The driven optical ring resonator as a model system for quantum optics

Physica B 175 (1991) 96-11(I North-Holland

The driven optical ring resonator as a model system for quantum optics R.J.C. S p r e e u w a n d J.P. W o e r d m a n Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands

A driven optical ring resonator has many features in common with a driven multilevel quantum system. Several such features have been demonstrated experimentally in the recent past. After a brief review of these results we present a speculative survey of further possibilities; this shows that the driven optical ring resonator has a surprising potential as a model system for quantum optics. When the ring resonator is suitably designed one may simulate spontaneous decay, implement the Jaynes-Cummings model and the kicked-top model, and possibly study quantum chaos. This wealth of possibilities is basically due to the macroscopic nature of the system.

1. Introduction In recent years optics has become a popular playground for testing of basic quantum mechanical concepts. Here we refer to questions dealing with quantum mechanical measurement theory and with the transition between quantum physics and classical physics [1-5]. In this paper we discuss classical optical experiments displaying features which are usually associated with quantum physics. It is an interesting question how far the optical implementation of quantum mechanics can be pushed for the experimental configurations which we consider here, and we will follow this route at some length. For completeness we note that the analogy between optics and quantum mechanics is a topic with a long history, which continues to appear in many different contexts [6-12]. However, in order to narrow down the scope of the paper we will avoid discussions which more properly belong to the foundations of quantum mechanics or to philosophy. We see it in fact as our main aim to point out how this kind of "analogy-thinking" leads to new experimental possibilities for investigating current issues in quantum optics. Our favorite optical system in this context is the optical ring resonator. The static properties of such a resonator are reviewed in section 2, in 0921-4526/91/$03.50

the language of coupled-mode theory. Dynamical properties are reviewed in section 3; this includes phenomena such as Rabi oscillations, Zener tunneling, Bloch oscillations, multiphoton transitions and the Bloch-Siegert shift. The experimental results reviewed in sections 2 and 3 establish in a sense the bona fide character and practical convenience of our model system. An important advantage of the macroscopic nature of the system is that all parameters are accessible to direct experimental control, including parameters that in the equivalent microscopic quantum system would be difficult or impossible to vary. Next we proceed to terra incognita, i.e. we propose experiments addressing theoretical issues in quantum optics which have not (or incompletely) been investigated so far. This refers either to problems in which the transition between classical and quantum behavior is essential, or to problems which are classical in nature but may lead to chaos. Specifically, in section 4 we discuss the effect of dissipation on the driven optical resonator; this may be used to simulate spontaneous decay of the equivalent two-level atom. In section 5 we discuss how the well known Jaynes-Cummings model, which describes the resonant interaction between a twolevel system and a single mode of the electromagnetic field, can be realized using the

© 1991 -Elsevier Science Publishers B.V. (North-Holland)

R. J. C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

driven optical ring resonator when an electrical feedback loop is added. In section 6 we show that the "kicked top", again a popular model in quantum optics, may be implemented by installing a (different) electrical feedback loop in the optical resonator. Finally, in section 7 we give a summary and an outlook.

2. The optical ring resonator as a coupled-mode system In this section we discuss an optical resonator in the language of coupled-mode theory. A ring resonator will turn out to be particularly useful from our point of view; later in the section we will briefly comment on the merits of the F a b r y Perot type resonator. Consider a ring resonator with clockwise (cw) and counterclockwise (ccw) traveling waves propagating along the ring (fig. 1). The eigenfrequencies of the cw and ccw modes are degenerate. We consider only one transverse and longitudinal mode. The degeneracy of the cw and ccw modes can be lifted by varying a control parameter S which acts in a nonreciprocal way on the modes. For this we may use the Sagnac effect which is produced by rotation of the ring. Crossing of the eigenfrequencies as a function of S is shown in fig. 2(a); it represents an optical level crossing. The cw and ccw modes are coupled due to partial reflection by a dielectric perturbation (amplitude reflectivity r) somewhere along the ring; this can be realized by introducing a glass plate in a conventional ring cavity or an air gap in a fiber-optic ring cavity [13, 14]. The equations of motion of the complex traveling wave amplitudes a I and a 2 in the presCW

r

Fig. 1. Ring resonator with clockwise and counterclockwise traveling-wave modes. The modes are coupled by a dielectric perturbation with amplitude reflectivity r.

97

{o)

S

(b)

Fig. 2. (a) level crossing of uncoupled eigenfrequencies as a function of the control parameter S; as an example, when we use the Sagnac effect the control parameter is the mechanical rotation frequency of the ring resonator sketched in fig. 1. (b) Avoided level crossing when the two modes are coupled.

ence of coupling follow from elementary optical considerations and can be written in matrix notation [15], dA

d~-

iliA

(1)

with A the state vector (a~, a 2 ) and H the dynamical matrix /I %,

H=

Wi2 ) -a "

(2)

For the lossless configuration sketched in fig. 1, the matrix H is Hermitian, with the mode detunings _+A as its diagonal elements and the couplings WI2 = W~l as its off-diagonal elements. In this Hermitian case the coupling is called conservative, which refers to the fact t h a t laiI2+ la212, i.e. the total energy of the two modes, is conserved. The coupling rate is given by IW12[ = W = r(c/L), where L is the circumference of the cavity. Note that eq. (1) is analogous to the Schr6dinger equation of a two-level system. As an alternative for the complex state-vector A one may choose to work with the real three-dimensional "Bloch vector" r = (r 1, r2, r3), with r I = 2 R e P12 ,

r 2 = - 2 Im 012 , r3

=

Pll

--

P22

,

(3)

98

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

where Pi~ = aiaj are the components of the density matrix. Note that the components of the Bloch vector can be considered as the expectation values of the spin components of a spin-½ particle with spinor A, i.e. ri= (ori)= At(riA [16]. This description leads to the optical Bloch equations (see eq. (6)). If several dielectric perturbations are present along the ring, their amplitude reflectivities (~WI2) should be added. Thus, such perturbations may interfere constructively or destructively depending on their separation along the ring. A simple example of this is the inference between the reflections from the front and rear dielectric interfaces when a glass plate is used as coupling element. Note that eq. (1) is not restricted to optical modes. In fact eq. (1) represents the canonical form of the equations of motion of any pair of linearly coupled harmonic oscillators, if the variation of the oscillator amplitudes is negligible on the time scale of the oscillator period [17]. Of course, in our case the harmonic oscillators are the cw and ccw modes of the electromagnetic field in the ring resonator. Diagonalization of H (eq. (2)) yields two normal modes with eigenfrequencies 0.)+ = + V A

2 -}- W 2 .

(4)

A plot of eq. (4), as given in fig. 2(b), shows that the coupling leads to an avoided level crossing: the eigenfrequencies are pushed apart, opening up a frequency gap ~12 = 2W. Instead of using the cw and ccw traveling waves as a basis for the field in the ring resonator we may equally well use the standing waves cw +ccw. In that case the dynamical matrix becomes

so that coupling and detuning are seen to have reversed their roles [15]. As has been demonstrated experimentally [13, 18] we may play essentially the same game by + using the polarization modes (i.e. x, y or cr , o'-) instead of the propagation modes (cw, ccw). In this case the Bloch-vector approach would

lead to a description of polarization in terms of the Poincar4 sphere. The two polarization modes may be coupled by means of a birefringent element, either linear or circular, and detuned (relative to each other) by means of a controllable birefringence, such as an electro-optic or magneto-optic (Faraday) modulator. The control parameter S is then an electric voltage or a magnetic field. Although many configurations are possible, they are all described by eqs. (1) and (2). One example is shown in fig. 3. Again, by using a suitable basis transformation, e.g. using x -+ y instead of x, y as a basis, the same birefringent element may be interpreted as a coupling instead of a detuning element, and vice versa. Also, when dealing with two polarization modes a ring resonator is not really necessary; a FabryPerot type resonator will do as well (see fig. 3). Interesting possibilities arise if we consider the case of four coupled modes in a ring resonator: two propagation modes (cw and ccw), coupled by partial reflection, and two polarization modes (x and y) coupled by birefringence. Recently the four-level case has been studied theoretically and experimentally [18]. The complexity of this case is such that a full analysis requires grouptheoretical methods; the group involved is U(2, 2). However, the useful properties of some highsymmetry configurations of the four-level case can also be explored heuristically. This applies for instance to configurations which allow simulation of Sagnac-detuning of two counterpropagating modes (cw, ccw) by means of the Faraday effect, as demonstrated in [14] (see also fig. 4). So far, the discussion has been restricted to a

Iv

/ TEOM1

TEOM2

Fig. 3. Polarization implementation of two-mode system. T E O M stands for transverse electro-optic modulator. In this case the ring configuration is not essential, also a linear configuration (Fabry-Perot cavity) can be used, with mirrors at positions indicated by the dashed lines.

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

Iccw. or-> Fig. 4. Simulation of Sagnac effect by means of a Faraday rotator. QW stands for quarterwave plate, QW1 and QW2 are mutually orthogonal. The (cw, ~+) and (ccw, ~r ) modes are mutually detuned by the Faraday effect and coupled by the partial back-reflector r.

single, be it four-fold degenerate resonance of the ring resonator. Clearly, the ring has a series of resonances, separated along the frequency axis by the free spectral range c/L. When varying the detuning parameter S this gives rise to a manifold of level crossings, instead of just a single crossing, so that, when introducing a coupling W a manifold of avoided crossings results, a so-called optical band structure. The analogy with electronic band structure has been extensively discussed elsewhere [14, 19, 20].

3. Dynamical aspects of the optical ring resonator The optical ring resonator can be driven by varying one of the control parameters (coupling or detuning) as a function of time. As discussed in section 2, coupling and detuning can be implemented by means of a variety of optical elements. In this section we will restrict the discussion to the two-mode case as described by eqs. (1) and (2) and consider only harmonic driving. As an example, when we substitute W12 = W21 = W0 sin S2t in eq. (2) we obtain the Hamiltonian of an optically driven two-level atom with a transition frequency 2A, the coupling strength with the optical field being described by the Rabi frequency W0. As another example, when we substitute A = A0 sin J2t in eq. (5) we have again the Hamiltonian of an optically driven two-level atom, this time with transition frequency 2W and Rabi frequency A0. Thus by a suitable choice of the basis, the Hamiltonian of the driven optical

99

resonator may always be mapped on that of a two-level atom driven by a classical optical field. In this sense the driven ring resonator may be called an "optical atom". Harmonic variation of W or A can be accomplished by harmonically driving an electro-optic or magneto-optic (Faraday) modulator inside the ring. For instance, the partial backreflector which couples a cw and ccw mode can be modulated by implementing it as a Fabry-Perot with an electro-optic phase modulator between its mirrors (see fig. 11). Note that we have now three harmonic oscillators in the problem: the two optical mode oscillators, referred to in section 2, and one electrical oscillator, e.g. an LC circuit, which drives the intracavity modulator. In practice, the mode oscillators have an optical frequency (10~4-10 ~5 Hz) and the electrical oscillator a radio frequency ( ] 0 6 - 1 0 7 H z ) . We have demonstrated experimentally a number of effects which are well known for driven two-level atoms [13]. For a classification of the effects it is useful to distinguish the regimes ~'~ <~ ~12' ~ ~ ~12 and ~ ~> ~2~2, where S2~2 is the transition frequency of the equivalent two-level atom. Assume now that in fig. 5 the system is prepared, at t = 0, in an eigenstate (top of the mountain hyperbola) and that, for t > 0 , the control parameter S becomes a harmonic (O) function of time. The response of the system becomes adiabatic in the limit ~2 ~ O~2 (keeping the "amplitude of the driving field constant); the representative dot then sweeps back and forth along the mountain hyperbola in fig. 5. In atomic physics this corresponds to the AC polarization set-up in an atom when driven by a low-frequen-

SocoS ~Dt

S

Fig. 5. Adiabatic response of optical two-level system upon an harmonic driving field. The dot represents the instantaneous state of the system and "rolls" back and forth over the mountain hyperbola.

I00

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

cy eletromagnetic field; in solid-state physics it is closely related to so-called Bloch oscillation in an electronic band structure and for this reason the same name is sometimes used for the adiabatic response upon (harmonic) driving of the optical resonator. Upon increasing the value of 12 the response acquires some nonadiabaticity: Zener tunneling of light occurs across the gap 12~2. In the limit 12 > 1212 Zener tunneling dominates the response. Experimentally we have observed both Bloch oscillation and Z e n e r tunneling [13]. The resonant condition (/~ - 12~2) takes a special place. For a driven two-level atom, this condition leads to Rabi oscillation, i.e. periodic population transfer between levels 1 and 2 at the Rabi "flipping" frequency. We have observed this Rabi oscillation for the resonantly driven optical resonator, in the time domain as well as in the spectral domain (Autler-Townes doublet splitting) [13]. An important point to make is that for our optical implementation of the driven two-level atom, the Rabi frequency (i.e. either W0 or k0) can be easily made as large as the transition frequency 12~2, or even larger, which is rather unusual in two-level atom physics. This implies that we may grossly violate the so-called rotating-wave approximation (RWA), which is an exceptional situation in quantum optics; see sections 5, 6 for further discussion. We have already observed consequences of violation of the RWA in our experiments [13]; we aim to demonstrate in the near future other consequences such as the Bloch-Siegert shift and multiphoton transitions (subharmonic resonance), i.e. resonant behavior at 12 = 12~2/ (2n + 1).

4. Dissipation and spontaneous decay A natural question to ask is whether one can simulate spontaneous decay of the optical atom. In order to be specific, let us assume that we have prepared the ring resonator in the upper level by injection of resonant light for t ~< 0. At t = 0 the injection is switched off and we study the subsequent behavior, searching for spontaneous decay of the excited optical atom. Of

course, in the absence of dissipation a decay cannot be expected; there is no coupling to a fluctuating bath such as the electromagnetic modes of free space in the case of a real atom. T h e r e f o r e we have to include dissipation, i.e. losses in the model. Formally, this corresponds to considering a non-Hermitian dynamical matrix H for the two-mode system described by eqs. (1) and (2) [15]. The losses can be due to either intracavity absorption or to transmission through the mirrors. In the first case we couple to the degrees of freedom of a material heat bath and in the second case to the electromagnetic modes of free space. We first deal with two optical modes which have equal losses. Consider for instance a cw and ccw mode of a ring resonator, with mode frequencies to _+ A, the detuning 2A~1212 being produced by the Sagnac effect and the losses being due to outcoupling through the mirrors. This clearly models spontaneous decay of an open two-level system in the " R y d b e r g " limit: the decay from both levels is equal, and exclusively to "outside" levels (fig. 6(a)). Simulation of spontaneous decay in a closed two-level, as shown in fig. 6(b), is less trivial. I n fact, the issue whether spontaneous decay can or cannot be realized in a classical two-level system such as the optical ring resonator, has been the subject of discussion at some recent conferences [21]. We give now a tentative proposal to illustrate how this might be done. Consider again a ring resonator with cw and ccw waves which are Sagnac-detuned by an amount 2A. It is now essential that the loss of one mode (say the cw wave) is converted to a gain of the other mode (say the ccw wave). This can be done if we insert in the ring an anisotropic partial reflector, which has different reflectivities for the cw and ccw

(o)

/

(b)

CCW

Fig. 6. (a) Spontaneous decay of open two-level system (cw, ccw); (b) idem for closed two-level system (cw, ccw).

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

(absorption)

white noJ/se

B R1

~

R2 -

101

/k~lP1

T 1

c

E0M

CW

T2 -

Fig. 7. Optical element with reflection, transmission and (possibly) absorption. R~ and 7", are reflection and transmission coefficients for the intensity. Fig.9. Sagnac-detuned cw and ccw modes form an optical two-level system. E O M stands for electro-optic modulator and A M P for optical amplifier.

Fig. 8. Anisotropic reflector. A wave which comes in from the left is (partially) back-reflected, whereas a wave which comes in from the right has zero backreflection. Output coupling at beamsplitters A and D (curly arrows) represent dissipation.

waves. We need in fact a device as indicated in fig. 7, with intensity reflectivies R, > 0 and R 2 = 0. Such a one-way mirror can be constructed if we allow dissipation. As an example, it follows from thin-film optics [22] that a thin metal layer (thickness d and complex index n + i k ) on a glass substrate (real index ng) has zero reflectivity from the glass side if d = A(ng- 1)/41Tnk, which corresponds to d = 12nm for silver at a = 633 nm [23]. In this paper, however, we will consider in particular an implementation as shown in fig. 8. Dissipation is now due to outcoupling at beam splitters A and D (curly arrows). A light wave coming in from the right is not reflected back. A light wave coming in from the left is partially reflected back via two pathways: A D C B A and A B C D A . It is easily verified that the two contributions interfere constructively (this is the same as in a fiber-optic loop reflector [24]). If we insert this device in a ring resonator (fig. 9) the cw mode is converted into the ccw mode but not vice versa. In the optical feedback loop of fig. 9 we have introduced an electro-optic

modulator E O M which is stochastically driven ("white noise"). This transforms a (small) part of the feedback light, which is initially at o~ + A, to the " p r o p e r " frequency, namely w - a. The intensity loss of the mode conversion (cw, oJ + A)-+ (ccw, w - A) is compensated by an optical amplifier AMP1. Note that in this scheme intensity (and not field amplitude) is transferred from the " u p p e r " to the "lower" mode. This corresponds to transfer of population during quantum mechanical spontaneous decay. Finally, another optical amplifier (AMP2) has been inserted in the main ring. This is a unidirectional amplifier; such a device may be constructed by combining a conventional amplifier with an optical diode. For the ccw wave it has a gain which is set to compensate the outcoupling losses at beamsplitters A and D; in this way the ccw wave is made loss-free so as to model the nondecaying ground level. For the cw wave AMP2 has unity transmission. In a Bloch equation description of two-level atom physics, already alluded to in section 2, dissipation is introduced by adding semiphenomenological relaxation terms to the "conservative" Bloch equations,

d dt

(r) (ri) ( rJ2 ) r2 r3

= ~

x

r2 r3

-

r2/T 2 (r 3 _ r;q)/Tl

,

(6)

where the r i have been defined in eq. (3) and where r~ q stands for the equilibrium value of the inversion. The term with the cross product describes the conservative part of the evolution, including that due to the driving field. The vector

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

102

g~ is generally time dependent, but becomes constant if one makes the RWA and performs a basis transformation to the so-called rotating coordinate frame [16]. The spontaneous decay, discussed at length above, contributes to T~ and T 2. One may ask also whether collisional dephasing, which contributes to the T 2 terms exclusively, may be simulated. We expect that this can be done by stochastic modulation of the level splitting. Inhomogeneous broadening, such as Doppler broadening, may be simulated by using a nondegenerate manifold of transverse modes of the resonator. In the optical Bloch equations the coupling of a two-level system with its environment is described by three parameters: T x and T 2, as discussed above, and the equilibrium population r3 q involving the temperature. We see no possibility to simulate the latter; this would require an ensemble of "optical atoms".

5. Jaynes-Cummings model Let us assume for a moment that we can neglect all effects of dissipation in the driven ring resonator, implying, e.g. that spontaneous decay does not occur. We deal then with an optical two-level system which is resonantly coupled with a driving (RF) oscillator and not with a dissipative continuum. This idealized situation corresponds in quantum optics to a single twolevel atom which is coupled to a single electromagnetic mode of a lossless resonant cavity. First we briefly review the theory of this archetype model of quantum optics. When we quantize the field, the Hamiltonian of the system consists of three parts, representing the field, the atom, and the interaction [25]: YC = h~2e*e

+

hf212S z + hg(S+ + S_)(e + e t) ,

(7) where J2 is the frequency of the field, S2tz the transition frequency of the two-level atom, g the atom-field coupling constant (proportional to the electric-dipole transition matrix element), e* and e the creation and annihilation operators of the field and Si the pseudospin operators which correspond to atomic polarization and inversion.

When the driving field is not too strong (i.e. the Rabi frequency should be much smaller than the transition frequency) one may neglect the energy-nonconserving processes S+e t, in which a photon is created as the atom makes an upward transition, and S e, in which a photon is annihilated as the atom makes a down transition. When mapped on a fictitious spin-½ system, the neglect of the nonresonant terms S+e* and S e corresponds to the neglect of the counterrotating magnetic field in a magnetic resonance experiment; therefore, this neglect has become known as the rotating-wave approximation (RWA). It leads to the so-called Jaynes-Cummings Hamiltonian [25], Y( = h~2ete + ]~f~12Sz ÷ hg(S+e + S_e t) ,

(8)

well known in cavity quantum electrodynamics (cavity QED). The Jaynes-Cummings model has drawn much theoretical interest since it has the virtue that it allows exact analytical solutions [26-37]. Also, an important stimulus for interest in the model is that it has been recently realized experimentally, by coupling a Rydberg atom to a high-Q microwave cavity [38]. In that experiment the dissipation rate (set by spontaneous emission and cavity loss) was smaller than the electric-dipole coupling rate, so that the freespace behavior of the atom was significantly changed. When resonantly driven (S2-J2,2), a system described by the Jaynes-Cummings Hamiltonian, eq. (8) will show a Rabi-type back and forth trading of energy between the atom and the field mode; the total unperturbed energy, i.e. (hS2ete + hS212Sz) is conserved. If the number of photons in the field mode is large, 1, the field may be treated classically, leading to the so-called semi-classical JaynesCummings model. However, the coupling of field dynamics and atomic dynamics survives a classical treatment of the field only if the atomfield coupling constant g is enlarged artificially [27, 28, 31-33]. This can be realized, in principle, if one considers, instead of a single two-level atom, an ensemble of N such atoms (N~> 1); in that case the transition dipole of the ensemble

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

can be described classically. (Of course, typical quantum features, such as collapse and revival of the exchange dynamics due to a quantum spread of Rabi frequencies do not survive in the semiclassical limit.) Returning now to the driven optical ring resonator we note that it is in principle described by the Jaynes-Cummings Hamiltonian (eq. (8)) if we quantize the RF oscillator (e.g. a LC resonant circuit) which drives the intracavity modulator [39]. However, in practice the RF oscillator is of course completely in the classical limit, and in that case the equivalent of the reaction of the atom on the field, so essential for the Jaynes-Cummings exchange dynamics, seems to be absent due to the unavoidable presence of dissipation. As an example, for an optical ring resonator driven by a RF modulated electro-optic crystal, reaction would imply that the optical field in the ring affects the RF field. Of course, this effect is present in principle; for instance an eletro-optic crystal, which is inside the capacitor of the LC circuit of the RF oscillator, may have its RF dielectric constant infinitesimally changed by the optical intensity. Observation of this reaction would require extremely low-noise conditions, i.e. extremely small optical loss in the ring (think of the fluctuation-dissipation theorem!) [40]. Nevertheless, there is a way to realize the semi-classical Jaynes-Cummings model by means of the driven optical ring resonator. The basic idea is, as in the case of a real two-level atom, to increase the coupling constant. The macroscopic nature of our system allows experimental realization of this idea, contrary to the case of a real atom where an increase of the coupling constant is doomed to remain a theoretical artifice. Figure 10 shows our proposition for an optical realization of the semi-classical Jaynes-Cummings model. The optical twolevel system consists of the cw and ccw modes of a ring resonator which are Sagnac-detuned by an amount ~~12 = 2A. The system is driven by harmonic modulation (12) of the coupling element, which is a partial backreflector. It consists, as shown in fig. 10, of a longitudinal electro-optic phase modulator L E O M inside a Fabry-Perot.

103

CW CCW

-~--

CW

CCW

CCW

•

;c..cc.

Fig. 10. Implementation of Jaynes-Cummings model using the Sagnac-detuned cw and ccw traveling wave modes as a two-level system. The electrical feedback loop affects the amplitude of the driving field in such a way that the JaynesCummings model is realized.

The key point is that we use an electrical feedback loop to simulate the reaction of the optical atom on its own resonant radiation field, i.e. on the RF oscillator. We couple a small fraction of the cw and ccw amplitudes out of the ring resonator. These traveling-wave amplitudes are combined by means of a coupler (or beamsplitter), leading to the amplitudes c w - c c w and cw + ccw. The amplitude cw + ccw is photo-electrically detected, leading to a photocurrent (cw + ccw) 2. The product term cw. ccw, which is modulated at frequency 12 and which corresponds to the polarization of the optical atom, is amplified by an AC-coupled amplifier; the DC terms cw 2 and ccw: are blocked. The product term cw.ccw is then added to the original driving field. A polarization implementation of the same feedback scheme is shown in fig. 11. The equation of motion of the mode amplitudes for the configurations of figs. 10 and 11 is a generalization of the linear case considered so far (el. eqs. (1) and (2)) and has the generic form

d (A,)= d-t A 2 zl - i ( W2,(A2, A , ) cos oJt

Wt2(A 1 , 7 2 ) c°s

wt](Al] '\A2J (9)

These coupled-mode equations are the classical

'

104

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

Q TEOM1

Y

[

IEOM2

x+y

E (x+y)2

Fig. 11. Implementation of Jaynes-Cummings model using a polarization implementation of the optical two-level system. TEOM stands for transverse electro-optic modulator. The electrical feedback loop affects the amplitude of the driving field in such a way that the Jaynes-Cummings model is realized.

limit of the operator equations in the Heisenberg representation. The nonlinearity W~j=Wi~(A i, A j) is implemented by the electrical feedback loop and has been chosen (see figs. 10 and 11) such that the equation of motion (eq. (9)) gives exactly the Heisenberg equations of motion corresponding with the Jaynes-Cummings model (without the RWA, i.e. eq. (7)). The loop should be much faster than the dynamics of the ring resonator, i.e. the bandwidth of the loop should be much larger than the Rabi frequency. This condition is easily satisfied since in recent experiments the Rabi frequency was typically a few 100 kHz [13]. The importance of having an experimental realization of the semi-classical JaynesCummings model is that it will hopefully allow an experimental verification of the theoretical prediction [26, 27, 28, 29, 30] that this system shows chaotic dynamics if it is driven so strongly that the RWA is not valid (i.e. if cos wt in eq. (9) cannot be replaced by ei~'). This prediction can be seen as a consequence of the fact that the unperturbed energy is no longer a constant of the motion if the RWA is dropped, increasing the number of degrees of freedom from 2 (the atomic inversion and the relative phase of the atomic polarization and the field) to 3, which is known to be a necessary (though not sufficient) condition for chaos. Since we are able, as dis-

cussed in section 3, to drive the optical two-level systems with sufficient strength to violate indeed the RWA, the set-up sketched in figs. t0 and 11 is ideally suited for a first experimental verification of chaos in the non-RWA extension of the semi-classical Jaynes-Cummings model. Particularly interesting is the experimental possibility of going from a situation where the RWA is exact to one where it is violated, for the same strength of the RF driving field. This transition can be made by starting from unperturbed states that are circularly polarized and then manipulating the polarization of the RF field, for instance by driving the intracavity transverse electro-optic modulator with a rotating RF field (using 4 electrodes) instead of an oscillating RF field (using 2 electrodes). Several extensions of the Jaynes-Cummings model have been reported in the theoretical literature, including versions with multiphoton excitation [36] and multilevel atoms [37]; also effects of detuning [32, 35] intensity-dependent coupling [36] and cavity damping have been addressed [34]. These extensions can probably be realized experimentally by adding suitably designed feedback loops to the optical ring resonator.

6. Simulation of driven quantum systems

The quantum treatment of systems which show chaotic motion in the classical limit is a popular topic of theoretical study. The motivation of these studies is to see how the character of classical chaos is changed if the system is made more and more quantum mechanical by changing a suitable control parameter. This field is loosely referred to as "quantum chaos". Although there seems to be a consensus that proper quantum chaos does not exist there is quite some debate on the nature of the transition between the classical and the quantum cases [41, 42]. Favorite theoretical models which are used in this context concern relatively simple quantum systems which freely evolve, except for kicks or reflections at sharp time or space coordinates. Examples are the kicked rotor [43] and the

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

kicked top [44-46]; the magnitude of the rotor or top angular momentum (in units of h) is the control parameter mentioned above. Only very few experimental data are available for verification of this theoretical model work; we point in particular to the experiments on microwave ionization of H atoms prepared in Rydberg states [47]. It is not obvious, however, that simple Hamiltonians as mentioned above are applicable in a case like this. Therefore, a recent proposition by Haake et al. [48] for precise experimental realization of the kicked-top Hamiltonian deserves special attention. Haake et al. consider two optical modes in a "suitable" nonlinear optical medium. In the context of the present paper the question then naturally arises whether one could also use two modes of an optical resonator equipped with a suitable feedback loop to produce the required nonlinearity; we will show here that this is indeed the case. The time-dependent Hamiltonian of a periodically driven top with angular momentum operator J = (Jx, jr.,,, Jz) can be written as

H(t) = J~ + W(t)Jx,

(10)

with W(t) a periodic function of time. The magnitude of the angular momentum is a constant of motion, j 2 = j ( j + l ) . Periodic kicking is described by

w(t) =

a(t

-

nT).

(11)

105

N = a*a + b*b is related to the angular momentum operator J through

j2

2

2

2

=J~ + J ~ + J ~ =

1

I

~N(~N+I).

(13)

The total number of photons in the two modes corresponds to twice the angular momentum quantum number j and is likewise a conserved quantity. We make now a connection with our previous treatment of the driven optical ring resonator by quantizing the two optical modes:

H = h ( o - a)a*a + h(w + a)b*b + hW(ab* + a ' b ) ,

(14)

where either the detuning ~ or the coupling W is a function of time; we will restrict ourselves to the latter case, W= W(t). In order to avoid confusion with the Jaynes-Cummings Hamiltonian introduced in section 5, note that in the present case, as described by eq. (14), a classical field drives two coupled quantized modes, whereas in the Jaynes-Cummings case, as described by eq. (8), a quantized field drives two coupled classical modes. (It is a bit paradoxical that we need two classical coupled optical modes in order to simulate a quantum mechanical twolevel system!) Slightly rearranging terms in eq. (14), and using that ( a * a + b * b ) = c o n s t a n t , yields

tt

Following Haake et al. [48] we introduce the Schwinger representation of angular momentum in terms of two Bose oscillators,

j~ = 1 (ab* + a ' b ) , 1

JY = TII (ab* - a ' b ) ,

(12)

J~ = ½ ( a ' a - b ' b ) , and we interpret the Bose oscillators as quantized modes of the electromagnetic field. (Note the similarity with the definition of the Bloch vector in eq. (3).) The photon number operator

H = hA(a+a - b+b) + hW(t)(ab t + a ' b ) .

(15)

When using eq. (12) this Hamiltonian can be seen to be identical with the nonlinear kickedtop Hamiltonian (eq. (10) if we substitute A o: (a*a - b'b). Therefore, a driven two-mode ring resonator can be mapped on a driven top if the mode detuning A is made to be proportional to the "inversion". An experimental implementation using two polarization modes is shown in fig. 12. The quartic terms (a'a) 2 and (b'b) 2, which appear in the Hamiltonian when we make the substitution just mentioned for A in eq. (15), reflect that the optical mode oscillators have been made anharmonic by the feedback loop.

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

106

w

2 Y

TEOM1

TEOM2

1

0

),2 Fig. 12. Implementation of kicked-top model using a polarization implementation of the optical two-level system. TEOM stands for transverse electro-optic modulator and W(t) represents the driving field. The electrical feedback loop makes the detuning proportional to the "atomic inversion"

"s

Fig. 13. Diabatic limit of sinusoidal driving of a two-mode system with transition frequency Y212= 2W. Projection on the W = 0 eigenmodes shows that the driving of the equivalent two-level atom has a kicked character.

x 2 y2.

In a classical description the Hamiltonian eq. (15) corresponds to a coupled-mode equation which has the generic form

d (AAI2) = _ i ( A ( A 1 , A 2 )

dt

W(t)

W(t)

~{Al~

- a ( A 1, A2)]\A2]" (16)

This may be compared with the JaynesCummings variety of the same equation (see eq. (9)). The classical driven top described by eq. (16) shows chaotic dynamic behavior if the driving and the nonlinearity satisfy certain criteria [44]. In the case of the two-mode optical ring resonator chaotic behavior would mean a chaotic distribution of the intensity between the two modes, the total intensity being constant. In the classical limit chaos does not occur for monochromatic driving if the RWA is valid. One expects that the more nonmonochromatic the driving is, the easier the dynamics becomes chaotic, kicked driving (eq. (11)) being the extreme example [48]. Kicked driving of the ring resonator can be realized by using a large bandwidth amplifier for applying high-voltage pulses to TEOM 1 in fig. 12. Chaos is also expected for monochromatic driving if the RWA is violated, which, as discussed in section 3, can easily occur for the driven optical resonator. Very strong violation of the RWA occurs in the diabatic

regime; as can be seen from fig. 13 this corresponds, in a sense, to kicked driving. A categorization can be made, as already mentioned in section 3, in the regimes Y2>> ~ 2 , / 2 - ~12 and Y2 ~ ~212; from the chaos point of view the "highfrequency" regime ~ >> Y2~2seems to be particularly interesting [47]. Of course, the real challenge is to investigate experimentally the quantum aspects of these issues. The interesting regime is that where the total number of photons in the ring resonator, N = 2 j ~ 10-104 . Smaller values of N would imply complete dominance of quantum fluctuations over the universal (i.e. j-independent) properties. For larger values of N quantum effects are presumably difficult to detect. In fact, in experiments on coherence and noise properties of lasers, it has turned out that quantum fluctuations can dominate technical fluctuations for photon numbers up to 103-105, or even -108 if Herculean technical efforts are spent [49-53]. Clearly, quantizing an optical mode oscillator has practical relevance, contrary to quantizing an RF oscillator as was done in section 5. The dynamics of a top with j2 =h21.(j + 1) which is periodically kicked at time intervals T has been predicted to be classically chaotic on time scales ~>Tln ], and to become quasiperiodic on time scales /> Tj [44]. Classical chaos is thus expected to be "dead" after j kicks; this quantum suppression of classical chaos can be seen as a consequence of quantum interference

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

[54, 55]. An optical realization of the kicked top should therefore be designed such that the "window" between the two types of dynamics is as large as possible, i.e. N(=2j) should be as large as possible. Clearly, the problem of measurement-induced back action noise still has to be solved. Values of N in the range 10-104 correspond to such small intensities that bulk-optical nonlinearities are clearly inadequate from a practical point of view. In our case, however, we profit from the macroscopic nature of the ring resonator: we can insert a high-gain amplifier in the feedback loop responsible for the nonlinearity. As always, this amplifier should have a bandwidth much larger than that of the internal (Rabi) dynamics. This approach towards realization of a driven quantum top might well be feasible experimentally. So far we have encountered two examples of a driven nonlinear two-mode system, namely the Jaynes-Cummings model described by eq. (9), where the nonlinearity was in the coupling, and the driven optical top described by eq. (16), where the nonlinearity was in the detuning. As in the linear case, coupling and detuning can be interchanged by a suitable basis transformation. For completeness we note that nonlinear coupling of two modes occurs also in other contexts in nonlinear optics [56-58]. As a generalization of the driven nonlinear two-mode problem, one may simultaneously modulate coupling and detuning i.e. W--W(t) and A = A(t). In the atomic analogue this would correspond to simultaneously driving an atomic transition with a laser field and modulating its transition frequency by means of the Stark or Zeeman effect. As another generalization one may consider a driven four-level system with nonlinear couplings (see section 2). This would allow study of chaos in a four-dimensional state space. As still another generalization one may introduce dissipative nonlinearities, either in coupling or detuning, instead of considering only conservative nonlinearities as was (implicitly) done so far. (We remind the reader that the label "conservative" implies that [All 2 + IA2[2 is con-

107

stant.) An example of the dissipative case is encountered when we consider the two-mode laser instead of the two-mode passive cavity discussed so far; the nonlinearity is then due to gain saturation and the equation of motion for the mode amplitudes is

dt

A2

-i

)(AI)

A + i G ( A x, A 2 )

W21

-A+iG(Az,

A 1

A 2

(17) where G is the amplitude-dependent gain. Upon modulating one of the intracavity parameters, and thus, for instance, Wij or A, the system may show dynamical chaos, an example of "laser chaos" [59]. Thus laser chaos is seen to be dissipative chaos, contrary to the conservative or Hamiltonian chaos encountered in the context of the Jaynes-Cummings or kicked-top model.

7. Conclusions

In summary, we have given a view upon the optical ring resonator as a coupled-mode system, employing both the propagation and polarization degrees of freedom. We have also introduced various possibilities to drive such a system and discussed the corresponding varieties of dynamical behavior. It has been shown how introduction of dissipation should enable us to mimic spontaneous decay of the corresponding twolevel atom; the essential clue here is the addition of an optical feedback loop to the main optical resonator. More generally, it has been indicated how much of the two-level atom physics described by the optical Bloch equations can be realized with a suitably designed optical resonator. Subsequently, situations were addressed in which the resonator was endowed with an artificial nonlinearity by means of an electrical feedback loop. It was shown that this may lead to experimental implementation of the semi-classical Jaynes-Cummings model and thus to pos-

'

108

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

sibility of experimental study of the chaotic dynamics predicted for this case (if the RWA is violated). Using such artificial nonlinearity one may also implement the driven top, a popular model system in the context of quantum chaos. Finally, a connection was made with the field of (dissipative) laser chaos. The artificial nonlinearities which we use in our approach have in fact been known for a long time in a rather different context: so-called "hybrid nonlinear optics" [60-63]. The generic example of this is the insertion of an electro-optic phase modulator inside a Fabry-Perot, the modulator being driven by a voltage derived from the intensity of light transmitted by the Fabry-Perot [60]. The refractive index inside the Fabry-Perot becomes then intensity-dependent as if the Fabry-Perot was filled with a X (3~ nonlinear optical medium. The general motivation of hybrid nonlinear optics is to produce a bistable device for use as an optical switch; also chaotic dynamics have been studied [63]. Although our motivation is different, namely implementation of popular model systems of quantum optics, the advantages of using an artificial, feedbackinduced nonlinearity are the same: (i) very low optical power requirements, (ii) no limitation by material response times, (iii) possibility of electrical control, (iv) possibility of tailoring resonator characteristics. Very interesting possibilities may also arise if we incorporate a time delay in the electrical feedback loop. If the time delay is much larger than the time scales of the internal degrees of freedom of the ring resonator this may lead to simulation of the dynamics of an extended system, i.e. in our case, a string of optical atoms [64, 65]. One might then think of, for instance, the analogue of self-induced transparency. The consequences of dissipation have only been briefly touched upon in the present paper, namely for mimicking spontaneous decay. Dissipative quantum dynamics (or the quantum theory of damping) is in fact a field of almost exclusively theoretical study [66-75] and our system possibly offers an experimental entrance to some aspects thereof. There seems to be a con-

sensus that dissipation suppresses quantum effects and makes classical pictures more valid. A proviso must be made here: as discussed in section 4 we see no possibility of introducing the concept of a temperature in our system since we deal with intrinsic nonequilibrium situations. Our system allows, in principle, exact realization of popular model Hamiltonians of quantum optics and may thus be considered as a dedicated optical analogue computer. As such it may serve a useful role, certainly as long as microscopic realizations are so scarce as is presently the case. One may ask Whether the optical character of our dedicated analogue computer is essential or just convenient from a practical point of view; i.e. could one equally well use "acoustic" or "electrical" instead of "optical" atoms? We think that in principle the other varieties could also be used as long as one is interested only in classical or semi-classical features (although it is not so easy to think of practical hardware implementations). However, simulation of true quantum features, such as discussed in section 6, will require the optical variety. In conclusion then, we feel that the driven optical ring resonator, when viewed upon as in the present paper, forms a gold mine for quantum optics. We hope that further experimentation will contribute to our insight in the transition between classical physics and quantum physics.

Acknowledgements We gratefully acknowledge useful discussions with E.R. Eliel, M.P. van Exter, F. Haake and F.T. Arecchi.

References [1] Z.Y. Ou, X.Y. Zou, L.J. Wang and L. Mandel, Phys. Rev. Lett. 65 (1990) 321. [2] L.J. Wang, X.Y. Zou and L. Mandel, Phys. Rev. Lett. 66 (1991) 1111. [3] P.G. Kwiat, W.A. Vareka, C,K. Hong, H. Nathel and R.Y. Chiao, Phys. Rev. A 41 (1990) 2910.

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

[4] W.M. Itano, D.J. Heinzen, J.J. Bollinger and D.J. Wineland, Phys. Rev. A 41 (1990) 2295. [5] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47 (1981) 460. [6] W.T. Scott, Erwin Schr6dinger: An Introduction to his Writings, (University of Massachusetts Press, Amherst, 1967) Ch. 3. [7] W. Pauli, Z. Phy. 80 (1933) 573. [8] P. Ehrenfest, Z. Phy. 78 (1932) 555. [9] R. Peierls, Surprises in Theoretical Physics (Princeton University Press, Princeton, 1979) p. 10. [10] D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972) p. 100. [11] J. Krug, Phys. Rev. Lett. 59 (1987) 2133. [12] R.E. Prange and S. Fishman, Phys. Rev. Lett. 63 (1989) 704. [13] R.J.C. Spreeuw, N.J. van Druten, M.W. Beijersbergen, E.R. Eliel and J.P. Woerdman, Phys. Rev. Lett. 65 (1990) 2642. [14] R.J.C. Spreeuw, J.P. Woerdman and D. Lenstra, Phys. Rev. Lett. 61 (1988) 318. [15] R.J.C. Spreeuw, R. Centeno Neelen, N.J. van Druten, E.R. Eliel and J.P. Woerdman, Phys. Rev. A 42 (1990) 4315. [16] L. Allen and J.H. Eberly, Optical Resonance and TwoLevel Atoms (Dover, New York, 1987). [17] G. Weinreich, J. Acoust. Soc. Am. 62 (1977) 1474. [18] R.J.C. Spreeuw, M.W. Beijersbergen and J.P. Woerdman, Phys. Rev. A, submitted. [19] J.P. Woerdman and R.J.C. Spreeuw, in: Analogies in Optics and Micro-Electronics, eds. W. van Haeringen and D. Lenstra (Kluwer, Dordrecht, 1990) p. 135. [2(I] D. Lenstra, L.P.J. Kamp and W. van Haeringen, Optics Commun. 61) (1986) 339. [21] Research Conference on Quantum Optics, Davos, I-5 October 1990. Int. Conf. on Quantum Optics, Hyderabad, 5-10 January 1991. [22] H.A. Macleod, Thin Film Optical Filters (Hilger, London, 1969). [23] T. Inagaki, J.P. Goudonnet, P. Royer and E.T. Arakawa, Appl. Opt. 25 (1986) 3635. [24] D.B. Mortimore, IEEE J. Lightwave Technology LT-6 (1988) 1217. [25] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 (1963) 89. [26] R. Graham and M. H6hnerbach, Phys. Lett. A 101 (1984) 61. [27] P.W. Milonni, J.R. Ackerhalt and H.W. Galbraith, Phys. Rev. Lett. 50 (1983) 966. [28] R.F. Fox and J. Eidson, Phys. Rev. A 34 (1986) 482. [29] J. Eidson and R.F. Fox, Phys. Rev. A 34 (1986) 3288. [30] R.F. Fox and J. Eidson, Phys. Rev. A 36 (1987) 4321. [31] L. Vahala, Phys. Rev. A 42 (1990) 1813. [32] A. Kujawski and M. Munz, Z. Phys. B 66 (1987) 135. [33] M. Jelefiska-Kuklifiska and M. Ku~, Phys. Rev. A 41 (1990) 2889.

109

[34] M. Munz and A. Kujawski, Europhys. Lett. 13 (1990) 11/3. [35] M. Hillery and R.J. Schwartz, Phys. Rev. A 43 (1991) 1506. [36] A. Rosenhouse, J. Mod. Optics 38 (1991) 269 and references therein. [37] K.-D. Harms and F. Haake, Z. Phys. B 79 (1990) 159. [38] G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [39] D. Marcuse, Principles of Quantum Electronics (Academic Press, New York, 1980) p. 55. [40] J. Gea-Banacloche, Phys. Rev. A 35 (1987) 2518. [41] F. Haake, Quantum Signatures of Chaos (Springer, Berlin, 1991). [42] F.M. Izrailev, Phys. Rep. 196 (1990) 299. [43] G. Casati, J. Ford, I. Guarneri and F. Vivaldi, Phys. Rev. A 34 (1986) 1413. [44] F. Haake, M. Kug and R. Scharf, Z. Phys. B 65 (1987) 381. [45] M. Kug, R. Scharf and F. Haake, Z. Phys. B 66 (1987) 129. [46] F. Haake and D.L. Shepelyanski, Europhys. Lett. 5 (1988) 671. [47] E.J. Galvez, B.E. Sauer, L. Moorman, P.M. Koch and D. Richards, Phys. Rev. Lett. 61 (1988) 2011. [48] F. Haake, G. Lenz and F. Purl, J. Mod. Optics 37 (1990) 155. [49] A. Gfittner, H. Welling, K.H. Gericke and W. Seifert, Phys. Rev. A 18 (1978) 1157. [50] P. Lett, W. Christian, S. Singh and L. Mandel, Phys. Rev. Lett. 47 (1981) 1892. [51] D. Welford and A. Mooradian, Appl. Phys. Lett. 40 (1982) 865. [52] T.A. Dorschner, H.A. Haus, M. Holz, I.W. Smith and H. Statz, IEEE J. Quant. Electron. QE-16 (1980) 1376. [53] J.L. Hall, C. Salamon and D. Hils, J. Optics Soc. Am. B 3 (1986) supplement P80-P81 (Proc. XIV Int. Quantum Electron. Conf., 9-13 June 1986, San Francisco). [54] S. Fishman, D.R. Grempel and R.E. Prange, Phys. Rev. Lett. 49 (1982) 51)9. [55] D.R. Grempel, R.E. Prange and S. Fishman, Phys. Rev. A 29 (1984) 1639. [56] A.B. Aceves and S. Wabnitz, Phys. Lett. A 141 (1989) 37. [57] C.M. de Sterke and J.E. Sipe, Phys. Rev. A 43 (1991) 2467. [58] N.J. Doran and D. Wood, Opt. Lett. 13 11988) 56. [59] F.T. Arecchi, in: Instabilities and Chaos in Quantum Optics, eds. F.T. Arecchi and R.G. Harrison (Springer, Berlin, 1987) p. 9. [60] P.W. Smith and E.H. Turner, Appl. Phys. Lett. 30 (1977) 280. [61] P.W. Smith, E.H. Turner and P.J. Maloney, IEEE J. Quantum Electron. QE-14 (1978) 207. [62] P.W. Smith, E.H. Turner and B.B. Mumford, Opt. Lett. 2 (1978) 55.

110

R.J.C. Spreeuw, J.P. Woerdman / The driven optical ring resonator

[63] F. Mitschke and N. Fl/iggen, Appl. Phys. B 35 (1984) 59. [64] F.T. Arecchi, G. Giacomelli, A. Lapucci and R. Meucci, unpublished. [65] F.T. Arecchi, private communication. [66] E.G. Harris, Phys. Rev. A 42 (1990) 3685. [67] P. Ao and J. Rammer, Phys. Rev. Lett. 62 (1989) 3004. [68] J. Huang, S.I. Chu and J.O. Hirschfelder, Phys. Rev. A 40 (1989) 4171. [69] R. Landauer, Phys. Rev. B 33 (1986) 6497.

[70] P.G. Wolynes, J. Chem. Phys. 86 (1987) 1957. [71] A.M. Eevine, W. Hontscha and E. Pollak, Phys. Rev. B 40 (1989) 2138. [72] F.H.M. Faisal and J.V. Moloney, J. Phys. B 14 (1981) 3603. [73] H.C. Baker and R.L. Singleton, Phys. Rev. A 42 (1990) 10. [74] W.R. Frensley, Rev. Mod. Phys. 62 (1990) 745. [75] G. Dattoli and A. Torte, Phys. Rev. A 42 (1990) 1467.