Artificial Gauge Fields and Spin-Orbit Couplings in Cold Atom Systems

# Artificial Gauge Fields and Spin-Orbit Couplings in Cold Atom Systems

Zhang, Junyi Département de Physique, l’École Normale supérieure, 24 rue d’Ulm, 75005, Paris, France
September 25, 2019

## I Introduction

### i.1 Berry Phase

The dynamics of quantum physics was represented by unitary operators. All the observables are connected to their expectation values of the corresponding operator sandwiched by state vectors and their dual. In most cases, physical realities depend only on the modular square of the wave function (by Born’s probabilistic interpretation Born1954 ) rather than the wave function itself. Therefore it is not important if the wave function is multiplied by some phase factor. Nonetheless, sometimes the results do depend on the phase factors. One used to argue that those results depending on the phase factor are not gauge invariant, thus they are not observable directly, and these factors can be gauged out by a proper gauge transform. Therefore, the effects of the phase factor has been long neglected. While historically, many effects, as A-B effect ABeffectES  ABeffect  ABExpTonomura1  ABExpTonomura2 , and nuclear motion in molecules Mead1979  MeadRevModPhys were predicted and observed which are clear clues that quantum phase does play some “observable” roles in physical reality. It was Berry Berry1984 who pointed out the importance of phase factor in a adiabatic circular evolution BornFock  Messiah  BOApprox of a quantum state, which is now often called Berry phase. (It is also called Mead-Berry Phase. One also uses geometric phase as synonym.)

Simon Simon1983 pointed out the mathematical structure of the Berry phase. He attributed Berry’s idea to the underlying holonomical structure of vector bundle. This idea stimulated understanding the topological characters of quantum Hall effect TKNN , which developed to another exciting field in condensed matter physics. Wilczek and Zee WilczekZee generalized the ideas of Berry and Simon to non-Abelian gauge fields and proposed possible methods to observe these effects ZeeQuadrupole .

In cold atom systems, as the motion of the atom can be considered adiabatic, then a nontrivial phase factor is induced by light-atom couplings which were well known in quantum optics. Here we shall follow ideas of Berry, Simon, and Wilczek and Zee to develop the formulations of gauge structure induced by adiabatic evolution. The coupling between light and beam will be analyzed in the following section by introducing Jaynes-Cumming’s model. The internal degrees of freedom play a role of pseudo-spin. The induced artificial non-Abelian gauge fields will naturally induce the spin-orbital coupling in a system, which has important and interesting features.

In Ref. 8, Berry first considered a non-degenerate state evolving adiabatically under a time dependent Hamiltonian that is parameterized by a circuit. In addition to the dynamical phase factor , the state acquires another phase factor , where , is the parameter. If the evolution path is not closed, this phase factor can be gauged away. While if the Hamiltonian gets back to its initial value, then this additional phase accumulated along the circuit is

 (1)

which is time independent under adiabatic approximation. Since , is real. By Stokes theorem, we have

 γ(C)=−Im∮C⟨ψ(R)|∇R|ψ(R)⟩⋅dR=−Im∬S(∂S=C)dS⋅∇×⟨ψ(R)|∇R|ψ(R)⟩=−Im∬S(∂S=C)dS⋅∇×A, (2)

which is gauge invariant. Simon Simon1983 point out that the integrand is in fact a two form relating to Chern class, which characterize the topology of space of . It also indicates the differential structure of the bundle Simon1983  WuYang .

In analog to the gauge structure of electromagnetic field, Yang and Mills generalized gauge to non-Abelian case YangMills , which is fundamental to the standard model of particle physics. Wu and Yang also studied nonintegrable phase factor in both Abelian and non-Abelian cases WuYang . Similarly, Berry and Simon’s idea was also generalized to non-Abelian case WilczekZee . Ref. 14 considered the “phase factor” of a degenerate sub-manifold with dimension larger than one, where gauge potential becomes a matrix. In Ref. 14 they resort to some symmetry of the system to guarantee the degeneracy that is key to the appearance of non-Abelian gauge field. However, in cold atom system, we can achieve the degenerate submanifold by dark states of atoms interacting with the lights and therefore the non-Abelian gauge fields emerges naturally.

### i.2 Jaynes-Cummings’s Model

In 1960s, Jaynes and Cummings introduced a model to describe the interactions of a two-level atom with cavity modes of electromagnetic field, which is now named after them JCModel1963  Cummins1965 . This model proves to be simple but precise enough to describe the actual experiments of Cavity Quantum ElectroDynamics (Cavity QED or CQED). It also provides a toy model for studying the artificial gauge fields in cold atom systems DalibardArtificialGauge .

According to Janes and Cummings (Ref. 18), we may consider a system of a two-level atom and singled mode of a cavity. The free electromagnetic field subjected to the boundary condition of the cavity can be quantized as

 A=A(^ϵa(t)eip⋅x+^ϵ∗a†(t)e−ip⋅x), (3)

where and are complex polarization of the field, and are creation and annihilation operators of the photon, and is the normalization factor depends on the geometry and boundary conditions of the cavity. We denote and for the internal degree of freedom for the two-level atom. Thus the noninteracting bases are .

The atom coupled to the electromagnetic field through the interaction Hamiltonian of the form

 Hint=−^D⋅E, (4)

where is the dipole operator acting on the states of the atom 111Firstly, in Ref. 18, they calculated results for both quantized fields and semiclassical approaches. Here we may well start from the most general quantized forms, but most consequences also work in semiclassical scheme, so we omit the over the fields, which will not cause any ambiguity from the context. Secondly, this form of coupling is connected to the form of by Fierz-Pauli transformation FierzPauli  CCTQEDV_2 .. For brevity, we may assume that the entries of the dipole operator is of the form or . (In general the dipole matrix is of the form . )

There are two kinds of couplings: one couples and ; the other couples and . The former is related to the process of de-excitation of the atom by emitting a photon or excitation by absorbing one photon. The later is in the contrast. It seems to be a violation of the energy conservation for the second term but it is not the case as we do not take the motions of the atom in real space in to account 222Comments in the lectures by Prof. ZHANG Li of IASTU, China and Prof. Jean-Michel Raimond of UPMC, France. According to Ref. 18, the second kind of coupling is negligible (as the detuning is not far away from the resonance it is equivalent to the seminal Rotating Wave Approximation (RWA) HarocheRaimond ). With RWA, the coupling Hamiltonian decoupled into blocks, with the ground state unshifted, coupling to , to … Then the Hamiltonian of the coupled block is of the form

 (Ee+(n−1)ℏω√nℏα√nℏα∗Eg+nℏω)=(E1V12V21E2), (5)

where is the coulpling of the two quasi-degenerate levels in the submanifold (when detuning is not large). Therefore the system can be solved exactly by diagonalizing the block Hamiltonian. The new eigenenergies are

 E±=12((E1+E2)±√(E1+E2)2−4(E1E2−|V|2))=(E1+E2)2±12√(E1−E2)2+4|V|2=(Ee+Eg)2+(n−12)ℏω±12√(Ee−Eg−ℏω)2+4nℏ2|α|2. (6)

We can also obtain the time evolution of the states with this Hamiltonian by solving time-dependent Schrödinger equation

 iℏ(˙a˙b)=(E1V12V21E2)(ab), (7)

where is the amplitude of state and is that of . In Ref. 18, they used this model for beam maser, so they considered an atom originally in excited state decaying to the ground state , where, specially, n=0 corresponds to the amplitude of spontaneous emission. Eliminate and its derivative, we obtain 333Take the derivative of the first line of Eq. 7 with respect to time; substitute second line for , and first line for .

 −ℏ2¨a=iℏ(E1+E2)˙a−(E1E2−|V|2)a. (8)

Let (test solution), and substitute for Eq. 8, we have

 (ℏω)2−ℏω(E1+E2)+(E1E2−|V|2)=0. (9)

It is easy to observe that this is exactly the secular equation we have solved for the eigen-energies, therefore , and the general solution is where and are to be determined by initial conditions.

Since at , the atom is in excited state, i.e. . Substitute back to Eq. 7, we obtain the initial conditions for their first derivatives . We achieve the solutions

 a=e−i(E1+E2)t2ℏ(cosΩ′t/2+iΔℏΩ′sinΩ′t/2),b=e−i(E1+E2)t2ℏ(−2iV∗ℏΩ′sinΩ′t/2),|a|2=cos2Ω′t/2+Δ′2ℏ2Ω′2sin2Ω′t/2,|b|2=4|V|2ℏ2Ω′2sin2Ω′t/2, (10)

where is called detuning and is the generalized Rabi frequency.

 |a|2≈cos2Ω′t/2,|b|2≈sin2Ω′t/2, (11)

i.e. the atom oscillates between and of frequency . When , take the limit of , by Theorem 1 (see Appendix)

 |b|2=4|V|2ℏ2Ω′2sin2Ω′t/2=2|V|2Tℏ2πδ(ωeg−ω)=2πℏ|V|2Tδ(Ee−Eg−ℏω), (12)

which is exactly Fermi’s golden rule. In fact, in Ref. 18, Jaynes and Cummings proposed their model to calculate the noise figures of maser beyond Fermi’s golden rule.

Now we introduce the semiclassical version of the Jaynes-Cummings model that inherits most features of the full quantum version and is easier to apply to real systems of CQED and cold atoms. The only difference is that now electromagnetic field is classical, i.e. . Then the interaction Hamiltonian in Eq. 4 is

 (13)

where is called Rabi frequency. As for the Hamiltonian of the two-level atom, by choosing a proper point of , we may write it as

 Hatom=12ℏωeg^σz, (14)

where is the energy difference of two internal levels. The total Hamiltonian is time dependent. Since the system is not far from resonance, we can change to a “rotating” frame to eliminate the time dependence by choosing an “interaction” picture properly.

 H=12ℏωeg^σz+ℏΩcos(ωt+ϕ)^σx=12ℏω^σz+12ℏ(ωeg−ω)^σz+ℏΩcos(ωt+ϕ)^σx=H0+H1, (15)

where and . Then

 HI=eiH0t/ℏH1e−iH0t/ℏ=12ℏΔ^σz+ℏΩcos(ωt+ϕ)×ei12ωtσz(^σ++^σ−)e−i12ωtσz=12ℏΔ^σz+ℏΩcos(ωt+ϕ)(eiωt^σ++e−iωt^σ−)=12ℏΔ^σz+12ℏΩ[e−iϕ^σ++eiϕ^σ−+ei(2ωt+ϕ)^σ++e−i(2ωt+ϕ)^σ−]≈12ℏΔ^σz+12ℏΩ[e−iϕ^σ++eiϕ^σ−]=ℏ2(Δe−iϕΩeiϕΩ−Δ)=ℏΩ′2(cosθe−iϕsinθeiϕsinθ−cosθ) (16)

where , , , and .

The first term of always commutes with . We used and , or Baker-Campbell-Hausdorff formula in the second step. Omitting the fast oscillating terms in the fourth step is so called RWA. The interacting Hamiltonian can also be written as , where , which has an easy geometrical interpretation. In the interaction picture, the Bloch vector corresponding to the state rotates around axis in direction of by angular velocity . For , the result of semiclassical approach agrees with the one obtained by full quantum method.

So far, we neglected the orbital motion of the atom in the real space. The phase of the laster and detuning angle are space dependent. When the atom moves in the light field, they will influence its orbital motions. We shall show bellow that this kind of influence can be described as a vector potential in adiabatic limit. For two level system, with initial condition in one of the internal eigenstate, this vector potential is an Abelian gauge potential; the atom motion was modified as if there is a magnetic field.

## Ii Artificial Gauge fields in Cold Atoms

### ii.1 Tow Level Atom in a Light Beam

As we have shown in the previous sections, two-level atoms can be described by Jaynes-Cummings’s model. We may denote {, } a basis of the two-dimensional Hilbert space of the internal degree of freedom. The coupling of the internal degree of freedom with the light field under RWA can be described by a 2 by 2 matrix of the form

 U=ℏΩ′2(cosθe−iϕsinθeiϕsinθ−cosθ), (17)

where is the generalized Rabi frequency, and are two position dependent angle parameters (as shown in Eq. 16). Therefore the total Hamiltonian of the atom moving in the light beam is

 H=(P22M+V)^I+U, (18)

where is the mass of the atom, is the total momentum operator and is the identity in internal space.

At any point , has two eigenstates

 |χ1⟩=(cos(θ/2)eiϕsin(θ/2)),|χ2⟩=(−e−iϕsin(θ/2)cos(θ/2)), (19)

with eigenvalues and respectively 444For brevity of notions we may denote simply for here as long as there is no ambiguity of the meaning of Rabi frequency; and in most cases we consider in this article are of near resonance . . A remark concerning state bases of and deserves to be emphasized here. We have shown in the previous section, when the frequency of the laser is near resonance and dipole coupling is strong, the off diagonal elements in Eq. 17 drive the internal states of the atom oscillating back and forth with absorbing and emitting a photon in resonance. Therefore, and are no longer convenient basis to describe the internal state of the atoms. On the contrary, states given in Eq. 19 are eigenvectors of , thus we can diagonalize by a unitary transform from the basis of to , and then the total Hamiltonian is diagonal in internal space. This perspective is also called “Dressed-Atom Approach” Dalibard1985 . When the coupling decrease to zero, the dressed states come back to and .

However, from Eq. 19, we may observe these dressed states are position dependent. When the atom is coupling with the field, they are not degenerate (separated by a finite gap ). Under adiabatic limit, if the atom is in one of the dressed eigenstate and the orbital motion of atom is slow enough, it shall always remain in the submanifold of that state, while position-dependent projections will contribute to the orbital wave function a phase factor in addition to the ordinary dynamical phase factor, which leads to the gauge field at last.

Now let us expand the total wave function in term of these local basis as

 |Ψ(r,t)⟩=∑j=1,2ψj(r,t)|χj⟩, (20)

where are time dependent wave functions 555In case the clumsy summing signs, we may adopt Einstein’s convention below; while the summing signs are sometimes written explicitly for carefulness. Both sub- and superscripts are used for convenience; since we do live well in three dimentional euclidean space with standard metric, they are in fact the same thing. . Apply the total Hamiltonian (Eq. 18) to our total wave function, we have our time dependent Schrödinger equation

 iℏ∂∂t|Ψ(r,t)⟩=H|Ψ(r,t)⟩=[(P22M+V)^I+U]|Ψ(r,t)⟩. (21)

Since the eigenstates of are position dependent, may act on both part of the wave function, differentiating with respect to . The second term represents trap potential that is diagonal both in internal space and real space (we shall assume that and experience same trap potential). The third term acts on the internal eigenstates and gives the corresponding eigenenergies.

In the first step, we shall calculate the total momentum operator acting on the total wave function .

 P|Ψ(r,t)⟩=−iℏ∇(∑j=1,2ψj(r,t)|χj⟩)=−iℏ∑j=1,2[∇ψj(r,t)]|χj⟩+ψj(r,t)(∇|χj⟩)=−iℏ∑j,l=1,2[∇ψj(r,t)]|χl⟩⟨χl|χj⟩+ψj(r,t)|χl⟩⟨χl|∇|χj⟩=∑j,l=1,2[(pδlj−Alj)ψj(r,t)]|χl⟩ (22)

where is of the form of momentum operator but not acting on the spinor, and is the gauge potential, or in a sense of mathematics, the “connection”.

Then we can calculate the kinetic energy.

 P22M|Ψ(r,t)⟩=12MP⎛⎝∑j,l=1,2[(pδlj−Alj)ψj(r,t)]|χl⟩⎞⎠=12M∑j,l,m=1,2({pδml[(pδlj−Alj)ψj(r,t)]}|χm⟩−{Aml[(pδlj−Alj)ψj(r,t)]}|χm⟩)=12M∑j,l,m=1,2({(pδml−Aml)[(pδlj−Alj)ψj(r,t)]}|χm⟩), (23)

where still only acts on the “orbital” part. Now, projecting it to the eigen space of , and rewrite the formula in the form of matrix,

 (24)

where is a matrix vector, is a two component wave function of orbital part, and the subscript at down right corner indicates the n-th component of the wave function.

The second term and the third term of Eqn. 21 is easy to obtain and project to

 (25)

where . Now let us calculate matrix vector . According to our definition , the a-th component of is

 ^Aa=iℏ⟨χl|∂a|χj⟩=iℏ(⟨χ1|∂a|χ1⟩⟨χ1|∂a|χ2⟩⟨χ2|∂a|χ1⟩⟨χ2|∂a|χ2⟩).

We shall always use for the indices of vector components, and for indices of matrix. Its entries are

 iℏ⟨χ1|∂a|χ1⟩=iℏ(cosθ2e−iϕsinθ2)⎛⎝⎛⎝−sinθ2eiϕcosθ2⎞⎠[∂aθ2]+(0eiϕsinθ2)[i∂aϕ]⎞⎠=−ℏsin2(θ2)[∂aϕ]=ℏ2(cosθ−1)[∂aϕ]iℏ⟨χ2|∂a|χ2⟩=iℏ(−eiϕsinθ2cosθ2)⎛⎝⎛⎝−e−iϕcosθ2−sinθ2⎞⎠[∂aθ2]+(e−iϕsinθ20)[i∂aϕ]⎞⎠=ℏsin2(θ2)[∂aϕ]=ℏ2(1−cosθ)[∂aϕ]iℏ⟨χ1|∂a|χ2⟩=iℏ(cosθ2e−iϕsinθ2)⎛⎝⎛⎝−e−iϕcosθ2−sinθ2⎞⎠[∂aθ2]+(e−iϕsinθ20)[i∂aϕ]⎞⎠=−iℏe−iϕ[∂aθ2]+iℏe−iϕcosθ2sinθ2[i∂aϕ]=−ℏ2e−iϕ(i∂aθ+(sinθ)∂aϕ)iℏ⟨χ2|∂a|χ1⟩=(−iℏ⟨∂aχ1|χ2⟩)∗=(iℏ⟨χ1|∂a|χ2⟩)∗. (26)

So

 ^Aa=⎛⎜⎝−ℏsin2(θ2)[∂aϕ]−ℏ2e−iϕ(i∂aθ+(sinθ)∂aϕ)ℏ2eiϕ(i∂aθ−(sinθ)∂aϕ)ℏsin2(θ2)[∂aϕ]⎞⎟⎠=−ℏ2((sinϕ)∂aθ+(sinθcosϕ)∂aϕ)^σx+ℏ2((cosϕ)∂aθ−(sinθsinϕ)∂aϕ)^σy−ℏsin2(θ2)[∂aϕ]^σz (27)

who is Hermitian.

Furthermore

and

 ^A2=3∑a=1^Aa2=(ℏ2sin2(θ2)|∇ϕ|2+ℏ24|∇θ|2)^I^A22jk=^Ajl⋅^Alk (28)

Thus, with Eq. 20, and projecting the left hand side of Eq. 21 to , we have

 (29)

where adiabatic assumption has been used666So far, every thing is exact. The adiabatic assumption just suggests that under this projection, if initial state is one of the energy eigenstates, it will always remain in this submanifold.. On the other hand, with the help of Eqn. 24 and  25, we have

 ⟨χn|H|Ψ(r,t)⟩=⟨χn|[(P22M+V)^I+U]|Ψ(r,t)⟩=⟨χn|P22M|Ψ(r,t)⟩+⟨χn|V|Ψ(r,t)⟩+⟨χn|U|Ψ(r,t)⟩=((p^I−^A)22MΦ(r,t))n+(V(r)Φ(r,t))n+(ℏΩ2σzΦ(r,t))n. (30)

Combining Eq. 29 and  30, we arrive at

 (31)

It is obvious to observe that the last two terms on the right hand side of Eqn. 31 are diagonal, while carefulness is needed to calculate the first term, since does not commute with . Without lose of generality, we may observed the first component of the wave function. Although is diagonal, itself is not, thus two components may be well coupled through the off-diagonal entries of .

We shall now calculate the first term explicitly 777One should pay attention to the calculations. is not only diagonal matrix, but also a differential operator who acts both on wave function and . We shall well keep their order during the calculation..

Notice that

 (32)
 (p^I−^A)22M(ψ1(r,t)ψ2(r,t))=12M(p2−p⋅^A−^A⋅p+^A2)(ψ1(r,t)ψ2(r,t))=12M⎛⎝(p−A11)2−A11⋅A11+(^A2)a11−(p⋅A12+A12⋅p)−(p⋅A21+A21⋅p)(p−A22)2−A22⋅A22+(^A2)a22⎞⎠(ψ1(r,t)ψ2(r,t)), (33)

With Eq. 26 or Eq. 28, we have diagonal term of Eq. 33 are

 (p−Aii)22M+W,

where . The off-diagonal terms

 (34)

couple the two components of the wave function .

If we further assume that initially , and take the adiabatic limit, we can arrive at Schrödinger equation for the wave function in subspace

 iℏ∂ψ1∂t=[(p−A)22M+V+ℏΩ2+W]ψ1, (35)

where and plays a role of artificial Abelian gauge field.

A remark concerning the adiabatic approximation deserves here. By definition, off-diagonal elements of connection 888, see Eq. 8 of Ref. 8. Thus the coupling term in Eq. 34 is of order of that vanishes as first order in adiabatic limit. Since the initial value of is zero, its amplitude also vanishes as first order, and its contribution to through the off-diagonal terms vanishes as second order under adiabatic limit, which validates our adiabatic approximation. Therefore projecting to one of the submanifolds, we arrive at the non-Abelian gauge theory as shown in Eq. 35. This estimation also gives a criteria for adiabatic approximation in real experiments. The typical orbital energy changes of order that should be much smaller than the gap of the dressed-states. This means that our atom should move “slow” enough, which is only a “naïve” sense of adiabatic approximation.

A further remark is that, in fact, our toy two-level dressed-atom model did exhibit full non-Abelian characters that is obvious from Eq. 31 (an example of the transformation of the connection is verified in Proposition 4 in appendix). However, our adiabatic limit guarantees that if initially our system is in an eigenstate of dressed state, it shall always remain in that sub-manifold. But if the gap of two states are not large enough, this simple projection will no longer be valid, while the non-Abelian gauge features in Eq. 31 still hold. More generally, if our system is initially in a sub-manifold spanned by several quasi degenerate states whereas this sub-manifold is well separated from other sub-manifolds by large gaps, which enables us to apply adiabatic projection, then we can construct artificially non-Abelian schemes in our atom-light systems.

Unfortunately, the scheme of our toy model is not very practical. The excited state of the atom may spontaneously decay. The collisions between the atom will also cause some problems. In the following sections, we shall introduce several more practical schemes for realizing artificial gauge fields in cold atoms, and we shall also analyze carefully the problems we may encountered. Nevertheless, the idea is just a simple generalization that shares almost all the key factors that we have shown in our toy model.

### ii.2 One, Two, Three to N…

In previous section, we coupled one ground state to an excited state by dipole interaction, where the bare states of atom are dressed by the light field. When we project to the sub-manifold of dressed states with adiabatic approximation, we obtain an artificial Abelian gauge field induced by the space-dependent Rabi frequency and detuning. We can simply generalize our scheme by coupling two quasi-degenerate ground states to another state. The scheme was shown in Fig. 2, where two state and are coupled to by two laser beams. The laser couplings are subjected to selection rules, so we can choose proper polarization of lasers to control them separately. As shown in previous section, the total coupling Hamiltonian under RWA can be written as

 HI=ℏ2⎛⎜⎝−2δκ∗1 0κ10 κ20κ∗2−2δ⎞⎟⎠, (36)

where are complex space dependent Rabi frequencies, are detunings of photon excitation with respect to the Raman resonance DalibardArtificialGauge . Now we consider the resonant case, i.e. . Rewrite this Hamiltonian in Dirac bra-ket notation

 HI=ℏ2(κ1|e⟩⟨g1|+κ2|e⟩⟨g2|)+h.c.=ℏ2(κ|e⟩⟨B|)+h.c.,

where , and . By this notation, it is obvious that the interaction only couples and together. But now we have a Hilbert space spanned by three independent state vector. So the third state

 |D⟩=(κ2|g1⟩−κ1|g2⟩)/κ, (37)

who is orthogonal and uncoupled to and subspace has an eigenenergy of zero, which is called Dark State. The other two eigenstates span the same subspace of . In this submanifold, the coupling Hamiltonian reduces to a matrix that is completely same as the one we solved in the previous section. By diagonalize the Hamiltonian, we find they are with eigenenergies respectively. And is called Bright State.

Since is well separated from the other two eigenstates by an energy gap of , by adiabatic approximation and projection, it plays the same role as the dressed states in previous section. On the other hand, is orthogonal to , i.e. no population in is , thus atom state is not affected by spontaneous emission. In fact, this remarkable property was already well-known in quantum optics such as subrecoil cooling, Electromagnetically Induced Transparency (EIT) and STImulated Raman Adiabatic Passage(STIRAP). These applications rely on the robustness of with respect to the decoherence caused by spontaneous emission DalibardArtificialGauge .

By little laborious calculation as we have done in previous section, we obtained effective Equation of Motion for orbital part projected to sub-manifold

 iℏ∂ψD∂t=[(p−A)22M+V+W]ψD, (38)

where and are the effective vector and scalar potential induced by the space-dependent dark state.

For pedagogical purposes, we do the calculations in detail again, but by the language of differential form for comparison. The antisymmetrical properties and compact form of exterior differential operator will save us from the tedious component notations and simplify our calculations.

First, project our total wave function to the eigen basis of dressed atom as in Eq. 18

 |Ψ(r,t)⟩=∑j=D,±ψj(r,t)|χj⟩. (39)

Since acts on both part of the wave function, as shown in Eq. 22, we can easily arrive at the EoM of the form in analog to Eq. 31, but the wave function has three components and all the matrices are . In analog to Eq. 33, the off-diagonal terms coupling to the other states are neglected due to the adiabatic approximation. Diagonal terms of and contribute to the vector and scalar potential.

Mathmatically, adiabatic assumption enables us to project the total wave function to the local basis of eigenstates of atom-light couplings. This in a sense define a frame bundle over the adiabatic-parameter manifold (here the adiabatic parameter, also called base space or base manifold in mathematics, is the real position, dimension for the cases discussed in this article). Local frame is -dimensional ( for our toy model, for -scheme, and for the N-pod scheme to be discussed). is the derivative operator. As it acts on the orbital part, it becomes as a differential operator, acting on the wave function, giving a 1-form corresponding to a “vector”. acts on the frame basis by natural derivative of wave function with respect to the adiabatic parameter defining a connection over the frame bundle , and is the connection matrix. The connection matrix is frame basis dependent. A transform of basis will induce a transform of connection matrix which corresponds to the gauge transform physically. Here the frame is a basis of orthonormal wave functions, and we would only consider a transform to another orthonormal basis, i.e. a Unitary Transform. More specifically, in most cases, we need only consider transform. Attaching this group to every point of the adiabatic=parameter manifold gives a principal bundle , correspondingly the structure group also called gauge group in physics. A local gauge transform is a smooth section of the principal bundle inducing a bundle morphism from to itself, i.e. a transform over every fibre of . Physically and mathematically, is gauge dependent or frame dependent and it does not transform as a vector but as 999A more familiar example to physicists might be the Christoffel in general relativity.. Mathematically one can define a homogeneously transforming 2-from associated to the connection

 Ω=dω+ω∧ω, (40)

which is called curvature. Similarly, we can also define a physical quantity corresponding to curvature

 B=dA+1iℏA∧A, (41)

which is gauge field101010This gives an intuitive clue to the definition of non-Abelian gauge field. In fact, Yang and Mills YangMills had tried tedious calculations and conter-intuitive tests, and finally find this generalization to non-Abelian gauge field. However by the analog here, this definition is quite nature, because an observable physical quantity should be gauge independent and transform homogeneously. In simple case, is a 1-form rather than a matrix of 1-form, the second term of vanishes naturally due to the antisymmetrical property of wedge and exactly gives the ordinary electromagnetic field..

Now let us be back from this long deviation to mathematical jargons.111111Although physicists who are not familiar with differential geometry may consider these strange words of mathematics bizarre and unnecessary, the author find it helpful for both mathematician and physicists to learn the language of each other’s. Therefore we include this short mathematics-physics dictionary here to span a bridge between mathematics and physics. In this article, we demonstrated calculations by both ordinary physicists using language and mathematician’s language. It would bring us more convenience by using them properly in specific applications. Readers who need more references in differential geometry may resort to Ref. 60 and  61. For convenience we shall rewrite dark state as

 |D⟩=κ2|g1⟩−κ1|g2⟩√|κ1|2+|κ2|2=eiϕ2|g1⟩−|ζ|eiϕ1|g2⟩√1+|ζ|2, (42)

where . Since now we have no other eigenstates in this degenerate space, we can multiply Eq. 42 by a phase factor to make our life easier. This local gauge transform does not affect curvature, but do changes connection coefficients. For the consideration of consistency, we shall keep our notation, try to calculate the more complex version. is the covariant derivative of the frame basis. Since we have chosen our basis as the dressed states for every point, then the covariant derivative is naturally defined in this way implied by adiabatic assumption (Physically, it is natural for us to assume , states of bare atom, as a flat frame).

 ∇|D⟩=|∇D⟩=−|ζ|d|ζ|1+|ζ|2|D⟩+ieiϕ2dϕ2|g1⟩−eiϕ1(i|ζ|dϕ1+d|ζ|)|g2⟩√1+|ζ|2⟨D|=e−iϕ