the structureof an actual piece of dielectric material, but it does takethe next step in that direction.As a model for a mirror of finite width we take thesame basic model as considered in Section II except thatwe replace the δ -function at x = 0 by a uniform slab ofdielectric of width 2 M centred at x = 0. The correspond-ing permittivity function is given by ε ( x ) = ε (1 + h ) − M ≤ x ≤ Mε − L < x < − M and M < x < L ∞ elsewherewhere n r = √ h is the refractive index inside the slab.Imposing the boundary conditions U n ( − L ) = U n ( L ) =0, the normal modes can be written as: U n ( x ) = A sin[ k ( x + L )] − L ≤ x ≤ − MB sin[ n r kx + φ ] − M ≤ x ≤ MC sin[ k ( x − L )] M ≤ x ≤ L . The parallel component of the electric field is continuousat the mirror surface, so that provides us with two moreboundary conditions upon the mode function, namelycontinuity of U n ( x ) at x = ± M . Another two conditionsare provided by the continuity of the parallel componentof the magnetic field at the mirror surfaces (assumingthe mirror is non-magnetic so that µ = µ ) which, whencombined with Faraday’s law ∇ × E = − ∂ B /∂t , lead tothe continuity of the first derivatives of U n ( x ) at x = ± M . Implementing these boundary conditions we findthat the allowed wave numbers k satisfy the followingequation − n r sin(2 n r kM ) sin (cid:20) k (cid:18) L − M + ∆ L (cid:19)(cid:21) × sin (cid:20) k (cid:18) L − M − ∆ L (cid:19)(cid:21) + n r cos(2 n r kM ) sin[ k ( L − M )]+ sin(2 n r kM ) cos (cid:20) k (cid:18) L − M + ∆ L (cid:19)(cid:21) × cos (cid:20) k (cid:18) L − M − ∆ L (cid:19)(cid:21) = 0 , (73)which is the equivalent of Eq. (7), but for the finite mirrorcase. The mirror resonances occur when 2 M kn r = lπ ,where l = 1 , , , . . . , and so the distance in k -space be-tween mirror resonances is π/ (2 M n r ). This is muchlarger than the distance between cavity resonances π/L providing L (cid:29) M n r .The transmission T = | t | of the finite mir-ror, given by Eq. (B1) in Appendix B as (cid:16) ( n r − n r sin (2 n r kM ) (cid:17) − , is shown as a func-tion of wave number by the dotted line in Fig. 11. Theseare the usual transmission fringes of a Fabry-Perotresonator. By contrast, the power transmission of the δ -function mirror, given by Eq. (B2) as (cid:0) α k / (cid:1) − ,is the monotonic function of wave number illustratedby the solid line in Fig. 11. The two have the sametransmission when k α = ( n r − n r sin (2 n r kM ) . (74)In general, this equality is satisfied only over narrow wavenumber regions where the two curves cross, but in Fig.11 we have chosen a value of α such that they have simi-lar transmissions over a wider range of wave numbers inthe vicinity of k = 800. In this region, therefore, the δ -function mirror can be a simple model for the more com-plicated finite mirror case. Of course, the phase shiftsproduced by the two mirrors are not equal, as one cansee from Eqns (B1) and (B2), and indeed the finite mirrorphase changes rapidly in the vicinity of the Fabry-Perottransmission peaks. At the transmission minima, how-ever, the rate of change of the phase shift with k is muchslower. Therefore, over the range of wave numbers near k = 800 where they have roughly equal values of T , the5
805 806 807 808 809 810 811 812 813-0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 W a v e N u m b e r k ( U n it s o f / L ) Length Difference (cid:54)
L (Units of L)
Delta Function MirrorFinite Mirror
Figure 12. (Color online) Comparison between the wave num-ber structure produced by a mirror of width 2 M (solid bluecurves) and the δ -mirror (dashed red curves) inside a dou-ble cavity. This plot is for the case when we are far froma mirror resonance. In order to make the comparison wematched the transmission probabilities T for each mirror overthe relevant range of wave numbers as best we could (see Fig.11). Up to a relative vertical shift, the agreement is verygood, including the magnitude of the gap 2∆ at the avoidedcrossings. In the calculation for the δ -function mirror we set α/ ( L − M ) = 0 . M/ ( L − M ) = 0 . n r M/ ( L − M ) = 0 . two mirrors produce phase shifts that do not change ap-preciably and are equivalent to small fixed displacementsof the mirror positions.Fig. 12 shows the wave number curves for these twocases. Since we are comparing the transmission functionof a δ -mirror to that of a mirror of width 2 M , we have re-duced the total cavity length for the δ -mirror case so that L → L − M . This means that the optical path lengthoutside the mirror is the same in both cases. Neverthe-less, the different phase shifts imposed upon the lightwhen it traverses the two different types of mirror pro-duces a small vertical shift in the wave numbers. Thetwo sets of curves are otherwise almost identical, includ-ing the gap 2∆ at the avoided crossings.A very different wave number structure occurs in theresonant case when an integer multiple number of halfwavelengths fit inside the mirror, as shown in Fig. 13. Atthe bottom of the figure we have the familiar horizontalavoided crossings where an even and an odd mode withthe same value of the index n are coupled. However, as weincrease the wave number the light approaches resonancewith the mirror and we find that the amplitude B of thelight inside the mirror becomes large. Intriguingly, we seethat the avoided crossings become vertical, meaning thatwe can adiabatically transfer between localized modeswith different values of n .
595 600 605 610 615-0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 W a v e N u m b e r k ( U n it s o f / L ) Length Difference (cid:54)
L (Units of L)
Figure 13. The wave number structure produced by a mir-ror of width 2 M inside a double cavity. This plot includeswave numbers for which there is a mirror resonance. At thebottom we see the familiar avoided crossing structure whichcouples even and odd modes with the same index n , but forhigher wave numbers we pass into resonance and the structurechanges to one where there are vertical avoided crossings thatallow for the possibility of adiabatically coupling localizedcavity modes with different values of the index m . The pa-rameters used to make this plot were M/(L-2M)=0.0005005,and 2 n r M/ ( L − M ) = 0 . X. EXPERIMENTAL FEASIBILITY
There are many different ways to realize the doublecavity model discussed here in an experiment. The keychallenge in all of them is likely to be achieving adia-batic transfer before the light leaks out of the system,e.g. through the end mirrors, which hitherto we assumedto be perfect, but which in reality never can be. Remark-ably, this challenge has already been overcome in the ex-periment [15] performed in a 6 µ m diameter ring resonatormade out of silicon where ultrafast electro-optical controlof the refractive index allows the effective optical lengthof the cavity to be changed on time scales shorter than 10ps, which is the lifetime of a photon in their resonator. Inthis way, the wavelength of light injected into the cavitywas adiabatically shrunk by up to 2.5nm before leakingout. The change in refractive index was achieved by shin-ing a laser onto the silicon thereby creating free chargecarriers which, via plasma dispersion, strongly modifiedits optical properties. The same electro-optical effect canalso be generated electrically [56].In order to make a double cavity out of this systemwe imagine coupling two ring cavities together. It is notnecessary to displace a mirror: the difference in cavitylengths ∆ L = L − L can be controlled by changingthe effective optical lengths of each cavity in a concertedfashion via the electro-optical effect; see Appendix C.The degree to which the light can be localized on one6 × -4 -1 × -4 × -4 × -4 A m p lit ud e R a ti o Length Difference (cid:54)
L (Units of L) |B/A|
Figure 14. A zoomed-in plot of the relative amplitudes versusmirror displacement ∆ L . It is clear that one only needs smalldisplacements to achieve large amplitude transfers. Here α/L = 0 .
3, and n = 128. side of the double cavity or the other is set by the di-mensionless parameter α/L , as discussed in Section IV.For a central “mirror” such that α/L = 0 .
01, the maxi-mum amplitude ratio that can be achieved when n = 128is approximately | A/B | ≈
10, and so the probability offinding a photon on one side versus the other is 100 : 1.For a double cavity of length L = 100 µ m we would thenrequire α = 1 µ m, and at a wavelength of 780nm Eq. (8)says that this corresponds to a central “mirror” with areflectivity of R = 0 .
94. This degree of coupling is readilyachieved.There are some advantages to using a more reflectivecommon mirror. Not only does the maximum localiza-tion become larger, but also a significant degree of local-ization can be achieved for a much smaller value of ∆ L .This is illustrated in Figure 14 for α/L = 0 . L = 10 − L (the maximum am-plitude ratio that can be achieved with α/L = 0 . n = 128 is 240:1).Another possible way to realize the double cavitymodel is to displace the mirrors, e.g. by piezo-electricmotors. These are routinely used to control mirror posi-tions in order to actively stabilize the lengths of ultrahighfinesse optical cavities against vibrations [49]. However,it is an obstacle that only rather low mirror speeds can beachieved. Piezoelectric motors can accurately move smalldistances (nm), but the current fastest ones of which weare aware only reach speeds of v = 0 . λ/
4, i.e. ∆ L needs to change by λ/
2. This gives us a transfer time of t = λ/ v ≈ v = 1 . α/L = 0 .
01 with a cavity of length L = 100 µ m, the cavity mode corresponding most closelyto light with a wavelength of 780 nm has n = 128. Insert-ing these numbers into Eq. (14) gives ∆ / (cid:126) ω av = 3 × − .Combining this with Eq. (30), we see that at a velocityof v = 1 . (cid:126) ϑ n / ∆ = 2 × − and thusthe transfer is perfectly adiabatic, i.e. the Landau-Zenertransition probability P LZ given by Eq. (27) is utterlynegligible. Non-adiabatic transitions only become signif-icant in the ultrafast electro-optical case: if we assumethe same parameters as for the Fabry-Perot cavity, ex-cept that the transfer takes 10 ps, and thus a rate ofchange of cavity length of d∆ L/ d t = 39 ,
000 m/s, we find (cid:126) ϑ n / ∆ = 6 .
8, and a non-adiabatic transfer probabilityof exp[ − π ∆ / ( (cid:126) ϑ )] = 0 .
4. In Appendix F we make theobservation that such a rapidly changing refractive indexprobably necessitates the use of a modified Maxwell waveequation for its proper theoretical description.At the relatively slow transfer speeds of piezo-electricmotors the leakage of light out of the end mirrors be-comes important. To estimate this effect we note that anumber of recent experiments have realized Fabry-Perotmicrocavities with lengths below 200 µ m and finesses ofup to F = 10 [51–54]. The decay rate of the electric fieldamplitude in a cavity is given by κ = cπ/ d F . Being op-timistic and assuming that for cavity of length 100 µ m wecan achieve a finesse F = 10 , one finds κ = 5 × s − ,and hence the field decays to only exp[ − κt ] = 0 . ≈
60% probability of escaping the cavity by the endthe transfer.The situation improves when the central mirror ismade more reflective, as mentioned above. For α/L =0 .
3, which is still an order of magnitude less reflectivethan the end mirrors, and with a total change of the cav-ity length difference of ∆ L = 0 . L (i.e. moving be-tween the points ∆ L = ± . L either side of ∆ L = 0)for a cavity of length L = 100 µ m, the transfer time is t = ∆ L/ v = 6ns when v = 1 . XI. DISCUSSION AND CONCLUSIONS
The optical double cavity is a closely related systemto the double well potential in quantum mechanics, al-though, as discussed in Appendix A, the common centralmirror in the cavity case does not act as a tunneling bar-rier. The wave numbers (frequencies) of the global modesinside the double cavity form a net of avoided crossings asa function of the difference in length of the two cavities,and this permits the adiabatic transfer of light from oneside of the common mirror to the other. Alternatively, ifthe transfer is halted right at an avoided crossing the light7
Figure 15. (Color online) By simultaneously moving all 3 mir-rors, it is possible to double the effective speed of the commonmirror, i.e. quadruple the rate of change of the relative cavitylength ∆ L ( t ) = L ( t ) − L ( t ). is left in a superposition of being in both cavities. Thedouble cavity is arguably, therefore, the simplest possibleprototype for a quantum network which can transfer pho-tons on demand from one node to another or can placethem into superpositions of being on different nodes.Using a δ -function model for the common mirror, wehave obtained analytic results such as those given in Eqns(9) and (10) for the wave numbers close to an avoidedcrossing. An expression for the maximum possible trans-fer ratio of light is given in Eq. (23), and depends onthe transmission properties of the central mirror, as pa-rameterized by the dimensionless quantity α/L . Whilstthe δ -function model gives an adequate description formany purposes, we saw that when the central mirror ismodelled as a dielectric slab of finite thickness there isthe possibility of mirror resonances which dramaticallychange the structure of the net of avoided crossings forthe global wave numbers. In the vicinity of a mirror res-onance the horizontal avoided crossings become verticalavoided crossings, suggesting a way to adiabatically moveup the ladder of cavity resonances.At first sight, the Schr¨odinger and Maxwell wave equa-tions seem to give incompatible descriptions of the timeevolution of the electromagnetic field because they arefirst- and second-order in time, respectively. The first-order time evolution found in quantum theory is a conse-quence of its hamiltonian structure, which is deeply con-nected with the conservation of probability. The stan-dard method for quantizing the Maxwell wave equation ∂ x E − ( n r /c ) ∂ t E = 0, where n r ( x ) is the refractive index,is to give it hamiltonian structure: a separation of vari-ables of the form E ( x, t ) = (cid:80) n B n ( t ) U n ( x ) leads to thetwo equations ∂ x U n + n r U n = 0 and ∂ t B n + ω n B n = 0,where ω n = c k n is the separation constant. The firstequation gives the normal (global) modes, and the sec-ond is the equation of motion for a simple harmonic os- cillator. This can be transformed into hamiltonian formby introducing the variables q = (cid:112) (cid:126) / mω ( B (cid:63) + B ) and p = i (cid:112) m (cid:126) ω/ B (cid:63) − B ) whose time evolution is given byHamilton’s equations ˙ x = ∂ p H and ˙ p = − ∂ x H , whichare first order in time. The final step is, of course, to usethe commutator [ x, p ] = i (cid:126) . This prescription is straight-forward to apply in cases where the refractive index isindependent of time, but when it depends on both timeand space the Maxwell wave equation does not separate,there are no normal modes, and a simple harmonic os-cillator equation for the field amplitudes is harder to ob-tain. The time dependent problem can be tackled innumber of ways, e.g. by working in a time-independentbasis. However, for slow time dependence it can be eas-ier to work in the adiabatically evolving basis U n ( x, t ),given by the instantaneous solutions to Maxwell’s waveequation at time t , so that electric field can be written E ( x, t ) = (cid:80) n c n ( t ) U n ( x, t ) exp[ − i (cid:82) t ω n ( t (cid:48) ) dt (cid:48) ].In this paper our primary concern was not the quan-tization of the electromagnetic field, but rather the ap-plication of a mathematical solution, namely, that of theLandau-Zener problem, to the transfer of classical elec-tromagnetic field modes in a double cavity using a time-dependent mirror (or, equivalently, time-dependent re-fractive indices, see Appendix C). In order to apply theLandau-Zener model we need a first-order-in-time equa-tion of motion. Our approach, which is closely relatedto using the adiabatic basis mentioned above, was to usea ‘paraxial approximation in time’ (slowly varying enve-lope approximation) to reduce the Maxwell wave equa-tion to the mathematical form of the time-dependentSchr¨odinger equation, given by Eq. (55). This reduc-tion holds in the regime ∆ / (cid:126) ω av → ϑ ≡ (cid:126) ϑ/ ∆ , τ i ≡ ∆ t i / (cid:126) and τ f ≡ ∆ t f / (cid:126) are held constant. ϑ is the quantitywhich appears in the Landau-Zener formula (27), and t i and t f are the initial and final times, respectively.˜ ϑ and τ appear in both the full Maxwell equation andthe Schr¨odinger-like equation, but ∆ / (cid:126) ω av only appearsin the full Maxwell equation. The reduction, therefore,does not affect the (lower order) Schr¨odinger dynamics.This process may be likened to the reduction of quantummechanics to classical mechanics by letting (cid:126) →
0, whilstholding the classical mechanics (i.e. the action) fixed [55].One consequence of the reduction to the Schr¨odinger-like equation (55) is that energy is artificially conserved,as discussed in Section VI. In Section VII we saw thatas a by-product of our approach we were able to quan-tize the hamiltonian in the regime ∆ / (cid:126) ω av →
0, andtherefore promote the Schr¨odinger-like equation to a trueSchr¨odinger equation for the quantum amplitudes of theelectromagnetic field in this regime. One of the lessonsour treatment of the time-dependent problem teachesus is that outside of the weak-coupling regime one can-not simply take the solutions from the time-independentMaxwell’s wave equation and plug them into the Landau-Zener theory in order to obtain time-dependent proper-ties of the electromagnetic field.8There is a simple physical interpretation of the factthat the reduction from second- to first-order-in-time dy-namics holds in the regime ∆ / (cid:126) ω av →
0. In this regimethe gap is small, which corresponds to weak coupling be-tween the two cavities, as is clear from Eq. (65), whichsays that the transmission probability T ∝ ∆ when T is small. When the transmission is small the timeevolution of the amplitudes A L ( t ) and A R ( t ) becomesslow, their higher time derivatives can be neglected, andhence the reduction to first-order-in-time is valid. Aspointed out in Section VIII, the weak-coupling require-ment ∆ / (cid:126) ω av → Q factors as high as Q = 3 × have been demonstrated [58] and this givesa photon lifetime of 0 . ACKNOWLEDGMENTS
We gratefully acknowledge B. M. Garraway, J. Larson,S. Scheel, J. E. Sipe and A.-C. Shi for discussions. Fund-ing was provided by the Ontario Ministry of Researchand Innovation, and NSERC (Canada); by EPSRC, andthe Royal Society (U.K.); and by the HIP project of theEuropean Commission.
Appendix A: The connection to the double wellproblem in quantum mechanics
In this appendix we compare and contrast classicallight waves obeying the Maxwell equations to quantummatter waves obeying the Schr¨odinger equation. Thesedifferences are well known, but we include them here forcompleteness. In particular, in the main body of this pa-per we have alluded to the existence of a close connectionbetween the problem of light in a double cavity, and theproblem of a quantum particle in a double well poten-tial. In fact, as we shall see in this appendix, the truecorrespondence is not to a quantum particle tunnelingbetween two classically allowed regions through a classi-cally forbidden potential barrier , but rather to a quantumparticle passing between two classically allowed regionsseparated by a potential well , all three regions being clas-sically allowed. Consider on the one hand the Maxwell wave equationobeyed by plane light waves with vacuum wave number k = ω/c propagating in one dimension in a medium withrefractive index n r ( x )d E d x + k n r ( x ) E = 0 . (A1)On the other hand, consider the time-independentSchr¨odinger equation for a particle with energy E = (cid:126) k / m and subject to a potential V ( x )d ψ d x + k (cid:18) − V ( x ) E (cid:19) ψ = 0 . (A2)An obvious difference between the two is that in theMaxwell case the eigenvalue k multiplies the spatiallydependent term n r ( x ), whereas in the Schr¨odinger equa-tion k does not multiply V ( x ), despite the suggestiveway we have written it here. Nevertheless, the two equa-tions are equivalent if we can identify n r ( x ) = 1 − V ( x ) E . (A3)This equation forms the basis of matter wave optics. As-suming that the refractive index obeys n r >
1, we seethat the analogy between and light waves and matterwaves only holds if V ( x ) <
0. Thus, the potential “bar-rier” presented by the mirror in the double cavity is, infact, a potential well if we consider the problem in quan-tum mechanical terms. This explains why we find theodd adiabatic solution lies below the even adiabatic so-lution (see Figure 3), in contradiction to our experienceof the double well problem in quantum mechanics.In the specific case of the δ -function mirror, which isdescribed by the dielectric function ε ( x ) = ε n r ( x ) = ε (1 + αδ ( x )), we haved E d x + k (1 + αδ ( x )) E = 0 (A4)whilst the equivalent problem in quantum mechanics hasa particle of energy E subject to a potential V ( x ) = βδ ( x ), giving rise to the Schr¨odinger equationd ψ d x + k (cid:18) − βδ ( x ) E (cid:19) ψ = 0 . (A5)The two equations can be considered as equivalent if α = − β/E .Other differences between the Maxwell andSchr¨odinger cases include the normalization conditionsand also the interpretation of the wave number as anenergy. Starting with the normalization, the wave func-tion ψ in the above equations obeys (cid:82) L − L | ψ ( x ) | d x = 1,whereas the electric field E obeys Eq. (3). This differencearises, of course, because ψ is a probability amplitudeand probability must be conserved, whereas E has nosuch interpretation. Turning now to the wave number9 k , if we were solving a quantum mechanical problem,the curves in Figures 2 and 3 showing k versus mirrordisplacement ∆ L would become energy curves. Nosuch connection exists in the Maxwell case. Indeed,in quantum mechanics energy and wave number areintimately connected, e.g. in free space E = (cid:126) k / m ,but on the other hand a quantum wave has an energyindependent of its amplitude. The exact converse istrue for classical light waves: their energy is completelyindependent of wave number, but depends rather on thesquare of their amplitude. A wavelength dependence ofthe energy of light waves only enters the story when thelight wave is quantized via the photon energy E = (cid:126) ck .The problem discussed in this paper is purely linearsince we have not included atoms in the cavities whichcan couple to the light field, nor have we considered anynonlinear optical elements. However, if a nonlinearityis introduced into the quantum mechanical double wellproblem then the resulting system has a close connectionto the Josephson junction [60]. A number of authors haveexploited this analogy between coupled optical cavities(or fibres) with nonlinearity and the Josephson junctionproblem to predict effects such as self trapping of the lightin the cavities [19–21]. If an array of cavities is consid-ered, then the possibility exists to realize a bosonic Hub-bard model, for which the self-trapping effect developsinto a full-blown bosonic version of the Mott-Insulatorquantum phase transition [61, 62]. Appendix B: Transmission amplitudes for losslessdielectric mirrors
In this appendix we collect together some results onthe transmission functions of δ -function and finite widthmirrors. The problem of light scattering in one dimen-sion from a finite mirror with a homogeneous refractiveindex is essentially the same as that of quantum particlesscattering from a square potential well [63] (see also Ap-pendix A). For concreteness, consider an incoming wave A exp( ikx ), a reflected wave B exp( − ikx ), and a trans-mitted wave C exp( ikx ). The thickness of the mirror istaken to be 2 M , and it is assumed to be made of a losslessdielectric material with refractive index n r . The trans-mission amplitude t is found to be t ≡ CA = exp( − i M k ) exp( i arctan[ n r +12 n r tan 2 M kn r ]) (cid:113) ( n r − n r sin (2 M kn r ) . (B1)The transmission probability T for light is given by T ≡| t | , and this is related to the reflection probability R by R = 1 − T .Performing a similar calculation for a δ -function mir-ror, as specified by the dielectric function ε ( x ) = ε (1 + αδ ( x )), we find t = 11 − ikα/ i arctan[ kα/ (cid:112) k α / . (B2) This latter result can also be obtained from the transmis-sion amplitude for a finite mirror as given above if we let M →
0, and n r → ∞ , but keep the quantity n r M fixed.This procedure allows us to identify the parameter α asbeing given by α = 2 M n r . (B3)This only holds providing 2 M kn r (cid:54) = mπ where m =0 , , . . . , i.e. avoiding resonances whereby an integernumber of half wavelength fit inside the mirror. Appendix C: A double cavity made of two coupledwaveguides with controllable refractive indices
In this appendix we consider a double cavity formedfrom two coupled waveguides, each made of a dielectricwhose refractive index can be controlled, e.g. via theelectro-optic effect. The optical lengths of the two cavi-ties can be independently controlled by modulating therefractive index in each waveguide, and this allows theoptical modes to be transferred between the two waveg-uides without moving the mirror. Consider the situationwhere each waveguide has the same length L/
2, and onehas the refractive index n and the other has refractiveindex n . The original double cavity model, as depictedin Fig. 1, can be converted into the waveguide problemby substituting L → n L L → n L . (C2)To conserve the total optical length during the transfer,we choose n = n + η (C3) n = n − η . (C4)Here, n is the unperturbed refractive index of the waveg-uides and η is the change due to the modulation. Eventhough the substitutions (C1) and (C2) capture the mainrelations between the two models, there remain sometechnical differences which originate with the secondboundary condition given below Eq. (5). We find thatin the coupled waveguide case the transcendental equa-tion (6) for the global wave numbers becomestan( n k m L/
2) = n n tan( n k m L/ αk m /n ) tan( n k m L/ − . (C5)Although the arguments of the tangent functions are ex-actly as given by the above substitutions, the factors in-volving the refractive indices n and n that appear out-side of the tangents are not. However, we note that inpractice the magnitude of the perturbation η that is re-quired to change the optical length ∆ L = L − L → ηL by a significant amount can be small. Indeed, according0to Section IV, complete transfer of a mode is achievedwhen k ∆ L ≈ π , and if we take λ = 780 nm and L = 100 µ m we then find we require η = 0 . n which by definition must begreater than unity. To a good approximation one canthen replace Eq. (C5) bytan( n k m L/
2) = tan( n k m L/ αk m /n ) tan( n k m L/ − . (C6)Note that we have kept the full refractive index depen-dence in the arguments of the tangent functions where itis crucial. This transcendental equation is obtained fromEq. (6) by precisely the substitutions (C1) and (C2), plusthe replacement αk → αk/n . This latter replacementcan be understood by examining the transmission proper-ties of a δ -function mirror that lies between two dielectricmedia, one with refractive index n , and the other withrefractive index n . We find that t = 2 n n + n − ikα (C7)so that T = | t | = n n + n ) + k α ≈
11 + k α n , (C8)cf. Eq. (B2). In the last step we substituted the defini-tions (C3) and (C4) and assumed that η (cid:28) n . Thus,when surrounded by a medium of refractive index n , thetransmission of the δ -function mirror is obtained fromthat in vacuum by putting αk → αk/n , as above. Thismeans that to maintain the same transmission in the twocases one should increase either k or α by a factor of n .With a little care most of formulae given in the mainbody of this paper can be modified for the waveguidecase. For example, in the key Eqns (11), (12), (13) and(14), the speed of light c in needs to be replaced by thespeed c/n in the guide. Appendix D: The diabatic basis
The solution of the transcendental equation (7) allowsus to construct global modes which exist throughout thedouble cavity. The global modes are the eigenvectorsof the time-independent Maxwell wave equation in thedouble cavity and need to be recalculated for each valueof the difference between the cavity lengths ∆ L . Thesemodes give the so-called adiabatic basis as specified byEqns (28) and (29), and we denote this basis by U e ( x )and U o ( x ).However, as discussed in Section V, it is convenient tobe able to express the double cavity problem in a ba-sis which we refer to as the “LZ diabatic basis”, or the“diabatic basis” for short. We define the diabatic basisas the basis in which solutions to the time-independentMaxwell/Schr¨odinger wave equation can be expressed in the form given by the LZ hamiltonian matrix (24). Inother words, the diabatic energies ± E ( t ) and eigenvectorsare those found from the LZ hamiltonian when ∆ = 0.We shall denote the diabatic basis vectors φ L ( x ) and φ R ( x ).For a δ -function mirror the diabatic basis does not ingeneral coincide with the entirely localized basis, as de-fined by the wave numbers k n = 2 πn/ ( L ± ∆ L ) foundwhen α → ∞ . This can easily be seen by comparing theLZ hamiltonian (24), for which the diabatic energies crossexactly halfway between the two adiabatic solutions, tothe entirely localized solutions which cross on the loweradiabatic curve (see Figure 3).Actually, we are in the slightly curious situation of al-ready knowing the adiabatic basis and needing to findthe diabatic basis. This is the converse of the usual situa-tion. Indeed, the adiabatic solutions are in a sense betterdefined because they correspond to the basis which diag-onalizes the hamiltonian and is therefore unique. Mean-while, the diabatic basis is somewhat arbitrary becauseit corresponds to a particular “undiagonalization” of thehamiltonian. Nevertheless, this particular undiagonaliza-tion into the specific LZ form (24) can easily be achievedby rotating the states correctly.We begin from the diagonalized LZ matrix (i.e. LZhamiltonian in the adiabatic basis) H ad = (cid:20) E + ( t ) 00 E − ( t ) (cid:21) (D1)where E ± = ±√ E + ∆ are the adiabatic energies(global wave numbers), where the plus sign correspondsto the solution which is even at ∆ L = 0. We want totransform this matrix into the LZ matrix in the diabaticbasis H LZ = (cid:20) E ( t ) ∆∆ − E ( t ) (cid:21) . (D2)In order to go between (D1) and (D2) we perform asimilarity transformation. Let H ad = S − H LZ S , where S = (cid:20) cos θ sin θ − sin θ cos θ (cid:21) . (D3)We find that the matrix elements of S are given bycos θ = (cid:115) √ E + ∆ + E √ E + ∆ = (cid:118)(cid:117)(cid:117)(cid:116) E + + (cid:113) E − ∆ E + (D4)andsin θ = − (cid:115) √ E + ∆ − E √ E + ∆ = − (cid:118)(cid:117)(cid:117)(cid:116) E + − (cid:113) E − ∆ E + (D5)where the expressions after the second equality in bothequations are written in terms of the adiabatic quantities1 (cid:45) (cid:45)
10 0 10 200.00.20.40.60.81.0 Τ (cid:200) d à L (cid:144) d Τ (cid:200) , (cid:200) d à R (cid:144) d Τ (cid:200) Figure 16. (Color online) Time derivatives of the diabaticamplitudes, as obtained by numerical solution of Eq. (55).The green curve (that takes the value one at τ ≡ t ∆ / (cid:126) = − A R / d τ , and the red curve gives themodulus of d ˜ A L / d τ . The values of the parameters are set to˜ ϑ ≡ (cid:126) ϑ/ ∆ = 1. obtained by solving the transcendental equation (7). Inparticular, the value of ∆ is found from ∆ = E + | ∆ L =0 .The relationship between the two sets of basis statesis then φ R ( x ) = cos θ U e ( x ) + sin θ U o ( x ) (D6) φ L ( x ) = − sin θ U e ( x ) + cos θ U o ( x ) (D7)and conversely U e ( x ) = cos θ φ R ( x ) − sin θ φ L ( x ) (D8) U o ( x ) = sin θ φ R ( x ) + cos θ φ L ( x ) . (D9)The orthogonality relations satisfied by φ L/R can befound by substituting Eqns (D8) and (D9) into Eq. (3).One finds 1 ε (cid:90) L − L ε ( x ) φ L/R ( x ) φ L/R ( x ) dx = 1 (D10)1 ε (cid:90) L − L ε ( x ) φ L ( x ) φ R ( x ) dx = 0 . (D11) Appendix E: Time variation of the diabatic modes
In this Appendix we check the validity of the approxi-mation made in Section VI whereby the time-dependenceof the diabatic mode functions φ L and φ R is neglected incomparison to the time dependence of the diabatic ampli-tudes A L and A R . If the time dependence of the diabaticmodes is taken into account then the right hand side ofEq. (33) becomes ¨ A L φ L +2 ˙ A L ˙ φ L + A L ¨ φ L plus an identicalterm where L → R . The largest terms we are neglectingin Section VI are the cross terms ˙ A L/R ˙ φ L/R .Even for quite modest values of the reflectivity, the di-abatic modes become strongly localized on one side of the (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Τ R e (cid:72) Ã L (cid:76) Figure 17. (Color online) Early time behavior of the real partof ˜ A L . The solid red curve gives the numerical solution of Eq.(55) and the dashed black curve gives the approximate ana-lytic solution given by Eq. (E6). The values of the parametersare the same as those in Fig. 16. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Τ I m (cid:72) Ã L (cid:76) Figure 18. (Color online) Early time behavior of the imag-inary part of ˜ A L . Two curves are plotted, one giving thenumerical solution of Eq. (55) and the other giving the ap-proximate analytic solution as given by Eq. (E6), but the dif-ference between the two is hardly visible for the times shown.Only for times greater than τ = −
22 do differences really be-come apparent. The values of the parameters are the same asthose in Fig. 16. mirror or the other allowing us to accurately approximatethe diabatic mode functions by the completely localizedmode functions given by φ L ( x ) ≈ (cid:114) L + ∆ L sin (cid:20) nπ (cid:18) xL + ∆ L + 1 (cid:19)(cid:21) (E1)and a similar expression for φ R . The cross terms referredto above can then be shown to lead to new terms in theequations of motion (i.e. once the mode functions havebeen integrated out) of the form˙ A L (cid:90) − L ˙ φ L φ L d x = ˙ A L v (cid:90) − L ∂φ L ∂ ∆ L φ L d x ≈ − vL ˙ A L (E2)and a similar term for φ R . This needs to be comparedwith ¨ A L/R . Removing the diagonal terms via the trans-2formation (49) to give the slowly rotating amplitudes, wefind that, assuming ∆ L (cid:28) L , the quantities that mustbe compared are − iβ L/R ˙˜ A L/R vs 2( ˙˜ A L/R − iβ L/R ˜ A L/R ) vL (E3)where the left hand side refers to the existing calculationin the main part of this paper and the right hand sidegives the new terms. In the case when v = 1 m/s and L =100 µ m, the first term on the right hand side is clearlynegligible in comparison to the other terms. To find themagnitudes of the remaining terms let us go back to theSchr¨odinger-like equations (55). Taking the derivative ofthese equations with respect to time we find, after somemanipulation, the following two uncoupled equations foreach amplitude d ˜ A L d τ = − i ˜ ϑτ d ˜ A L d τ − ˜ A L (E4)d ˜ A R d τ = i ˜ ϑτ d ˜ A R d τ − ˜ A R (E5)where we have defined the dimensionless quantities τ ≡ t ∆ / (cid:126) and ˜ ϑ ≡ (cid:126) ϑ/ ∆ . In his original paper [36], Zenermapped these equations onto the Weber differential equa-tion, and used the known asymptotic results for Weberfunctions for τ → ±∞ to obtain the transfer probabilityquoted in Eq. (27). Here we are interested in a slightlydifferent problem, namely the solutions of equations (E4)and (E5) for finite times. In particular, referring to Fig-ure 16, we see that the time derivative of ˜ A L takes itssmallest values at the beginning of the time evolution.These are the times we therefore need to worry aboutmost since it is then that it is most likely that the mag-nitude of the right hand side of Eq. (E3) may exceed themagnitude of the left hand side.At early times, when A L ≈ A R ≈
0, we find thatEquation (E4) is approximately satisifed by˜ A L ( τ ) = 1 + i ˜ ϑ ln[ τ /τ ] (E6)+ i ϑ exp[ i ˜ ϑτ / (cid:16) Ei[ − i ˜ ϑτ / − Ei[ − i ˜ ϑτ / (cid:17) where Ei( z ) is the exponential integral function [64], and τ is the initial time at which ˜ A L = 1. The real andimaginary parts of this solution are compared to the nu-merical results in Figures (17) and (18). This solutionyields the following time derivatived ˜ A L d τ = iτ ˜ ϑ (cid:16) − exp[ i ˜ ϑ ( τ − τ ) / (cid:17) . (E7)In terms of the dimensionless time τ , the comparisongiven in Eq. (E3) becomes (after dropping the negligible first term on the right hand side)d ˜ A L d τ vs ˜ A L vL (cid:126) ∆ (E8)From Eq. (E6) we see that the magnitude of ˜ A L is ap-proximately one, whereas the magnitude of d ˜ A L / d τ isapproximately 1 /τ ˜ ϑ . Also, in the main part of the textwe argued that a reasonable value of ∆ / (cid:126) is 10 s − . Wetherefore see that for the case considered in this paper(initial times τ = −
25 and ˜ ϑ = 1), the time dependenceof the diabatic mode functions can be ignored. Appendix F: The Maxwell wave equation in amoving dielectric
Leonhardt and Piwnicki [48] have derived a modifiedversion of Maxwell’s wave equation (cid:18) ∇ − n r c ∂ ∂t − n r − c v . ∇ ∂∂t (cid:19) E ( x , t ) = 0 , (F1)in order to describe light propagating in a medium whichis moving with velocity v . This equation, which is valid indispersionless dielectrics with refractive index n r ( x, t ) = (cid:112) ε ( x, t ) /ε , is correct to first order in | v | /c . In our case,the moving dielectric medium is the common mirror, andour aim here is to estimate the size of the last term onthe left hand side because we have not included it in ourtreatment. The maximum size it can take is of order k v/c , as can be seen by taking 2 n r ( n r −
1) = 1 andletting both derivatives act on the phase φ = n r kx − ωt of the electric waves. To justify our neglect of thisterm we must check that it is much smaller than thesmallest term we keep in our reduction of Maxwell’s waveequation to a first-order-in-time equation, as described inSection VI. Referring to Eq. (51), we see that the smallestterm we keep is β d ˜ A/ d t , where β ≈ ω av . In Section VIwe argue that this term has a magnitude of β d ˜ A/ d t ≈ ω av ∆ / (cid:126) = 10 s − , and thus ( β/c )d ˜ A/ d t ≈ m − .This assertion is supported by the results plotted in Fig.16, which show that | d ˜ A/ d τ | = ( (cid:126) / ∆) | d ˜ A/ d t | ≈ v ≈ k v/c ≈ m − , which is considerably smaller thanthe smallest retained term. However, the other case dis-cussed in Section X concerned a waveguide whose refrac-tive index is changed on a timescale of 10 ps, which isequivalent to a mirror moving at speeds v ≈ ,
000 m/s.Under these circumstances we expect the modificationsto Maxwell’s wave equation to become important. [1]
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