10. Cavity QED and the Jaynes-Cummings Model¶
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Chapter 10 Cavity QED and the Jaynes–Cummings Model Now we consider the simplest fully quantum model for the atom–field interaction: a two-level atom and a single mode of the electromagnetic field. As we will discuss, this model applies to an atom interacting with the field of an optical cavity in the ‘‘good-cavity’’ limit. 10.1 Single Cavity Mode The uncoupled Hamiltonian for a two-level atom and a single mode of the optical field is
, (10.1) where the ground state has zero energy, \omega0 is the atomic transition frequency, and \omega is the cavity resonance frequency corresponding to the field mode. The dipole form of the atom–field interaction Hamiltonian is
(10.2) where the atomic dipole operator is
=: dge
. (10.3) Here, \sigma := |g\rangle \langle e| is the atomic lowering operator, and dge := \langle g|d|e\rangle is the dipole matrix element for the atomic transition. The (Heisenberg-picture) electric field mode of the cavity from Eq. (8.56) is
r ¯h\omega 2ϵ0 f(r) a(t) + f∗(r) a\dagger(t) , (10.4) where f(r) is the normalized spatial mode profile, and a is the mode annihilation operator. Thus, the interaction Hamiltonian becomes HAF = − r ¯h\omega 2ϵ0
dge \cdot
. (10.5) Then defining the atom–field coupling energy
r ¯h\omega 2ϵ0 dge \cdot f(r) (10.6) (cavity QED coupling constant)
Chapter 10. Cavity QED and the Jaynes–Cummings Model (g is called the cavity QED coupling constant, and 2g is called the one-photon Rabi frequency, as we will see below), the interaction Hamiltonian becomes HAF = ¯h
. (10.7) At any given location r, we may choose the phase of the atomic dipole such that g is a real and positive. In this case, the atom–field Hamiltonian becomes HAF = ¯hg
. (10.8) In the rotating-wave approximation, we drop the energy nonconserving terms (corresponding to fast-rotating terms in the Heisenberg picture), as we discussed before in Section 5.1.5.2, so that HAF = ¯hg
, (10.9) thus keeping only terms corresponding to photon annihilation with atomic excitation, and atomic lowering with photon creation. Note that in view of the normalization Z
(10.10) if the mode is uniform over an optical cavity volume V (also called the quantization volume), then
\sqrt V . In this case we can write the coupling constant (10.6) in terms of the mode volume as
r \omega 2ϵ0¯hV , (10.11) where ˆ\epsilon is the polarization vector of the field mode at the location of the atom. Thus, in general, it is common to define the coupling constant
r \omega 2ϵ0¯hV , (10.12) and then write the spatial dependence as
(10.13) where f ′(r) is a dimensionless mode profile, with maximum modulus of order unity. The coupling constant increases with decreasing cavity volume: this reflects the fact that locally, the electric field for a single photon increases as the confinement of the photon increases. The total Hamiltonian is, dropping the vacuum-field energy, and regarding the atom to be fixed at position r,
. (Jaynes–Cummings model) (10.14) This Hamiltonian defines the Jaynes–Cummings model:1 it is the model for an atom interacting with a single, nearly resonant cavity mode within the RWA, ignoring any dissipation process such as spontaneous emission or any input or output from the cavity. 10.2 Dynamics To investigate the dynamics of the Jaynes–Cummings model, we will decompose the state in terms of the joint eigenstates of HA and HAF:
\infty X n=0 h cg,n|g, n\rangle + ce,n|e, n\rangle i . (10.15) 1E. T. Jaynes and F. W. Cummings, ‘‘Comparison of quantum and semiclassical radiation theories with application to the beam maser,’’ Proceedings of the IEEE 51, 89 (1963).
10.2 Dynamics Putting this form of the state vector into the Schrödinger equation
(10.16) and projecting with \langle e, n| and \langle g, n + 1| gives the coupled pairs of equations
\sqrt
\sqrt n + 1 g ce,n, (10.17) where we have used the following form of the field annihilation operator: a = \infty X n=0 \sqrt
(10.18) The above structure of the Jaynes–Cummings model is important: only pairs of eigenstates are coupled, and thus the Hamiltonian is block diagonal, in 2 \times 2 blocks, making it simple to diagonalize analytically.2 Physically, the meaning here is that an excited atom can emit a photon into the cavity and reabsorb it, but that is the extent of the energy transfer. (Of course, then the vacuum amplitude cg,0 is not coupled to any other amplitude, since no absorption can occur in the absence of light quanta.) The above pair of equations (10.17) is formally equivalent to the semiclassical equations of motion for the atom–field interaction (the Rabi-flopping problem in the rotating frame), with Rabi frequency 2g\sqrtn + 1 and detuning
(10.19) which is just the usual field detuning from the atomic resonance. Thus, we have already solved this problem. For example, an atom initially in the state |g, n + 1\rangle , coupled to a resonant field, has the populations
n + 1 gt = 1 1 −cos \sqrt n + 1 gt
n + 1 gt = 1 1 + cos \sqrt n + 1 gt . (10.20) (Rabi flopping) Thus, the Rabi oscillations for n + 1 energy quanta occur at angular frequency 2\sqrtn + 1 g.3 In particular, for a single photon, the Rabi oscillations occur at frequency 2g: hence g is referred to as the single-photon Rabi frequency (though note the difference in convention of a factor of 2). For an off-resonant field, the Rabi oscillation proceed at the generalized Rabi frequency ˜Ωn = p
(10.21) (quantized generalized Rabi frequency) as in the semiclassical Rabi problem. We can also consider this to be a crude model for spontaneous emission. The atom, initially in the state |e, 0\rangle , oscillates to the state |g, 1\rangle . This model has the rather odd result that a spontaneously emitted photon will be reabsorbed, because only a single frequency is present. For a general superposition of states, the Rabi oscillations occur for each pair of states at their respective Rabi frequencies. This can lead to complicated beating behavior between the many frequencies involved. For example, if we consider an initially excited 2It turns out to be possible to diagonalize the Hamiltonian also without the rotating-wave approximation, though the solution is considerably more complicated; see D. Braak, ‘‘Integrability of the Rabi Model,’’ Physical Review Letters 107, 100401 (2011) (doi: 10.1103/PhysRevLett.107.100401). 3The quantization and \sqrtn + 1 dependence of the Rabi frequency has been observed with Rydberg atoms by M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, ‘‘Quantum Rabi Oscillation: A Direct Test of Field Quantization in a Cavity,’’ Physical Review Letters 76, 1800 (1996) (doi: 10.1103/PhysRevLett.76.1800).
Chapter 10. Cavity QED and the Jaynes–Cummings Model atom, where the field is in a superposition of states, then we no longer expect perfectly harmonic dynamics. For example, if the initial field state is the coherent state
\infty X n=0 \alphan \sqrt n!
(10.22) then we find collapses and revivals of the atomic population.4 This phenomenon is illustrated in the plot below, which shows the excited-state population for an initially excited atom, and a cavity initially in a
\sqrt 10). gt ree n—oo= 10 We see the Rabi oscillations quickly collapse, but then recur later. If the mean photon number is larger,
\sqrt 20), the recurrences occur at later times, and more coherent recurrences are visible. gt ree n—o= 20 The decay of the Rabi oscillations still is not exponential, as we expect from the optical Bloch equations. The collapses and revivals are characteristic of a discrete spectrum, and in fact anytime we have a discrete spectrum, we have almost quasiperiodic behavior (when a finite number of states are significantly populated), 4The collapses and revivals were observed in a single-atom maser. See Gerhard Rempe, Herbert Walther, and Norbert Klein, ‘‘Observation of quantum collapse and revival in a one-atom maser,’’ Physical Review Letters 58, 353 (1987) (doi: 10.1103/PhysRevLett.58.353).
10.3 Dressed States and the Vacuum Rabi Doublet and we thus expect the state to eventually recur arbitrarily close to the initial condition. In this sense, spontaneous emission into a single field mode is reversible. The irreversible, exponential decay of the Rabi oscillations only occurs when the atom is coupled to a continuum of states, and we will show that this is the case later in Chapter 11. 10.3 Dressed States and the Vacuum Rabi Doublet We can also take advantage of the block-diagonal structure that occurs due to the RWA in the Jaynes– Cummings Hamiltonian to define dressed states for the coupled atom–quantized field system. Each of the 2 \times 2 blocks is again formally equivalent to the semiclassical Rabi problem, and thus the previous dressed-
and |g, n + 1\rangle are degenerate and coupled by a Rabi frequency 2g\sqrtn + 1. Thus, after diagonalizing the 2 \times 2 blocks, the eigenstates are the dressed states |+, (n)\rangle and |−, (n)\rangle , which have energies (n + 1)\omega \pm g\sqrtn + 1 (i.e., the splittings are 2g\sqrtn + 1). w = wº |e, 0Ò fi |+, (0)Ò |-, (0)Ò |+, (1)Ò |-, (1)Ò |e, 1Ò |g, 1Ò |g, 0Ò |g, 0Ò |g, 2Ò w = wº 2ooÆ o2oog 2og
higher energy than |g, n + 1\rangle for ∆< 0, and |g, n + 1\rangle having the higher energy for ∆> 0). The pairs of bare states are repeated every ¯h\omega in energy, corresponding to having an additional photon around. Also, as before, in the general case the dressed states are defined by the rotation-type superposition
(10.23) (dressed states) where the Stückelberg angles \thetan are defined as before by
∆
. (10.24) (Stückelberg angles) Notice that within the rotating-wave approximation, the ground state |g, 0\rangle is completely uncoupled and
splittings for adjacent pairs of dressed states are nearly equal, giving rise to the Mollow triplet in resonance fluorescence, as we discussed in Section 5.7.4.2. However, this picture breaks down for small photon numbers where the splittings depend strongly on n. In particular, if a single photon interacts with an unexcited atom, there are only two possible transitions in the dressed-state basis, giving rise to the vacuum Rabi doublet. This becomes manifest, for example, for spectroscopy of a cavity tuned to the atomic resonance, as a doublet in the cavity transmission spectrum as a function of the frequency of a weak input field.5 5The vacuum Rabi splitting or ‘‘normal-mode structure’’ was observed for a single atom (the same atom throughout the entire spectral measurement) in a high-finesse cavity by A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, ‘‘Observation of the Vacuum Rabi Spectrum for One Trapped Atom,’’ Physical Review Letters 93, 233603 (2004) (doi: 10.1103/PhysRevLett.93.233603); and P. Maunz, T. Puppe, I. Schuster, N. Syassen, P. W. H. Pinkse, and G. Rempe, ‘‘Normal-Mode Spectroscopy of a Single-Bound-Atom–Cavity System,’’ Physical Review Letters 94, 033002 (2005) (doi: 10.1103/PhysRevLett.94.033002).
10.3.2 Atom-Photon ‘‘Molecule’’¶
Chapter 10. Cavity QED and the Jaynes–Cummings Model 10.3.1 Photon Blockade Due to the \sqrtn + 1-dependence of the energy-level splittings, the atom-cavity system can act effectively as a highly nonlinear optical medium. One of the most striking demonstrations of this is the photon blockade effect.6 The idea here is that under the right conditions, only a single photon can occupy the cavity at once. Suppose that the incident field is tuned to one of the sidebands of the vacuum Rabi doublet (say the red sideband), for the case when the cavity and atomic resonances coincide. This corresponds to a probe frequency \omegap = \omega −g. |+, (0)Ò |-, (0)Ò |+, (1)Ò |-, (1)Ò |g, 0Ò 2oÆ o2oog 2og In this case, it is possible for a photon to enter the cavity, when the cavity state makes the transition |g, 0\rangle −\rightarrow |−, (0)\rangle . However, if another photon is to enter the cavity, the atom must make the transition to the state |−, (1)\rangle . But the splitting for the two-photon manifold is \sqrt 2 larger than for the one-photon manifold, and thus this second transition has a resonant frequency (2\omega − \sqrt
\sqrt 2 −1)g. Thus, the probe field is detuned by ( \sqrt 2 −2) g from the second transition. If g is large, much larger than the widths of these transitions (when we include dissipative effect), then the second transition is suppressed, and at most one photon occupies the cavity at once. The cavity output is thus antibunched, even for a coherent input light. 10.3.2 Atom-Photon “Molecule” Of course, the splittings of the dressed states are space-dependent, and this can lead to dipole forces when the cavity photon number is nonzero in steady state. In fact, for large g, the dipole forces can be appreciable even for a cavity photon number of around unity. The dipole forces are sufficiently strong that they can trap an atom inside the cavity. This is something really remarkable: a single atom is bound by the dipole force to the field due to a single photon. This is called the ‘‘atom-photon molecule,’’ since it represents a mechanically bound state of an atom and a photon.7 The key to getting such high coupling strengths is to make the cavity mode volume V very small, which from Eq. (10.12) we see makes g large: the cavities in these experiments were spherical-mirror Fabry–Perot cavities with mirror separations of only 10 µm. (They also had very high reflectance coatings, and achieved near-record finesses in the 105 range.) How does the atom get in there in the first place? Cesium atoms were cooled and trapped in a usual MOT, placed above the cavity, and then dropped. The density is low enough that only one atom at a time crosses through the cavity. You can actually see the atoms crossing through the cavity in real time, by tuning a weak probe laser to one of the vacuum-Rabi sidebands (normal modes or dressed-state transitions). 6L. Tian and H. J. Carmichael, ‘‘Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,’’ Physical Review A 46, R6801 (1992) (doi: 10.1103/PhysRevA.46.R6801); the term ‘‘photon blockade’’ was introduced and the effect clarified by A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, ‘‘Strongly interacting photons in a nonlinear cavity,’’ Physical Review Letters 79, 1467 (1997) (doi: 10.1103/PhysRevLett.79.1467). This effect was observed using a single atom trapped in a high-finesse microcavity by K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, ‘‘Photon blockade in an optical cavity with one trapped atom,’’ Nature 436, 87 (2005) (doi: 10.1038/nature03804). The name ‘‘photon blockade’’ is in analogy to the Coulomb blockade, where charge transport through small devices happens one electron at a time, since only one electron can occupy an intermediate structure. 7C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, ‘‘The Atom-Cavity Microscope: Single Atoms Bound in Orbit by Single Photons,’’ Science 287, 1447 (2000) (doi: 10.1126/science.287.5457.1447). In these experiments and others by the same group, the coupling rate is in the range of g/2\pi ∼120 MHz, compared to a cavity decay rate of \kappa/2\pi ∼40 MHz and a spontaneous emission rate of \Gamma/2\pi ∼5 MHz.
10.4 Refinements of the Model The splitting is zero when there is no atom in the cavity, and the probe is thus not transmitted. An atom crossing event is marked by a spike in the cavity transmission, at which time the probe intensity can simply be increased to trap the atom. 10.4 Refinements of the Model For completeness, we will give some modifications to the Jaynes–Cummings model that make it more realistic. For the moment, we will just state them without proof, since they are quite reasonable, and we will defer the derivations of these results until Chapter 12. The first modification that we can put in is atomic spontaneous emission, or the interaction of the atom with all the other modes, particularly outside the cavity. From the semiclassical treatment, we can simply tack on a Lindblad term to the master equation to obtain
(10.25) where the Hamiltonian H is given in Eq. (10.14). This is pretty straightforward, although since the cavity modifies the local vacuum modes, the decay rate \Gamma may not be the same as the free-space value. The enhancement of the spontaneous-emission rate by a resonant cavity is called the Purcell effect,8 but the spontaneous-emission rate can also be suppressed by an off-resonant cavity. Along the same lines, the cavity intensity also decays. The cavity is just a harmonic oscillator, and we can use a decay term of the same form as for spontaneous emission. The total master equation then becomes
(10.26) where \kappa is the decay rate of the cavity energy (it is also common to use the convention where \kappa is the field decay rate, which would be smaller by a factor of 2). Recall that we already analyzed the harmonic oscillator with this type of damping in Section 5.6.1.2, where we saw that it is consistent with the classical damped oscillator (here, the classical damped oscillator being the classical damped cavity). These damping rates can obviously have a strong effect on the cavity dynamics, particularly the Hamil- tonian dynamics that we analyzed above. For the Jaynes–Cummings model to give a good approximation to the true dynamics, the atom–cavity system must be in the regime of strong coupling, where g ≫\kappa, \Gamma. In this case the dissipation is relatively slow, and the dynamics are Hamiltonian for short times. Note that ‘‘strong coupling’’ is sometimes also used for g ≫\Gamma, where the atomic emission is primarily into the cavity mode and not into other vacuum modes. Finally, a source of energy is typically necessary for interesting atom–field interactions. Of course, the cavity can in principle be prepared in an arbitrary initial state, but damping will drive the system towards the vacuum steady state. It is also common to pump the cavity with an external classical field, to provide photons in steady state to drive the atom–field interaction. We can model this by adding a drive term to the Hamiltonian
+ ¯hE
, (10.27) where \omegaL is the frequency of the classical field, E = p \kappaP/¯h\omegaL, and P is the power of the driving laser. Note that this term has the same form as the semiclassical atom–field interaction in the optical Bloch equations. As we discussed before in Section 5.6.1.2, in the absence of the atom–cavity coupling, the field would settle down to a coherent state whose amplitude depends on E and the light–cavity detuning. 8E. M. Purcell, ‘‘Spontaneous Emission Probabilities at Radio Frequencies,’’ Physical Review 69, 681 (1946) (doi: 10.1103/PhysRev.69.674.2). This was first observed with Rydberg atoms in a superconducting cavity by P. Goy, J. M. Raimond, M. Gross, and S. Haroche, ‘‘Observation of Cavity-Enhanced Single-Atom Spontaneous Emission,’’ Physical Review Letters 50, 1903 (1983) (doi: 10.1103/PhysRevLett.50.1903).
Chapter 10. Cavity QED and the Jaynes–Cummings Model 10.5 Exercises Problem 10.1 A simple model of damping of an optical cavity is via the interaction of the cavity mode with a beam of ground-state two-level atoms,9 assuming that only one atom interacts with the cavity mode at a time, that the atoms interact briefly with the cavity mode, and that the rate at which atoms arrive is much faster than the cavity evolution. In this case, you will show that the reduced density operator for the cavity obeys the master equation
(10.28) where \rhoc(t) is the reduced density operator for the cavity. (a) Assume that the atom–cavity interaction occurs via the Jaynes–Cummings Hamiltonian. Assuming
operator for the cavity, compute an expression for \rho(t+\tau) to first order in \tau, and to second order in g\tau, since we will eventually take the limit as g becomes large. Assume that the cavity QED coupling rate g is constant during the interaction time \tau. Note: there are fewer terms to work out in the interaction picture, but the Schrödinger picture will work as well. (b) Then trace over the atomic state to obtain \rhoc(t + \tau). (c) Finally, assume that \tau is small, and that immediately after one atom leaves the cavity (after the interaction of strength g and time \tau), the next one immediately follows and does the same thing. Then take the limit \tau −\rightarrow 0 (with g −\rightarrow \inftyin some sensible way) and write down a differential equation for \rhoc(t). What is \kappa? Problem 10.2 A two-level atom initially in the superposition
\sqrt
(10.29) passes slowly across the mode volume of an optical cavity. Assume the cavity to contain photons in only one field mode, which is tuned far off the atomic resonance, and ignore things like spontaneous emission and cavity decay. Explain qualitatively why a measurement of the phase of the atom after the interaction acts as a measurement of the cavity photon number. Indicate how the phase shift should scale with the photon number, assuming a small atom–field coupling rate. Problem 10.3 In Problem 10.1, you worked out a simple model for the damping of a cavity due to a beam of two-level atoms. There, the resonant atom-cavity systems lead to energy dissipation in the absorptive regime. In this problem you will work out an example of cavity damping in the dispersive regime. Consider the same cavity, with a beam of two-level atoms crossing the cavity one at a time, treating the atom–cavity
(a) Recall that for very large detunings, we can identify the bare atom-cavity states |e, n\rangle |g, n + 1\rangle with their dressed counterparts. Use what you know about the semiclassical and Jaynes–Cummings dressed states to write down the ac Stark shifts for these states due to an atom–cavity coupling rate g. Show that your answer agrees with the ac Stark shifts we derived for the two-level atom interacting with a classical field. (b) As the atoms cross the cavity, the coupling rate g(t) varies with time (going from zero to a maximum and back to zero). Assume that g(t) changes slowly with time so that the dressed states are adiabatic 9For a literal realization of this setup for monitoring the state of a cavity field using a beam of Rydberg atoms, see Christine Guerlin, Julien Bernu, Samuel Deléglise, Clément Sayrin, Sébastien Gleyzes, Stefan Kuhr, Michel Brune, Jean-Michel Raimond, and Serge Haroche, ‘‘Progressive field-state collapse and quantum non-demolition photon counting,’’ Nature 448, 889 (2007) (doi: 10.1038/nature06057).
10.5 Exercises eigenstates, and then use the results of part (a) to argue that the atom–field interaction Hamiltonian can be replaced by the effective interaction
∆
. (10.30) (c) Suppose that each atom starts in the initial state
\sqrt
. (10.31) Show that under the effective interaction of part (b), the relative phase \theta changes according to the cavity state, and thus realizes a (nondemolition) measurement of the cavity photon number. Write down an expression for \delta\theta, assuming the cavity to be in state |n\rangle . (d) Finally, suppose one atom crosses the cavity in each time interval of duration \tau, with the above initial state, but with \theta random (what is the atomic density operator?). Assume \tau to be long enough that g(t) varies slowly, but fast compared to the cavity dynamics. Thus, you may trace over the atomic states and then formally take the limit \tau −\rightarrow 0 (keeping the lowest-order terms in the atom–field coupling necessary to obtain the simplest nontrivial result) to derive a master equation for the cavity state of the form
(10.32) Give a physical interpretation to the damping equations you find (especially regarding energy dissipa- tion).