Trabajo de Fin de Grado en Física Ultrastrong coupling between quantum light and matter Laura García-Álvarez Director: Prof. Enrique Solano Departamento de Química Física Facultad de Ciencia y Tecnología Universidad del País Vasco UPV/EHU Leioa, Junio del 01
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Acknowledgements I want to thank Prof. Enrique Solano for giving me the opportunity of participating in such a fascinating project, for his support and encouraging words. And, probably, one of the most important things, he has broadened my mind by showing how is the scientific work after the grade studies. My deep thanks to Dr. Guillermo Romero, who has been an essential person in this project. Thanks for teaching me so much about quantum technologies and interaction between light and matter and how to program in Matlab, and for being such a good and comprehensive supervisor, who has never hesitated to help me every time I had needed it. To the rest of the group: Simone, Urtzi, Unai, Jorge, Lucas, Antonio, Julen, Roberto and Daniel, for their help and for making me proud of belonging to it. To my friends who have been there since we started to study physics four years ago and for those whom I have met in the way: Andrea, Santi, Adri, Alba, Xabi, Lander, Egoitz and Jani, for making each day an enjoyable experience. From our discussions on physics theory to good laughs, these years with them have convinced me that the decision I took choosing this career was the correct one. Beyond physics, David, for his constant support and sense of humor, for calming me down in the worst moments and for being a good example of perseverance and effort. Finally, I will never be thankful enough to my family, for their invaluable support, patience and tolerance, and for trusting on me even when I did not. I would not have done this without all of them. 3
Contents Contents 4 1 Introduction 5 Two-level atoms 7.1 Atomic model of Bohr and two-level approximation................ 7. Spin model for a two-level system.......................... 8 3 Two-level atoms and a classical driving: Rabi model 10 3.1 Dipole approximation and r E Hamiltonian................... 10 3. A single two-level atom coupled to a single-mode classical field......... 11 3.3 Rotating-wave approximation RWA)........................ 13 4 Quantum nature of light 15 4.1 Brief discussion of field quantization........................ 15 4. Harmonic oscillator and a quantized single mode cavity: Fock states...... 17 4.3 Coherent states.................................... 18 4.4 Field quadratures................................... 0 5 Quantum Rabi model and Jaynes-Cummings model 1 5.1 The quantum Rabi model and the Jaynes-Cummings model........... 1 5. The uncoupled atom-field system.......................... 3 5.3 The coupled atom-field system........................... 4 5.4 The resonant atom-field coupling.......................... 5 5.4.1 Vacuum Rabi oscillations.......................... 6 5.4. Collapses and revivals induced by a coherent field............. 7 6 Ultrastrong and deep strong coupling regimes of the quantum Rabi model 9 6.1 Ultrastrong coupling regime of the quantum Rabi model............. 9 6.1.1 Experiments................................. 30 6.1. Simulations and applications........................ 34 6. Deep strong coupling regime of the quantum Rabi model............. 35 7 Conclusions 39 Bibliography 41 4
Chapter 1 Introduction The interaction processes between light and matter have always been a main subject of study in physics and occupy a special position in the attempts to understand nature, both classically and quantum mechanically. To describe these processes, we need to focus on the electromagnetic interaction that plays a key role in understanding the motion of electrons and nuclei, which are the constituents of matter at the electron-volt scale. It is also responsible of the atoms cohesion and allows to explain the absorption and emission of light. Hence, the study of the interactions between the electromagnetic field and matter is the foundation to describe the dynamics and structure of these systems. An important step forward for describing lightmatter interaction was made in 1960 with the invention of the laser. This extraordinary device has allowed us to manipulate radiation in experiments where large coherence and monochromaticity are required. In 1936, one of the first attempts to explain the results coming from experiments was the Rabi model RM), that describes the simplest interaction between light and matter [1]. In its semiclassical form, this just includes the dipolar coupling of a single-mode field and two atomic levels whose energy splitting is close to the mode frequency. Thus the problem is reduced to a pseudospin-1/ system driven by a classical radiation field. In most situations, the assumption of the field to be classical is valid. However, there are instances where the quantum aspects of radiation and the concept of photon are required to explain the experimental results. Cavity Quantum Electrodynamics [, 3], cavity QED in short, is a subfield of quantum optics that emerged in the 1970s. It allows us to study the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of light photons is significant. In the quantum version of the Rabi model, the application of the rotating wave approximation RWA) leads us to the Jaynes-Cummings model JCM), originally proposed in 1963 [4]. This toy model is a cornerstone in the theory of coherent interaction between a two-level system and a single bosonic field. In analogy with the semiclassical theory, the atom behaves in a good approximation as a two-level system. However, the field in the cavity is a quantum oscillator whose elementary excitations are photons. The discreteness of the field states affects dramatically the classical predictions and yields purely quantum effects, like the revivals of the atomic population inversion after its collapse, as well as the appearance of Jaynes-Cummings doublets. Up to now, the advantages of the JCM versus the quantum Rabi model were its integrability and suitability for almost all physical scenarios in light-matter interaction. However, the advent of new quantum technologies such as circuit QED [5, 6, 7, 8, 9, 10, 11, 1, 13] has allowed to reach new regimes of the coupling strength between light and matter, where the RWA is no longer applicable. In this sense, the most general description has to be done in terms of the quantum Rabi model, whose full analytical solutions were developed in 011 [14]. 5
The physics beyond the RWA has been studied recently in the so-called ultrastrong and deep strong coupling regimes USC/DSC) [15, 16, 0, 17]. These regimes are characterized by ratios between the coupling strength g) and the mode frequency ω) of g/ω 0.1 for the USC regime, and g/ω 1 for the DSC regime. Two key experiments have allowed us to access the ultrastrong coupling regime [18, 19]. The deep strong coupling regime has been theoretically explored since experiments cannot reach those parameter values. However, the quantum simulation of the quantum Rabi model has confirmed their predictions expected for the Hamiltonian governing this regime [1, ]. The aim of this work is to present the basic models describing the interaction between light and matter from the initial until the most recent ones. A historical trip will be made, passing from the classical to the quantum descriptions and analyzing the approximations in each case. Finally, we will present the newest USC and DSC regimes, which are essential for the development of quantum optics and new quantum technologies for information science. 6
Chapter Two-level atoms.1 Atomic model of Bohr and two-level approximation The atom is the fundamental building unit of each substance, and we will focus on it in order to describe the coupling between light and matter. Throughout the years, the view of the atom has changed, and new ideas and technologies have influenced the atomic model. Although the modern description of the atom comes from the quantum theory, we will focus in the atomic model of Bohr, since it contains the main features to study the light-matter interaction in the simplest way. In 1913, Bohr proposed his quantized shell model of the atom to explain how electrons can have stable orbits around the nucleus. He used atomic spectra to prove that electrons are placed in definite energy levels. This is the main feature of the theory, that in general is common for any quantum mechanical system or particle that is confined spatially. The coupling between light and matter can be treated thus, in a first step, by considering the quantized spectrum of an atom and the transitions that the electrons can make between the different levels. These transitions are carried out by emitting or absorbing electromagnetic radiation a photon) whose energy must be exactly equal to the energy difference between the two levels. The term light, here, refers to electromagnetic radiation of any wavelength, whether visible or not, which can be absorbed or emitted by the electrons in their transitions. The photon is the quantum of the electromagnetic interaction, and this quantum nature of light will be taken into account in following chapters. Now, if we examine the energy spectrum of atoms and compare the energy differences between the levels, we realize that the problem can be reduced by considering only one frequency of the light and the two levels that are connected by this energy. From the Bohr atomic model, we have that energy levels for hydrogen-like atoms are for nuclei with Z protons: E = Z R E n.1) where R E is a constant called the Rydberg energy and n can take on the values n = 1,, 3,.... It follows that: ) E = E f E i = Z 1 R E n 1 i n = ω..) f We notice that the anharmonicity of E for different levels let us restrict to just two levels with an energy difference ω, when we are using a laser of frequency ω. The other levels can be neglected because they are non-resonant with the frequency of the electromagnetic radiation and do not participate in the interaction. We have simplified the interaction between light and matter to the interaction between one mode of a certain frequency of the light and a two-level atom. 7
. Spin model for a two-level system The operator algebra of a two-level system can be described, without loss of generality, as a pseudo-spin S which components along an arbitrary direction in three-dimensional space can take only one of the two values ± /. This analogy will lead us to a useful geometrical representation of the Hilbert space of the system on a unitary sphere, known as Bloch sphere, see Fig..1. We notice that the most general observable of this system can be expressed as a linear combination with real coefficients of the unity operator I and of the three Pauli operators σ i = S i / i = x, y, z), which in the basis of the σ z eigenstates, are: σ x = 0 1 1 0 ) 0 i ; σ y = i 0 These operators satisfy the following commutation rules: ) 1 0 ; σ z = 0 1 )..3) [σ i, σ j ] = iε ijk σ k.4) where ε ijk is the Levi-Civita symbol, which is 1 if i, j, k) is an even permutation of 1,, 3), 1 if it is an odd permutation, and 0 if any index is repeated. The eigenvalues of the σ i operators are ±1. The eigenstates of σ z with eigenvalues +1 and 1 are 0 and 1, respectively. And the eigenstates of σ x and σ y can be expressed as linear combinations of the previous states: 0/1 x = 0 ± 1 )/ and 0/1 y = 0 ± i 1 )/. The most general traceless observable of a two-level system corresponds to a spin component along an arbitrary direction defined by the unit vector v with polar angles θ and φ Fig..1). It can be written in terms of the Pauli matrices as: ) cos θ sin θ e iφ σ v = cos θ σ z + sin θ cos φ σ x + sin θ sin φ σ y = sin θ e iφ. cos θ This observable has also eigenvalues ±1, with the corresponding eigenstates: ) ) θ θ 0 v = cos 0 + sin e iφ 1 ) ) θ θ 1 v = sin 0 cos e iφ 1.5) It must be noted that the most general spin state, a 0 + b 1, can be expressed as an eigenstate of the spin component along v with eigenvalue +1, where the direction is defined by the polar angles θ and φ which satisfy the relation tanθ/) e iφ = b/a. Now, we can introduce a geometrical representation which let us describe the Hilbert space of this system in a useful way. To accomplish this analogy we notice that the tip of the vector 0 v belongs to a sphere of radius unity, known as the Bloch sphere Fig..1). The orthonormal state 1 v in Hilbert space, corresponds to the point along the direction v, since 1 v = 0 v according to Eq..5). The Bloch sphere is a unit sphere, with each pair of antipodal points corresponding to mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the basis vectors 0 and 1, respectively; whereas 0 x and 0 y states are associated to points on the OX and OY axes, on the equator. The two-level atom that we are considering is thus equivalent to a spin- 1 by making the correspondence e 0 and g 1, where e and g are the excited and ground states of the atom, respectively. 8
Figure.1: Geometrical representation of the pure state space of a two-level quantum mechanical system through the Bloch sphere. We introduce the atomic raising and lowering operators: σ ± = 1 σ x ± iσ y )..6) In terms of the spin eigenstates along the OZ axis, they can be written: σ + = 0 1 ; σ = σ + ) = 1 0.7) The action of these operators is to make transitions between the states of the atom as follows: σ + 1 = 0 ; σ + 0 = 0 ; σ 1 = 0 ; σ 0 = 1..8) and they satisfy a fermonic commutation relation: {σ, σ + } = I..9) This relation comes from the fact that the atom carries at most one excitation. In the following chapters the spin- 1 algebra will be used in order to describe two-level systems. 9
Chapter 3 Two-level atoms and a classical driving: Rabi model This chapter is devoted to study the coupling of a two-level atom with a single mode of the electromagnetic field in the semi-classical approach. It is already known that a two-level atom description is valid if the two atomic levels are resonant or nearly resonant with the driving frequency field, while other levels are highly detuned. The problem is also drastically simplified by choosing only one mode of the driving field. This is a reasonable approximation since experimentally, there exist coherent sources of light, as a laser, where a single frequency with a narrow bandwidth can be selected, which is characterized by its high degree of spatial and temporal coherence. We will describe how the dipolar coupling plays a key role in the dynamics allowing the system to exhibit Rabi oscillations in the atomic populations. 3.1 Dipole approximation and r E Hamiltonian The radiation field of a single frequency is treated as a plane electromagnetic wave. The Hamiltonian that describes an electron of charge e and mass m interacting with an external electromagnetic field is H = 1 m [p ear, t)] + eur, t) + V r), 3.1) where p is the canonical momentum operator, A and U are the vector and scalar potentials of the external field, respectively, and V is an electrostatic potential provided by the atomic binding potential. Let us consider the problem of an electron bound by a potential V to a force center nucleus) located at r 0, and we shall reduce the problem by using the dipole approximation. The whole atom is immersed in a plane electromagnetic wave described by a vector potential Ar 0 + r, t), where r stands for the electron position. By using the dipole approximation, k r 1, where k is the wave vector the field, this vector potential can be written as Ar 0 + r, t) = At) exp[ikr 0 + r)] At) expik r 0 ). 3.) This is a reasonable approximation because atomic radio a 0, typically 10 9 m, is small compared to the wavelength λ of the field, a 0 λ 1, and one can neglect the spatial variation of Ar + r 0, t) in the Hamiltonian. In this case one can write { m [ i e } Ar 0, t)] Ψr, t) + V r) Ψr, t) = i t 10. 3.3)
If we work in the radiation gauge, that is Ur, t) = 0 A = 0, 3.4) Equation 3.3) can be simplified by applying the gauge transformation χr, t) = e Ar 0, t)r and defining a new wave function Φr, t) as ) ie Ψr, t) = exp Ar 0, t)r Φr, t). 3.5) Finally, we obtain the Schrödinger equation where i Φr, t) = [H 0 er Er 0, t)]φr, t), 3.6) H 0 = p + V r) 3.7) m is the unperturbed Hamiltonian of the electron. If we use E = Ȧ, the total Hamiltonian reads H = H 0 + H 1 = H 0 er Er 0, t), 3.8) is given in terms of the gauge-independent field E. This interaction Hamiltonian is the first term in the multipole expansion of the charge-field interaction. If the atomic dipole operator has a vanishing matrix element on the transition resonant with the field, the expansion must be carried out to higher orders in a 0 λ. Selection rules involving parity and angular momentum determine whether any of this terms has a non-zero matrix element between the levels of relevant transition. 3. A single two-level atom coupled to a single-mode classical field The simplest case that can be treated is a single two-level atom interacting with a singlemode field. This problem is the basis for preparing arbitrary quantum states in any physical implementation. From the theoretical point of view, a quantum state can be achieved by applying rotations on the Bloch sphere induced by classical fields, be resonant or quasi-resonant with the atomic transition. Let e and g represent the upper and lower level states of the atom, so they are eigenstates of the unperturbed part of the Hamiltonian H 0 with eigenvalues ω e and ω g, respectively. Then H 0 = ω g g g + ω e e e. 3.9) By defining ω 0 = ω e ω g and shifting the origin of the energy we can write in terms of Pauli matrices in Eqs..3): H 0 = ω 0 σ z. 3.10) Let us consider the classical time-dependent field Et) = ie 0 e r e iωrt+ϕ) e re iωrt+ϕ)), 3.11) that will interact with the atom. This classical field is characterized by its real amplitude E 0, its angular frequency ω r, e r the complex unit vector describing its polarization, and ϕ its phase. In the two-level basis spanned by vectors { e, g }, the interaction Hamiltonian H 1 = er Et) 3.1) 11
can be written as H 1 = de iϕ 0 e a g e + de iϕ 0 e a e g )Et), 3.13) where we have used the completeness relation g g + e e = 1. Assuming that e and g are levels of opposite parity, the odd-parity d = er dipole operator can be expressed in terms of the atomic basis states, and only the off-diagonal terms do not vanish since g d g = e d e = 0. In this case, one can write d by using the atomic raising and lowering operators σ + = e g and σ = g e, thus obtaining d = de iϕ 0 e a g e + e iϕ 0 e a e g ) = de iϕ 0 e a σ + e iϕ 0 e aσ + ), 3.14) where de iϕ 0 e a = g d e is the matrix element of the atomic transition, and e a stands for the complex unit vector describing the atomic polarization. Finally, the whole Hamiltonian describing the dynamics of the systems reads H = ω 0 σ z ide 0 ea e r e iϕ 0 e iωrt+ϕ) σ e a e re iϕ 0 e iωrt+ϕ) σ +e ae r e iϕ 0 e iωrt+ϕ) σ + e ae re iϕ 0 e iωrt+ϕ) σ +). 3.15) We will introduce the interaction picture in order to investigate the effects of the interaction, that will induce transitions between states g and e. We define the state Ψ I in the interaction picture as Ψ I t) = U 0 t) Ψ It), where U 0 t) = exp i ) H 0t 3.16) represents the evolution operator. For operators representing observables, in particular the time-dependent potential, we define the Hamiltonian in the interaction picture as V I t) = U 0 t) H 1 U 0 t). 3.17) The differential equation that characterizes the time evolution of a state in this picture, can be deduced by taking the time derivative of Eq. 3.16) with the full Hamiltonian, that is i t Ψ It) = e ih 0t/ H 1 e ih 0t/ e ih 0t/ Ψt), thus obtaining t Ψ It) = i V It) Ψ I t). 3.18) A formal solution of this equation is where U I t) = I i t 0 Ψ I t) = U I t) Ψ I 0) 3.19) dt 1 V I t 1 ) + i ) t t1 dt 1 dt V I t 1 )V I t ) +... 3.0) 0 0 is time evolution operator in the interaction picture, and this series is known as the Dyson series. The interaction Hamiltonian corresponding to Eq. 3.15) reads V I = ide 0 ea e r e iϕ 0 e iωr+ω 0)t+ϕ) σ e a e re iϕ 0 e iωr ω 0)t+ϕ) σ +e ae r e iϕ 0 e iωr ω 0)t+ϕ) σ + e ae re iϕ 0 e iωr+ω 0)t+ϕ) σ +), 3.1) 1
where the following relations have been used e i ω 0 t σz σ ± e i ω 0 t σz = σ ± e ±iω 0t. 3.) From the interaction Hamiltonian of Eq. 3.1), one can define the classical Rabi frequency, Ω r, that quantifies the strength of the interaction between the atom and the electromagnetic field Ω r = d E 0e ae r e iϕ 0, 3.3) where the phase ϕ 0 can be adjusted to make Ω r real positive. This parameter establishes the time scale at which the electron experiences coherent population inversion in the transition g e, known as classical Rabi oscillations. 3.3 Rotating-wave approximation RWA) Some terms in the Hamiltonian of Eq. 3.1) are negligible under certain conditions, that depend on the physical realization and frequencies we are dealing with. Experimentally, one observes that fast rotating terms proportional to expω 0 + ω r ) do not influence the result of measuring observables if ω 0 + ω r Ω r is satisfied see Fig. 3.1). This is the case of physical scenarios like cavity QED [, 3], trapped ions [3], and circuit QED [5, 6, 7]. Under this parameter regime, we can directly apply the rotating-wave approximation RWA) that neglect fast oscillating terms compared to the Rabi frequency Ω r. It is direct from the Dyson series in Eq. 3.0) that these terms will lead to a negligible contribution to the state amplitudes at time t. In this case, the total Hamiltonian after applying the RWA reads V I = i Ω r σ + e iωr ω 0)t+ϕ) Ω rσ e iωr ω 0)t+ϕ) ). 3.4) The atom-field interaction of Eq. 3.1) can be described by a time-independent Hamiltonian after a unitary transformation. We write H 0 = ω r σ z / + σ z /, where we introduce the atom-field detuning: = ω 0 ω r. 3.5) We use an interaction representation with respect to ω r σ z /, so the operators will change according to Eq. 3.). In this case, we obtain the Hamiltonian H = σ z ide 0 ea e r e iϕ 0 e iωrt+ϕ) σ e a e re iϕ 0 e iϕ σ +e ae r e iϕ 0 e iϕ σ + e ae re iϕ 0 e iωrt+ϕ) σ +). 3.6) The two terms proportional to exp±iω r t) are fast oscillating terms and can be neglected employing the RWA. The Hamiltonian in terms of the classical Raby frequency of Eq. 3.3) is: H = σ z i Ωr σ + e iϕ Ω rσ e iϕ), 3.7) where ϕ depends on the phase offset between the classical field and the atomic transition dipole. It can be tuned by sweeping the phase of the classical field. We can adjust the phase ϕ 0 in Eq. 3.3) to make Ω r real positive, and write H in terms of the Pauli matrices: H = Ω r σ n, 3.8) 13
a) b) c) 1 1 1 0.8 0.8 0.8 0.6 0.5 0.6 P e RWA) P g RWA) 0.6 0.5 P e NRWA) 0.4 0.48 0.48 0.5 0.5 0.4 P g NRWA) 0.4 0. 0. 0. 0 0 0.5 1 1.5 0 0 0.5 1 1.5 0 0 0.5 1 1.5 r t/ r t/ r t/ Figure 3.1: Numerical comparison of probabilities P e t) and P g t) calculated with the Hamiltonian of Eq. 3.3) approximated by using RWA and the total Hamiltonian of Eq. 3.15) for different values of Ω r. a) Ω r = 1, the RWA is valid. b) Ω r = 10, the RWA is still valid, but its approximation character is appreciated. c) Ω r = 100, the RWA dynamics diverge from the real behavior. For all cases the initial state is e, ω r = 00 and = 0.1. All frequencies are given in arbitrary units. with σ being the formal vector of components σ x, σ y and σ z, and: Ω r = + Ω r, 3.9) n = u z Ω r sin ϕ u x + Ω r cos ϕ u y Ω r. 3.30) Here u x, u y and u z are the unit vectors in the Bloch sphere space, with axes OX, OY and OZ, not to be confused with the real space unit vectors ˆx, ŷ and ẑ. The Hamiltonian of Eq. 3.8) describes the precession at the angular frequency Ω r of the spin around the axis along n, in the frame rotating at angular frequency Ω r around OZ. A far off-resonant field, Ω r, does not affect appreciably the atomic state since the spin projection along OZ is nearly constant, and there is no any energy transference between the electric field and the atom. As one gets closer to resonance, the direction n swings away from the polar direction, and the circular trajectory of the tip of the Bloch vector widens, bringing it periodically farther and farther from its initial position. At the resonant case, = 0, the Bloch vector rotates at frequency Ω r around an axis in the equatorial plane, which direction is determined by the phase ϕ. In particular, by choosing the phase ϕ = 0 and considering the resonant case = 0), the Hamiltonian of Eq. 3.8) can be rewritten as: H = Ω rσ y, 3.31) where σ y = σ + σ )/i. Moreover, by considering the resonant case and ϕ = π, the Hamiltonian now reads H = Ω rσ x, 3.3) where σ x = σ + + σ. In this manner, by controlling the phase ϕ we can make arbitrary local rotations on a single pseudo-spin particle. Figure 3.1 shows the dynamical behavior of the atomic population probabilities P e t) and P g t) of finding the atom in the excited and ground state, respectively), as a function of the dimensionless time τ = Ω r t/π, considering the Hamiltonian in Eq. 3.3) and the total Hamiltonian in Eq. 3.15). 14
Chapter 4 Quantum nature of light 4.1 Brief discussion of field quantization Quantum-mechanical properties of the electromagnetic field are not present in a classical treatment. We start quantizing this field by considering the classical field equations without charges and currents. The Maxwell equations are E = 0 B = µ 0 ɛ 0 E t E = B t B = 0 4.1) where µ 0 and ɛ 0 are the magnetic permeability and electric permittivity of free space, and µ 0 ɛ 0 = 1 c. Maxwell s equations are gauge invariant when no sources are present. We will choose the Coulomb gauge or the radiation gauge in Eqs. 3.4), where both fields E and B may be written in terms of a vector potential, Ar, t), in this way E = A, 4.) t B = A. 4.3) Substituting Eq. 4.3) into Eq. 4.1) we find that Ar, t) satisfies the wave equation Ar, t) = 1 c Ar, t) t 4.4) In the classical theory, the solution of this equation can be expressed separating the vector potential into two complex terms Ar, t) = k ck u k e iω kt + c k u k eiω kt ) 4.5) We deal with a discrete set of variables rather than the whole continuum because it is more convenient. We are allowed to do that because we describe the field restricted to a certain volume of space and expand the vector potential in terms of a discrete set of orthogonal mode functions, which are u k. These vector mode functions, which correspond to the frequency ω k, will satisfy the Helmholtz equation ) + ω k c u k r) = 0 4.6) provided the volume contains no refracting material. These functions are also required to satisfy the transversality condition, u k r) = 0 4.7) 15
Moreover, the mode functions form a complete orthonormal set, because the inner product of them is u k r)u k r)dr = δ kk 4.8) V The mode functions depend on the boundary conditions of the physical volume under consideration. We can choose periodic boundary conditions corresponding to travelling-wave modes or conditions appropriate to reflecting walls which lead to standing waves. From this one can verify that the solutions may be written u k = f k r)ê λ) e ik r, 4.9) where each component of the wave vector k takes discrete values, since a finite volume V is considered. We use a simple normalization, with u = 1 at the points where the field amplitudes are maximum, then it can be written in terms of a scalar function f k with maximum value 1 and a unit vector ê λ). This vector is the unit polarization vector, and by using Eq. 4.7) is required to be orthogonal to k. This way, there are two possible values of ê λ), so we can describe it by two polarization index λ = 1,. The frequency of each mode is determined by the dispersion relation ω k = c kx + ky + kz. We can rewrite Eq. 4.5) as Ar, t) = ) 1/ f k r) a k,λ e ik r e iωkt + a k,λ e ik r e iω kt ω k ɛ 0 V k,λ, 4.10) where the normalization factors have been chosen such that the amplitudes a k and a k are dimensionless. The mode index k in Eq. 4.10) describes the three Cartesian components of the propagation vector k and the mode index λ, the polarization index. Classically, the vector potential Ar, t) has been decomposed in finite Fourier series. The quantization of this system can be accomplished by associating operators to the variables. The vector potential Ar, t) must be real because it describes a photon, therefore this implies that the associate operator  must be hermitian, that is,  = Â. The vector potential in terms of operators is given by Âr, t) = ) 1/ ) f k r) â k,λ e ik r e iωkt + â k,λ e ik r e iω kt 4.11) ω k ɛ 0 k,λ Its hermiticity leads us to ) 1/ ) f k r) â k,λ e ik r ωkt) + â k,λ e ik r ω kt) = ω k ɛ 0 V k,λ ) 1/ ) f k r) â e ik r ωkt) ω k ɛ 0 V k,λ + â eik r ω kt) k,λ, 4.1) k,λ we find that â k,λ = â k,λ, finally obtaining a hermitian solution for the vector potential as Âr, t) = k,λ ) 1/ ) f k r) â k,λ e ik r ωkt) + â e ik r ω kt) ω k ɛ 0 V k,λ. 4.13) Since photons are bosons, the appropriate commutation relations to choose for the operators â k and â k are the boson commutation relations [â k,λ, â k,λ ] = [â k,λ, â k,λ ] = 0, [a k,λ, â k,λ ] = δ k,k δ λ,λ. 4.14) 16
The corresponding quantized form of E in terms of A can be obtained by substituting Eq. 4.13) in Eq. 4.): Êr, t) = i k,λ ) 1/ ωk ) f k r) â k,λ e ik r ωkt) â e ik r ω kt) ɛ 0 V k,λ. 4.15) This expression corresponds to the Heisenberg picture and provides a direct link between quantum and classical physics, but it is useful for the next chapters to introduce the electric field in the Schrödinger picture: Er, t) = ie 0 ur)a u r)a ), 4.16) where the normalization factor has been absorbed into E 0, with the dimension of an electric field. The dynamical behavior of the electric-field amplitudes is described by the quantized Hamiltonian of the free radiation field, which can be deduced by considering: H = 1 ) ɛ 0 E + 1µ0 B dr. 4.17) Substituting Eq. 4.15) for E and the equivalent expression for the magnetic field B, the Hamiltonian may be reduced by making use of transversality and orthonormal conditions in Eqs. 4.7) and 4.8) to: H = ω k a k a k + 1 ), 4.18) k where we have subsumed polarization and wave vectors labels into the single subscript k. We observe that the dynamics of the electric-fields amplitudes may be described by an ensemble of independent harmonic oscillators with frequencies ω k, therefore the quantum states of each mode may be analyzed independently of one another. The operators a and a that we have introduced are the photon annihilation and creation operators, and the Hamiltonian represents the sum of the number of photons in each mode multiplied by the energy of a photon in that mode, plus 1 ω k representing the energy of the vacuum fluctuations in each mode. 4. Harmonic oscillator and a quantized single mode cavity: Fock states A field mode in a cavity is treated as a one-dimensional harmonic oscillator, since we may restrict to a single frequency by choosing a suitable geometry of the quantum cavity. Lets solve then the Hamiltonian of Eq. 4.18) for this particular case: H = ω a a + 1 ). 4.19) The Hamiltonian has the eigenvalues ω n + 1 ), where n are the non-negative integers n = 0, 1,... ). The associated eigenstates to this non-degenerate spectrum are denoted as n and are known as Fock states or number states. They are eigenstates of the photon number operator N = a a such that: N n = n n. 4.0) The action of the operators a and a on the Fock states yields: a n = n n 1, a n = n + 1 n + 1. 4.1) 17
The ground state or vacuum state of the oscillator is defined by: a 0 = 0, 4.) and the remaining states can be generated from this one by successive application of the creation operator: n = a n n! 0 4.3) The number states form an orthonormal basis of the Hilbert space: n m = δ n,m, n n = I. 4.4) The energy of each n is the corresponding eigenvalue of H, E n = ωn + 1/), and may be interpreted as the energy of n quanta ω plus the energy of the vacuum state, ω/. It must be noticed that in the Hamiltonian of Eq. 4.18) representing the electromagnetic field, the energy n k ω k, since there is no upper bound to the frequencies in of the ground state is infinity, 1 the sum over electromagnetic field modes. This conceptual difficulty of quantized radiation field theory do not lead to any divergence in practice because experiments measure a change in the total energy and finite quantities result from a subtraction of infinities. These vacuum fluctuations also appear when the electromagnetic field is considered, since even though the ground state of each field mode has electric and magnetic fields with zero mean values, the energy densities are finite. This fact leads us to an energy argument which can be used to justify the normalization factor in the electric field introduced above in Eq. 4.15). The electromagnetic energy in the n-photon state is ωn + 1/), so it implies: n ɛ 0 E d 3 r n = ωn + 1/), 4.5) where the integral is over the whole space. Nevertheless, this condition can be simplified in our treatment of the field stored in a cavity, since we can define the effective mode volume as: V = ur) d 3 r 4.6) and the condition results in: The field amplitude is thus: n ɛ 0 E 0VN + 1) n = ωn + 1/). 4.7) E 0 = ) ω 1/, 4.8) ɛ 0 V which represents the r.m.s. electric field amplitude of the vacuum for the mode and depends only on the frequency and on the cavity geometry. 4.3 Coherent states The classical description of the field involves a well-defined amplitude and phase, so that the Fock states are not the most suitable representation, since they have zero average electric field and indefinite phase. The closest states to this classical picture are called coherent states and have an indefinite number of photons, which allows them to have a more accurately defined phase. The product of the uncertainty in amplitude and phase for a coherent state is the minimum allowed by the Heisenberg s principle. 18
0.14 0.1 0.1 p n) 0.08 0.06 0.04 0.0 0 0 5 10 15 0 5 30 Figure 4.1: Numerical estimation of the photon number statistical distributions. Coherent field with n = 3 photons on the average. n These coherent states can be obtained using the unitary displacement operator: where α is an arbitrary complex number. We can take advantage of the Glauber identities: Dα) = expαa α a), 4.9) e A+B = e A e B e [A,B]/, 4.30) which holds when both A and B commute with [A, B], in order to rewrite the displacement operator and combine displacements: Dα) = e α / e αa e α a, 4.31) Dα)Dβ) = Dα + β)e αβ α β)/. 4.3) It may be proved that the displacement operator satisfies: D 1 α) = D α) = D α), D0) = I, D α) a Dα) = a + α, D α) a Dα) = a + α. 4.33) From this properties can be derived that coherent states are eigenstates of the annihilation operator: a α = α α, 4.34) where the eigenvalue is a complex number since a is non-hermitian. The coherent states contains an indefinite number of photons, as we see in their Fock state representation α = n c n n. The c n coefficients are obtained by expanding the exponentials in the Eq. 4.31) in power series: α = e α n 19 α n n! n, 4.35)
and the photon number distribution, p α n) = c n, obeys a Poisson law: α α n p α n) = e n!, 4.36) which average photon number is n = α a a α = α. The coherent states are normalized but they are not orthogonal, unless α β 1: 4.4 Field quadratures β α = e α β. 4.37) In quantum physics, the mechanical harmonic oscillator is solved expressing the dimensionless position and momentum operators in terms of a and a, in the following way: X 0 = a + a and P 0 = i a a. 4.38) In analogy with this case, one introduces the two Hermitian dimensionless linear combinations of creation and annihilation operators, known as field quadratures operators: X φ = ae iφ + a e iφ. 4.39) It is followed that X 0 and X π/ correspond, in the case of a mechanical oscillator, to X 0 and P 0. Hence the two-dimensional phase space coordinates may be taken in a general way as {X φ, X φ+π/ }. Any two-orthogonal field quadratures, which are associated to canonically conjugate quantities, satisfy the following commutation and uncertainty relations: [X φ, X φ+π/ ] = i and X φ X φ+π/. 4.40) It is possible to describe all quantum states of a system in a phase space. Each point in this phase space corresponds to a unique state of a system, and a phase diagram may be obtained by plotting the quadratures of a system against each other as functions of time. 0
Chapter 5 Quantum Rabi model and Jaynes-Cummings model In this chapter, the Jaynes-Cummings model will be introduced as a result of the Rabi model where we have used the rotating-wave approximation RWA). This model describes how quantization of the radiation field affects the predictions for the evolution of a state of a two-level system, which is the typical situation in cavity quantum electrodynamics, when a single atom is coupled to a cavity mode. It shows purely quantum effects as the revival of the atomic population inversion after its collapse [4] or the Lamb shift [5]. 5.1 The quantum Rabi model and the Jaynes-Cummings model We consider the coupling of a two-level system, modeled as a spin- 1, with the radiation field described as a quantum harmonic oscillator. The complete Hamiltonian of the atom-cavity interaction is the quantum version of the semiclassical Hamiltonian in Eq. 3.15) and can be expressed as: H = H 0 + H ac = H 0 ˆd Ê, 5.1) where H 0 is the free Hamiltonian of the atom and the cavity and H ac is the interaction term. This last term is the quantized form of the semiclassical Hamiltonian in Eq. 3.1), which can be written by taking into account the dipole operator of Eq. 3.14) and the quantized field mode of Eq. 4.16): H ac = de iϕ 0 e a σ + e iϕ 0 e aσ + ) ie 0 e c a e ca ). 5.) The expansion of the scalar product involves four terms and a coupling constant proportional to de 0, which depends on the strength of the interaction between the atomic dipole and the field mode. The Hamiltonian describing the whole system is then: H = H 0 + H 1 = ω 0 σ z + ω c a a + 1 ) ide 0 e iϕ 0 e ae c σ + a e iϕ 0 e a e cσ a + e iϕ 0 e a e c σ a e iϕ 0 e ae cσ + a ), 5.3) where ω 0 and ω c are the frequencies associated to the atomic transition and the field mode respectively. This is the general way to describe quantized matter-radiation interaction, which goes under the name of Rabi model. As we could see in section 3.3 under certain conditions we can employ the RWA to discard anti-rotating terms, σ a and σ + a. The first correspond to a transition from the upper level e to the lower level g, together with the annihilation of a photon. The second describes 1
the reverse process: emission of a photon by an atom performing a transition from g to e. Under the conditions of the approximation these terms correspond to highly non-resonant processes, which can be appreciated if the system dynamics is described in the interaction picture considering Eqs. 3.16-3.0). In this case: V I t) = ide 0 e iω c ω 0 )t e iϕ 0 e ae c σ + a e iωc ω 0)t e iϕ 0 e a e cσ a +e iωc+ω 0)t e iϕ 0 e a e c σ a e iωc+ω 0)t e iϕ 0 e ae cσ + a ). 5.4) The RWA allows to neglect the terms multiplied by the factors e ±iωc+ω 0)t which are fast rotating in time. A more mathematical approach for the RWA is to consider the Dyson series expansion of the time evolution operator in the interaction picture of Eq. 3.0), which rules the time evolution of the state vector in the interaction picture. Considering the first order in the series expansion: U I t) I de 0 e iϕ 0 e ae c σ + a e iωc ω 0)t + e iϕ 0 e a e ω c ω cσ a eiωc ω0)t 0 ω c ω 0 ) +e iϕ 0 e a e c σ a e iωc+ω 0)t ω c + ω 0 + e iϕ 0 e ae cσ + a eiωc+ω0)t ω c + ω 0 5.5) The integration provides terms as ω c + ω 0 ) 1 connected to the operators σ a and σ + a, whereas terms like ω c ω 0 ) 1 are linked to σ + a and σ a. The major contributions to the time evolution operator come from the last terms, which are dominant under the resonance condition ω c ω 0 and the assumption that de 0/ ω c+ω 0 1. With the RWA, the interaction term of the Hamiltonian reduces to: H ac = i Ω 0σ + a Ω 0σ a ), 5.6) where the vacuum Rabi frequency Ω 0 is introduced, formally identical to the semiclassical Rabi frequency in Eq. 3.3), where the amplitude E 0 is now given by Eq. 4.8). Finally, the fully quantized Jaynes-Cummings model concerns a two-level system interacting with one radiation mode. Under the dipole approximation and the rotating wave approximation, this system is described by the following Hamiltonian: H = ω 0 σ z + ω c a a + 1 ) i Ω 0σ + a Ω 0a σ ). 5.7) We assume, for the sake of simplicity, that e ae c e iϕ 0 is real and positive, therefore Ω 0. If this were not the case, we make it real positive by multiplying it by a proper phase term e iϕ, and the two terms of H ac are then, as in Eq. 3.7), multiplied by e ±ϕ. The frequency Ω 0 quantifies the strength of the atom-field coupling. Hence, the atom-cavity coupling reduces to: H ac = i Ω 0 σ+ a σ a ). 5.8) The two terms describe a single quantum exchange between the coupled systems, a two-level atom and a field mode, where one of them the field) can store an arbitrary number of quanta, while the other the atom) saturates with a single excitation.
5. The uncoupled atom-field system Without any interaction between the atom and the cavity field mode Ω 0 = 0), the Hamiltonian of the uncoupled system is: [ ω0 H = H 0 = σ z + ω c a a + 1 )]. 5.9) The eigenstates of Eq. 5.9) are the tensor products g, n g n atom in the ground state and n photons) and e, n e n atom in the excited state and n photons) of atomic and cavity energy states, where n with n = 0, 1,... ) denotes the Fock basis states of a bosonic field mode. One can distinguish two situations, the resonant and the non-resonant case, by considering the atom-cavity detuning: c = ω 0 ω c. 5.10) -At resonance c = 0), Eq. 5.9) becomes σz H 0 = ω 0 + a a + 1 ) = ω 0 σ + σ + a a) = ω 0 N, 5.11) where the operator N = σ + σ +a a represents the total number of atomic and field excitations. It must be noted that even when we consider the coupled system described by Eq. 5.7), at resonance, the total number of excitation quanta is conserved: [H 0, H ac ] = 0 and then [H 0, H] = 0, 5.1) meaning that H 0 and H have common eigenvectors; and [N, H] = 0, 5.13) hence, N is a constant of motion. Then, we choose eigenstates of the operator N with the same eigenvalue, for instance n + 1, therefore we consider g, n + 1 and e, n, which are degenerate. They are also eigenstates of H 0 with the eigenvalue ω 0 n + 1). The only exception is the ground state of the whole system g, 0, for which H 0 g, 0 holds. -In general, for a non-resonant condition c 0), the Hamiltonian of the uncoupled system can be written as σz H 0 = ω c + a a + 1 ) + c σ z. 5.14) Now, the two system eigenstates g, n + 1 and e, n have different eigenvalues: [ H 0 e, n = E e,n e, n = n + 1)ω c + c H 0 g, n + 1 = E g,n+1 g, n + 1 = [ n + 1)ω c c leading to n uncoupled Hilbert subspaces Ξn) of dimension x. ] e, n, 5.15) ] g, n + 1, 5.16) 3
5.3 The coupled atom-field system Introducing the coupling term given by Eq. 5.8) in the Hamiltonian of the system, the previous separated systems atom and field mode) become a unique entangled system, which is no more diagonal. The excitation number, N, being conserved, it implies that the atom-field couplig H ac connects only states inside a single subspace or bidimensional manifold) Ξn). One notice that: e, n H ac g, n + 1 = i Ω 0 n + 1, 5.17) is the matrix element that describes the process of absorption, which corresponds to the transition g, n + 1 e, n ; and g, n + 1 H ac e, n = i Ω 0 n + 1, 5.18) is the matrix element that describes the emission process, which corresponds to the transition e, n g, n + 1. For this reason, it is sufficient to consider only the eigenspace of H 0 at a given number of photons n: Ξn) : { e, n, g, n + 1 } n = 0, 1,,... ). 5.19) Within each of these subspaces, the Hamiltonian of the system is considered the restriction of the Jaynes-Cummings Hamiltonian to the nth doublet. By introducing the n-photon Rabi frequency: Ω n = Ω 0 n + 1, 5.0) H n can be written in matrix form as: where: and therefore: H n = V n = H n = ω c n + 1)I + V n, 5.1) e, n H 0 e, n e, n H ac g, n + 1 g, n + 1 H ac e, n g, n + 1 H 0 g, n + 1 c iω n iω n c ) = cσ z + Ω n σ y ). The diagonalization can be accomplished by noticing the similitude with the Hamiltonian of the semiclassical treatment in Eq. 3.8). In this case: R n = c + Ω n, 5.) n = c u z + Ω n u y R n. 5.3) where R n is called generalized Rabi frequency from which, at resonance c = 0), one gets the Rabi frequency R n = Ω n. The direction n makes with the axis OZ of the Bloch sphere the angle θ n defined by: tanθ n ) = Ω n. 5.4) c This mixing angle let us express the eigenstates of the atom-field system by noticing the similitude between the 0, 1 θn,ϕ spin states whose Bloch vectors point along the θ n, ϕ = π/ direction on the Bloch sphere see section.) and our basis, { e, n, g, n + 1 }. From this, we obtain the two energy eigenvalues: [ E ±,n = n + 1)ω c ± R ] n, 5.5) 4 ),
At resonance, the degeneracy of the two energy states of the system is lifted and the energy separation is given by E) n = Ω n. Moreover, the energy separation of the two levels is higher as the number of photons increases, E) n n + 1. The corresponding eigenstates of the eigenvalues in Eq. 5.5) are: +, n = cosθ n /) e, n + i sinθ n /) g, n + 1,, n = sinθ n /) e, n i cosθ n /) g, n + 1. 5.6) These states are general entangled states of the global system atom-field. They are also called dressed states, as the bare states are now dressed by means of the interaction with photons. The interaction Hamiltonian H ac produces a unitary transformation on the bare basis states { e, n, g, n+1 }, which is a rotation of an angle θ n / in Ξn). In particular, at resonance condition one obtains: ±, n = 1 e, n ± i g, n + 1 ), 5.7) which is a maximally entangled state, with θ n = π/, from the relation of Eq. 5.4). Now, if c, the mixing angle θ n 0 and the uncoupled or bare states are recovered. Within each subspace Ξn), we consider the variation of the dressed energies as a function of c. For large detunings, the dressed eigenvalues tend to the bare ones, as the atomic and field subsystems were uncoupled. At zero detuning, the uncoupled energy levels cross, whereas the coupling removes this degeneracy with the the minimum distance between the dressed states being the coupling energy Ω n. We deepen now the resonant c = 0) limiting case. 5.4 The resonant atom-field coupling At the resonance condition c = 0, the generalized Rabi frequency R n coincides with the Rabi frequency Ω n, and the dressed states are given by Eq. 5.7): ±, n = 1 e, n ± i g, n + 1 ), 5.8) which are separated by Ω n. We consider inicially t = 0) an excited atom injected in a cavity with the field mode in a Fock state with n photons, that is, the initial pure state Ψ0) n = e, n. One can expand this state on the dressed states basis: Ψ0) = 1 +, n + i, n ). 5.9) The evolution of this state will be described in the interaction representation with respect to the constant term ω c n + 1)I in Eq. 5.1). The evolved state of the system can be written as: Ψt) = 1 Ωn i +, n e t + i, n e i Ωn t), 5.30) which can be written by reverting to the uncoupled basis as: ) ) Ωn t Ωn t Ψt) = cos e, n + sin g, n + 1. 5.31) From this, the probabilities of finding the atom in e or g are: ) P e,n t) = cos Ωn t = 1 [1 + cosω nt)], ) P g,n+1 t) = sin Ωn t = 1 [1 cosω nt)]. 5.3) 5
P e,n 1 0.8 0.6 0.4 0. P g,n+1 0 0 0.5 1 1.5.5 3 n t/ Figure 5.1: Plot of Rabi oscillations for the probability amplitudes P e,n and P g,n+1 in units of the dimensionless time Ω n t/π. The initial state of the system is Ψ0) = e, n. The levels occupation probabilities oscillate at the Rabi frequency Ω n = Ω 0 n + 1. It means that at this frequency the two-level atom and the field mode exchange a single photon, process known as Rabi oscillations. This phenomenon can be understood as a transition between the two dressed states, which are the energy levels to be considered in the Jaynes-Cummings model. As we can see in Fig. 5., at times Ω n t = kπ with k = 0, 1,... ) the state of the system is the basis eigenstate ± e, n or ± g, n + 1, if we consider k even or odd, respectively. At intermediate times Ω n t = k + 1)π/ the system is an entangled state, as for instance 1 e, n + g, n + 1 ) when Ω n t = π/. 5.4.1 Vacuum Rabi oscillations We examine now the particular case with the initial state Ψ0) = e, 0, where the cavity is in the vacuum state with zero photons n = 0), and the atom is in its excited state e. The probabilities for the uncoupled basis states { e, 0, g, 1 } are deduced from Eq. 5.3) with n = 0, obtaining: ) P e,0 t) = cos Ω0 t = 1 [1 + cosω 0t)], ) P g,1 t) = sin Ω0 t = 1 [1 cosω 0t)], 5.33) where Ω 0 is the vacuum Rabi frequency. It is the transition frequency for the dressed states ±, 0 = e, 0 ± i g, 1 )/ in the subspace Ξ0), with energies: E ±,0 = ω c ± Ω ) 0. 5.34) 6
1 0.8 P e t) 0.6 0.4 0. 0 0 10 0 30 40 50 0 t/ Figure 5.: Numerical calculation of quantum collapse and revivals: computed probability P e t) of finding the atom in state e versus the interaction time t in units of π/ω 0. The cavity contains initially an n = 3 photon coherent field. The energy separation between the two dressed levels is E) 0 = Ω 0, and this particular case is known as vacuum Rabi oscillations. This oscillation can also be view as an oscillatory spontaneous emission, i.e. an atom initially in its exited state emits a photon while undergoing a transition to its ground state. In free space, we consider the coupling to a continuum of available modes of the field, hence this process is an irreversible spontaneous emission where the photon escapes and the atom remains in its ground state. Inside a cavity, this becomes a reversible process, with an oscillatory behavior in time, because there is only one mode of the field accessible and the photon emitted by the atom remains trapped. The irreversible decay process is replaced by a coherent oscillation. 5.4. Collapses and revivals induced by a coherent field In the general case, n 0, but resonance conditions, we have deduced in Eq. 5.3) the probability to find the atom again in one of the bare states it was prepared. Now, we will consider an initial coherent state α = n c n n of the cavity field. For a coherent state which average photon number is n = α, we have the c n coefficientes and the photon number distribution, p α, given in Eqs. 4.35) and 4.36). Taking into account Eq. 5.31), the system state in this case at time t is: Ψt) = n Ωn t c n cos ) e, n + c n sin Ωn t ) g, n + 1, 5.35) and the probability of measuring the atom in e is: [ P e t) = 1 1 + e ] α n α cosω n t) n! n. 5.36) 7
The Rabi oscillation in a coherent field is an average over all possible Fock states n, weighted by the probabilities p α n) of finding the corresponding photon number n in the initial coherent state. It is possible to demonstrate that for α 1 the probability P e is: P e t) 1 [ ] 1 + cosω 0 α t)e Ω 0 t, 5.37) and at short times, the Rabi oscillation occurs at frequency close to Ω 0 n, and they continue until a collapse time given by the inverse of the Rabi frequency, t c 1/Ω 0, where the oscillations associated to different values of n become uncorrelated. A purely quantum feature is the revival effect of Rabi oscillations, which occurs when the contributions of the different photon numbers come back into phase and lead again to largeamplitude oscillations in P e, with a period of time t r π n/ω 0 > t c see Fig. 5.). The duration of the revivals increases progressively and the effect gets more and more ruined since the revivals are never complete. This phenomenon is general for fields with irrational frequency components contributing to the signal, hence it depends directly on the quantization of spectrum itself. 8
Chapter 6 Ultrastrong and deep strong coupling regimes of the quantum Rabi model Recently, new regimes have been explored in the field of coherent interaction between a twolevel system and a single bosonic quantized field mode, for which RWA is no longer applicable and it is necessary to abandon the Jaynes-Cummings model. In order to describe more clearly the interaction between an atom and a laser field, we will consider a different representation of the general quantum Rabi Hamiltonian, in which we express the atom-field Hamiltonian in terms of the canonical momentum p and the vector potential A instead of position r and field E operators see section 3.1). Using this formulation, the quantum Rabi Hamiltonian equivalent to Eq. 5.3) is: H = ω 0σ z + ωa a + gσ + + σ )a + a ). 6.1) where ω 0 and ω are the frequencies associated to the atomic transition and the field mode respectively, and we have rescaled the origin of energies by avoiding the constant ω/. The coupling constant Ω 0 in the previous representation is renamed here as g. The physical predictions derived from these two formulations concerning the interaction Hamiltonians r E and p A are equivalent. The conditions we had established for the JC model to be valid were extracted from the Dyson series expansion of the time evolution operator in Eq. 5.5), and are written in this new g ω+ω 0 formulation as 1. Since two years ago, new regimes are been studied both theoretically and experimentally, where g becomes a significant fraction of ω, and the counter-rotating terms are expected to manifest, providing direct evidence for the breakdown of the Jaynes-Cummings model. 6.1 Ultrastrong coupling regime of the quantum Rabi model The ultrastrong coupling regime emerges approximately for 0.1 g/ω 1. The term ultrastrong was coined firstly in 005 in a theoretical proposal describing a planar microcavity photon mode strongly coupled to a semiconductor intersubband transition [15], and was progressively accepted and analyzed in other areas. In 007, the coupling was studied theoretically in artificial Josephson atom coupled to a transmission line resonator [16] and in a superconducting flux qubit coupled to the center conductor of a coplanar waveguide transmission-line resonator [0]. 9
When the coupling strength reaches these larger values, the anti-resonant or non-rotating wave terms σ a and σ + a ) in the Hamiltonian become important, and we cannot neglect them like in the JC Hamiltonian. The first term, σ a, correspond to a transition from the upper level e to the lower level g, together with the annihilation of a photon, that is, the destruction of two excitations. The second term, σ + a, describes the emission of a photon by an atom performing a transition from g to e, hence, the creation of two excitations. However, the quantum Rabi model does not violate energy conservation. This quantum Rabi Hamiltonian had no analytical solution until 011 [14], and physicists have worked with the approximated integrable JC model since for most cases predicts right results. The development of novel quantum technologies such as circuit Quantum Electrodynamics QED) has allowed to reach larger coupling strengths in experiments, and the JC model with analytic solutions breaks down. Despite the non analyticity of the general quantum Rabi Hamiltonian, important counter-intuitive predictions can be deduced. Unlike the JC model, the total excitation number N = σ + σ + a a is no longer conserved, since [N, H] 0 when the anti-rotating terms are taken into account. Parity is conserved, since the parity operator Π = σ z 1) a a commutes with the Hamiltonian, [Π, H] = 0. This fact is important and it will be considered in the deep strong coupling DSC) regime. Regarding the ground state, it depends on the interaction strength. In these new regimes, the standard vacuum g, 0, that was not changed by light-matter interaction in the Jaynes- Cummings model, is no longer the ground state. Instead, we consider a ground state without neglecting counter-rotating terms: G = α g, 0 + β e, 1 + γ g, + δ e, 3 + ε g, 4 +.... 6.) It is remarkable that the ground state contains photons, G a a G > 0, contrasting with the predictions of the JC Hamiltonian. In this case, the parity operator has a definite value, G Π G = +1. The ground state is the lowest energy state and the photons in the ground state cannot escape the cavity, producing a photonic cloud around the qubit, even in an open transmission line. The design of faster quantum gates in order to increase operation speed before decoherence affects in circuit QED needs unavoidably from these new regimes. The usual step to make a faster gate was to increase coupling, but quantum gate protocols are based on JC model, and this model breaks down in the limit g/ω 0.1. The challenge of ultrafast quantum gates remains on this new regimes, but also new methods of implementing the operations must be developed [6]. 6.1.1 Experiments We will describe some experiments where normalized coupling rates g/ω r of up to 0.1 were reached [18]. This strong coupling is achieved at microwave frequencies by using superconducting artificial atoms with large dipole momenta coupled to on-chip cavities. The qubit consists of three nanometre-scaled Josephson junctions [7] interrupting a superconducting loop, which is threaded by an external flux-bias φ x. The flux quantization for suitable junction sizes, lead us to reduce the qubit potential to a double-well potential, where the two minima corresponds to states with clockwise and anticlockwise persistent macroscopic currents ±I p. In the quit eigenbasis, the qubit Hamiltonian reads: H q = ω qσ z 6.3) where ω q = + I p δφ x ) / 6.4) 30
is the qubit transition frequency, which can be adjusted by the external flux bias. The term is the minimum energy gap between two states which are a linear superposition of the degenerate ground states of each potential minimum, when their depth coincide at δφ x = φ x φ 0 /, denoting by φ 0 = h/e the flux quantum. The two-level approximation is justified for the qubit because of its anharmonicity, derived from the non-linearity of the Hamiltonian describing the Josephson junctions. a) b) 0 µm 5 mm Figure 6.1: Quantum circuit and experimental set-up. a) Optical image of the superconducting waveguide resonator light blue rectangle). b) Red rectangle: SEM image of the galvanically coupled flux qubit. T. Niemczyk et al., NPhys 6, 77 010). The cavity is modeled by a coplanar waveguide resonator, whose effective model is obtained by coupling inductances and capacitors. After considering boundary conditions and quantization, the resonators modes are described as harmonic oscillators: H n = ω n a na n + 1 ), 6.5) where ω n is the resonance frequency and n is the resonator-mode index and the creation and annihilation operators, a n and a n, acts in the nth resonator mode. It must be noted that the higher mode frequencies are not integer multiples of the fundamental resonance frequency, ω 1, because of the inhomogeneous transmission line geometry [0]. The coupling constant is enhanced by a Josephson junction which is shared between the flux qubit and the resonator, and mediates most of the inductive coupling of this superconducting qubit galvanically connected to the strip. The strength of the coupling is determined by the local inductance M = L j + L, which includes both the Josephson inductance of the coupling junction, L j, and the inductance of the shared edge between the centre conductor and the qubit, L. A largel j is used and, although it dominates M, has a negligible influence on the vacuum current I n ω n /L r in the resonator because the total resonator inductance L r L j, L. From this, the coupling constants of the qubit to the nth cavity mode may be written as g n = MI n I p [0]. This quantum circuit is located at the base temperature of 15 mk in a dilution refrigerator in order to achieve the superconducting phase of the materials. Then, the Hamiltonian of the quantum circuit can be written as: H = H q + n [H n + g n a n + a n )cos θ σ z sin θ σ x )], 6.6) where the flux dependence is included in sin θ = / ω q and cos θ, and the counter-rotating terms of the Rabi Hamiltonian appears explicitly if the Pauli operator σ x is expressed as a combination of raising and lowering operators, σ + and σ. 31
1 7.8 0 /π GHz) 7.78 ωrf 7.74 5 0 5 m δφx Φ0 Figure 6.: Cavity transmission third mode) as a function of δφ x and probe frequency ω rf /π. Comparison of the spectrum with the results predicted by Hamiltonian in Eq. 6.6) black lines). T. Niemczyk et al., NPhys 6, 77 010). A large number of anticrossings results from the multimode structure of the system, and it is necessary to incorporate the first three resonator modes to explain the spectroscopy data showing the dressed qubit transitions. The values of the normalized strength constants g n /ω n for each mode are 11.%, 11.8% and 7.3%, respectively, and lead us to a physics beyond the Jaynes-Cummings model since significant deviations appear. The comparison of the energy-level spectra of the Hamiltonian in Eq. 6.6) and the Jaynes- Cummings one with the experimental data shows that, depending on δφ x, there are significant deviations from the Jaynes-Cummings model predictions which can be explained by the complete Hamiltonian. The analysis will be accomplished by using the notation q, N 1, N, N 3 = q N 1 N N 3, where q = {g, e} denote the qubit ground or excited state, and N n = { 0, 1,,... } represents the Fock state with photon occupation N in the nth resonator mode. Lets study the origin of the anticrossing see Fig. 6.), where ω 3 ω q, and the dominant contributions to the eigenstates Ψ ± come from approximate symmetric and antisymmetric superpositions of the degenerate states ϕ 1 = e, 1, 0, 0 and ϕ = g, 0, 0, 1. The transition from ϕ 1 to ϕ may be explained as the annihilation of two excitations, one in the first mode and one in the qubit, while, simultaneously, creating one excitation in the third mode. Such a process require counter-rotating terms, which only are present in the complete Hamiltonian of Eq. 6.6), but not in the Jaynes-Cummings approximation, where only eigenstates with an equal number of excitations can be coupled. Another important experiment in the ultrastrong coupling regime also guide us to the addition of counter-rotating terms in the description of the Jaynes-Cummings model, in order to explain a quantum Bloch-Siegert shift in a qubit-oscillator system [19]. The Bloch-Siegert 3
shift is an energy shift in the level transition that appears for atoms very strongly coupled to single photons, and up to now, it was deduced from semiclassical systems. When the RWA is invoked, the resonance between the mode field and the two-level system occurs when the field frequency is identical to the spin transition frequency, however, it is an approximation. In 1940 Bloch and Siegert showed that the dropped parts oscillating rapidly can give rise to a shift in the true resonance frequency of the dipoles [8]. In the literature, the Bloch-Siegert shift and the dynamical Stark shift are sometimes wrongly identified. Although in both of them the effective Hamiltonian keeps terms up to the second order in the coupling constant, in the Bloch-Siegert shift the terms we take into account are the fast rotating ones, and in the dynamical Stark shift, the co-rotating ones. This difference is critical, for instance, in the optical range, where the Bloch-Siegert shift, which goes proportional to g /ω 0 + ω r ), is neglected compared to the Stark shift, which goes with g /ω 0 ω 0 ). The experimental set-up consists of a four-josephon-junction flux qubit, galvanically attached to an LC resonator with a coupling wire. This resonator is made of two capacitors linked by two superconducting wires, and by taking into account its inductance and capacitance, the corresponding resonance frequency ω r = 1/ L r C r /) can be calculated. At 0 mk the resonator will be mostly in its ground state, with zero-point current fluctuations I rms = ω r /L r. As we have seen above, the flux qubit with an externally applied magnetic flux of φ φ 0 /, behaves effectively as a two-level system, which Hamiltonian can be written as: H q = 1 ɛσ z + σ x ), 6.7) where ɛ = I p φ φ 0 /), and operators are expressed in the basis of persistent current states ±I p, { +, }, in the qubit loop. is the tunnel coupling between the two persistent current states. The interaction between the qubit and the resonator can be described by a coupling of dipolar nature H int = ga + a)σ z in the basis of the persistent current states, and the photon creation and annihilation operators in the basis { n } of Fock states of the resonator. The strength of the coupling depends on the inductance of the coupling wire, L k, which is made larger than the rest of them, and leads us to describe it as g = I p I rms L k. By this method, normalized coupling rates g/ω r of approximately 0.1 have been achieved, which bring us into the ultrastrong coupling. In the basis of the eigenstates of the qubit, { g, e }, the complete Hamiltonian reads as Eq. 6.6) with a single mode field: H = ω q σ z + ω r a a + 1 ) + ga + a)cos θ σ z sin θ σ x )], 6.8) where ω q = ɛ + and tan θ = /ɛ. Pauli matrices can be rewritten in terms of raising and lowering operators σ + and σ, showing explicitly the corotating and counter-rotating terms in the Hamiltonian: H = ω q σ z + ω r a a + 1 ) + g[cos θ σ z a +a) sin θ σ + a+σ a +σ + a +σ a)]. 6.9) In this regime, counter-rotating terms cannot be neglected, and their effect may be evaluated by a unitary transformation H = e S He S, with S = γσ + a σ a) and γ = g sin θ/ω q +ω r ) to eliminate counter-rotating terms. If we keep terms up to second order in γ, the effective Hamiltonian is: H = ω q σ z + ω r N + 1 ) + ω BS [σ z N + 1 ) 1 ] + gn)a σ + aσ + gn) 6.10) 33
Figure 6.3: Bloch-Siegert shift. Spectrum in the proximity of the resonator frequency, and comparison of the Eq. 6.9) predictions solid black line) and the Jaynes Cummings model dashed green line) predictions. P. Forn-Díaz et al., PRL 105, 37001 010). with the number operator N = a a. The Bloch-Siegert shift is described by the term proportional to ω BS = g sin θ/ω q + ω r ). The coupling constant has been renormalized to gn) = g sin θ [1 Nω BS /ω q + ω r ] and the term g cos θ σ z a + a) from Eq. 6.8) has been neglected since as to second order it only adds a global phase. In the basis of the bared states { g, n + 1, e, n }, the effective Hamiltonian in Eq. 6.10) is box diagonal. The box corresponding to n photons has eigenvalues: λ n,± = ω r n + 1) ± δn+1 + 4g n+1 ω BS ; λ 0,g = δ, 6.11) where δ = ω q ω r is the detuning, and δ n+1 = δ + ω BS n + 1) and g n+1 = g sin θ n + 1[1 n + 1)ω BS /ω q + ω r )]. The eigenvalues corresponding to + and are related to the qubit in its ground and excited state, respectively. If we compare, for both JC and USC models the difference between the eigenvalues, λ 1,g λ 0,g, with the measured spectral peaks, we find that the RWA is no longer applicable see Fig. 6.3). That is, the Bloch-Siegert shift appearing here force us to consider the USC regime. 6.1. Simulations and applications In 011, a quantum simulation of the USC/DSC dynamics in a standard circuit QED setup was proposed [1], whose implementation has been recently accomplished [9]. The treatment makes use of a superconducting qubit strongly coupled to a microwave-resonator mode, which is orthogonally driven by two classical fields. The key idea is tuning the parameters of the external drivings in a suitable way such that, while the value of the coupling constant g is fixed, we can reach values of the effective parameters ideally in the USC/DSC. The initial Hamiltonian describing the complete dynamics of the qubit-cavity system driven by two classical fields includes the RWA, but as we will see, it will simulate an ultrastrong regime. The possible coupling between the orthogonal driving and the resonator field has been disregarded since experimentally it is easy to avoid. If we choose one of the drivings relatively strong, for instance Ω 1, and consider the interaction picture of the frame rotating with the frequency of that driving, the following effective 34