Cold Trapped Ions as Quantum Information Processors
Abstract
In this tutorial we review physical implementation of quantum computing
using a system of cold trapped ions. We discuss systematically all the aspects for
making the implementation possible.
Firstly, we go through the loading and confining
of atomic ions in the linear Paul trap, then we describe
the collective vibrational
motion of trapped ions. Further, we discuss interactions of the ions with
a laser beam. We treat the interactions in the travellingwave and
standingwave configuration for dipole and quadrupole transitions. We
review different types of laser cooling techniques associated with trapped
ions. We address Doppler cooling, sideband cooling in and beyond
the LambDicke limit, sympathetic cooling and laser cooling using
electromagnetically induced transparency. After that we discuss the problem
of state detection using the electron shelving method. Then
quantum gates are described. We introduce singlequbit rotations,
twoqubit controlledNOT and multiqubit controlledNOT gates. We also
comment on more advanced multiqubit logic gates. We describe how
quantum logic networks may be
used for the synthesis of arbitrary pure quantum states. Finally, we discuss
the speed of quantum gates and we also give some numerical estimations for
them. A discussion of dynamics on offresonant transitions associated with
a qualitative estimation of the weak coupling regime and of the LambDicke
regime is included in Appendix.
PACS numbers: 03.65.Ud, 03.67.Lx, 32.80.Pj, 32.80.Ys
Contents
I Introduction
Although trapped ions have found many applications in physics NIST , they caused a turning point in the evolution of quantum computing when the paper entitled Quantum computation with cold trapped ions was published by Cirac and Zoller in 1995 955 . This proposal launched also an avalanche of other physical realizations of quantum computing using different physical systems, from high finesse cavities to widely manufactured semiconductors sam . Through the years we have learnt a lot, but also revealed many peculiarities, about the physical realization of quantum computing which has led to many discussions concerning the conditions under which we could in principle implement quantum computing in certain quantum systems.
Before we give the list of requirements for the physical implementation of quantum computing we will introduce the fundamental terminology to appear throughout this paper. We will follow the definitions in Ref. 004 .

A qubit is a quantum system in which the logical Boolean states 0 and 1 are represented by a prescribed pair of normalized and mutually orthogonal quantum states labelled as and . These two states form a computational basis and any other (pure) state of the qubit can be written as a superposition
(1) for some and such that . It can be shown that we may choose and . A qubit is typically a microscopic system, such as an atom, a nuclear spin or a polarized photon, etc. In quantum optics a twolevel atom with a selected ground and excited state represents a qubit. Hence the notation and is used for the computational basis instead of and . For instance, some qubits can serve for logic operations or the storage of information. Then we refer to logic qubits. Some others can be used especially for sympathetic cooling of logic qubits and we may call them cooling qubits. Some further qubits can be used as a quantum channel for transferring the information between distinct logic qubits and then we refer to them as to a quantum data bus.

A quantum register of size refers to a collection of qubits.

A quantum gate is a device which performs a fixed unitary operation on selected qubits in a fixed period of time.

A quantum network is a device consisting of quantum gates whose computational steps are synchronized in time.

A quantum computer (processor) can be viewed as a quantum network or a family of quantum networks.

A quantum computation (computing) is defined as a unitary evolution associated with a set of networks which takes a initial quantum state (input) into a final quantum state (output) and can be interpreted in terms of the theory of information processing.
For the moment we presume that the following five requirements (termed DiVincenzo’s checklist) should be met in order to realize quantum information processing on a quantum system divin . Actually, there are two more requirements for the case of the transmission of qubits in space (flying qubits). However, it appears that all these requirements are necessary but not sufficient for successful experimental realization of a quantum processor loss .

The system must provide a well characterized qubit and the possibility to be scalable in order to create a quantum register.

We must be able to initialize a simple initial state of the quantum register.

Quantum gate operation times must be much shorter than decoherence times. The quantum gate operation time is the period required to perform a certain quantum gate on a single qubit or on a set of qubits. The decoherence time approximately corresponds to the duration of the transformation which turns a pure state of the qubit into a mixture .

We need a set of quantum gates, to perform any unitary evolution operation that can be realized on the quantum system. It has been shown that any unitary evolution can be decomposed into a sequence of single qubit rotations and twoqubit controlledNOT (CNOT) gates 958 .

The result of a quantum computational process must be efficiently read out, i.e. the ability to measure distinct qubits is required.
Now we introduce briefly the physical system under consideration. Cold trapped ions is a quantum system of atomic ions confined in a linear trap. We assume an anisotropic and harmonic trapping potential. The ions are laser cooled to a very low temperature, beyond the Doppler cooling limit, reaching the recoil cooling limit russia . Hence the term cold trapped ions. The ions form a linear crystal and oscillate in vibrational collective motional modes around their equilibrium positions. In their internal structure, depending on the choice of atomic species, we distinguish distinct atomic levels. The ions are individually addressed with a laser or a set of lasers in the travellingwave or standingwave configuration. We can detect the internal state of ions using optical detection devices. Further, we address briefly the requirements for the physical implementation of quantum computing (mentioned above) using cold trapped ions.

The qubit is represented by a selected pair of internal atomic states denoted as and . This selection is discussed in detail in Sec. IV. The quantum register is realized by ions forming the ion string in the linear trap, namely the linear Paul trap, which is reviewed in Sec. II. A selected collective vibrational motional mode (normal mode) is used as the quantum data bus. The vibrational motion of the ions is treated in Sec. III.

Different laser cooling techniques can be used for the proper initialization of the motional state of the ions. They are described in Sec. V. The initial internal state where all the ions are in the state can be reached by optical pumping to atomic states fast decaying to the ground state (Sec. V and VI).

The influence of the decoherence on the motional state of the ions is suppressed by laser cooling to ground motional states of the normal modes. The internal levels of the ions representing the qubit states and are selected such that they form slow transitions with excited states of long lifetimes. A very detailed discussion of the decoherence bounds of trapped atomic ions can be found in Ref. 985 .

Singlequbit quantum rotations can be realized on any ion and twoqubit controlledNOT and multiqubit controlledNOT quantum gates can be applied between chosen ions due to the possibility of individual addressing with laser beams. The implementation of quantum gates is discussed in Sec. VII.

The result of a computational process on cold trapped ions is encoded into the final state of the internal atomic states. This information can be very efficiently read out using the electron shelving method addressed in Sec. VI.
Ii Ion trapping
Due to the charge of atomic ions, we can confine them by particular arrangements of electromagnetic fields. For studies of ions at low energy two types of traps are used. (i) Penning trap uses a combination of static electric and magnetic fields and (ii) Paul trap confines ions by oscillating electric fields. Paul was awarded the Nobel Prize in 1990 for his work on trapping particles in electromagnetic fields 901 . The operation of different ion traps is discussed in detail in Ref. ghosh . For the purpose considered in this paper we will discuss only one trap configuration: the linear Paul trap (FIG. 1). We will follow Ref. ghosh and LesH for the mathematical treatment.
The linear Paul trap is basically a quadrupole mass filter, which is plugged at the ends with static electric potentials. An electric potential
(2) 
oscillating with the radiofrequency is applied between two diagonally opposite rod electrodes. The electrodes are coupled together with capacitors so that the potential (2) is constant as a function of the coordinate. The other two rod electrodes are grounded. The resulting potential at the trap axis (parallel with the direction) has the form
(3) 
where is the distance from the trap centre to the electrode surface. In this field the (classical) equations of motion for an ion of the mass and charge are
(4) 
or rewritten in the components
(5)  
(6)  
(7) 
After the substitution
(8) 
Eq. (5) and (6) take the form of the Mathieu equation
(9)  
(10) 
The Mathieu equation can be solved, in general, using the Floquet solution. However, typically we have , then the approximate stable solution of Eq. (9) and (10) are
(11)  
(12) 
where
(13) 
and , , , are constants determined by initial conditions. We see from Eq. (11) and (12) that the motion of a single trapped ion in the radial direction is harmonic with the amplitude modulated with the frequency . The harmonic oscillation corresponding to the frequencies and is called the secular motion, whereas the small contribution oscillating at is termed the micromotion nagerl ; roos . We can eliminate the micromotion under certain conditions roos . For instance, well chosen voltages on additional compensation electrodes (not shown in FIG. 1) null the micromotion. Then the ion behaves as if it was confined in a harmonic pseudopotential in the radial direction given by
(14) 
Typically, and hence , so the radial frequencies and are degenerated. Then Eq. (14) reduces to
(15) 
where the radial trapping frequency is given by
(16) 
In experiments nagerl ; roos ; 003 ; blatt1 ; blatt2 , typical operating parameters are , , , so we achieve the radial frequency for Calcium ions . In nature, of Calcium consists of this isotope. To provide confinement along the direction, static potentials and are applied on the ring electrodes. Ideally, . Numerical calculations show that the potential near the trap centre at the trap axis is harmonic with the approximate axial trapping frequency given by
(17) 
where is the distance from the trap centre to the ring electrode and is a geometric factor describing how much of the static field from the ring electrodes is present along the trap axis nagerl . Typical parameters are for and blatt1 ; blatt2 . The resulting pseudopotential for ions confined in the linear Paul trap in all three directions takes the form
(18) 
where the radial trapping frequency is given by Eq. (16) and the axial trapping frequency is defined by Eq. (17). For values of experimental parameters given above, we can calculate the depth of the potential well in the axial direction ()
(19) 
and in the radial direction ()
(20) 
The potential well in the radial direction is almost several times deeper than along the trap axis, i.e. there is a strong binding in the radial direction. Therefore we will not take into account radial oscillations of the ions in our further considerations.
Finally we briefly mention how ions are loaded into the trap. We will follow the account of practical procedures in Ref. roos . Before starting the loading process, the trapping potentials are turned off for a while in order to get rid of any unwanted trapped residual ions. The atomic oven producing Calcium atoms is switched on and heats up. This takes about a minute. Then we turn on the electron gun ionizing neutral Calcium atoms directly in the trapping volume. Cooling lasers are directed on the ion cloud containing several hundreds of ions with a diameter of about 200 m. The ion cloud gradually relaxes into a steady state where the radiofrequency heating (from the electrodes) is balanced by laser cooling. The number of trapped ions is reduced by turning off the cooling. At low ion numbers, the ions undergo a phase transition and form a linear crystal structure. Therefore, we refer to the ion crystal or to the ion string or eventually to the ion chain. The loading process itself takes normally about a minute.
Iii Collective vibrational motion
iii.1 Equilibrium positions
We have learnt that the ions form a linear crystal structure in the linear Paul trap after the loading process. We will assume a string of trapped ions. Due to the strong binding we can neglect the radial oscillations. However, if a large number of ions is confined in the trap, the radial vibrations become unstable and the ions undergo a phase transition from a linear shape to an unstable zigzag configuration. The relation
(21) 
determines a critical value for the ratio of the trapping frequencies for a given number of trapped ions . When exceeds the critical value , the ions are exposed to a zigzag motion. The experimental values of the constants in Eq. (21) are and . For experimental details and the theoretical treatment we refer to Ref. zigzag .
Further, we describe the collective vibrational motion of the ions. We will follow the treatment given by James in Ref. 987 . The ions are exposed to the harmonic potential (18) due to the trap electrodes and also to the repulsive Coulomb force from each other. Taking into account all the assumptions given above, the potential energy of ions confined in the linear Paul trap is given by the expression
(22) 
where is the position of the th ion numbering them from left to right with the origin in the trap centre, is the mass of the ion with the charge , is the axial trapping frequency (17) and is the permitivity of the vacuum.
Assuming that the ions are cold enough, we can write for the position of the th ion
(23) 
where is the equilibrium position and expresses small vibrations around . The ions placed in the equilibrium positions minimize the potential energy. Hence these positions are determined by the condition
(24) 
where and . We introduce a scaling factor by the relation
(25) 
and the dimensionless equilibrium position as . Then one can rewrite Eq. (24) to the form
(26) 
is a trivial case (). We can find the analytical solution of Eq. (26) for two and three ions:
Numerical calculations are necessary for . For the Calcium ions and the trap frequency , we may calculate the equilibrium positions as
The minimum value of the distance between two neighbouring ions in the trap occurs at the centre of the ion crystal, because the outer ions push the inner ions closer together. It has been calculated from numerical data that this minimum distance is given approximately by the relation oxf ; 987
(29) 
However, slightly different numerical results may be found in Ref. 985 . The relation (29) happens to be important when one considers individual ion addressing with a laser beam. Quantum statistics of the ion ensemble is not considered here because the spatial spread of the zeropoint wavefunctions of the individual ions is of the order of 10 nm and the wavefunction overlap is then negligible oxf .
iii.2 Normal modes
The (classical) Lagrangian of the ions in the trap is given by the formula
(30) 
where we have expanded the potential energy (22) in a Taylor series about the equilibrium positions. In the expansion we have omitted the constant term and the linear term which is zero [see Eq. (24)]. Higher order terms have been also neglected. However, they may cause a crosscoupling between different vibrational modes which becomes a source of decoherence 985 . The partial derivatives in Eq. (30) can be calculated explicitly and we obtain the expression
(31) 
where
(32) 
It follows from Eq. (32) that . The values of are given by Eq. (III.1) for and for , whereas they have to be calculated numerically for .
The dynamics of the trapped ions is governed by the Lagrange equations
(33) 
with the Lagrangian given by Eq. (31). We will search for a particular solution of Eq. (33) in the form
(34) 
where are constants. Substituting Eq. (34) into (33) we get the condition for in the form
(35) 
where is the Kronecker symbol and denotes the determinant. The equation (35) has in general up to real and nonnegative solutions . The frequencies are characteristic parameters of the system. They depend only on its physical features (not on initial conditions). A general solution of Eq. (33) is a superposition of particular solutions (34) and we may write
(36) 
where
(37) 
By definition we will require the vectors
(38) 
to be the eigenvectors of the matrix defined in Eq. (32), i.e.
(39) 
and also to be orthogonal and properly normalized
(40) 
We will number the eigenvectors in order of the increasing eigenvalues . It can be shown that the first two eigenvectors always have the form
(41)  
(42) 
We should emphasize that Eq. (41) and (42) (they characterize two basic collective motional modes) are not dependent on the number of the ions in the trap. Next eigenvectors must be, in general, calculated numerically. Substituting Eq. (41) into (40) we get the relation
(43) 
We can determine analytically the eigensystem for two and three ions:
(44)  
(45)  
(46) 
For larger , the eigenvectors and eigenvalues must be computed numerically. The numerical values for up to ten ions can be found in Ref. 987 .
Substituting Eq. (36) into (31) we get a new expression for the Lagrangian
(47) 
where
(48) 
The Lagrangian (47) has split into uncoupled terms, where [Eq. (37)] refer to the normal modes and defined in Eq. (48) are termed the normal frequencies. Finally, the position of the th ion in the trap can be rewritten in terms of Eq. (36) using (23) to the form
(49) 
where denotes the real part and are constants given by initial conditions. The collective vibrational motion of trapped ions determined by the eigenvector [Eq. (41)] refers to the normal mode called the centerofmass (COM) mode
(50) 
and corresponds to all of the ions oscillating back and forth as if they were a rigid body. The motion determined by the next eigenvector , [Eq. (42] refers to the breathing mode
(51) 
It corresponds to each ion oscillating with the amplitude proportional to its equilibrium distance from the trap center. The COM motional mode can be excited in experiments by applying an additional AC voltage on one of the ring electrodes. For exciting the breathing motional mode, a 300times higher voltage must be applied 003 . Higher motional modes require gradient field excitation due to the nontrivial configuration of the ions in the ion string. However, in the limit of large ion trap dimension in comparison with the ion crystal dimension, the electrode electric fields are almost uniform across the ion crystal and the COM mode is very susceptible to heating due to these fields. Therefore, it seems to be more advantageous to use rather the breathing mode, which is much less influenced by uniform fields, as the quantum data bus. This will be discussed in more detail later on in the section on sympathetic cooling (Sec. V.3). On the other hand, the ions can be easily addressed with a laser beam in the COM mode, while higher modes require accurate bookkeeping when addressing distinct ions in the ion crystal 987 ; symp1 .
iii.3 Quantized vibrational motion
The normal modes are uncoupled in Eq. (47), so the corresponding canonical momentum conjugated to is and one may write the (classical) Hamiltonian
(52) 
The quantum motion of the ions can be considered by introducing the operators
(53)  
(54) 
with the corresponding commutation relations
(55) 
The Hamiltonian operator associated with the external (vibrational) degrees of freedom of the trapped ions is then expressed as follows
(56) 
where and are the usual annihilation and creation operators referring to the th normal mode. We use the standard notation for the number states associated with the collective vibrational motion of the ions
(57) 
where refers to the state of the th normal mode and denotes the number of vibrational phonons in this mode. The states form the complete and orthonormal basis
(58) 
We can quantize the motion of the ions by applying Eq. (53) to the relation (36) and expressing the displacement operator of the th ion in the timeindependent picture
(59) 
where [see Eq. (48)]
(60) 
We can easily calculate from Eq. (41) that for the COM mode applies
(61) 
and for the breathing mode [Eq. (42)]
(62) 
Although we have not considered the radial vibrations due to the strong binding of the ions in the radial direction, a detailed treatment of the ion motion in the trap would require the extension to all three dimensions. Then Eq. (23) has to be replaced with
(63) 
where denotes the equilibrium position of the th ion in the 3D space and is its displacement from the equilibrium position. We can write
(64) 
where , , are the equilibrium positions of the th ion and x, y, z are unit vectors in the 3D space. The free Hamiltonian associated with the vibrational motion in the 3D space reads
(65) 
and the displacement operators in Eq. (64) are given as follows
(66) 
where the numerical factors in general have to be determined numerically.
Iv Laserion interactions
Information is encoded in internal (atomic) states, while it is transferred via external (motional) states of the ions. We can manipulate these states due to laserion interactions. It can be accomplished in the travellingwave and standingwave configurations. We will address in detail both approaches in what follows. However, we should first comment on the selection of the two internal atomic levels to form the qubit. There are three possibilities 987 :

We can employ a ground and metastable fine structure excited state. This applies for ions with zero nuclear angular momentum [FIG. 2(a)]. In this case we refer to the single beam scheme and we can drive transitions on optical frequencies. This configuration is used, for example, by the group in Innsbruck using Calcium ions innsbruck ; 0011 .

We can also choose two sublevels of a ground state within the hyperfine structure (ions with nonzero nuclear angular momentum) [FIG. 2(b)]. The spacing of such two sublevels is in the range of GHz. Thus, a twobeam Raman scheme via a third virtual level is required in order to resolve the individual sublevels. Experiments in this configuration with Beryllium ions were performed in Boulder NIST ; 985 ; exp .
We have to mention also other active groups running experiments towards quantum logic with trapped ions. For instance (in alphabetical order) IBM Almaden using IBM , Imperial College (, ) imperial , JPL in Los Angeles () JPL , Los Alamos National Laboratory () lanl ; lanl , Oxford University () oxf ; oxford , University of Aarhus () aarhus , University of Hamburg (, ) hamburg and University of Mainz () mainz .
We can use dipole and quadrupole transitions. Theoretically, the difference is only in the interaction constants as we will see later on in this section. On the other hand, in experiments quadrupole transitions have much longer lifetimes (one second for Calcium ions) comparing to fast decaying dipole transitions (). Experiments on an octupole transition in an Ytterbium ion has also been realized. The predicted theoretical lifetime in this system is of the order of oct . However, in this case one deals with very weak transitions with very stringent demands on the laser sources used in the experiment (although they are of major interest as potential ion trap clocks). Moreover, weak transitions have to be driven with a very intense laser which enhances the possibility for offresonant excitations. From now on we will describe in this paper all experimental procedures for Calcium ions (FIG. 3).
In the following we will deal with the single beam scheme, i.e. transitions being driven by a single laser beam. We will not treat here the Raman scheme. The derivation of the Hamiltonian in this scheme can be found in Ref. raman . We just mention that the final Hamiltonian in the Raman scheme has the same form as the one in the single beam scheme, except for differences in coupling constants and for atomic frequencies which are Stark light shifted. In the Raman scheme the resulting effective light field has the direction (frequency) determined by the difference of the wavevectors (frequencies) of the two participating laser beams, where each beam is represented (in a semiclassical approach) with a monochromatic travelling wave. Finally, the single beam scheme requires a very high laser frequency stability, while in the Raman scheme we only need to control the relative frequency stability between the two laser beams which is technically less demanding. With the Raman scheme we can also ensure the relative wavevector of the two beams to be parallel to the trap axis which suppresses the coupling to radial motional modes. On the other hand, the Raman scheme can introduce significant Stark light shifts 985 .
In the rest of the paper we will use the standard atomic level notation , where is the principal quantum number, is the spin angular momentum, is the orbital angular momentum and is the total angular momentum of electrons. For the fine structure case the notation is where is the projection of onto the quantization axis. In the case of the hyperfine structure we denote where is the total angular momentum of the atom (electrons + nucleus) and is the projection of onto the quantization axis.
Let us consider that the ion has two internal levels, denoted (lower) and (upper) with corresponding energies and , where the transition frequency is . Then the free Hamiltonian associated with the internal degrees of freedom is given by
(67) 
where and . Finally, we can write the total free Hamiltonian for the th ion of ions confined in the trap communicating via one of the collective vibrational modes [see Eq. (56)]
(68) 
where we have omitted constant terms , and dropped down the index denoting a vibrational mode. The motional mode used for manipulations (especially quantum logic operations) with the ions is called the quantum data bus because, as we will see later, it serves to transfer the information between distinct ions within the ion crystal (representing a quantum register). We will consider for this purpose only the COM mode or the breathing mode .
Further, we assume a powerful laser, i.e. the interaction with the ions has no influence on the laser photon statistics. Therefore, we will employ a semiclassical description of the laser beam. We will consider the laser beam in the (i) travellingwave and (ii) standingwave configuration.
iv.1 Travellingwave configuration
There are two different ways for addressing the ions. We can set the laser beam at a fixed position and shift the ion string by a very slight variation of the DC voltage on the ring electrode. On the other hand, we can fix the ion string and scan the laser across the string. In this case an acoustooptical modulator is used for laser beam deflection 003 .
Let us approximate the laser beam as a monochromatic travelling wave (FIG. 4). We can write
where is the real amplitude, is the polarization vector with , is the laser frequency, is the wavevector with , is the speed of light, is the position vector and is the phase factor. The full Hamiltonian for the th ion is given by
(70) 
where the interaction Hamiltonian (assuming a hydrogenlike atomic configuration) expanded to second order (neglecting magnetic dipole interaction) has only two terms
(71) 
The electric dipole (DP) term is defined as follows
(72) 
summing over . We refer to Eq. (72) as the dipole approximation. The electric quadrupole (QD) term reads
(73) 
where the sum is applied over and we refer to Eq. (73) as the quadrupole approximation. We denote to be the electron charge, is the internal position operator associated with the position of the valence electron in the th ion and is the external position operator corresponding to the position of the th ion in the trap.
For the present we will consider only the dipole term (72), regarding to the situation when the dipole interaction is present and the quadrupole contribution (73) is then negligible. Later we will also comment on the quadrupole interaction. If we consider consider only a single motional mode, we get from Eq. (59) for the external position operator of the ion
(74) 
Then we can sandwich the internal position operator with the unity operator and rewrite Eq. (72) to the form
(75) 
where , , , , , , , with and defined by Eq. (60). In Eq. (75) we consider that , because we assume spatial symmetry of the wavefunctions associated with the internal atomic states and . The schematic configuration is depicted in FIG. 4. It is useful to transform to the interaction picture defined by the prescription
(76) 
The Hamiltonian (75) after the transformation to the interaction picture (IV.1) reads
(77) 
where and we have neglected rapidly oscillating terms at the frequency compared with lowfrequency terms at . In practice , therefore to a good degree of approximation for times of interest, highfrequency terms average to zero louisell . This approximation is called the rotating wave approximation (RWA). In Eq. (77) we substitute for the COM mode or for the breathing mode. The laser coupling constant introduced in Eq. (77) is defined by the relation
(78) 
However, for a dipole forbidden transition when , the dipole term (72) does not contribute () and the key role is played by the weaker quadrupole interaction. In that case the laser coupling constant in the Hamiltonian (77) reads
(79) 
where all parameters are defined in Eq. (IV.1).
Next, let us assume the detuning of the laser frequency from the atomic frequency for the vibrational frequency in the form
(80) 
and apply the BakerCampbellHausdorff theorem louisell Eq. (77). Then we can write
(81) 
If the laser is tuned at the frequency such that , the spectral line is termed the th blue sideband. For the line is called the carrier and for refers the th red sideband because the laser is red (blue) detuned from the atomic frequency (FIG. 5).
When the constant is sufficiently small we can assume that there are no excitations on offresonant transitions (weak coupling regime). Then the level structure of the ion can be considered as a series of isolated twolevel systems 972 . Precisely what is meant by sufficiently small is detailed in Appendix A. Assuming the weak coupling regime, we can neglect offresonant terms () and rewrite Eq. (81) for to the form
(82) 
and for