Solid State Physics IV -Part II : Macroscopic Quantum Phenomena

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1 Solid State Physics IV -Part II : Macroscopic Quantum Phenomena Koji Usami (Dated: January 6, 015) In this final lecture we study the Jaynes-Cummings model in which an atom (a two level system) is coupled to a cavity field (a harmonic oscillator). The coupled system exhibits a hybridized anharmonic energy-level structure called the Jaynes-Cummings ladder. The splitting of the nominally degenerate lowest two excited states is due to one and only one photon exchanged between two agencies, an atom and a cavity, and is called the vacuum Rabi splitting. We shall see how these interesting quantum phenomena modify under the realistic circumstances with ubiquitous dissipations. III. TWO LEVEL SYSTEM AND CAVITY QED C. Cavity QED and circuit QED. Jaynes-Cummings model Let us consider an atom (a two level system) with energy hω A interacting with Purcell-enhanced electromagnetic mode in a cavity, whose angular frequency ω c is about the same as ω A. The celebrated Jaynes-Cummings model [1] is devised to depict this situation with an Hamiltonian ) ( H JC = hω }{{ Aˆσ z + hω } c â â + 1 atom }{{} cavity field i hg (ˆσ + â ˆσ â ), (1) }{{} interaction where hω µ c ϵ 0 V g h ωc = µ ϵ Ω 0 () 0 hv is called the vacuum Rabi (angular) frequency. Note that the 1D dipole moment µ in Eq. () and the D dipole moment of the circular Rydberg atom µ n are related by µ µ n. We can notice that the electric dipole interaction term in Eq. (1) connects 0 A n c state and 1 A n 1 c state. Within this manifold, we have [ 1 H n = h ω A + ( ) n 1 ωc ig ] n ig n 1 ω A + ( ) n + 1. (3) ωc For the resonant case in which ω A = ω c ω 0, the Hamiltonian Eq. (3) becomes [ nω0 ig ] n H n = h ig. () n nω 0 The eigenenergies are obtained to be with respective eigenstates E ± n = hω 0 n ± hg n, (5) ψ (n) ± = 1 ( 0 A n c i 1 A n 1 c ). (6) Electronic address: usami@qc.rcast.u-tokyo.ac.jp

2 The anharmonic energy level structure for atom-cavity system given by Eq, (5) is called the Jaynes-Cummings ladder, where the nominally degenerate energy eigenstates 0 A n c and 1 A n 1 c are hybridized to become non-degenerate energy eigenstates ψ (n) ± whose energies are split by hg n. In particular the splitting occurs even when n = 1, that is, the relevant manifold spanned by 0 A 1 c and 1 A 0 c. The energy splitting hg = hω 0 is called the vacuum Rabi splitting, with which the atom and the cavity field exchange one and only one photon at the rate of Ω 0. For the general case in which ω A ω c, the Hamiltonian Eq. (3) becomes [ nωc H n = h ig n ig n nω c + and the eigenenergies are ], (7) with eigenstates E ± n = hω c n ± h + g n, (8) (( ψ (n) ± = 1 ) A± ± + g n 0 A n c ig ) n 1 A n 1 c, (9) where A ± = + g n ± + g n. (10) 5. Weak coupling regime and strong coupling regime [] To be more realistic we shall now incorporate the dissipations both for the atom and the cavity field and see the conditions in which the spectacular quantum phenomena, such as the vacuum Rabi splitting, predicted by the Jaynes- Cummings model can be observed. By adding the extra terms responsible for the dissipations to the Jaynes-Cummings Hamiltonian Eq. (1) we have ( H = hω Aˆσ z + hω c â â + 1 ) i hg (ˆσ + â ˆσ â ), }{{} i h dω Γ A π i h dω κ π H JC ( ˆσ +ˆb(ω) ˆσ ˆb (ω) ) dω + π hωˆb (ω)ˆb(ω) (â ĉ(ω) âĉ (ω) ) dω + π hωĉ (ω)ĉ(ω), (11) where Γ A represents the atomic decay rate (the Einstein A coefficient) to the free-space continuum field ˆb(ω) and κ represents the cavity field decay rate to the continuum field ĉ(ω). Now let us suppose that the atomic state rests predominantly on 0 0 as is the case where all the relevant field amplitudes are very small and thus ˆσ z = 1. (1) Then what we have to do is to find the equations of motion for the cavity field â and the atomic dipole ˆσ. For â we have â = ī h [H, â] = iω câ + gˆσ dω κ ĉ(ω), (13) π and for ˆσ we have ˆσ = ī h [H, ˆσ ] = iω A â + g ˆσ }{{} z â dω Γ A π ˆb(ω) 1 = iω A â gâ Γ A dω π ˆb(ω). (1)

3 3 Since the bath modes ˆb(ω) and ĉ(ω) can be formally solved to be ˆb(ω, t) = e iω(t t 0 )ˆb(ω, t0 ) + Γ A t ĉ(ω, t) = e iω(t t 0)ĉ(ω, t 0 ) + κ t t 0 dτe iω(t τ)ˆσ (τ) (15) t 0 dτe iω(t τ) â(τ). (16) These are plugging into Eqs. (13) and (1) to have the coupled equations of motion in a rotating frame at a frequency ω 0 : ˆσ â = i (ω c ω 0 ) â + gˆσ κ â κĉ(t) (17) where ĉ(t) and ˆb(t) are the time-domain continuum (input) field operators: = i (ω A ω 0 ) ˆσ gâ Γ A ˆσ Γ Aˆb(t), (18) ĉ(t) = ˆb(t) = dω π ĉ(ω)e i(ω ω0)t (19) dω π ˆb(ω)e i(ω ω0)t. (0) Neglecting the input noise fields ĉ(t) and ˆb(t), we have the following coupled equations: [ ] [ ] [ ] â κ = (1 + iθ) g â ˆσ g Γ, (1) A (1 + i ) ˆσ where Θ = ω c ω 0 κ/ and = ω A ω 0 Γ A / are the cavity detuning parameter and the atomic detuning parameter, respectively. To gain some insights without mess let us consider the case where Θ = = 0. The matrix in Eq. (1) can be diagonalized for the respective hybridized eigenoperators with the eigenvalues (κ λ ± = κ + Γ ) A ΓA ± g, () where the real parts of the eigenvalues λ ± represent the decays while the imaginary parts represent the detunings from ω 0. We shall now examine some limiting cases. Weak coupling regime: First, consider the weak coupling regime g κ, Γ A. There are then two symmetric limits. The one is the broad cavity limit (bad cavity limit), κ Γ A, in which the eigenvalues becomes λ + Γ ) A (1 + g = Γ A (1 + C) (3) κγ A λ κ ) (1 g = κ (1 C), () κγ A where C = g κγ A is called cooperativity and is omnipresent whenever the two dissipative systems couple. Note that the cooperativity coincides with the Purcell factor: ( ) µ ω C = g hϵ 0 V = ( ) = 3 Q κγ A π V λ3 = F p. (5) ω Q ) ( π V ω µ π c 3 3 In the weak coupling regime λ + is more or less the eigenvalue of the atomic operator ˆσ and expresses the wellknown Purcell-enhancement of the spontaneous emission due to the presence of the cavity, while λ is more or less the eigenvalue of the cavity field operator â and expresses the lesser known atom-inhibited cavity decay. The other limiting case within the weak coupling regime is the broad atomic response limit, Γ A κ, with the eigenvalues ω hϵ 0V λ + κ (1 + C) (6) λ Γ A (1 C). (7)

4 Now λ + is more or less the eigenvalue of the cavity field operator â and expresses the atom-enhanced cavity decay, while λ is more or less the eigenvalue of the atomic operator ˆσ and expresses the cavity-inhibited spontaneous emission. Strong coupling regime: When the coupling strength g becomes greater than the cavity decay rate κ and the atomic decay rate Γ A, that is, g > κ, Γ A, the eigenvalues in Eq. () becomes complex values λ ± = κ + Γ A ± iω 0, (8) where Ω 0 = ( ) κ ΓA g. (9) Decay rates: Re Λ Π Hz Coupling strength: g Π Hz FIG. 1: The real part of the complex eigenvalues λ ± (the decay rates of the eigenoperators consisting of â and ˆσ ) as a function of the coupling strength g. Here κ/π = MHz and Γ A /π = 1 MHz. We can see at g = 50 khz the decay rates merged entering the strong coupling regime Detunings: Im Λ Π Hz Coupling strength: g Π Hz FIG. : The imaginary part of the complex eigenvalues λ ± (the detunings of the eigenoperators consisting of â and ˆσ ) as a function of the coupling strength g. Here κ/π = MHz and Γ A/π = 1 MHz. We can see at g = 50 khz the nominally degenerate detunings bifurcate and further increase of g causes the vacuum Rabi splitting, a hallmark of the strong coupling regime. Note that the dashed lines indicate the vacuum Rabi splitting in the case of no dissipations.

5 Figures 1 and depict the real and imaginary part of λ ± in Eq. (8) as functions of the coupling strength g, respectively. When the coupling strength g reaches the value κ Γ A / the nominally different decay rates of the eigenoperators are merged and become a unique one, Im[λ ± ] = κ Γ A, (30) while the nominally degenerate (zero) detunings bifurcate and further increase of g causes the vacuum Rabi splitting, a hallmark of the strong coupling regime where the spectacular quantum phenomena predicted by the Jaynes-Cummings model can be seen even with the dissipations! 5 6. Circuit QED It is apparent that the coupling strength g can be boosted by shrinking the cavity volume V. If we push the volume V towards the ultimately small one V λ x for the dipole moment of µ = e x with λ being the wavelength of the relevant field, we have the following limit on the coupling strength: ω g c = e x hϵ 0 λ x = e πϵ 0 hc ω = αω, (31) where α is the fine-structure constant. For the natural atoms x is roughly the Bohr radius ( 0.1 nm) and for the relevant optical transition λ is roughly 1µm and thus building such an ultimately small cavity is very hard, if not impossible. For the artificial atoms with Josephson junctions, on the other hand, x can be designed to be very large (typically 100µm). The relevant transition ω/π for the typical Josephson atom is around 10GHz (which should be far smaller than the superconducting gap energy) and λ is accordingly roughly 3 cm. These two facts puts the current micro-fabrication technology in a favorable position to build a (near-field) 1D transmission-line cavity which achieves the ultimate volume V λ x. We have been witnessing the spectacular developments of such circuit QED systems with an artificial Josephson atom and a microwave cavity. Report problems please send your report to me (usami@qc.rcast.u-tokyo.ac.jp) by 015//1 A. Chose one of the following themes, read the cited references (as well as others if it is needed), and write an essay on it. Josephson atom [3, ]: How can an artificial atom be constructed by an LC circuit with Josephson junctions? Ion trap [5, 6]: How can a cavity QED system be implemented with an ion trap? Collective enhancement [7 9]: How the magnetic dipole interaction, which is typically far smaller than the electric dipole interaction, can be boosted by the so-called collective enhancement? B. Write about what you think about my lecture (I will improve the next-year lecture upon these feedbacks). [1] S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, New York, 1997). [] H. J. Kimble, Structure and dynamics in cavity quantum electrodynamics in Cavity Quantum Electrodynamics, P. R. Berman ed. (Academic Press, Boston, 199).

6 [3] M. H. Devoret, in Les Houches Session LXIII, Quantum Fluctuations, pp (Elsevier, Amsterdam, 1997). [] M. H. Devoret, S. Girvin, and R. Schoelkopf, Ann. Phys. (Leipzig) 16, 767 (007). [5] J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, Phys. Rev. Lett. 70, 76 (1993). [6] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 81 (003). [7] R. H. Dicke, Phys. Rev. 93, 99 (195). [8] A. Imamoglu, Phys. Rev. Lett. 10, (009). [9] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. Lett. 113, (01). 6