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Quantum Rabi Oscillation A Direct Test of Field Quantization in a Cavity Torben Müller University of Mainz From a talk for the seminar on quantum optics by Prof. Immanuel Bloch (Dated: June 23rd, 2004) Since Planck’s hypothesis, the quantization of radiation is a universally accepted fact of nature. However, another simple fact granted in all quantum field descriptions, i.e., the discreteness of the energy of the radiation stored in a cavity mode, has escaped direct observation up to some years ago. In 1995, M. Brune et al. (Département de Physique de l’Ecole Normale Supérieure, Paris) reached to observe the Rabi Oscillation of circular Rydberg atoms in vacuum and in small coherent fields stored in a high Q cavity. Their measured signal exhibited discrete Fourier components at frequencies proportional to the square root of successive integers. This provided a direct evidence of field quantization in the cavity. The weights of the Fourier components yielded the photon number distribution in the field. I. INTRODUCTION The study of the Jaynes-Cummings Hamiltonian, which describes the the ideal coupling of a two-level atom to a single quantized mode of the e.m. field, indicates that a signature of the discrete nature of the field quanta could be provided by the observation of a single atoms’s Rabi nutation in a weak radiation field. So, at first, we are going to treat the atom-field interaction fully quantum mechanically, then discuss the dynamics of the atomfield model for various states of the field and finally obtain an elementary picture of the Cummings collapse and revivals due to the quantum granularity of the field. In the last part of this article we will give a brief survey on the experiment with circular Rydberg atoms in small coherent fields stored in a high Q cavity done by M. Brune et al. at ENS in 1995. II. atom-field Hamiltonian consisting of three parts: The two-level unperturbed Hamiltonian Hatom , the freefield Hamiltonian Hf ield and the interaction Hamiltonian Hint : Htot = Hatom + Hf ield + Hint The two-level atom Hamiltonian is just given by Hatom = Hint ~ = −e~r · E (1) ~ is the electric field and e~r is the atomic dipolewhere E moment operator. The form of the interaction energy remains the same for ~ becomes the electric field operator quantized fields, but E Ê: 1 Hf ield = ~ωL (a† a + ) 2 (2) In order to describe the ideal coupling of a two-level atom to a single quantized field mode we need the total (4) (5) For the single mode field, the interaction Hamiltonian becomes Hint = ~(a + a† )(gσ+ + g ∗ σ− ) (6) where σ+ and σ− are the Pauli spin-flip matrices and the electric-dipole matrix element [1] g = − ℘εΩ sinKz 2~ (7) and µ εΩ = Hint = −e~r · Ê 1 ~ω0 |eihe| 2 where ω0 is the two-level frequency and |eihe| counts the number of atoms in the upper state |ei. The free-field Hamiltonian for a single mode can be described by the creation and annihilation operators a† and a and the light’s frequency ωL [1]: INTERACTION BETWEEN ATOMS AND QUANTIZED FIELDS Considering the interactions of atoms with optical fields semiclassically the interaction Hamiltonian between an atom and a classical field is given in the dipole approximation by (3) ~Ω ²0 V ¶ 21 (8) Without loss of generality for two-level systems at rest, we can choose the atomic quantization axis such that the matrix element g is real. 2 One of the basic approximations in the theory of twolevel atoms is the rotating wave approximation. We can understand how this approximation works with quantized fields by considering the various terms in the interaction energy (6): aσ+ corresponds to the absorption of a photon and the excitation of the atom from the lower state |gi to the upper state |ei. ¡ 1 1 ~ω0 |eihe| + ~ωL (a† a + ) 2 2 † +~g(aσ+ + a σ− ) (9) The unperturbed Hamiltonian H0 = Hatom + Hf ield satisfies the following eigenvalue equations: µ ¶¸ 1 1 H0 |e, ni = ~ ω0 + n + ωL |e, ni 2 2 · µ ¶¸ 1 1 H0 |g, ni = ~ − ω0 + n + ωL |g, ni 2 2 Ω = Htot = X Hn (12) n where Hn acts only on the manifold εn and is given in the {|e, ni, |g, n + 1i} basis by the matrix √ g n+1 ¡ ¢ n − 12 − ω0 2 (13) (14) (15) p (ω0 + ωL )2 + 4g 2 (n + 1) (16) The energy eigenvectors are |2ni = cosθn |e, ni − sinθn |g, n + 1i (17) |1, ni = sinθn |e, ni + cosθn |g, n + 1i (18) where the mixing angle θn is given by (14) and (15). We have seen in the preceding section that the various manifolds are √ g n+1 tan(2θn ) = ωL − ω0 (10) In this case |g, ni and |e, ni, the unperturbed Hamiltonian’s eigenstates, are so called atom-field states, i.e. a tensor product of the atom state and the field state. |g, ni means, that the atom is in the lower state and n photon are in the single mode. Accordingly, |e, ni: atom excited and n photons within the mode. The interaction energy Hint couples the atom-field states |e, ni and |g, n + 1i for each value of n, but does not couple other states such as |e, ni and |g, n − 1i (aσ− would, but is dropped in the RWA). Hence, we can consider the atom-field interaction for each manifold εn = {|e, ni, |g, n + 1i} independently and write Htot as a sum ω0 2 where we have introduced the quantized generalized Rabi flopping frequency · (11) − 1 E1n = ~nωL + Ω 2 1 E2n = ~nωL − Ω 2 a† σ− represents the absorption of a photon and the excitation of the atom. Htot = ¢ In order to get the eigenvalues of the total Hamiltonian, we can diagonalize this matrix and find the new eigenvalues: These combinations correspond to those kept in the rotating wave approximation. But those last two processes are not energy conserving and so dropped in this approximation. Consequently dropping the last two combinations, we obtain the total atom-filed Hamiltonian: 1 2 Hn = ~ √ g n+1 a† σ− conversely describes the emission of a photon and the de-excitation of the atom. aσ− correlates to the emission of a photon and the excitation of the atom from the upper state |ei to the lower state |gi. n+ (19) The states given by (17) and (18) are called dressed states, namely the eigenstates of the Hamiltonian describing the two-level atom interacting with a singlemode field. We refer to the eigenstates of the unperturbed Hamiltonian, i.e., not including the atom-field interaction as bare states. III. JAYNES-CUMMINGS MODEL - QUANTUM RABI OSCILLATION The dressed-atom picture developed in section II provides us with very useful physical insight into the dynamics of a two-level atom in a single field mode. Since the dressed states are the eigenstates of the two-level atom interacting with a single mode of the radiation field, we can use them to obtain the state vector of the combined system as a function of time. Writing the Schrödinger equation in the integral form |Ψ(t)i = e−iHtot t/~ |Ψ(0)i (20) 3 we insert the identity operator expressed in terms of the dressed-atom states |jni to find |Ψ(t)i = ∞ X 2 X e−iEjn t/~ |jnihjn|Ψ(0)i Rabi flopping in a fully quantized way and comparing the results to the corresponding semiclassical treatment [2], we find out that the equations of motion for the atom-field probability amplitudes have the same form. (21) n=0 j=1 where Ejn is given by (14) and (15). We have seen in the preceding section that the various manifolds εn are uncoupled. In matrix form and in an interaction picture rotating at the frequency (n + 21 )ωL , the dressed state amplitude coefficients inside one such manifold εn read C2n (t) C1n (t) = 1 e 2 iΩt 0 0 1 e− 2 iΩt C2n (0) (22) C1n (0) Example: In particular in resonance, what we have assumed in the last section and will assume for the following consideration, for a resonant atom initially in the upper level and n-1 photons within the mode 1 |Ψ(0)i = √ (|2ni + |1ni) 2 (23) IV. COLLAPSE AND REVIVAL √ The quantum Rabi flopping frequency g n + 1 explicitly shows that different photon number states |ni have different quantum Rabi flopping frequencies. Up to now, we have only considered field states with discrete photon numbers, so called Fock states. But usually, the field presents a dispersion of photon numbers. If it is thermal, the probability P(n) of finding n photons in a mode is exponential, while for a coherent field 2 |αi = e−|α| /2 X αn √ |ni n! n (28) it is Poissonian P (n) = |hn|αi|2 the time derivation is given by: 2 ´ 1 ³ |Ψ(t)i = √ |2nie−iE2n t/~ + |1nie−iE1n t/~ (24) 2 = e−|α| we obtain the probabilities ¡ √ ¢ |Ce,n−1 (t)|2 = cos2 g n + 1t (26) ¡ √ ¢ |Cg,n (t)|2 = sin2 g n + 1t (27) This show how the atom Rabi flops between the upper and the lower levels within a manifold εn , i.e., the atom interchanges an energy-quantum with the radiation field. Note: Rabi oscillation even should be observed, if the photon number n is equal to zero, i.e., Rabi oscillation in a vacuum field. We see that both the dressed-atom and the bare-atom approaches lead to Rabi flopping as they must, and in general anything you can study with one basis set you can study with the other. Furthermore treating X α2n n n! (29) where the average photon number is given by Transforming this state given in the dressed-atom basis into the bare-atom basis ³ ´ 1 |Ψ(t)i = |g, ni e−iE2n t/~ + e−iE1n t/~ 2 ³ ´ 1 + |e, n − 1i e−iE2n t/~ + e−iE1n t/~ (25) 2 /2 n̄ = |α|2 (30) The coherent state is characterized by its minimal uncertainty 4φ 4 n ≥ 1, i.e., minimal fluctuations of phase and of amplitude. Classical fields are usually described by coherent fields. In particular, we consider an initially excited atom interacting with a field initially in a coherent state. Combining the coherent state photon number probability (29) with the single-photon state result (26), we have the probability for an excited atom regardless of the field state: P2 = X P (n)|c2n |2 n = e−|α| 2 /2 X α2n n n! √ cos2 [g nt] (31) For sufficiently intense field and short enough times t ¿ |α|/g, this sum can be shown to reduce to P2 ' 2 2 1 1 + cos(2|α|gt)e−g t 2 2 (32) 4 Intuitively this result can be understood by noting that the range of dominant Rabi frequencies in (31) is from Ω = g[n̄−4n]1/2 to Ω = g[n̄+4n]1/2 and the probability in (31) dephases in a time tc such that t−1 = g[n̄ + 4n]1/2 − g[n̄ − 4n]1/2 ' g c (33) which is independent of n̄. Here we have used the property of a Poisson distribution 4n = n̄1/2 . Hence, regarding (32), the Rabi oscillations are 2 2 damped with a Gaussian envelope, e−g t , independent 2 of the photon number n̄ = |α| , a result sometimes called the Cummings collapse. This collapse is due to the interference of Rabi floppings at different frequencies. For still longer times, the system exhibits a series of revivals. Because the photon numbers n are discrete in the quantum sum (31), the oscillations rephase in the revival time nature of the field, so that the atomic evolution is determined by the individual field quanta. Eventually, the revivals, which are never complete and get broader and broader, overlap and give a quasi-random time evolution. It is rather surprising that while the coherent state is the most classical state allowed by the uncertainty principle, it leads to result qualitatively different from the classical Rabi flopping formula. In contrast, the very quantum mechanical number state |ni has the nice semiclassical correspondence. The number state and the classical field share the property of a definite intensity which is needed to avoid the interferences leading to a collapse. The indeterminacy in the field phase associated with the number state, but not with the classical field, is not important for Rabi flopping since the atom and the field maintain a precise relative phase in the absence of decay processes. In contrast, the coherent state field features a minimum uncertainty phase, but its minimum uncertainty intensity causes the atom-field relative phase to diffuse away. The remarkable fact is that these collapse and revivals have recently started to be observed experimentally in cavity QED experiments. V. FIG. 1: Cummings collapse and revival for a coherent field with n̄ = 10 photons [3] t−1 ' 4π/g r = 4πn̄[n̄ + 4n]1/2 − g[n̄1/2 tc (34) as illustrated in FIG. 1. This revival property is a much more unambiguous signature of quantum electrodynamics than the collapse: Any spread in field strengths will dephase Rabi oscillations, but the revivals are entirely due to the grainy EXPERIMENTS As we have seen, the study of the Jaynes-Cummings Hamiltonian indicates that a signature of the discrete nature of field quanta could be provided by the observation of a single atom’s Rabi nutation in a weak radiation field. But however, the discreteness of the energy of the radiation stored in a cavity mode has up to now escaped direct observation. Obviously, a detector more subtle than an ordinary linear photodetector counting clinks is required. But in 1995, M. Brune et al. (Département de Physique de l’Ecole Normale Supérieure, Paris) reached to observe the Rabi Oscillation of circular Rydberg atoms in vacuum and in small coherent fields stored in a high Q cavity. Their measured signal exhibited discrete Fourier components at frequencies proportional to the square root of successive integers. This provided a direct evidence of field quantization in the cavity. The weights of the Fourier components yielded the photon number distribution in the field [3]. The setup of this experiment, given in FIG. 2 , is cooled to 0.8 K. Rubidium atoms , effusing from oven O, are prepared by a time resolved process into the circular Rydberg state e (principle quantum number 51) in the box B. At a repetition rate of 660 Hz, 2 µs long pulses of Rydberg atoms start from B with a Maxwellian velocity spread (mean velocity 350 m/s). The atoms cross the cavity C made of two niobium superconducting mirrors (diameter 5 cm, radius of curvature 4 cm, mirror separation 2.75 cm). This cavity, whose axis is vertical, sustains the two TEM900 modes with orthogonal linear polarization and transverse Gaussian profiles. the lower frequency is tuned 5 FIG. 2: Sketch of the experimental setup [3] into resonance with the e to g transition between adjacent Rydberg states with principle quantum numbers 51 and 50 (51.099 GHz), see FIG.3. The other mode serves for stabilizing and fine tuning. The mode Q factor is 7 · 107 , corresponding to a photon lifetime Tcav = 220µs, which is longer than the atom-cavity interaction time. A very stable source S is used to inject continuously into the cavity a small coherent field with a controlled energy varying from zero to a few photons. The atoms are detected one by one after the cavity by state selective field ionization (detector D) and the transfer rate from e to g is measured (Ionization-energy for n=51 is lower than for n=50) [3]. In circular Rydberg atoms, the valence electron is confined near the classical Bohr orbit. These atoms have a long radiative lifetime (32 and 30 ms for e and g), which makes atomic relaxation negligible during the atom transit time across the apparatus, and are strongly coupled to radiation (very large dipole moment: 1250 u.a.). The m=49 to m=50 transition represents a two-level system (in a weak electric field). However, the preparation is somewhat complex, 53 photons: see FIG. 3 In order to get a time resolved signal the control of the atom-cavity interaction time t is essential by velocity selection based on optical pumping: see FIG. 4. The signal is recorded corresponding to three sequences of interaction time between 0 and 90 µs. Finally, the three parts are then combined and checked that they merge smoothly. The signals are presented in FIG. 5 [3]. FIGS. 5 (A) to 5(D) show the Rabi nutations for increasing field amplitudes. Figure 5(A) presents the nutation in cavity vacuum (with very small corrections due to thermal field effects). Four oscillations are observed. The signal exhibits the reversible spontaneous emission and reabsorption of a single photon in an initially empty cavity mode, an effect predicted by the Jaynes Cummings model, as we have seen before. When a small coherent field is injected [FIGS. 5(B) and 5(C)], the signal is no longer sinusoidal, as it would be for an atom interacting with a classical field. In FIGS. 5(C) and 5(D), after a first oscillation, a clear collapse and revival feature is observed. FIGS. 5(a) to 5(d) show the Fourier transform of the nutation signal, obtained after symmetrization with respect √ to√t = 0. Discrete peaks at frequencies ν=47 KHz, ν 2, ν 3, and even 2ν are clearly observable, revealing directly the quantized nature of the field up to three photons. The frequency ν is in good agreement with the expected value. The total area of the Fourier transform curve remains constant, as required by P(n) normalization. The time dependent signals are fitted √ by a sum of damped sinusoids, with frequencies ν n + 1, n varying from 0 to 5 [solid lines in FIGS. 5(A) to 5(D)]. The agreement is also very good. From the relative weights of the terms in these fits, the photon number probabilities are determined, shown in FIGS. 5(α) to 5(δ). When no field is injected [FIG.5(α)], this distribution fits the thermal radiation law (solid line) with very small average photon number n̄=0.06(±0.01), corresponding well with the value deduced from the cavity temperature. With an injected coherent field [FIGS. 5(β) to 5(δ)], there is a very good agreement with the experimental data and the Poisson law (solid lines), providing an acurate value of the mean photon number in each case: 0.40(±0.02), 0.85(±0.04) and 1.77(±0.15). This experiment can also be viewed as a measurement of the atom-cavity spectrum, deduced from the JaynesCummings Hamiltonian. The excited levels of this system are organized in doublets, separated by one field quantum. The splittings of doublets corresponding to √ √ increasing energies are precisely hn, hν 2 , hν 3 ,... . The Rabi nutation is thus a quantum beat signal, resulting from the coherent excitation and detection of linear superpositions of all these levels. The spectral component at frequency n, the only one to be excited if the field is in the vacuum state, reveals the splitting of the first manifold, already observed in direct spectroscopic investigations (vacuum Rabi splitting). This resonant experiment dramatically shows once more that circular Rydberg atoms are very sensitive probes of millimeter wave fields, able to measure not only the mean field intensity with subphoton sensitivity, but also to determine accurately its statistics [3]. [1 ]Elements of Quantum Optics, P. Meystre, M. sargent III (Springer, Third Edition, 1999) [2 ] Licht-Atom Wechselwirkung im Zwei-Niveau System, I. Bloch (Lecture Notes, University of Mainz, 2003) [3 ] Quantum Rabi Oscillation: A Direct Test of Field Quantization in a Cavity, M. Brune et al., Phys. Rev. Lett., 76, 11 (1996) [4 ] Cavity Quantum Electrodynamics Group, LKB, ENS, Paris 6 FIG. 3: Preparation of circular states and circular Rydberg states [4] FIG. 4: Velocity selection based on optical pumping [4] 7 FIG. 5: (A), (B), (C), and (D): Rabi nutation signal representing Pe,g , for fields with increasing amplitudes. (A) No injected field and 0.06(±0.01) thermal photon on average; (B), (C), and (D) coherent fields with 0.40(±0.02), 0.85(±0.04) and 1.77(±0.15) photons on average. The points are experimental [errors bars in (A) only for clarity]; the solid √ lines √ correspond to theoretical fits (see text). (a), (b), (c), (d) Corresponding Fourier transforms. Frequencies ν=47 KHz, ν 2, ν 3, and even 2ν are indicated by vertical dotted lines. Vertical scales are proportional to 4, 3, 1.5, and 1 from (a) to (d). (α), (β), (γ), (δ) Corresponding photon number distribution inferred from experimental signals (points). Solid lines show the theoretical thermal (α) or coherent [(β), (γ), (δ)] distributions which best fit the data. [3]