# Atom-wave diffraction between the Raman-Nath and the Bragg regime:

Effective Rabi frequency, losses, and phase shifts.

###### Abstract

We present an analytic theory of the diffraction of (matter) waves by a lattice in the “quasi-Bragg” regime, by which we mean the transition region between the long-interaction Bragg and “channelling” regimes and the short-interaction Raman-Nath regime. The Schrödinger equation is solved by adiabatic expansion, using the conventional adiabatic approximation as a starting point, and re-inserting the result into the Schrödinger equation to yield a second order correction. Closed expressions for arbitrary pulse shapes and diffraction orders are obtained and the losses of the population to output states otherwise forbidden by the Bragg condition are derived. We consider the phase shift due to couplings of the desired output to these states that depends on the interaction strength and duration and show how these can be kept negligible by a choice of smooth (e.g., Gaussian) envelope functions even in situations that substantially violate the adiabaticity condition. We also give an efficient method for calculating the effective Rabi frequency (which is related to the eigenvalues of Mathieu functions) in the quasi-Bragg regime.

###### pacs:

03.75.Be; 32.80.Lg; 32.80.Wr; 03.75.Dg## I Introduction

### i.1 Background

Diffraction by a point scatters light or matter waves into all directions. A two-dimensional grating produces a few diffraction orders at those angles where the scatter from all of the grating adds coherently. Bragg diffraction by an infinite three-dimensional lattice can produce a single diffraction order, which happens when the scatter from all layers adds constructively, as described by the Bragg condition. When this happens for a higher scattering order (“high-order Bragg diffraction”) virtually all incident radiation can be scattered into this high order, in contrast to the two-dimensional case. By quasi-Bragg diffraction, we refer to the intermediate regime where the infinite lattice assumption is no longer valid but approximately true. Using the nomenclature of, e.g., Keller , this regime is the transition between the short-interaction Raman-Nath regime and the long-interaction Bragg (weak potential) and “channelling” (strong potential) regimes. In this region, the Bragg condition softens and there may be significant scattering into other than the desired orders. Moreover, couplings between the nonzero diffraction orders may lead to phase shifts of the diffracted waves Buchner , which is undesirable in many applications.

In this work, we present an analytic treatment of such quasi Bragg scattering. We will find that by prudent choice of the scattering potential and its envelope function, behavior very similar to Bragg scattering, in particular very low losses and phase shifts, can be obtained for scatterers that substantially violate the assumptions of the simplified theory.

Bragg diffraction famously provides us with the basic knowledge of the structure of crystals, including proteins. It is also important for many technical applications, like acousto-optic modulators (AOMs) Hobbs , distributed Bragg reflectors (DBR) in diode Voges and fiber lasers as well as photonic bandgap crystals vanDriel . Moreover, Bragg diffraction is a basic method for making surface acoustic wave (SAW) filters in radio frequency technology. In atomic physics, Bragg diffraction is a special case of the Kapitza-Dirac effect Kapitza ; Altshule ; Martin ; Berman ; Batelaan .

Bragg scattering is used as a tool for experiments with Bose-Einstein condensates (BECs) Gupta ; Gupta2 ; Morsch . For example, Kozuma et al. Kozuma have shown experimentally that thirteen subsequent first-order Bragg diffractions of a BEC can still have good efficiency. More exotic applications include the generation of a collective frictional force in an ensemble of atoms enclosed in a cavity, due to Bragg scattering of a pump light an a self-organized atomic density grating Black , much in the same way as stimulated Brillouin scattering by self-organized acoustic waves in optical fibers Voges .

Moreover, Bragg diffraction can act as a beam splitter for matter waves Martin ; Altshule ; Meystre ; Gupta ; Gupta2 ; Bernet ; Wu ; Borde ; Malinovsky . The highest order diffraction so far achieved with matter waves seems to be by Koolen et al. Koolen , who obtained up to eighth-order Bragg diffraction. Atom interferometers based on Bragg diffraction include the one by Giltner et al. Giltner ; Giltner2 , who built a Mach-Zehnder atom interferometer using up to third order diffraction. Miller et al. Miller achieved high contrast in a two-pulse geometry with first-order diffraction and a sufficiently short time between pulses. Torii et al. Torii have used first order Bragg diffraction in a Mach-Zehnder geometry with a Bose-Einstein condensate. In addition, Rasel et al. Rasel have built a Mach-Zehnder atomic-beam interferometer based on Raman-Nath scattering.

More generally, atom interferometers can be used for measurements of atomic properties Ekstrom ; Miffre ; Schmiedmayer , the local gravitational acceleration Peters , the gravity gradient, Newton’s gravitational constant Stuhler , tests of the equivalence principle Jason and the fine-structure constant via Wicht ; Biraben ; Paris . For planned experiments in space, see Coq ; Yu .

While not all the atom interferometers just cited use Bragg diffraction, high-order Bragg diffraction offers several interesting possibilities for atom interferometers: (i) it makes the atom interact with photons at once, which may increase the sensitivity of the interferometer by a factor of Paris relative to the photon transitions. (We note here that other possibilities exist for using high-order transitions in atom interferometers, like applying multiple low-order pulses Giurk , operation in the Raman-Nath regime Dubetsky or the magneto-optical beam splitter Pfau ) (ii) Since Bragg diffraction theoretically allows coherent momentum transfer with an efficiency close to one, it allows the insertion of many -pulses for additional momentum transfer, which increases the signal in photon recoil measurements Paris (up to -pulses based on two-photon adiabatic transfer were used in Wicht , transferring photon momenta; if these had been 5-th order Bragg pulses, they would have transferred 300). (iii) If losses can be neglected, Bragg diffraction is basically a transition in a 2-level system. Thus, many of the techniques developed for standard beam splitters based on Raman transitions can be taken over. For example, several beam splitters addressing different velocity groups respectively can be performed simultaneously Paris ; PLL .

### i.2 Overview of the existing theory

A summary of the material that is the basis of this work can be found in the textbook by Meystre Meystre .

A lot of attention has been paid on the theory of the long-interaction time (channelling or Bragg) regimes on the one hand and the Raman-Nath regime on the other hand. Keller et al. Keller give a brief account of the most important results. They have been derived using various formalisms: Berman and Bian Berman use a pump-probe spectroscopy picture, focussing on applications as beam splitters in atom interferometers. The phase-shift of the diffraction process has been studied by Büchner et al. Buchner , limited to first and second order diffraction. Giltner et al. Giltner ; Giltner2 have reported an atom interferometer based on Bragg diffraction of up to third order and give the effective Rabi frequency in the long-interaction regime, our Eq. (15). A similar derivation was given by Gupta et al. Gupta ; Gupta2 .

While most of this work (as well as ours) is concerned with on-resonance transitions, Dürr and Rempe Duerr99 have considered the acceptance angle (i.e., linewidth) of diffraction. They restrict attention to the case of square-envelope pulses. Wu et al. Wu give a numerical study of the latter, again for the case of (a pair of) square pulses. Stenger et al. Stenger describe the line shape of Bragg diffraction with Bose-Einstein condensates within the Bragg regime.

More general types of diffraction have been studied, such as using chirped laser frequencies Malinovsky in the adiabatic regime where the chirp is sufficiently slow. This regime is similar to Bloch oscillations Peik . Band et al. Band have considered the loss due to atom-atom interactions, which is relevant in experiments with Bose-Einstein condensates (BECs). Blackie and Ballagh Blackie explore the use of Bragg diffraction in probing vortices in BECs.

On the other hand, the picture is still incomplete in the quasi-Bragg regime we are concerned with. Dürr and Rempe have analytically calculated corrections to the effective Rabi frequency for large potential depth Duerr in the case of second-order scattering with a square envelope function. Champenois et al. Champenois consider both the Raman-Nath and the Bragg regime using a Bloch-state approach; for the Bragg regime, however, their treatment is restricted to the first diffraction order and square envelopes. Bordé and Lämmerzahl Borde give and exhaustive treatment of square-envelope scattering that is based on matching the boundary conditions at the beginning and end of the pulse. This method, unfortunately, cannot readily be generalized to smooth envelope functions.

The Mathieu equation formalism is a powerful tool for studying diffraction with arbitrary potential depth and interaction times, as demonstrated by Horne, Jex, and Zeilinger Horne . However, like the work of Bordé and Lämmerzahl, the Mathieu equation formalism assumes constant envelopes and cannot readily be generalized to smooth envelope functions. As we shall see (and has already been pointed out, see, e.g., Keller ), smooth envelope functions are of particular interest because they allow high-efficiency scattering into a single order even in the quasi-Bragg regime. The mathematical properties of the Mathieu functions themselves have been explored by many workers. Of relevance for this work is the power-series expansion of Mathieu functions reported by Kokkorakis and Roumelotis Kokkorakis .

### i.3 Motivation

In the experimental applications, it is often desirable to make the interaction time as short as compatible with certain requirements on the efficiency and parasitic phase shifts. This can be achieved by operating in the quasi-Bragg regime. For example, in atomic physics, long interaction times increase losses due to single-photon excitation and also systematic effects in atom interferometers. For fifth order Bragg scattering of cesium atoms in a standing light wave, satisfying the adiabaticity criterion requires interaction times s, see Sec. I.5.2. This exceeds the time available in experiments under free-fall conditions, and would give rise to huge losses by single-photon excitation. Moreover, the very sharp Bragg condition in the pure Bragg regime means that scattering happens only if the incident waves are within a very narrow of the velocity distribution of a thermal sample, which may mean that a large fraction will not be scattered at all. However, operation in the quasi-Bragg regime requires a theoretical calculation of the losses and phase-shifts encountered, and strategies to minimize them. This is best done by analytic equations for these parameters, which allow one to easily see which parameters have to take which values.

Unfortunately, in this regime the population of the diffraction orders is particularly difficult to calculate Keller and so the existing analytic theory of the quasi-Bragg regime is restricted to square envelope functions or low orders, or both. Moreover, even then, using the known formalisms it is hard to obtain power-series approximations of parameters like the effective Rabi frequency in terms of the interaction strength.

The aim of this paper is to present an analytic theory of the quasi-Bragg regime that allows the treatment of arbitrary scattering orders and envelope functions of the scattering potential. To do so, we develop a systematic way of obtaining more and more accurate solutions of the Schrödinger equation that starts from the usual adiabatic approximation. This allows us to calculate the population of the diffraction orders, including losses to unwanted outputs and phase shifts. We can thus specify the minimum interaction time and the maximum interaction strength that yield losses which are below a given level, and consider the influence of the pulse shape. It will turn out that efficient scattering can be maintained with interaction times that substantially violate the adiabaticity criterion, in agreement with experiments Keller . For example, we show that fifth order scattering of Cs atoms still has negligible losses and phase-shifts for interaction times on the order of 10 s if the pulse shape of the light is appropriately chosen. This minimum interaction time even decreases for higher scattering order. We will restrict attention to on-resonant Bragg diffraction, neglecting any initial velocity spread that the atoms may have. This can certainly be a good assumption for experiments using Bose-Einstein condensed atoms, but also for a much wider class of experiments: The minimum interaction time that will still lead to low losses will turn out to be roughly given by , where is the recoil frequency and the Bragg diffraction order, see Eq. (74). Such short pulses have a Fourier linewidth that is on the order of . This means that a velocity spread on the order of the recoil velocity cannot be resolved, even though the velocity selectivity increases due to the multiple photons scattered. The preparation of atoms having a velocity spread of 1/100-1/10 of the recoil velocity is standard practice in atomic fountains.

While our theory will be stated in the language of atomic physics, with an eye on applications in atom interferometry, it can be adapted to Bragg diffraction in all fields of physics.

### i.4 Outline

This paper is organized as follows: In Sec. I.5, we describe our basic Hamiltonian and the conventional theory for the Bragg and the Raman-Nath regime. In Sec. I.6, an exact solution for rectangular envelope functions will be presented that uses Mathieu functions. In Sec. II, we find a general form for the corrections of the effective Rabi frequency which is valid for arbitrary scattering orders. In Sec. III we present our method for calculating the phase shifts and losses, adiabatic expansion. In Sections IV and V, we consider square and Gaussian envelope functions and give a practical example of high-order Bragg scattering of Cs atoms.

### i.5 Problem

In the remainder of this Sec. I, we define the basic problem and review the basic theory of the adiabatic and the Raman-Nath case (as described, e.g., in Meystre ) and of the Mathieu equation approach. This is to define the notation and for the reader’s convenience. The reader already familiar with this may want to proceed to the following sections, which describe the new results of this paper.

Consider scattering of an atom of mass by a standing wave of light along the direction. Ignoring effects of spontaneous emission, the Hamiltonian describing the interaction of the atoms with the standing wave having a wavenumber is (in a frame rotating at the laser frequency )

(1) |

where and is the detuning. The Rabi frequency may in general be time-dependent. For the purpose of this introduction, we will assume it to be constant. In the later sections of this paper, we shall be interested in the effects of different pulse shapes, however. Substituting

(2) |

into the Schrödinger equation yields the coupled differential equations

(3) |

where the dot denotes the time derivative. For large compared to the linewidth of the excited state (and thus also ) and the atoms initially in the ground state, we can adiabatically eliminate the excited state:

(4) |

This equation with its periodic potential is invariant under a translation by an integer multiple of . Applying the Bloch theorem, we can look for solutions having constant quasi-momentum; in particular, we can restrict attention to the case of vanishing quasi-momentum. For constant , this is a Mathieu equation for which exact solutions are known; this formalism will be described in Sec. I.6 and developed further in Sec II. If we let

(5) |

and use , we obtain

(6) |

where we have introduced the two-photon Rabi frequency

(7) |

and the recoil frequency

(8) |

This can only hold if for all

(9) |

[Since this equation couples only odd or even momentum states, respectively, we can and will look for solutions that have either the odd or even terms zero. In view of this the use of both even and odd indices may seem unnecessary, but will have advantages when we consider Bragg diffraction.] The theoretical description of Bragg diffraction is relatively simple in the short-interaction limit (the Raman-Nath regime) and in the case of an infinite scatterer, the Bragg regime.

#### i.5.1 Raman-Nath Regime

The Raman-Nath regime is defined as the case of very short interaction time, so that the kinetic energy term is negligible against the resulting energy uncertainty. Equation (9) reduces to

(10) |

which we have simplified by shifting the energy scale by . Since these equations only couple states which differ by an even multiple of the momentum , we can restrict attention to even indices . They can be satisfied by Bessel functions:

(11) |

At , this solution has all atoms in the zero momentum state . For , the probability to find the atom to have a transverse momentum is . The Raman-Nath approximation holds provided that (because then a high energy uncertainty justifies our neglect of the kinetic energy). Clearly, the transfer efficiency for any particular is limited. For example, the maximum probability to find the atom in the ground state after scattering 2 photons is approximately 0.34.

#### i.5.2 Bragg Regime

For the Bragg regime, we take into account the kinetic energy term and work in configuration space. We now assume initial conditions and for . To simplify, we subtract a constant offset from the energy scale. Eq. (9) now reads

(12) | |||||

Energy conservation will favor transitions from , if the processes are sufficiently slow. This is the result of the adiabatic elimination of the intermediate states ( and ): If

(13) |

for all , we can assume that the th equation is always in equilibrium with . Then, for example,

(14) |

Relations like this can be used to successively eliminate all intermediate states. With

(15) |

we obtain

(16) |

(where we have removed a constant light shift term). This can be readily solved:

(17) |

For a time-varying ,

(18) |

This is an exact solution of the adiabatic equations of motion Eqs. (I.5.2) for real , as can be verified by insertion. If the integral appearing in the trigonometric functions is equal to (a “-pulse”) , all of the population ends up in the final state; if it is (a “-pulse”), half of it.

While operation in the Bragg regime is lossless, it requires relatively long interaction times. In the previous section, we used the condition , which translates into

(19) |

This is for , but drops rapidly, e.g., for .

For later use, we consider the case of complex , where is the argument of . Hermiticity then requires us to use in the second of Eqs. (I.5.2). The solution of these equations for constant is

### i.6 Mathieu equation approach

For constant , we can apply the method of separation of variables to Eq. (4). We are looking for a solution of the form

(20) |

to obtain

(21) |

with being the separation constant. The second equation is the Mathieu equation. By using ,

(22) |

it can be brought to the standard form

(23) |

This is an eigenvalue equation in . The eigenfunctions can be expressed by Fourier series

(24) |

The eigenvalues associated with the even functions are denoted , those associated with odd functions are denoted Abramowitz . Insertion of these series into the Mathieu equation allows to determine the eigenvalues and the Fourier coefficients Abramowitz ; Rhyshik . This is tedious, but tables Abramowitz and standard numerical routines allow to find numerical values easily.

In this paragraph, we use units with . For an atom initially in a pure momentum state with , we express the wave function as a series of Mathieu functions:

(25) |

This is possible, because the functions and form a complete orthogonal set Abramowitz . The coefficients of the expansion are thus given by the Fourier coefficients of the Mathieu functions. Once the they are known, we can write down the amplitude of finding the atom with a momentum at a later time:

(26) |

(The has a minus in the exponent because this is a reverse Fourier transform.) While in general this is a very complicated function of , for low values of only and will be large. If we neglect all others,

(27) |

The plus sign is for , the minus sign for . Thus, the atoms will oscillate between and with an effective Rabi frequency .

For an explicit example, let the two photon Rabi frequency be for and 0 otherwise. Suppose further that for the atom is in an initial state having momentum . The coefficients are most easily obtained by numerical calculation of the Fourier integral.

The amplitude of finding the atom with a momentum at a later time is given by Eq. (I.6):

(28) |

For the population in the initial state, we find

(29) | |||||

#### i.6.1 Losses

The solution (Fig. 1) oscillates quickly around the mean as given by Eq. (I.6). The frequencies of oscillation are relatively large compared to the effective Rabi frequency and depend on the pulse amplitudes. Thus, observing them requires very accurate timing and control over the pulse amplitudes. Especially, because of interference fringes in optical setups, it is hard to achieve an amplitude stability better than about 1%. Thus, these oscillations may be hard to observe in practice and the transfer efficiency very close to one at some of their peaks is not useful in practice. Most of the time, the population that can practically be achieved will thus be more close to the mean value as given by Eq. (I.6). In our previous example, this reaches a maximum of 0.914, i.e., 8.6% of the population are not transferred to the final state.

#### i.6.2 Phase shifts

In this picture, the initial and final momentum states have the same energy. If they were freely propagating (no interactions with neighbor states), their wave function should thus exhibit the same phase factor . The interactions, however, cause a difference of the phase which can be seen by considering the ratio of the amplitudes

(30) |

which can be calculated in a straightforward way. The phase is most conveniently discussed in terms of , by subtracting the phase of which is expected in the pure Bragg regime, compare Eq. (17). As shown in Fig. 2, this is an oscillating function of time. Around the time for a -pulse, the amplitude of the oscillation of the relative phase changes are given by . However, since the phase is an oscillatory function, there are instances where the phase is larger or vanishes exactly.

For the practical application of this in atom interferometry, where this relative phase adds to the phase to be measured, two remarks of caution are appropriate: (i) As mentioned before, making use of the theoretically exact vanishing of the phase shift at particular times requires very accurate timing and control over the pulse amplitudes. (ii) In certain interferometer geometries, equal parasitic phases of subsequent beam splitters cancel out. However, since this depends sensitively on very small amplitude changes of the pulses (that would affect the times of the zero-crossings of the wiggles in Fig. 2), the cancellation is impaired. Thus, in practice, it may be impossible to rely on this exact vanishing or cancellation.

## Ii Efficient method for calculating the effective Rabi frequency

The adiabatic elimination process yields a simple equation for the effective Rabi frequency . However, it is inappropriate in the quasi-Bragg regime. Higher order corrections, that tend to reduce , will have to be taken into account. In this section, we shall determine these corrections for arbitrary scattering orders. A calculation for second order scattering has been published previously in Duerr .

The natural approach to determine is via the eigenvalues in the Mathieu equation formalism. This approach can yield to any desired accuracy (for constant ) by calculating the eigenvalues of the matrix representing an appropriately large subset of the infinite set of equations. In this section, we are looking for an efficient iterative method to calculate , for constant as well as time-varying . This is, at the same time, a method for calculating the difference of the eigenvalues of the Mathieu equation. This method is based on an extension of the idea of adiabatic elimination.

We express the equation of motion Eq (I.5.2) as the matrix equation

(31) |

where the vector contains the

(32) |

In analogy to the above solution in the Mathieu equation formalism, we are interested in a solution that is slowly varying in time, in which the population is mainly consisting of and . The evolution of the other states is governed by the equation

(33) |

where is the vector with removed, is with rows and columns of removed, and

(34) |

Suppose for now that adiabatically follows , i.e., . thus can be expressed as functions that are linear in , and thus (but not necessarily ):

(35) |

Let us define

(36) |

that is,

(37) |

In

(38) |

we apply Eq. (37) to replace the and obtain the analogy to Eq. (I.5.2),

(39) |

where

(40) |

An analogous computation leads to the expansion for ,

in analogy to Eqs. (I.5.2), where is the effective Rabi frequency. The leading order in of obtained this way is identical to the one obtained in previous section, see Eq. (15).

Although initially the fast varying was set to zero by the adiabaticity assumption, we now take into account a slowly varying part due to the adiabatic following, which has similar time scale as the initial and final states and thus cannot be ignored in Eq.(35). In the remainder of this section, refers to this slowly varying part only. To first order, can be approximated from :

(41) |

Inserting into Eq. (35),

(42) |

Since is a function of and , the time derivatives as well as still unknown corrections of . Therefore, care must be taken to properly separate various orders of corrections. We expand and as and , where () are an order of magnitude larger than () and are functions of and : contain

(43) | |||||

We expand

(44) | |||||

Eq.(33) thus becomes

(45) |

Since the time derivative decreases one order of magnitude for each increase in , the -th order in Eq.(45) is

(46) |

Thus,

(47) | |||||

We are now ready to describe an iterative procedure to obtain to any desired order: We start from as defined in Eq. (35). Each component is known as a linear combination of , and therefore are also known.

Now suppose that are known for . In the last Eq. (II), each can be expressed as a component of which, by Eq. (43), is known as a linear combination of . We insert this into Eq. (II) to obtain , again as a linear combination of . The coefficients of this linear combination are the . The process can now be iterated to obtain the next higher order.

The efficiency of this method is, in part, due to the fact that each order can be computed using the same inverse matrix . (Moreover, is a tridiagonal matrix, which helps in computing the inverse.) to -th order is thus obtained by plugging into Eq.(38) and finding the coefficient of as in Eq.(39).

Note that and in principle are infinite-dimensional. However, for obtaining the effective Rabi frequency to order , it is sufficient to include the initial and final states, states in between, and nearest neighboring states on each side, i.e., . Including more states yields the same result.

With this method, we explicitly calculate the effective Rabi frequency for Bragg diffraction orders of . They can be given as power series in and :

(48) | |||||

At first, this results in a list of numerical values for the coefficients and for each . However, closed expressions as function of can be found, which are listed in appendix A.

Eq.(48) also allows us to give validity conditions for the simple adiabatic elimination method presented in Sec. I.5.2. For this to be a good approximation, the corrections should be much less than 1. Thus, we obtain

(49) |

For large , the first of these conditions translates into , which is actually larger than the one given by the adiabaticity condition .

### Population in other states

Summing up the population of states other than at the end of a pulse (where ), we obtain

(50) | |||||

The population lost into other states after the pulse is switched off, when , vanishes. The method presented in this chapter is not suitable for obtaining those losses, because the states other than have been assumed to adiabatically follow their neighbors, and the losses are a non-adiabatic phenomenon.

However, as long as , the effect of the losses on and thus can be neglected.

This iterative method, although powerful for calculating , does not approach an exact solution. It can be seen from Eq.(39) that this method gives no wiggles in the sinusoidal change of the initial or final state population, while there are fast variations in the exact solution of a square pulse as shown in Mathieu function section. However, it approaches the solution for the initial and final state as averaged over the high-frequency wiggles, Eq. (I.6) and thus predicts the correct effective Rabi frequency.

## Iii Adiabatic expansion

To investigate the losses, corrections to the adiabatic method must be calculated. We will relabel the results of the adiabatic method as . They represent the first order adiabatic approximation. We now want to calculate corrections to the population of the states,

(51) |

where to first order only the initial and final state are nonzero. For calculating the second order, we insert into Eqs. (I.5.2). Inserting from Eq. (17), we obtain population of the levels next to the initial and final states to second order

(52) | |||||

. These are all states for which . The process can be iterated: From the , corrections can be obtained, and so forth. Here and throughout, we shall drop the superscript as long as no confusion arises.

These inhomogenous equations can be solved by standard methods, such as variation of the constant or a Green’s function:

(53) | |||||

### iii.1 Losses

We are mainly interested in the population . For that, we can take out a phase factor and note that . The absolute squares of these give the population in the neighboring states. These are closed, analytic expressions for the losses arising in the second order. As an example for a third order correction,

(54) | |||||

### iii.2 Phase shifts

To 3rd order, both contribute to :

(55) | |||||

[The has been omitted because it is one throughout the integration range.] Since this is a complex number, the phase of