12. Coupled-Mode Theory¶
PDF pages 511–532
12.1.1 Classical Field in a Single Cavity¶
Chapter 12 Coupled-Mode Theory 12.1 Cavity QED 12.1.1 Classical Field in a Single Cavity To start, let’s just consider the simplest case of a classical field in a linear resonator. The resonator consists of two mirrors of field reflection coefficients 1 and r, respectively, surrounding a region of vacuum of length L. ro=o1 r Eo(+) We will assume an approximately monochromatic field of frequency \omega. Following the field one round trip around the resonator, we can write
(12.1) where E(+)(t) is a slowly varying amplitude for the field, since the fast optical time dependence is written out explicitly, and
c (12.2) (cavity round-trip time) is the round-trip time of the cavity. Note that we have dropped the polarization of the field, since in our idealized setup it is an invariant. We will assume that the cavity is ‘‘good,’’ or not very lossy, so that |r| \approx 1. In this case, E(+)(t + \taurt) is almost the same as E(+)(t), and we can expand E(+)(t + \taurt) as
(12.3) to first order in \taurt. Putting this into Eq. (12.1), we obtain the rate equation
\taurt E(+)(t) (12.4) for the slowly varying field amplitude. For a steady-state solution (i.e., for a stable mode to exist), we must
(12.5) which only happens if |r| = 1, and also for
(12.6) (resonance condition)
12.1.2 Classical Coupled Modes of Two Cavities¶
Chapter 12. Coupled-Mode Theory where \phi is the phase of the reflection, r = |r|ei\phi, and q is some integer. It is only for a completely closed cavity, |r| = 1, where a stable mode exists, and this is precisely the condition that we assumed when quantizing the field. Otherwise, the rate equation (12.4) leads to an exponentially decaying mode (at least within the approximation of a slowly varying amplitude). 12.1.2 Classical Coupled Modes of Two Cavities Now we can extend this treatment to the case of two cavities, which are weakly coupled by a beam splitter where the modulus |r| of the reflection coefficient is close to unity.1 ro=o1 ro=o1 E1 (+) E2 (+) r12,ot12 r21,ot21 We will assume, as for the two-level atom, that only two modes of the individual cavities are approximately ‘‘resonant,’’ and so we can treat only these two modes. This, of course, implies that the lengths L1 and L2 of the two cavities are appropriately mismatched. The outer mirrors of the double cavity are perfectly reflecting, so that the total system is closed. This will allow us to quantize the coupled fields. The coupling through the beam splitter is described by field reflection and transmission coefficients r12 and t12, as seen by the field E(+) in cavity 1, and coefficients r21 and t21, as seen by the field E(+) in cavity 2. The (lossless) beam splitter induces a unitary transformation on two input fields to generate the output fields. Thus, if the operator for the beam splitter is written U = t21 r12 r21 t12 , (12.7) so that the output fields are written in terms of the input fields as " E(+) 1\leftarrow E(+) 2\rightarrow # = U " E(+) 2\leftarrow E(+) 1\rightarrow # (12.8) (with arrows indicating directions of the input and output traveling waves), then it must also have the general unitary form U = t r −r∗ t∗ , (12.9) with |r|2 + |t|2 = 1, so that r = r12 = −r∗
- (These relations are derivable classically by examining a beam incident on the beam splitter, and the time-reversed process; the resulting relations are
perturbation on the isolated cavity modes. We can thus write coupled equations for the fields in basically the same way as for the single cavity: E(+)
(t)e−i\omega2t E(+)
(t)e−i\omega1t. (12.10) Here \taurt1 and \taurt2 are the round-trip times 2L1/c and 2L2/c for the two respective cavities, and we have counted the accumulated phase from the fields starting at the beam splitter at t = 0. Performing the 1For more details on the classical theory of mode coupling in composite resonators, see Robert J. Lang and Amnon Yariv, ‘‘Local-field rate equations for coupled optical resonators,’’ Physical Review A 34, 2038 (1986) (doi: 10.1103/Phys- RevA.34.2038).
12.1 Cavity QED same expansion (assuming weak coupling between the cavities), these coupled equations reduce to the rate equations d dt " E(+) E(+) # =
\taurt1 t21e−i∆t \taurt1 t12ei∆t \taurt2
\taurt2 " E(+) E(+) # , (12.11) where ∆:= \omega2 −\omega1 is the detuning between the two cavity modes. We can simplify our notation a bit by writing d dt " E(+) E(+) # = −i \delta\omega1 \chi12e−i∆t \chi21ei∆t \delta\omega2 " E(+) E(+) # , (12.12) where we have defined the frequency offsets
\taurt1 ,
\taurt2 , (12.13) and the field coupling coefficients
\taurt1 ,
\taurt2 . (12.14) (classical field-coupling coefficients) To self-consistently treat the double cavity, we would find the modes by letting \omega1, \omega2 −\rightarrow \omega, where \omega is the eigenfrequency to be found. Then the time derivative vanishes in Eq. (12.12) for eigenmodes of the system, which implies that the determinant of the matrix must also vanish, leading to the condition
= t21t12 (12.15) (again, with \omega1,2 −\rightarrow \omega) that determines the allowed frequencies \omega. However, in the perturbative limit that we want to use here, we keep the original frequencies \omega1 and \omega2. We also note that in the perturbative regime, the transmission coefficients t12 and t21 are O(ϵ), where ϵ is some small perturbation parameter, and with this definition, the frequencies \delta\omega1 and \delta\omega2 are O(ϵ2) if the resonance conditions for the individual cavities are satisfied, since |r| = p 1 −|t|2 \approx 1 −|t|2/2. Thus, we will simply ignore the diagonal elements of the evolution matrix so that d dt " E(+) E(+) # = −i\chi12e−i∆t −i\chi21ei∆t " E(+) E(+) # . (12.16) In effect, since \delta\omega1,2 have small imaginary parts for any small coupling of the cavities, in neglecting these we are explicitly making the approximation that the isolated-cavity modes are still well defined but coupled together. As a last simplification, we can transform into a rotating frame by defining ˜E(+)
ei∆t, ˜E(+)
, (12.17) so that d dt " ˜E(+) ˜E(+) # = i∆ −i\chi12 −i\chi21 " ˜E(+) ˜E(+) # , (classical coupled-mode equations) (12.18) and thus we have eliminated the explicit time dependence in the problem. Formally, we see that the dy- namics of the two modes are formally equivalent to those of the amplitudes of a two-level atom driven by a classical field (without spontaneous emission), where we identify 2|\chi12| with the Rabi frequency Ω, as we see by comparison to Eqs. (5.25). We thus expect Rabi oscillations of the field between the two cavities, characteristic of a pair of coupled harmonic oscillators.
12.1.3 Quantization of the Coupled Modes¶
Chapter 12. Coupled-Mode Theory 12.1.3 Quantization of the Coupled Modes We can now write down our quantum description of the coupled cavities by simply identifying the field variables as operators, as we discussed in Chapter 8: E(+) \alpha (r, t) −\rightarrow − r
2ϵ0
(12.19) It is more convenient to write the modes in this case as E(+) \alpha (r, t) −\rightarrow − r
2ϵ0V\alpha f ′
(12.20) where the f ′ \alpha(r) are dimensionless mode functions, whose amplitude is of order unity (and of equivalent form for the two cavities), as we discussed in Chapter 10. Then we can interpret the rate equations (12.18) as Heisenberg equations for the field operators a1(t), a2(t), where the operators are also understood to be slowly varying (with the optical time dependence factored out). After solving for the operators, we obtain d dt ˜a1 ˜a2 = i∆ −ig12 −ig21 ˜a1 ˜a2 , (quantum coupled-mode equations) (12.21) where the twiddles on the operators remind us that they are in the rotating frame, where the e−i∆t time dependence is suppressed, and we have defined the quantum mode-coupling coefficients
r \omega2V1 \omega1V2 ,
r \omega1V2 \omega2V1 . (12.22) Note that we have dropped the spatial dependence of the modes, as consistent with the classical treatment. We will also assume the two cavities to be mode-matched (as is consistent with a two-mode treatment), which amounts to taking V1 = AL1 and V2 = AL2, where the area A is the same for both cavities, and thus V1/V2 = L1/L2. Further, the ratio of frequencies here is an artifact of perturbation theory—had we taken the self-consistent approach described above, we would have had a pair of eigenvalues, giving the frequencies
coefficients, to obtain
r L1 L2 ,
r L2 L1 , (quantum mode-coupling coefficients) (12.23) where, in view of the definitions (12.14), we can write g12 = ict21 2\sqrtL1L2 , g21 = ict12 2\sqrtL1L2 . (quantum mode-coupling coefficients) (12.24) The Heisenberg equations here are precisely those that arise from the general Heisenberg equation ˙˜a = −i ¯h[˜a, ˜H], (12.25) if we take the Hamiltonian H to be the sum of the free Hamiltonian in the rotating frame,
1˜a1, (free-field Hamiltonian, rotating frame) (12.26) and the interaction Hamiltonian, ˜H12 = ¯h g21˜a1˜a\dagger
, (12.27) also in the rotating frame. Note from Eqs. (12.24) that with our definition of the coefficients, this Hamiltonian is in fact not Hermitian. This comes from the usual conventions for the reflection and transmission coefficients
12.1.4 Cavity Driven by a Classical Field¶
12.1 Cavity QED¶
12 to obtain a Hermitian interaction Hamiltonian. This interaction Hamiltonian is our main result. In particular, note that by redefining the relative phase of the two modes (with the above phase convention in mind), we may always assume a real coupling coefficient, g21 = g12 ≡g \in R, ˜H12 = ¯hg ˜a1˜a\dagger
. (mode-coupling Hamiltonian, rotating frame) (12.28)
interaction Hamiltonian. Notice that the interaction here has exactly the same form as for the atom–field interaction in the Jaynes–Cummings model, as in Eq. (10.9), where we can identify a1 with the atomic dipole operator \sigma and a2 with the quantized field mode a. Note that we have implicitly made a rotating- wave approximation in the classical setup, since we ignored an coupling from E(−) in one cavity to E(+) in the other, since these couplings are far off resonance (as is consistent with the single-mode approximation). Of course, the coupling coefficient g is not related to the one-photon Rabi frequency from the Jaynes–Cummings model, except in the way it appears in the interaction Hamiltonian. Notice that if we transform out of the rotating frame, the free Hamiltonian is trivial, H0 = 0, (12.29) (free-field Hamiltonian, nonrotating frame) while explicit time dependence returns to the interaction Hamiltonian: H12 = ¯hg a1a\dagger
1e−i∆t . (mode-coupling Hamiltonian, nonrotating frame) (12.30) This means that we are now in the interaction picture with respect to the free evolution of the fields (which was equivalent to factoring out the free time dependence of the optical fields). Transforming into the Schrödinger picture yields the alternate free Hamiltonian
a\dagger 1a1 + 1
a\dagger 2a2 + 1 , (12.31) as well as an interaction Hamiltonian that is equivalent to the rotating-frame Hamiltonian: H12 = ¯hg a1a\dagger
. (12.32) In fact the rotating frame and the Schrödinger picture are equivalent here except for offsets of the bare energy levels. We now have essentially the same situation as in the classical case, but now the fields are represented by operators. a1 a2 g g We also have a symmetric coupling between the two fields, representing a Hermitian interaction Hamiltonian. 12.1.4 Cavity Driven by a Classical Field As an application of the above formalism, we will derive a model for driving a (slightly) lossy cavity with an external, classical field. In this case, cavity 1 will be our cavity, so we will change notations by a1 −\rightarrow a, and we will regard it as being driven by cavity 2. Cavity 2 has a classical field, and with many photons around we may neglect the fluctuations in photon number compared to the mean photon number, making
Chapter 12. Coupled-Mode Theory the replacement a2 −\rightarrow \alpha, assuming cavity 2 is in the coherent state \alpha. While technically the classical field is circulating in the right-hand cavity, this is not necessary—since we will eliminate all the parameters of the classical cavity except the circulating power, we can equally well think of the classical field as an incident traveling wave. a a We should then go into the interaction representation with respect to the free dynamics of cavity 2 to obtain the free Hamiltonian for cavity 1,
, (12.33) where we have changed notations \omega1 −\rightarrow \omega, and the interaction Hamiltonian becomes H12 = ¯hE
. (classical driving Hamiltonian for cavity field) (12.34) Here, we have defined the driving amplitude
(12.35) (classical-field driving amplitude) and we have changed notations \omega2 −\rightarrow \omegaL, where \omegaL is the ‘‘external drive frequency’’ for cavity 1, and again by appropriate definition of relative phase we may assume \alpha to be a real number (hence E is also real). Notice that our setup here is formally equivalent to the two-level atom interacting with a classical field, as we might guess from our remarks above. This correspondence carries through if we identify a with the atomic operator \sigma, \omega with the atomic resonance frequency, \omegaL with the frequency of the classical field, and 2E with the Rabi frequency. Also, the interaction Hamiltonian is of the form x cos \omegaLt, as we expect for a forced harmonic oscillator (here, forced by in the incident field). 12.1.4.1 Cavity Decay Rate
c|t21| 2\sqrtL1L2 . (12.36) Now we can define the power P of the ‘‘external’’ field in cavity 2, which is the energy per unit time, or the product of the photon number and photon energy divided by the round-trip time:
\taurt2 . (12.37) Then we can eliminate \alpha (dropping its phase to focus on the magnitude) to obtain E = r P\taurt2 ¯h\omega2 c|t21| 2\sqrtL1L2 = r P ¯h\omega2 \sqrtc|t21| \sqrt2L1 = r P ¯h\omega2 |t21| \sqrt\taurt1 , (12.38)
12.1.5 Cavity Decay¶
12.1 Cavity QED where we used \taurt1 = 2L1/c and \taurt2 = 2L2/c. We can then define the cavity decay rate2
\taurt1 , (12.39) (cavity decay rate) which defines the rate at which energy escapes cavity 1 (in the absence of cavity 2), being the intensity transmission coefficient divided by the round-trip time (to give the rate of energy transport through the
E = r \kappaP ¯h\omega , (12.40) (classical-field driving amplitude) where now the frequency \omega refers to the driven cavity (cavity 1). Of course, this coupling rate can be complex, with the phase of the input field \alpha. In this form, the coupling rate is independent of the details of the auxiliary cavity (cavity 2), and is thus a general result, assuming the input power is mode-matched with the output port of the cavity. Often, in a real Fabry–Perot cavity, there are multiple loss channels that contribute to \kappa, in which case an alternate value \kappa′ should be used in the above formula, which is the decay rate that the cavity would have, assuming that the input port of the driving field gives the only contribution to \kappa. It is common to define the finesse of a cavity by
s 1 −\sqrtPs , (12.41) (cavity finesse)
the cavity we have set up here). The finesse measures how ‘‘good’’ a cavity is, with a large finesse indicating low loss (and well-resolved resonance lines). In the good-cavity limit, we can then write F \approx 2\pi |t21|2 , (12.42) so that the decay rate becomes
2\pi
FSR F . (12.43) (decay rate in terms of finesse) In the last expression, we defined the free spectral range FSR := 1/\taurt, which is the frequency spacing between adjacent modes in the Fabry–Perot resonator. Note that the result here is valid for the asymmetric cavity that we started with, where the output coupler is the only source of loss (for example, the factor of 2 disappears for a symmetric cavity). 12.1.5 Cavity Decay The other application of the coupled-mode formalism that we will discuss is the quantum theory of the cavity decay. Since the interaction Hamiltonian (12.32) has the same form as the atom–field interaction in the rotating-wave approximation, we will make heavy use of the Weisskopf–Wigner treatment of atomic spontaneous decay in Chapter 11, and our calculation here will essentially just be mapping the cavity problem onto the atomic problem. First, we will again consider cavity 1 to be the cavity we are modeling, and cavity 2 will contain the ‘‘output field.’’ However, we will take the limit as cavity 2 becomes large, to get the free-space limit of the output field. The single-mode approximation for cavity 2 will break down, and we must consider coupling to many modes of cavity 2. Cavity 2 will be initially in the ground state, and always 2Here we are defining \kappa to be the energy decay rate of the cavity, in analogy with the population decay of the two-level atom. It is also common to define the decay rate of the field using the same symbol, which would differ by a factor of 2 from the definition here.
Chapter 12. Coupled-Mode Theory ‘‘approximately’’ in the ground state so that the radiated energy never re-enters the cavity. The interaction is given by Hint = ¯h X q gq aa\dagger
, (12.44) where a is the annihilation operator for cavity 1, and the aq are the annihilation operators for the output field modes. Thus, as in atomic spontaneous emission, the cavity will by damped by coupling to a ‘‘bath’’ of harmonic oscillators. We can write the coupling coefficient, combining Eqs. (12.39) and (12.24), as gq = c|t21| 2\sqrtL1L2 = |t21|
= r \kappa \taurt2 , (12.45) where \omega is the frequency of the decaying mode of cavity 1, \omegaq is the frequency of the qth output mode, and \kappa is defined as above (note that \kappa may depend on q via the frequency dependence of the mirror reflectance).
couples only to states with one less photon in the cavity. Thus it also suffices to consider the state restricted to the manifold given by the superposition
X q cq|n −1, 1q\rangle , (12.46) where again the 1q denotes a single photon present in the qth output mode. This analysis is valid for any other initial state, since any Fock state in the superposition is coupled to a separate set of states from the other Fock states. Writing down the equations of motion for the coefficients, transforming into the rotating frame, and decoupling the equations as in the atomic case leads to
X q n|gq|2 Z t dt′ ˜cn(t′)e−i(\omegaq−\omega)(t−t′)
\taurt2 X q Z t dt′ ˜cn(t′)e−i(\omegaq−\omega)(t−t′). (12.47) The spacing of the output modes is the free spectral range ∆\omegaq = 2\pi/\taurt2 of the output cavity, and thus in the limit where cavity 2 becomes large, we change the sum to an integral according to X q
Z \infty d\omegaoutf(\omegaout), (12.48) where \omegaout refers to the output frequency, so that
2\pi Z \infty d\omegaout Z t dt′ ˜cn(t′)e−i(\omegaout−\omega)(t−t′). (12.49) Carrying out the integrals as in the atomic case amounts to setting cn(t′) −\rightarrow cn(t), and then introducing an extra factor of \pi:
2 ˜cn. (12.50) (cavity-decay dynamics) This is consistent with |cn|2 decaying at the rate n\kappa, as we might expect from the classical definition (12.39) of \kappa. a k Thus, we have arrived at the situation where we have eliminated the right-hand cavity by pushing the end mirror away to infinity, and obtained irreversible decay from the remaining cavity into the continuum at rate \kappa.
12.2 Input–Output Formalism 12.1.5.1 Master Equation Still proceeding along the lines of the atomic case, from the above decay equation, the equation of motion for the nth state population is
(12.51) When tracing over the output field states, the only state coupled to |n\rangle is |n −1\rangle , and thus, the n −1 level must take up the decayed population
(12.52) Finally, transforming out of the rotating frame, using the appropriate combination of coefficients, and tracing out the field gives the equation of motion for the coherence
\rho(n)(n−1). (12.53) Our setup was for short times, but in the Markovian approximation (where the emitted field does not act back on the cavity), we simply evolve for a short time and repeat the argument, so that these equations of motion are always valid. Of course, then we must consider all couplings of the above forms between the density matrix elements. Then the equations of motion are precisely those generated by the Lindblad-form master equation
(12.54) (cavity-decay master equation) where again, ignoring the zero-point energy offset,
(12.55) and the Lindblad superoperator is
. (12.56) Recall that the operator form of the cavity-decay master equation has the same form as for atomic sponta- neous emission (i.e., the optical Bloch equations), under the identifications \sigma −\rightarrow a, \omega0 −\rightarrow \omega, and \Gamma −\rightarrow \kappa. 12.2 Input–Output Formalism The input–output formalism3 is an extension of the above formalism for treating the evolution of systems in the Heisenberg picture, particularly when coupled to a continuum. The difference is that we will now keep explicit track of the inputs and outputs (in the case of a cavity, the input and output fields) via Heisenberg-picture operators. To set this up for a cavity of resonance frequency \omega, we begin with the interaction (12.44) Hint = ¯h X q gq aa\dagger
, (12.57) where the coupling to the qth mode outside the cavity is gq = r \kappa \taurt2 , (12.58) 3M. J. Collett and C. W. Gardiner, ‘‘Squeezing of intracavity and traveling-wave light fields produced in parametric am- plification,’’ Physical Review A 30, 1386 (1984) (doi: 10.1103/PhysRevA.30.1386); C. W. Gardiner and M. J. Collett, ‘‘Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,’’ Physical Review A 31, 3761 (1985) (doi: 10.1103/PhysRevA.31.3761); C. W. Gardiner and P. Zoller, Quantum Noise, second enlarged edition (Springer, 2000); M. Ley and R. Loudon, ‘‘Quantum theory of high-resolution length measurement with a Fabry–Perot inter- ferometer,’’ Journal of Modern Optics 34, 227 (1987); R. Graham, ‘‘Quantum Langevin equation and input-output fields for arbitrary linear response,’’ Zeitschrift für Physik B 76, 265 (1989) (doi: 10.1007/BF01312694); D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994), Chapter 7.
12.2.1 Quantum Langevin Equation¶
Chapter 12. Coupled-Mode Theory where again \kappa is the cavity decay rate (which could depend on q via the frequency dependence of the output coupler), and \taurt2 is the round-trip time of the ‘‘external cavity,’’ in which the exterior field is quantized. Before, we passed over to the continuum limit after setting up a calculation, but now it will be convenient to do so right away. The idea was that we made the replacement X q |gq|2 −\rightarrow \taurt2 2\pi Z \infty
\taurt2 = Z \infty
2\pi . (12.59) This is equivalent to making the replacement gq −\rightarrow p \kappa/2\pi when passing the sum over to an integral over the external-mode frequencies \omega′, P q −\rightarrow R \infty 0 d\omega′. Making these replacements in the interaction Hamiltonian, Hint = ¯h \sqrt 2\pi Z \infty d\omega′p
, (cavity coupling to external modes) (12.60) where in the continuous limit, we have changed notation aq −\rightarrow b(\omega′) (b for the external ‘‘bath’’ modes), and we are explicitly indicating any frequency dependence of the decay rate, due to frequency dependence of the transmission coefficient (which must ultimately converge to unity as \omega′ −\rightarrow \infty). Again, this interaction assumes the rotating-wave approximation in omitting terms like ab(\omega′) and a\daggerb\dagger(\omega′). In what follows, it is convenient to extend the lower limit of integration to −\infty: Hint \approx ¯h \sqrt 2\pi Z \infty −\infty d\omega′p
. (cavity coupling to external modes) (12.61) This is justified since only bath frequencies \omega′ near the cavity resonance \omega should be important, and \omega is much larger than the rates associated with the decay interaction. The bath modes themselves satisfy the commutation relations
(12.62) (bath-mode commutation relation) which is the continuum version of [aq, a\dagger q′] = \deltaqq′. Also, we have the free-evolution Hamiltonian for the cavity, or the ‘‘system,’’
(12.63) (free cavity Hamiltonian) and the Hamiltonian for the external bath modes Hext = ¯h Z \infty
Z \infty −\infty
(free external bath Hamiltonian) (12.64) if we drop the zero-point contributions and extend the integral again over negative frequencies. 12.2.1 Quantum Langevin Equation Now for an arbitrary system operator c, the Heisenberg equation of motion is
¯h[c, H]. (12.65) In particular, for the cavity annihilation operator a, we have
¯h[a, Hsys] −i ¯h[a, Hint]
i \sqrt 2\pi Z \infty −\infty d\omega′p
(12.66)
12.2 Input–Output Formalism¶
¯h[b, Hext] −i ¯h[b, Hint]
r
2\pi a(t). (12.67) To solve this latter equation, we can transform to a rotating frame, \partial t h
= −i r
2\pi a(t) ei\omega′t. (12.68) Integrating from some past time t0 to t, we obtain
r
2\pi Z t t0 dt′ a(t′) ei\omega′t′, (12.69)
r
2\pi Z t t0 dt′ a(t′) e−i\omega′(t−t′). (12.70) Putting this into Eq. (12.66), we find
i \sqrt 2\pi Z \infty −\infty d\omega′p
2\pi Z \infty −\infty
Z t t0 dt′ a(t′) e−i\omega′(t−t′). (12.71) To proceed, we now make the Markov approximation by ignoring the frequency dependence of the decay rate,
(12.72) Strictly speaking, this cannot be true, but can be a good approximation over the frequency range of interest— the resonance linewidth in the case of the optical cavity. Then we can name the integral in the second term of Eq. (12.71) such that it becomes \sqrt\kappa ain(t), where
i \sqrt 2\pi Z \infty −\infty
(12.73) (input field operator) is the input field operator, which we will interpret in just a bit. The last term of Eq. (12.71) then becomes −\kappa 2\pi Z t t0 dt′ a(t′) Z \infty −\infty
2 a(t), (12.74) where we have used Z \infty −\infty
(12.75) and Z t t0
2 , (12.76) since the delta function is ‘‘split,’’ with half the contribution of the exponential factor in Eq. (12.75) being picked up here. Putting these pieces together, we have the Heisenberg equation
(12.77) (quantum Langevin equation) called the quantum Langevin equation, since as we will see, the second term represents damping, and the last term represents quantum noise. This last term also represents an input to the system, since it represents the influence of the external modes b(\omega′) at time t0 in the past on the present system operator a(t).
12.2.2 Output Field¶
Chapter 12. Coupled-Mode Theory 12.2.1.1 Evolution of the Mean To get a bit more insight into the Langevin equation (12.77), recall the Schrödinger-picture description of cavity damping represented by the master equation (12.54). The master equation implies the equation of motion
a\daggerAa −1
(12.78) for an arbitrary system operator A. Setting A = a,
(12.79)
this case. This is consistent with our expectation, since the reservoir that leads to this master equation is in
that the expectation value damps exponentially away:
(12.80) Obviously, though, the operator a must have more to it than just the expectation value—in particular, it has fluctuations about the mean. Otherwise, for example, the commutator [a(t), a\dagger(t)] would decay to zero, but it must be unity for all times. The input ain thus acts as a quantum noise that represents the fluctuations of a—fluctuations that are required in the presence of damping to ensure that commutators are preserved. This is one manifestation of the fluctuation–dissipation relation (Section 14.3.8.1). This interpretation of the input operators as noise terms is reinforced by computing the commutator of ain(t), h ain(t), a\dagger in(t′) i = 1 2\pi Z \infty −\infty d\omega′ Z \infty −\infty d\omega′′ h
0(\omega′′) i e−i\omega′(t−t0)ei\omega′′(t′−t0) = 1 2\pi Z \infty −\infty d\omega′ Z \infty −\infty
= 1 2\pi Z \infty −\infty
(12.81) so that h ain(t), a\dagger in(t′) i
(12.82) (input-operator commutator) The input operator thus appears to have the character of white noise, since its correlation function is a delta function (i.e., the power spectrum is flat). We will formalize this notion better after defining some fundamental concepts in stochastic calculus. 12.2.2 Output Field We can proceed again as before, but instead of integrating Eq. (12.68) from a past time t0 to t, we can integrate from t to a future time t1, to obtain
r
2\pi Z t1 t dt′ a(t′) e−i\omega′(t−t′), (12.83)
i \sqrt 2\pi Z \infty −\infty d\omega′p
2\pi Z \infty −\infty
Z t1 t dt′ a(t′) e−i\omega′(t−t′). (12.84) Making the Markov approximation, defining the output field
i \sqrt 2\pi Z \infty −\infty
(12.85) (output field operator)
12.2.4 General Heisenberg Equations¶
12.2 Input–Output Formalism and carrying out the integrals as before, we find the alternate, time-reversed Langevin equation
2 a(t) −\sqrt\kappa aout(t). (12.86) (time-reversed Langevin equation) Here, the output operator aout(t) represents the coupling of the system to future bath modes, and thus we interpret this to be the system output. However, since the influence of aout(t) is in the future, this equation represents the backwards evolution of the system—hence the negative damping term. Using essentially the same calculation leading up to Eq. (12.82), we find that the output-field commu- tator h aout(t), a\dagger out(t′) i
(12.87) (output-operator commutator) is also a temporal delta function. Not surprisingly, the output operator aout(t) has the same spectral properties as the input ain(t). 12.2.3 Input–Output Relation To relate the input and output fields, we start by integrating Eq. (12.70) over all frequencies (in the Markov approximation), Z \infty −\infty
Z \infty −\infty
r \kappa 2\pi Z t t0 dt′ a(t′) Z \infty −\infty
= − \sqrt 2\piiain(t) −i
2 a(t), (12.88) which we can rewrite as i \sqrt 2\pi Z \infty −\infty
\sqrt\kappa 2 a(t). (12.89) In particular, this means that the combination of operators on the right-hand side commutes with any system operator c(t), since it is independent of the state of the bath at the same time. Similarly, we can integrate Eq. (12.83) over all frequencies, Z \infty −\infty
Z \infty −\infty
r \kappa 2\pi Z t1 t dt′ a(t′) Z \infty −\infty
= − \sqrt 2\piiaout(t) + i
2 a(t), (12.90) which we can rewrite as i \sqrt 2\pi Z \infty −\infty
\sqrt\kappa 2 a(t). (12.91) Again, the combination of operators on the right-hand side commutes with any system operator c(t). Com- parting Eq. (12.91) with Eq. (12.89), we find the important relation
(12.92) (input–output relation) for the input, output, and system fields. 12.2.4 General Heisenberg Equations Above, we derived the Heisenberg–Langevin equations for the cavity annihilation operator a(t). However, it is also useful to derive Langevin equations for an arbitrary system operator c(t). We leave the derivation as an exercise; the results are
¯h [c, Hsys] − n
− \kappa
in(t) [c, a] o (quantum Langevin equation) (12.93)
12.2.6 Example: Reflections from a Cavity¶
Chapter 12. Coupled-Mode Theory and
¯h [c, Hsys] − n c, a\dagger −\kappa
− −\kappa
out(t) [c, a] o . (time-reversed Langevin equation) (12.94) These obviously reduce to the previous Langevin equations if c −\rightarrow a. 12.2.5 Causality Suppose again that c(t) is some Heisenberg-picture system operator. Since we integrate Eq. (12.93) forward in time to find the evolution of c(t) in response to ain(t), we can see that c(t) only depends on ain(t′) in the past (t′ < t). Expressed as a commutator, this statement is
(t′ > t). (12.95) Similarly, integrating Eq. (12.94) gives the influence of aout(t) on c(t) in the past, so
(t′ < t). (12.96) But returning to Eq. (12.92), we see that both ain(t) and aout(t) give rise to the same commutator (up to a minus sign), but on different sections of the time axis. We can thus combine Eqs. (12.92), (12.95), and (12.96) to obtain the general commutators
(12.97) (input–output commutators) These commutators are a general statement of causality with respect to the system and input/output oper- ators. The general strategy for the input–output formalism, then, is to specify the input field ain(t) for a system, use the Langevin equation (12.93) to determine the influence of the input on a system operator c(t) [in particular, a(t)], and then use the input–output relation (12.92) to determine the system output aout(t). 12.2.6 Example: Reflections from a Cavity Let’s consider the same one-sided cavity as usual, now in the input-output formalism.4 ao(t) aouto(t) aino(t) k Further, let’s resolve the cavity-field operator a(t) into frequency components a(\omega′) via the usual Fourier transform:
\sqrt 2\pi Z \infty −\infty dt a(t) ei\omega′(t−t0). (12.98) We can resolve the input and output fields in the same way; for example, examination of Eqs. (12.73) and
(12.99) becomes
(12.100) 4Here, we are following M. J. Collett and C. W. Gardiner, op. cit.
12.2.7 Example: Cavity Transmission¶
12.2 Input–Output Formalism or¶
h
i a(\omega′). (12.101)
with the result
h
i [aout(\omega′) −ain(\omega′)], (12.102) or h
i
h
i aout(\omega′), (12.103) so that
ain(\omega′). (input–output relation for one-sided cavity) (12.104) This equation demonstrates that in steady state, the intensity reflected at each frequency is the same as the incident intensity (there can be no net flux). However, there is a phase shift that vanishes at large detunings away from the cavity resonance (so that the reflection is mostly directly from the output mirror), and that becomes a \pi phase shift exactly at resonance, when the output field is entirely radiated by the cavity. Note that we also see here the explicit phase convention for reflecting from the output mirror, for which the field acquires a factor of −1 (+1 for a reflection from the same mirror, but from inside the cavity). 12.2.7 Example: Cavity Transmission Now if we consider a two-sided cavity,5 the generalization is straightforward: we have input and output operators for each dissipation channel. We will use the notation a\leftarrow in (t) and a\leftarrow out(t) for the input and output to the left-hand-side mirror, a\rightarrow in (t) and a\rightarrow out(t) for the input and output to the right-hand-side mirror. ao(t) aá outo(t) aá ino(t) aÜ ino(t) aÜ outo(t) k/2 k/2 We can generalize the Langevin equation (12.77) by simply adding both inputs,
2 a(t) − r\kappa 2 [a\leftarrow in (t) + a\rightarrow in (t)] , (12.105) begin careful to add damping terms for each input. Here we assume a symmetric cavity, with decay rate \kappa/2 for each mirror, and thus a total decay rate of \kappa. Switching to frequency space as before, h
i
r\kappa 2 [a\leftarrow
in (\omega′)] , (12.106) and stipulating that there is no input from the right-hand side (a\rightarrow
h
i
r\kappa 2 a\leftarrow in (\omega′). (12.107) Now noting that the input–output relation (12.92) holds for each pair of input and output operators sepa- rately (since they represent separate baths), a\leftarrow out(\omega′) −a\leftarrow
r\kappa
out(\omega′) −a\rightarrow in (\omega′), (12.108) 5Again, we are following M. J. Collett and C. W. Gardiner, op. cit.
12.2.8 Example: Driven Cavity¶
Chapter 12. Coupled-Mode Theory again with a\rightarrow
out(\omega′) to obtain h
i a\rightarrow
2 a\leftarrow in (\omega′), (12.109) so that the transmitted field becomes a\rightarrow
\kappa/2
a\leftarrow in (\omega′). (12.110) (transmitted field) This is a Lorentzian response with a full-width at half-maximum of \kappa, and perfect transmission on resonance, a\rightarrow
in (\omega). Then from Eq. (12.108), a\leftarrow
\kappa/2
a\leftarrow in (\omega′), (12.111) and thus the output field becomes a\leftarrow
a\leftarrow in (\omega′), (12.112) (reflected field) which is, of course, whatever didn’t make it through the cavity. Returning to Eq. (12.107), we find that the internal field is given by
" p \kappa/2
a\leftarrow in (\omega′). (12.113) (transmitted field) This is again a Lorentzian response, with a maximum amplitude at resonance of − p¶
terms of intensities, this is
\kappa|a\leftarrow
(12.114)
(free spectral range), so
2\pi 2|a\leftarrow
FSR . (12.115) (cavity buildup) The usual classical result is that the cavity intensity builds up to a factor of F/2\pi over the input intensity (for large F); the factor of FSR/2 here emphasizes the different normalization of the input/output operators compared to the cavity operator, since the cavity operator is defined in a bounded region, while the input- output operators are defined on unbounded ones. 12.2.8 Example: Driven Cavity Suppose we have a one-sided cavity driven by a classical field. Then the cavity Hamiltonian is
, (12.116) and the Hamiltonian for the external driving laser at frequency \omegaL is, from Eq. (12.34), HL = ¯hE
. (12.117) The quantum Langevin equation (12.77) is thus modified to include the free evolution of both Hamiltonians, with the result
(12.118)
12.2.9 Example: Atomic Motion in an Optical Cavity¶
12.2 Input–Output Formalism Defining the rotating-frame operator
(12.119) (with a similar definition for ˜ain), the Langevin equation becomes
(12.120) With the external bath in the vacuum state, the expectation value of this equation is
(12.121) In steady state, \partial t\langle ˜a\rangle = 0, so
iE
(12.122) (cavity steady state) But due to the presence of the vacuum input field in Eq. (12.120), the field a(t) fluctuates about this value. In fact, we have already seen in Section 5.6.1.3 that the steady state of the damped harmonic oscillator is a coherent state. That is consistent with what we see here, since if the steady state is |\alpha\rangle , then by definition the coherent state satisfies a|\alpha\rangle = \alpha|\alpha\rangle , so that \langle a\rangle = \langle \alpha|a|\alpha\rangle = \alpha. The effect of the input is then to superpose vacuum fluctuations on the classical motion \alphae−i\omegaLt of the cavity steady state. On resonance, then, the steady-state amplitude is given by
2FP
(12.123) (resonant cavity steady state) in terms of the input power P, where we used E = p \kappaP/¯h\omega from Eq. (12.40). Recall that for a coherent state |\alpha|2 is the mean photon number, since \langle \alpha|a\daggera|\alpha\rangle = |\alpha|2. Finally, note that rather than include a driving Hamiltonian, we could have specified the classical driving field as part of the input field. That is, if the quantum Langevin equation without the contribution from HL is
(12.124) Then making the replacement ain(t) −\rightarrow iE
(12.125) leads to the same driven Langevin equation (12.118). That is, the input field is in a coherent state, with amplitude \alpha/2 [the same factor of 2 that we saw in Eq. (12.115) crops up here as well], and again ain(t) represents the input-field vacuum fluctuations. 12.2.9 Example: Atomic Motion in an Optical Cavity Now consider a single atom in a resonantly driven cavity. The cavity Hamiltonian is again
, (12.126) the Hamiltonian for the external driving laser exactly on resonance, \omegaL = \omega, is again HL = ¯hE
, (12.127) the free evolution of the atom is HA = p2
(12.128) and the Jaynes–Cummings Hamiltonian (10.9) for the atom–field interaction is
, (12.129)
Chapter 12. Coupled-Mode Theory where
(12.130) gives the spatial dependence of the cavity QED coupling rate due to the standing-wave field in the cavity along the longitudinal direction. With this collection of Hamiltonians, the quantum Langevin equation (12.77) becomes
(12.131) which becomes
(12.132)
general quantum Langevin equation (12.93) gives
2 \sigma(t) − \sqrt \Gamma\sigmain(t), (12.133) where we used \sigma2 = 0 and [\sigma, \sigma\dagger\sigma] = \sigma. Also, we are including dissipation of the atom via spontaneous emission, where \Gamma is the decay rate into modes other than the cavity mode (which is approximately the free-space rate, so long as the cavity does not subtend a large solid angle from the atom). This dissipation has the same form as for the cavity, but with the replacements \kappa −\rightarrow \Gamma and a −\rightarrow \sigma. In the rotating frame, this becomes
2 ˜\sigma − \sqrt \Gamma˜\sigmain(t), (12.134) where we now have the usual detuning
(12.135) of the laser field from the atomic resonance. Working in the limit of large detuning |∆| ≫\Gamma, \kappa, g, the atom is only weakly excited, and thus we can take \sigma\sigma\dagger \approx 1 and \sigma\dagger\sigma \approx 0, since the expectation values of these operators represent the populations of the ground and excited states, respectively. Thus, we have
i∆−\Gamma ˜\sigma(t) −ig cos(kx)˜a(t) − \sqrt \Gamma˜\sigmain(t) (12.136) for the atomic evolution. 12.2.9.1 Adiabatic Approximation Now to make the adiabatic approximation. The idea is that the time scale for atomic motion is much slower than any of the time scales |∆|, \Gamma, \kappa, g representing the cavity or internal atomic dynamics. Since these other processes are explicitly damped at the (fast) rates \kappa/2 and \Gamma/2, respectively, we can assume that they are always in quasi-equilibrium with respect to the atomic motion, so that \partial t˜\sigma = 0 and \partial t˜a = 0. This is sensible since to a good approximation, the atom will not respond to the fast fluctuations in ˜\sigma(t) and ˜a(t). Further to this end, then, we can set the input noise terms in Eqs. (12.132) and (12.136) to zero. This is equivalent to replacing ˜\sigma(t) and ˜a(t) by their expectation values \langle ˜\sigma(t)\rangle and \langle ˜a(t)\rangle , again since, to a first approximation, the atomic motion should not respond to the fast fluctuations, so long as we are careful about the procedure. Thus, Eq. (12.136) becomes
∆cos(kx)˜a(t), (12.137) and Eq. (12.132) becomes
∆cos2(kx)˜a(t) −\kappa 2 ˜a(t) (12.138)
12.2 Input–Output Formalism¶
−iE ig2
(12.139) As in Section 12.2.8, we can define
\kappa (12.140) as the free-cavity coherent-state amplitude. Then expanding Eq. (12.139) to lowest order in ∆−1,
−i\alpha 1 + i2g2 \kappa∆cos2(kx) \approx −i\alpha 1 −i g2 \kappa∆cos2(kx) , (12.141) so that we now have adiabatic expressions for both the atom and cavity lowering operators. The atomic Hamiltonian (12.128) involves the combination \sigma\dagger\sigma, so it is also useful to derive an adiabatic relation for this operator. Note that for weak excitation, as we implicitly saw in coherent vs. incoherent scattering from the two-level atom in Section 5.7.2,
\approx
\sigma\dagger
(12.142) This is because in the (far-detuned) weak-excitation limit, the spontaneous-emission rate from Eq. (5.272) for classical-field excitation with Rabi frequency Ωis \GammaΩ2/4∆2. The correction to factoring the expectation value in Eq. (12.142) is O(∆−2), and thus ignorable in this regime. From Eq. (12.137)
∆2 cos2(kx)a\daggera, (12.143) since nothing changes upon transforming out of the rotating frame. The atomic Hamiltonian (12.144), once transformed into the rotating frame, becomes Heff = p2
(12.144) since this Hamiltonian correctly generates the first term in the Heisenberg equation (12.134). But with Eq. (12.143), we have the effective Hamiltonian Heff = p2 2m −¯hg2 ∆cos2(kx)a\daggera (effective Hamiltonian for atomic motion) (12.145) for the atomic motion in response to the cavity field a\daggera in the adiabatic approximation. Now considering the motion of a\daggera under Heff, which now replaces HA and HAF, we can use Eq. (12.93) to obtain
˜a −˜a\dagger − n
+ \kappa
in(t) a o = iE ˜a −˜a\dagger
h
in(t)˜a i (12.146) where we used [a\daggera, a] = [a\dagger, a]a = −a and [a\daggera, a\dagger] = a\dagger[a, a\dagger] = a\dagger, with the same relations holding in the rotating frame, where a\daggera = ˜a\dagger˜a. In the adiabatic approximation, we again set the noise terms to zero on average (noting that, e.g., a(t) and ain(t) are statistically independent), and set the time derivative to zero, with the result a\daggera \approx iE \kappa ˜a −˜a\dagger
˜a −˜a\dagger 2i . (12.147) Now using Eq. (12.141),
(12.148)
Chapter 12. Coupled-Mode Theory Then the effective Hamiltonian (12.145) becomes Heff \approx p2 2m −¯h\alpha2g2 ∆ cos2(kx), (effective Hamiltonian for atomic motion) (12.149) which represents the average effective potential seen by the atoms. Of course, the operator nature of Eq. (12.145) shows that there will be fluctuations in the effective potential due to fluctuations in the cavity field, which are in turn due to cavity decay. Physically, think of it this way: each time a photon escapes the cavity at a random time, \alpha2 jumps discontinuously downward by 1, and this amplitude smoothly recovers due to the driving field until the next random jump.
12.3 Exercises 12.3 Exercises Problem 12.1 Derive the decay rate for an optical cavity in the high-finesse limit using Fermi’s Golden Rule. Assume that the decay occurs on the continuous family |n\rangle −\rightarrow |n −1, 1q\rangle of transitions. Problem 12.2 Derive the input–output Heisenberg equations of motion (12.93) and (12.94)
¯h [c, Hsys] − n
− \kappa
in(t) [c, a] o
¯h [c, Hsys] − n c, a\dagger −\kappa
− −\kappa
out(t) [c, a] o (12.150) for an arbitrary system operator c, for the system and bath interaction as we set up in class. Problem 12.3 Use the general Born–Markov master-equation formalism (Section 4.5) to derive the master equation for cavity decay (i.e., for a damped harmonic oscillator, coupled to a reservoir in the vacuum state). You should end up with the usual Lindblad master equation of the form
(12.151) To make things simpler, ignore the Lamb shift, and assume an equal coupling of the cavity to all reservoir modes (i.e., a frequency-independent decay rate). Problem 12.4 (a) Show for a damped cavity, where the cavity annihilation operator a satisfies the quantum Langevin equation
(12.152) that for a bath in the vacuum state, that
(12.153) (b) The above result might seem alarming, as it appears that operators are damping away to nothing. However, the noisy character of the input operator in the vacuum state compensates for the damping. To illustrate this, show explicitly that the commutator [a, a\dagger] is time-invariant in spite of the damping. Problem 12.5 Consider a single-mode optical cavity with mode annihilation operator a(t), with corresponding input and output operators ain(t) and aout(t), respectively, and decay rate \kappa. (a) Show that [a\dagger
in(t), ain(t′)]. What is the physical interpretation of this result? (b) Show that D a\dagger out(t) aout(t′) E
a\dagger(t) a(t′)
, assuming a vacuum input. What is the physical interpretation of this result?
physical interpretation of this result?