Vol.32 (Aug) 2022 | Article no.23 2022
In this paper, we review and discuss the main properties of the time-reversal operator T and its action in classical electromagnetism and in quantum mechanics. In particular, we discuss the relation (and differences) between time-reversal invariance and reciprocity.
In quantum field theory, global and local symmetry invariance play an important and defining role [1]. Some, such as rotation invariance, are continuous symmetries and are described by a Lie group [2, 3] while others, like time reversal symmetry, are discrete ones [4]. Continuous symmetries give rise to conservation laws and the existence of locally-conserved currents as exemplified by Noether’s theorem [1]. For example, the invariance under continuous time translation gives rise to the conservation of energy. However, a system described by a Hamiltonian or Lagrangian density which are invariant under a given symmetry can be found in a state which breaks this symmetry. This brings in the important concepts of explicit and spontaneous symmetry breaking [5, 6]. This happens for time crystals, a new class of systems that recently drew attention, where the invariance under time translation is broken in the same way as the symmetry under continuous space translations is broken in usual crystals [7, 8].
In this pedagogical article, we will narrow down and focus on time-reversal symmetry T, a discrete symmetry related to charge conjugation C and parity P through the celebrated CPT theorem [13]. We will detail some of the properties of T-invariant systems, discuss the notion of reciprocity and its link and differences with time-reversal invariance.
Before the advent of the relativistic theory of the electron [9], the asymmetry between particles with positive and negative electric charges was considered a fundamental one. The success of the relativistic quantum theory has been to understand, through the concept of antiparticle, that there was a symmetry relation between positive and negative charges. In some sense, this could already be seen in the Maxwell’s equations, where the change of sign of the particles preserves the form of the equations, provided that the currents and the electric and magnetic fields also have their signs changed. A unitary operator C has then been built, which realizes the conjugation between positive and negative charges Q→−Q. When the Hamiltonian is invariant under charge conjugation, CHC^{−1}=H, the scattering matrix satisfies [4] \(\mathrm {C}{\mathcal{S}} \mathrm {C}^{-1}={\mathcal{S}}\). The electromagnetic interaction and the strong interaction are C-invariant. On the opposite, the weak interaction breaks the charge conjugation invariance [10].
The parity operator P is a unitary operator which transforms spatial vectors into their opposite: r into −r and p into −p, but leaves invariant pseudo-vectors such as L=r×p. P is a symmetry operator which can be obtained from a mirror reflection followed by a rotation. It can be shown that, if the Hamiltonian of the system is invariant under parity, then the scattering matrix \({\mathcal{S}}\) has the same property. The parity operation is closely related to the orientation of space: it transforms a system of positive orientation into a system of negative orientation. Until 1955, it was believed that the laws of Physics were invariant under parity: Nature could not distinguish left from right. Of course, the existence of non-chiral objects was known, but they were considered as “accidents”. The experiments by Wu and co-workers [11], using the β-disintegration of polarised ^{60}Co, have shown that the weak interaction violates parity. Further investigations lead to the conclusion that this violation is maximal. The electromagnetic and strong interactions, as far as they go, are P-invariant.
Hence, the electromagnetic and strong interactions are invariant under C,P and CP. The status of the weak interaction is quite different: it violates both C- and P-symmetry. It also violates the CP-symmetry, although only slightly [12]. So none of the symmetries C,P or CP is a fundamental invariance of Nature. However, an important theorem states that any quantum field theory which is local and Lorentz-invariant describes phenomena which are invariant under the combined action of C,P and T: this is the celebrated CPT theorem [13]. As no violation of the CPT symmetry has yet been observed, and also because its two hypotheses seem reasonable, CPT is widely believed to be a fundamental symmetry of Nature. It is worth mentioning that the proof of the CPT-theorem is technical, and that there is no intuitive explanation for it that we know of. The observation of a CPT breaking phenomenon would lead to found quantum field theory on completely new grounds.
A consequence of the CPT-theorem is that T is not a fundamental symmetry of Nature: it is broken by the weak interaction. However, the strong and electromagnetic interactions are CP-invariant, and thus also T-invariant.
In the following, we will focus on the electromagnetic interaction, which dominates all others from the atomic to the macroscopic scale. The consequences of the T-invariance of this interaction have been studied in detail by Onsager [14]: it implies restrictions on the behaviour of a system. This study applies only if the whole system is taken into account, including all the electromagnetic sources and fields. However, there are many situations in which we are interested only in a subsystem, which may not be T-invariant. The aim of this article is precisely to report what remains of the T-invariance of a global system in one of its non T-invariant subsystems.
For a system consisting of particles, time reversal is the reversal of motion. Each particle occupies the same positions as with the usual direction of time, but in reversed order, and its velocity is reversed. When the particles possess an electric charge, they form densities of charge and current that are the sources of an electromagnetic field. The reversal of their motion creates a time-reversed electromagnetic field. The fact that both the time-reversed electromagnetic field and the motions of the particles are physically acceptable makes the theory of electromagnetism time-reversal invariant. By physically acceptable, it is meant that the reversed quantities obey the same equations as the non-reversed ones.
It could seem from this presentation that time reversal can be seen, for example, by filming the onward time evolution and then playing the movie back. This is not correct in every situation, mainly when there is retroaction of a particle on itself. We can see this from the radiation of an oscillating dipole: in the onward time direction, the dipole oscillates and radiates an electromagnetic field. When the movie is played back, one sees the electromagnetic field coming back to the dipole and making it oscillate. Obviously, the dipole should also radiate because of its forced oscillation, but this radiation field is absent from the movie.
If the system contains a macroscopic number of particles, it is practically not possible to know the detailed motion of each particle. The system is then described by a small number of average, macroscopically relevant quantities, and a large number of particle configurations give rise to the same set of macroscopic quantities (up to a reasonably good approximation). The configuration where all the particles get simultaneously their motion reversed is highly improbable and does not happen in practice. In this point of view, although the laws of physics are time-reversal invariant, the generic behaviour of macroscopic systems shows an arrow of time. There is no contradiction, and the lesson is that macroscopic irreversibility does not break the time-reversal invariance of the whole system.
However, a manifestation of the irreversibility is that the macroscopic quantities obey equations which are not time-reversal invariant, that is to say that are not conserved when time evolves backward. Typical examples are friction in the contact of two solids and dissipation of energy by a resistor in an electronic circuit. We use the example of the resistor to illustrate how non time-reversal invariant laws can arise from invariant ones.
When an electric field is applied to a resistor (a conductor), the charge carriers inside it experience a diffusive motion and interact with scatterers. The overall motion and interaction are time-reversal invariant. One is interested in the motion of the charge carriers to determine the density of current, but usually not in the degrees of freedom of the scatterers. So these degrees of freedom are averaged [15]. This results in a loss of information on the whole system and thus generates irreversibility. This procedure also provides the microscopic and local Ohm’s law, which breaks time-reversal invariance (see Section 2.4). In this sense, the motion of the charge carriers is not time-reversal invariant because the subsystem {charge carriers } is open.
Let a particle enter a time-reversal invariant system at point A, with velocity v_{A}. It propagates inside the system, and finally exits from it at point B with velocity v_{B}. If one time-reverses the evolution of the particle, it appears that a particle entering the system at point B with velocity −v_{B} will be exiting it at point A with velocity −v_{A}. This sym- metry relation that appears when exchanging the entrance for the exit of a system is called reciprocity. It is automatically satisfied by time-reversal invariant systems, but non time-reversal invariant systems can also be reciprocal.
Lord Rayleigh has established the reciprocity theorem for sound waves : if the system that is crossed by the wave consists of obstacles of any kind, then “a sound originating at A is perceived at B with the same intensity as that with which an equal sound originating at B would be perceived at A”. [16].
For an optical system, this corresponds to the law that if point B can be seen from A through the system, then A can be seen from B.
The reciprocal properties of various systems have been studied by Onsager in the frame of the linear response theory, bringing to light symmetry relations between response coefficients [14]. We exploit these results in Section 2, where time-reversal and reciprocity in classical electromagnetism are addressed.
To give a rigorous definition of reciprocity in quantum mechanics, it is necessary to study first the time-reversal operator T (Section 3) and its impact on the evolution of a quantum system (Section 4). Reciprocity in quantum systems is addressed in Section 5 where we focus on the reciprocal properties of sub-systems which are not time-reversal invariant. At last, we exhibit a reciprocal sub-system of a non-reciprocal system in the frame of multiple scattering of light from cold atoms in Section 6.
Maxwell’s equations (SI units) for the electromagnetic field read [17]:
where ρ and j are the microscopic electric charge and current densities and all fields are real quantities.
Importantly, Maxwell’s equations remain invariant if the densities and fields are transformed according to:
This set of transformations defines the action of time reversal T on the electromagnetic fields. The corresponding transformation for the scalar and vector potentials (defined through E=−∇V−∂_{t}A and B=∇×A) immediately follow:
The Poynting vector Π(r,t)=(E×B)/μ_{0} then transforms according to:
meaning, as it should, that the energy flow is reversed under time reversal.
Let us introduce the time-frequency Fourier transforms of the fields:
and let us rewrite Eq. (10) as
It is easily seen that the time-reversal transformation X(r,t)→±X(r,−t) (+ sign for even fields, − sign for odd fields) is simply achieved in Fourier space by the transformation \(\boldsymbol {\mathcal{X}}(\boldsymbol {r}, \omega) \to \pm \boldsymbol {\mathcal{X}}(\boldsymbol {r}, -\omega)\), that is by the physically-appealing swap of positive and negative Fourier components (further accompanied by a sign change for odd fields). Time reversal invariance in real space, expressed by the condition X(r,t)=±X(r,−t) is thus equivalent to the condition \(\boldsymbol {\mathcal{X}}(\boldsymbol {r}, \omega) = \pm \boldsymbol {\mathcal{X}}(\boldsymbol {r}, -\omega)\) in Fourier space. When the field X(r,t) is real, it is straigthforward to see from Eq. (11) that \(\boldsymbol {\mathcal{X}}(\boldsymbol {r}, -\omega) = \boldsymbol {\mathcal{X}}^{*}(\boldsymbol {r}, \omega)\). Hence, for real fields, time reversal in the Fourier domain is expressed as \(\boldsymbol {\mathcal{X}}(\boldsymbol {r}, \omega) \to \pm \boldsymbol {\mathcal{X}}^{*}(\boldsymbol {r}, \omega)\) thereby linking time reversal to complex conjugation [18]. Real fields are then time-reversal invariant when \(\boldsymbol {\mathcal{X}}(\boldsymbol {r}, \omega) = \pm \boldsymbol {\mathcal{X}}^{*}(\boldsymbol {r}, \omega)\) in Fourier space. Time-reversal invariant even real fields are thus characterized by real \(\boldsymbol {\mathcal{X}}(\boldsymbol {r}, \omega)\) Fourier components, while time-reversal invariant odd real fields are characterized by imaginary \(\boldsymbol {\mathcal{X}}(\boldsymbol {r}, \omega)\) Fourier components.
To derive Maxwell’s equations in continuous media, one usually breaks the charge and current densities into free and bound components, ρ=ρ_{f}+ρ_{b} and j=J_{f}+J_{b}. The bound charge ρ_{b}=−∇•P is described in terms of the dipole moment density P(r,t) (or polarization) of the material while the bound current density J_{b}=∂_{t}P+∇×M is described both in terms of P(r,t) and of the magnetization M(r,t) of the material [19]. Using the auxiliary fields D=ε_{0}E+P (displacement field) and H=B/μ_{0}−M (magnetizing field), the macroscopic Maxwell’s equations in continuous media read [17]:
Theses equations show that D and P transform like E while H and M transform like B under the time-reversal operation T.
To go further, we need constitutive relations, i.e. the expressions of P,M and J_{f} in terms of E and B. Here, for sake of simplicity, we shall restrict ourselves to the case of non-magnetic neutral homogeneous dispersive linear media [20] for which \(\boldsymbol {\mathcal{M}}(\boldsymbol {r},\omega) = \boldsymbol {0}, \rho _{f} = 0, \boldsymbol {\mathcal{J}}_{f}(\boldsymbol {r},\omega) = \underline {\sigma }(\omega) \boldsymbol {\mathcal{E}}(\boldsymbol {r},\omega)\) (Ohmic law) and \(\boldsymbol {\mathcal{P}}(\boldsymbol {r},\omega)=\varepsilon _{0} \, \underline {\chi }(\omega)\boldsymbol {\mathcal{E}}(\boldsymbol {r},\omega)\). Here the tensors \(\underline {\sigma }(\omega)\) and \(\underline {\chi }(\omega)\) are the complex conductivity and dielectric susceptibility of the material medium. In turn, we get \(\boldsymbol {\mathcal{B}} = \mu _{0} \boldsymbol {\mathcal{H}}\) and \(\boldsymbol {\mathcal{D}} = \varepsilon _{0} \, \underline {\varepsilon }_{r}(\omega) \boldsymbol {\mathcal{E}}\) where \(\underline {\varepsilon }_{r}(\omega) = \unicode{x1D7D9} + \underline {\chi }_{r}(\omega)\) is the (complex) relative (dielectric) permittivity constant of the medium. The (complex) refractive index tensor of the media is defined by \(\underline {N}_{r}(\omega)=\sqrt {\underline {\varepsilon }_{r}(\omega)}\).
Maxwell’s equations in such media read:
and the equation of propagation for \(\boldsymbol {\mathcal{E}}(\boldsymbol r,\omega)\) easily follows:
where we have introduced the generalized relative permittivity tensor
Energetic considerations show that dissipative processes are described by the non-Hermitian part of the tensor \(\underline {\varepsilon }\) while its Hermitian part describes non-dissipative (reactive) processes [20, 21].
Let us now see when the medium is time-reversal invariant. Taking the complex conjugate of Eq. (20) and requesting \(\boldsymbol {\mathcal{E}}^{*}(\boldsymbol {r},\omega)=\boldsymbol {\mathcal{E}}(\boldsymbol {r},\omega)\), we see that \(\boldsymbol {\mathcal{E}}(\boldsymbol {r},\omega)\) would obey Eq. (20) but with the substitution \(\underline {\varepsilon }(\omega) \to \underline {\varepsilon }^{*}(\omega)\). By consistency, this is only possible if \(\underline {\varepsilon }(\omega)\) is real. Since polarization and conduction are different phenomena, it is easy to see from Eq. (21) that this would imply that the imaginary part of \(\underline {\varepsilon }_{r}\) and the real part of \(\underline {\sigma }\) should both vanish. As a consequence time reversal invariance is broken in ohmic materials due to the presence of the Joule effect. Another example is provided by non-ohmic (\(\underline {\sigma } = 0\)) scattering media. In this case, a propagating electromagnetic wave would be depleted by scattering events giving rise to an attenuation of the field (Beer-Lambert law). It is worth noting that these two processes are energetically different: the Joule effect transforms the electromagnetic energy into thermal energy, whereas scattering distributes the electromagnetic energy into initially-empty field modes without changing its nature.
The important message of this Section is that the time-reversal invariance is broken by any phenomenon which gives an imaginary part to \(\underline {\varepsilon }(\omega)\). However, such a phenomenon is not necessarily associated to a dissipation of energy. Indeed, \(\underline {\varepsilon }\) can have an imaginary part, thereby breaking time-reversal invariance and still be Hermitian, \(\underline {\varepsilon } = \underline {\varepsilon }^{\dagger }\), so that electromagnetic energy is conserved. In this case, the refractive index tensor \(\underline {N_{r}}(\omega)\), being also Hermitian, can be diagonalized and has real eigenvalues. An important example of this situation is the Faraday effect, which we shall discuss later. In the presence of absorption (dissipation), these eigenvalues, when they exist, would become complex and have a non-zero imaginary part.
We conclude this Subsection with a consideration about Ohm’s law. Under time reversal, \(\boldsymbol {\mathcal{J}}_{f} = \underline {\sigma } \boldsymbol {\mathcal{E}} \to \boldsymbol {\mathcal{J}}^{*}_{f} = \underline {\sigma }^{*} \boldsymbol {\mathcal{E}}^{*}\). If the conductivity is real, the current j_{f} transforms like the electric field and does not change sign. However, the charge carriers at the origin of the current do reverse their motion. Thus, Ohm’s law breaks explicitly the time-reversal invariance of the Maxwell’s equations when the conductivity is real. This is no surprise, because it is related to the Joule effect. Microscopically, the Ohm’s law comes from the tracing out of the degrees of freedom of the scatterers of the charge carriers inside the conductor. As we will see in Section 3.6, this tracing out results in irreversibility. At last, the conductivity can be purely imaginary in reactive media, such as a plasma, and then the current changes sign under time reversal.
We now consider a medium embedded in external static fields. The equation of propagation in the Fourier frequency domain remains identical, provided \(\underline {\varepsilon }\) and \(\underline {\sigma }\) now depend on the external fields. These external fields modify the response of the medium to the propagating wave. In the following, we shall restrict ourselves to the case of an external magnetic field B_{0}. In the framework of the linear response theory of Onsager [14], the following reciprocity relations can be proven:
and thus
Each tensor can be decomposed into the sum of a symmetric and an antisymmetric tensor. The Onsager relations show that the symmetric part must be even in B_{0} while the antisymmetric part must be odd in B_{0}.
The notions of time reversal, energy conservation and reciprocity are related to the symmetries of the relative permittivity tensor \(\underline {\varepsilon }\).
In a vanishing external field B_{0}, the Onsager relations Eqs.(22)–(24) show that all permittivity tensors are symmetric. Decomposing \(\underline {\varepsilon } = \underline {\varepsilon }_{1} + i \underline {\varepsilon }_{2}\) into real and imaginary parts, we conclude that both \(\underline {\varepsilon }_{1}\) and \(\underline {\varepsilon }_{2}\) are Hermitian. We also see that \(\underline {\varepsilon }\) cannot be Hermitian unless \(\underline {\varepsilon }_{2}\) vanishes. Such a situation is associated to absorption, which transforms into gain under time reversal. Some precautions have to be taken when one considers the case of the depletion of the incident mode through scattering. If one plays the movie back, one sees all electromagnetic modes being depleted to the benefit of the initial one, which thus gains energy. We hence have the feeling of an “arrow of time”. However, this interpretation must be carefully considered. Indeed, in the case of elastic scattering, it is in principle possible to measure the complete electromagnetic field (amplitude and phase) going out of the scattering medium, to time reverse it and send it back into the medium. This has already been achieved experimentally with acoustic and electromagnetic waves [22, 23]. Time-reversal invariance is not broken: scattering can be reversed although it seems to break time reversal for a single propagating mode.
Up to this remark on the losses due to scattering, absorption breaks the time-reversal invariance (\(\underline {\varepsilon }\) is not real), does not conserve the electromagnetic energy (\(\underline {\varepsilon }\) is not Hermitian), but is reciprocal because \(\underline {\varepsilon }\) is symmetric. This last point results from the Onsager relation Eq. (23) with a vanishing magnetic field. If we can see you through an absorbing optical system, you can also see us: the wave is absorbed in the same way in both directions.
A monochromatic plane wave propagating along the external magnetic field B_{0} experiences a dielectric tensor \(\underline {\varepsilon }\) which, at lowest order in B_{0}, reads [21]:
where the real number g(ω) is the gyromagnetic factor and Φ the vector with the antisymmetric matrices [Φ_{k}]_{ij}=ε_{ijk} as components (ε_{ijk} is the Levi-Civita symbol). As one can see, the additional term linear in B_{0} is purely imaginary and thus breaks time-reversal invariance. It is easy to show that, when a linearly polarised electromagnetic wave propagates along B_{0} in the medium, its linear polarization state rotates around B_{0} during the course of propagation (Faraday effect).
Let us assume now that \(\underline {\varepsilon }(\omega)\) is real (no absorption) and discuss only the effect of B_{0}. Then, since ^{t}Φ_{k}=−Φ_{k}, we deduce:
Under time reversal, the motion of all particles are reversed, currents change sign and the magnetic fields do also change sign. The Faraday effect breaks time-reversal symmetry for the sub-system {propagating electromagnetic field }, but preserves it for the whole system {propagating field + external magnetic field }. This also holds for reciprocity. We shall take here the “subsystem” point of view. The Faraday effect breaks the time-reversal invariance because \(\underline {\varepsilon }(\omega,-{\boldsymbol {B}}_{0})\neq \underline {\varepsilon }(\omega,{\boldsymbol {B}}_{0})\). However, \(\underline {\varepsilon }(\omega,-{\boldsymbol {B}}_{0})\) is Hermitian: the Faraday effect does not induce energy dissipation. To sum up, the Faraday effect breaks time-reversal invariance, breaks reciprocity since \(\underline {\varepsilon }(\omega,{\boldsymbol {B}}_{0})\) is not symmetric, see Eq. (23), but conserves the electromagnetic energy. Although we can see you through an optical system subject to the Faraday effect, you might not see us. This idea has been exploited in optical diodes [25].
When time is reversed, the particles of a system occupy the same position as during the real evolution, but have reversed velocities. According to this, the orbital angular momentum L=r×p changes sign under time reversal. As the spin S is also an angular momentum, we demand that it gets also inverted under time reversal, for consistency. So the operator T should satisfy:
where F stands for the total angular momentum.
In quantum mechanics, the operator T is also required to conserve the absolute value of scalar products between state vectors |μ〉 and |ν〉:
where 〈Tμ| is the bra associated to the ket |Tμ〉. From the Wigner theorem [26], this implies that T is either linear and unitary, or anti-linear and unitary (anti-unitary). From the transformation rules Eq. (27), we deduce that T must be anti-linear. An elegant way to prove it makes use of the commutation relation [x,p_{x}]=i (we use units such that \(\hbar =1\)). This relation must be invariant under time reversal, and it is then easy to show that
which follows from T(i)T^{−1}=T[x,p_{x}]T^{−1}=[x,−p_{x}]. Another way could have been to consider the relation F×F=iF.
Anti-linear and anti-unitary operators are presented in the Appendix 7. As a anti-unitary operator, T verifies T^{−1}=T^{†} and 〈Tμ|Tν〉=〈ν|μ〉=〈μ|ν〉^{∗} where the star denotes the complex conjugation.
It seems also natural to require that two successive applications of T reproduce the initial state of a system, up to a global phase factor: T^{2}|ϕ〉=e^{iα}|ϕ〉. Using the anti-linearity of T, this leads to:
because T^{3}=T^{2}T=e^{iα}T=TT^{2}=Te^{iα}=e^{−iα}T⇒e^{iα}=±1
Finally, a system is said to be invariant under time reversal if, when we know a solution of the dynamical equations of the system, its time-reversal counterpart is also a solution of these equations.
Let us consider the product \(\mathrm {T} \mathcal{C}\) where \(\mathcal{C}\) is the complex conjugation operator associated to a given representation of the Hilbert space (see Appendix 7). \(\mathrm {T} {\mathcal{C}}\) is linear and unitary, because \(\langle {\mathrm {T} {\mathcal{C}}\mu |\mathrm {T} {\mathcal{C}}\nu }\rangle =\langle {{\mathcal{C}}\nu |{\mathcal{C}}\mu }\rangle =\langle {\mu |\nu }\rangle \). It is thus always possible to write T in the canonical form
where U is a unitary operator. As \(\mathrm {T}^{2}=\pm \unicode{x1D7D9}\) and T is anti-unitary, we deduce that T^{†}=±T and \(U{\mathcal{C}} U{\mathcal{C}}=UU^{*}=\pm \unicode{x1D7D9}\). The unitarity of U then implies ^{t}U=±U: U is either symmetric or antisymmetric.
An observable O of a system is said to be time-reversal invariant if there is an anti-unitary operator T which commutes with O: [T,O]=0. This is equivalent to require the condition
Expressing O in a basis {|μ〉} of the Hilbert space:
This expression shows that, if it is possible to build a T-invariant basis |Tμ〉=|μ〉, then we have the simple result TOT^{†}=O^{∗}. In such a T-invariant basis, O is invariant under time reversal if and only if it is real O^{∗}=O.
It is the simple case where it is always possible to build a T-invariant basis {|ψ_{m}〉} such that |Tψ_{m}〉=|ψ_{m}〉. Such a representation is called real. In a real representation, it is always possible to chose \(U=\unicode{x1D7D9}\) in the canonical form Eq. (31). T then reduces to the complex conjugation \({\mathcal{C}}_{\psi }\).
Indeed, starting from a ket |ϕ_{1}〉 we construct |ψ_{1}〉=|ϕ_{1}〉+T|ϕ_{1}〉. This latter vector verifies T|ψ_{1}〉=|ψ_{1}〉. Using a procedure analogous to the Gram-Schmidt orthonormalization, we build a ket |ϕ_{2}〉 orthogonal with |ψ_{1}〉, and its associated ket |ψ_{2}〉. It is easy to check that 〈ψ_{1}|ψ_{2}〉=0. This procedure can be continued to obtain an orthogonal basis of the Hilbert space, which is T-invariant [27]. In this representation, the operator \(U=\mathrm {T} {\mathcal{C}}_{\psi }\) verifies \(U=\unicode{x1D7D9}\). Simple examples of real representations are given by the basis {|r〉} or by the helicity basis {|±〉} associated with the operator S•p which is invariant under T.
In these real representations, any linear operator O is transformed into its complex conjugated O^{∗} by T. Thus O is time-reversal invariant if and only if O=O^{∗}, that is to say the matrix associated with O is real in any real representation.
This case is less intuitive, because it is no longer possible to build a real representation. One is not allowed to set \(U=\unicode{x1D7D9}\), and T does not reduce to the complex conjugation operator. There is, however, a compensation for this lack of simplicity: a ket |ψ〉 and its time-reversed counterpart |Tψ〉 are orthogonal (T has no eigenvectors in this case):
where Eq. (99) has been used.
This property explains the Kramer’s degeneracy: any eigenvalue of a time-reversal invariant Hamiltonian H=THT^{†}, with \(\mathrm {T}^{2}=-\unicode{x1D7D9}\), is twice degenerated:
If the Hilbert space has a finite dimension, then this dimension is even. The previous results show that one can build a basis:
Defining the complex conjugation \(\mathcal{C}_{\sigma }\) for this representation, we easily see that:
U is block-diagaonal in this representation. Each block represents U in the subspace {|σ_{m}〉,|Tσ_{m}〉} by the unitary and antisymmetric matrix (−iσ_{y}). Because U is block-diagonal, the time-reversal invariance of an observable O does not have the same straightforward interpretation than in the case \(\mathrm {T}^{2}=\unicode{x1D7D9}\). As we shall see later, the paradigmatic situation in which \(\mathrm {T}^{2}=-\unicode{x1D7D9}\) occurs is a spin one-half system.
As time reversal is closely related to the complex conjugation, we call two linear operators related by time-reversal symmetry complex conjugated operators. We call invariant operators O=TOT^{†} real operators. Getting further in the analogy with complex numbers, we call the operators satisfying TOT^{†}=−O imaginary operators.
In the standard representation of angular momentum {|Fm〉}, the matrix of the projection F_{z} of F along the quantization axis is diagonal and real, whereas the matrix associated with F_{x} is real, and the one associated with F_{y} is purely imaginary. This is related to the fact that the matrices of F_{±}=F_{x}±iF_{y} are real. Although this representation is not real, it can be made real for integral F because in this case \(\mathrm {T}^{2}=\unicode{x1D7D9}\) as is shown below.
The complex conjugation operator \({\mathcal{C}}_{F}\) associated to the standard representation transforms the components of the angular momentum in the following way:
Looking for T such that TFT^{†}=−F, we search a unitary operator U satisfying:
The rotation by angle π around the y-axis satisfies these conditions: \(\phantom {\dot {i}\!}U=e^{-i\pi F_{y}}\). This leads to
For a set of N particles of angular momentum F_{n},n∈[1;N], U is the tensor product \(\phantom {\dot {i}\!}\otimes _{n} e^{-i\pi F_{y,n}}\). The computation of T^{2} then leads to:
We recover the fact that a system containing bosons (integral spin) or an even number of fermions (half-integral spin) satisfies \(\mathrm {T}^{2}=\unicode{x1D7D9}\), and that \(\mathrm {T}^{2}=-\unicode{x1D7D9}\) for a system containing an odd number of fermions.
The basis vectors |ω〉 of a representation are usually defined as the eigenvectors of an observable Ω. Let us assume that Ω is either real or imaginary: TΩT^{†}=±Ω. Applying T to the eigenvalue equation Ω|ω〉=ω|ω〉, we deduce:
This property allows to determine the action of T on real or imaginary representations. For example:
spatial representation:
momentum representation:
standard representation |Fm〉 of the angular momentum:
spinor representation \(\left (\begin {array}{c} \psi _{\uparrow }\\ \psi _{\downarrow } \end {array} \right)\) for spin one-half particles:
helicity representation:
Photons with wave vector k and polarization ε⊥k:
In the formulae for spinors, σ_{y} stands for the Pauli matrix describing the y-component of a spin one-half.
We use the action of T on the |Fm〉 states to determine the transformation law of the irreducible tensor operators \(T^{(K)}_{Q}(F,F')\). These operators are useful to study the coupling of two angular momentums F and F^{′} [29]. Their expression reads:
with the Clebsch-Gordan coefficients 〈FmF^{′} −m^{′}|KQ〉, and they transform into
Using the relation
we deduce:
When F^{′}=F, K and Q are integers and Eq. (47) simplifies:
These tensors are very useful in the frame of multiple scattering, to compute the intensity of a multiply scattered wave. This intensity can be computed from Feynman-like diagrams in which the vertices couple identical spins [30].
The case of spin-j bosons is very simple:
where m=−j,•••,j and j is integer.
The case of spin-j fermions is more subtle as we will see:
where m=−j,•••,j but now j is half-integer. Here, Θ is the Heaviside step function, Θ(m)=0 for m<0 and Θ(m)=1 for m>0 (j being half-integer, the value m=0 is not allowed). The sign change encoded by the step function in fact relates to the action of the operator −iσ_{y}, present in the expression of T, on half-integer angular momentum states.
As an example, let us consider the case j=1/2:
There is a subtlety here. Indeed, from the preceding relations, one easily finds:
At this point, a naive reader might run into a contradiction by (erroneously!) using the relation \(\mathrm {T}^{\pm 2} = - \unicode{x1D7D9}\) and arrive at the wrong statement f_{↑}=−f_{↑}. This would be forgetting that the statement \(\mathrm {T}^{\pm 2} = - \unicode{x1D7D9}\) only applies to a single fermion. For a system of N fermions, one actually has \(\mathrm {T}^{\pm 2} = (-)^{N} \, \unicode{x1D7D9}\). Now, it is important to realize that the annihilation operator f_{↑} connects spaces differing by one particle. When considering the expression Tf_{↑}T^{−1}, one sees that T^{−1} acts on the right on a state with N fermions while T acts on the right on a state with N−1 fermions (because f_{↑} destroys one fermion). As a consequence, \(\mathrm {T}^{-2} = (-)^{N} \, \unicode{x1D7D9}\) and \(\mathrm {T}^{2} = (-)^{N-1} \, \unicode{x1D7D9}\). All in all, we have T^{2}f_{↑}T^{−2}=(−)^{2N−1} f_{↑}=−f_{↑} and the seemingly contradiction disappears. This result extends easily to higher half-integer spins.
For completeness, we give the action of T on the creation and annihilation operators \(a^{\dagger }_{{\boldsymbol {k}}\boldsymbol {\epsilon }}\) and a_{kε} for photons with momentum k and polarization ε⊥k:
Let us consider a system composed of two sub-systems A and B, and described by the Hilbert space \(\mathcal{H}=\mathcal{H}_{A}\otimes \mathcal{H}_{B}\). The time-reversal operator T acting on \(\mathcal{H}\) has the form \((U_{A}\otimes U_{B}){\mathcal{C}}\), where U_{A} and U_{B} are unitary operators acting on \(\mathcal{H}_{A}\) and \(\mathcal{H}_{B}\) respectively, and \({\mathcal{C}}\) is the complex conjugation operator in a given representation of \(\mathcal{H}\).
What can we say about the time-reversal properties of an observable O of the system, when we only look at the part A ? O reduces then to O_{A}=Tr_{B}(O), acting on \(\mathcal{H}_{A}\). The time-reversal operator transforms O_{A} into Tr_{B}(TOT^{†}). As T and \({\mathcal{C}}\) are anti-linear operators, the calculation of the trace must be done carefully: the cyclicity Tr(ABC)=Tr(CAB) holds for linear operators only:
The last step is achieved by computing explicitly the trace in the basis of the eigenvectors of O, using the fact that the eigenvalues of an observable are real.
One finds the “natural” result that O_{A} is transformed into \(\mathrm {T}_{A} O_{A}\mathrm {T}^{\dagger }_{A}\), and the time-reversal operator associated with the sub-system A is \(\mathrm {T}_{A}=U_{A}{\mathcal{C}}_{A}\).
This result remains true if the subsystem B is described by a time-reversal invariant density matrix.
In any case, taking the trace over the degrees of freedom of B results in a loss of information on the whole system {A+B}. One consequence is that if the whole system is time-reversal invariant, the effective description of A alone may break this invariance.
The evolution of a quantum system may be described in many equivalent representations or pictures. The most known is the Schrödinger picture. The Heisenberg and the interaction pictures are also widely used. We will first describe the action of the time-reversal operator T in each of those pictures. Then, we will enter into details of the interaction picture, which will allow us to study the properties of the scattering operator \({\mathcal{S}}\) under T [29].
In the Schrödinger picture, the state vector |Ψ_{S}(t)〉 evolves in time, following the Schrödinger equation i∂_{t}|Ψ_{S}(t)〉=H_{S}(t)|Ψ_{S}(t)〉. It describes a causal evolution, and |Ψ_{S}(t)〉 at time t can be deduced from |Ψ_{S}(t_{0})〉 at another time t_{0} : |Ψ_{S}(t)〉=U_{S}(t,t_{0})|Ψ_{S}(t_{0})〉 where U_{S}(t,t_{0}) is a unitary operator called the evolution operator.
The observables O_{S} which do not depend explicitly on time are constant in this picture.
The action of the time-reversal operator T on the evolution of the system in the Schrödinger picture can be studied by applying T to the Schrödinger equation i∂_{t}|Ψ_{S}(t)〉=H_{S}(t)|Ψ_{S}(t)〉, and replacing t by −t:
This result leads to adopt the following definition: a system is dynamically invariant under time reversal if there is a anti-unitary operator T such that
In this case, the form of the Schrödinger equation is not affected by the transformation “ t→−t”. If |Ψ_{S}(t)〉 is a solution, then |TΨ_{S}(−t)〉 too. This expression of the time-reversal invariance may seem different from the one we have given for an observable in Eq. (32). This only happens because we let the Hamiltonian depend on time t. For a time independent Hamiltonian, we recover the expected condition TH_{S}T^{†}=H_{S}.
In this picture, a quantum state is described by a constant vector state |Ψ_{H}〉 equal to the state vector in the Schrödinger picture |Ψ_{S}(t_{0})〉 at some initial time t_{0}. The ket vector in the Heisenberg picture is deduced from the ket vector in the Schrödinger picture by the transformation : \(|{\Psi _{H}}\rangle =U_{S}^{\dagger }(t,t_{0})|{\Psi _{S}(t)}\rangle \).
The observables of the Schrödinger picture transform into \(O_{H}(t)=U_{S}^{\dagger }(t,t_{0})O_{S}U_{S}(t,t_{0})\). The observables in the Heisenberg picture depend explicitly on time. Their evolution is governed by the Heisenberg equation
where H_{H}(t) is the Hamiltonian in the Heisenberg picture.
The action of the time-reversal operator T on the evolution of the system can be studied in the same way as in the Schrödinger picture : one applies T to the Heisenberg Eq. (59) and reverses time t→−t. It is then seen that TO_{H}(−t)T^{†} obeys the same equation as O_{H}(t), provided that
Hence, the evolution of a quantum system is dynamically invariant under time reversal if the condition Eq. (60) is fulfilled. This condition is very similar to Eq. (58).
This picture is intermediate between the Schrödinger and Heisenberg pictures. It is well adapted to describe systems which contain two or more interacting subsystems. They have a Hamiltonian of the form \(H_{S}=H_{S}^{(0)}+H_{S}'\) where \(H_{S}^{(0)}\) describes the subsystems without interaction and HS′ describes the interaction. The eigenstates of \(H_{S}^{(0)}\) are supposed to be known, as well as its associated evolution operator \(U^{(0)}_{S}(t,t_{0})\).
In the interaction picture, the kets are defined by \(|{\Psi _{I}(t)}\rangle =U^{(0)\dagger }_{S}(t,t_{0})|{\Psi _{S}(t)}\rangle \) and the observables by \(O_{I}(t)=U^{(0)\dagger }_{S}(t,t_{0})O_{S}U^{(0)}_{S}(t,t_{0})\).
The state vectors obey the Schrödinger equation i∂_{t}|Ψ_{I}(t)〉=HI′(t)|Ψ_{I}(t)〉 with the Hamiltonian \(H^{\prime }_{I}(t)\) and the observables follow the Heisenberg Eq. (59) with the Hamiltonian \(H^{(0)}_{I}(t)\).
The dynamics of such an interacting system, described in the interaction picture, is time-reversal invariant if the two following conditions are satisfied :
In this Section and in the following, we shall use the interaction representation. The time evolution of a vector |Ψ_{I}(t)〉 is given by:
Let U_{I}(t,t_{0}) be the Green’s function of Eq. (63). U_{I}(t,t_{0}) is the evolution operator of the system and obeys the integral equation \(U_{I}(t,t_{0})=\unicode{x1D7D9}-i\int _{t_{0}}^{t}H'_{I}(\tau)U_{I}(\tau,t_{0})\,d\tau \) with the condition \(U_{I}(t_{0},t_{0})=\unicode{x1D7D9}\). U_{I}(t,t_{0}) is unitary and its inverse is U_{I}(t_{0},t).
The action of T on U_{I}(t,t_{0}) is:
Substituting (t,t_{0},τ)→(−t,−t_{0},−τ) in the previous equation, we get
Let us assume that the system is T-invariant, as expressed by Eq. (61). Then, TU_{I}(−t,−t_{0})T^{†} obeys the same integral equation than U_{I}(t,t_{0}), with the same initial condition. Therefore these two operators are equal. Using the fact that U_{I}(−t,−t_{0})=[U_{I}(−t_{0},−t)]^{†}, we deduce:
We now apply this result to the scattering matrix \(\mathcal{S}\), which connects the state vector from time t_{0}=−∞ to t=+∞:
Any scattering problem can be described by the transformation of an initial state at t_{0}=−∞ to a final state at t=+∞ (these states are asymptotically free, that is to say eigenstates of H^{(0)} : the interaction is assumed to vanish at infinity). The scattering amplitude is therefore given by the corresponding matrix element of \({\mathcal{S}}\).
For a T-invariant system, Eq. (66) applies and leads to:
The scattering matrix can be expressed in terms of the transition matrix \(\mathcal{T}\): \(\mathcal{S}=\unicode{x1D7D9}-2i\pi \mathcal{T}\). In this case, Eq. (68) is equivalent to \(\mathrm {T}{\mathcal{T}}\mathrm {T}^{\dagger }={\mathcal{T}}^{\dagger }\).
Let us consider the vector state \(\left | \psi _{out} \right \rangle =\mathcal{S}\left | \psi _{in} \right \rangle \) resulting from the scattering of the initial state |ψ_{in}〉. We time-reverse |ψ_{out}〉 to get |ϕ_{in}〉=T|ψ_{out}〉, and then let the new initial state evolve to \(|{\phi _{out}}\rangle ={\mathcal{S}}|{\phi _{in}}\rangle \). How is |ϕ_{out}〉 related to |ψ_{in}〉? From the definition of |ϕ_{out}〉:
If the dynamics of the system is T-invariant, then \(\mathrm {T}\mathcal{S}\mathrm {T}^{\dagger }=\mathcal{S}^{\dagger }\). Using the unitarity of \({\mathcal{S}}\), we finally find:
The initial information contained in |ψ_{in}〉 spreads on all the accessible channels through scattering. In the absence of degradation or loss of information, it is possible to rebuild the reversed initial state from the whole outgoing state by time reversal.
Whereas the time-reversal invariance is a dynamical property, related to the Hamiltonian, reciprocity is a property of the amplitudes of transition of the evolution operator. Reciprocity is thus related to the scattering matrix \({\mathcal{S}}\) [24].
A system is said to be reciprocal if the amplitude of the transition from |a〉 to |b〉 is equal to that of the transition from |Tb〉 to |Ta〉:
Other definitions can be found in the literature [21, 31] and various definitions are given in the review by Potton [25] for optical systems. Although they take various mathematical forms, they all express the same physical phenomenon. We have chosen a pragmatic definition, which concerns only the system under study. It is formally equivalent to the definition given by van Tiggelen and Maynard [32].
An important relation results from the anti-linearity of T: for any linear operator O (see Eq. (103) in the Appendix):
Comparing with the definition of reciprocity in Eq. (71), we deduce that any system satisfying to \(\mathrm {T} \mathcal{S}^{\dagger }\mathrm {T}^{\dagger }=\mathcal{S}\) is reciprocal. If it is possible to build a real representation of the Hilbert space, this condition simplifies into \(^{t}{\mathcal{S}}={\mathcal{S}}\): \({\mathcal{S}}\) is symmetric. This explains the heuristic, and not very clear sentences such as “If I can see you, you can see me” accompanying lectures on reciprocity. This is also why the helicity basis, which is real, is so widely used when reciprocity is discussed for optical systems. Once again, this easy interpretation is only valid when \(T^{2}=\unicode{x1D7D9}\). For systems containing an odd number of fermions, one must come back to the general definition of reciprocity.
Systems depending on external parameters which change sign under time reversal are of particular interest. For example, this happens in the presence of an external magnetic field B_{0}. For such systems, one has \(\mathrm {T} {\mathcal{S}}^{\dagger }({\boldsymbol {B}}_{0})\mathrm {T}^{\dagger }={\mathcal{S}}(-{\boldsymbol {B}}_{0})\). So that:
Thus, these systems are not reciprocal. It is important here to notice that the external parameters, and their sources if any, are not part of the system. The set {system + external parameters } is reciprocal if all variables get time-reversed. Close attention has to be paid to which system or sub-system is considered.
This representation |Fm〉 is not real. Using the action of T on it, we derive :
When
the reciprocity condition \(\mathrm {T}\mathcal{S}^{\dagger }\mathrm {T}^{\dagger }=\mathcal{S}\) is fulfilled and the scattering is reciprocal. As this representation of the angular momentum is not real, the reciprocity condition is different from \(^{t}{\mathcal{S}}={\mathcal{S}}\).
We have shown in Section 4.4 that a system which is invariant under time-reversal symmetry satisfies the important relation \(\mathrm {T} {\mathcal{S}}\mathrm {T}^{\dagger }={\mathcal{S}}^{\dagger }\), from which \(\mathrm {T} {\mathcal{S}}^{\dagger }\mathrm {T}^{\dagger }={\mathcal{S}}\) follows.
Thus, time-reversal invariant systems are also reciprocal. This is illustrated by the beautiful experiments on acoustic waves by Fink and co-workers [22].
At first sight, this title could seem paradoxical because the reciprocity relation \(\mathrm {T}\mathcal{S}^{\dagger }\mathrm {T}^{\dagger }=\mathcal{S}\) implies \(\mathrm {T}{\mathcal{S}}\mathrm {T}^{\dagger }={\mathcal{S}}^{\dagger }\). However, this last equation is not the time-reversal invariance condition given in Eq. (58) but only an implication of it. It can be fulfilled although there is no time-reversal invariance.
A typical example is the propagation of a wave through an isotropic and homogeneous absorbing medium. Let us consider a wave propagating from r to r^{′}. Because of the absorption, the amplitude of the wave decreases. The action of T results in the propagation from r^{′} to r with an increasing amplitude. So the system {light + absorbing medium } is not time-reversal invariant. However, if the wave propagates from r^{′} to r, its amplitude will decrease and the attenuation will be exactly the same as during the propagation from r to r^{′}. This shows that the system {light + absorbing medium } is reciprocal.
A mathematically equivalent example is the study of an atom prepared in a excited state, if we only consider the time evolution of this excited state. The whole system {atom + vacuum fluctuations } is time-reversal invariant. But the evolution of the sub-system {excited state } is irreversible, though reciprocal. Indeed, this open system can be described by an effective Hamiltonian \(H=H_{0}-i(\Gamma /2)\unicode{x1D7D9}\), where H_{0} is hermitian and Γ is a positive coefficient. Such a system is reciprocal provided that H_{0} is T-invariant.
This example emphasizes the situations in which reciprocity may be of interest : the study of a (open) subsystem, which is part of a larger time-reversal invariant system. The subsystem is generally not time reversal invariant, and reciprocity may bring additional knowledge on it. It has to be noticed that, considering only the subsystem, the time reversal operator is only applied to this subsystem.
As far as we know, the general form of irreversible evolutions which preserve reciprocity remains an open question.
Let us consider a system consisting of two interacting subsystems A and B. The sub-systems were separately prepared and did not interact initially. A is represented by the Hilbert space \(\mathcal{H}_{A}\), of which a basis is {|a〉}, and B is represented by \(\mathcal{H}_{B}\), of which a basis is {|μ〉}. The scattering operator \({\mathcal{S}}\) acts on \(\mathcal{H}=\mathcal{H}_{A}\otimes \mathcal{H}_{B}\). Its matrix elements are \(\langle {b\nu }|{\mathcal{S}}|{a\mu }\rangle \). After A and B have interacted, the final density matrix of the whole system is \(\rho ^{(fin)}={\mathcal{S}}\rho ^{(in)}{\mathcal{S}}^{\dagger }\), where \(\rho ^{(in)}=\rho ^{(in)}_{A}\otimes \rho ^{(in)}_{B}\) is the initial density matrix. In the following, we only consider the subsystem A, the final density matrix of which is \(\rho ^{(fin)}_{A}=\text {Tr}_{B}[{\mathcal{S}}\rho ^{(in)}{\mathcal{S}}^{\dagger }]\) [33]. Assuming A was initially in a pure state |i〉, the probability to observe it in the state |f〉 at the end of the scattering process is
where the basis {|μ〉} diagonalizes \(\rho ^{(in)}_{B}\).
The composite system is reciprocal for A if
where T_{A} is the time-reversal operator restricted to A. This is equivalent to compare the two following experiments: (1) A is prepared in the state |i〉 and scattered by B, which was prepared in a certain way. A signal is detected for A in the state |f〉. (2) B is prepared in the same way as in (1). The system A is prepared in |Tf〉 and scattered by B. A signal is detected for A in the state |Ti〉. When the signals detected in (1) and (2) are identical, the system is said to be reciprocal.
A natural question is whether a sub-system of a globally reciprocal system can be reciprocal. We make use of the definition Eq. (77) of reciprocity. Using Eq. (76) and the reciprocity of the global system, we find
After the exchanges (Tμ,Tν)→(μ,ν) and μ⇔ν, this reduces to
We can draw two consequences from this expression:
A is reciprocal if \(\rho ^{(in)}_{B}\propto \unicode{x1D7D9}\), i.e. if B is prepared in a full statistical mixture.
A is reciprocal if B has no internal structure. Indeed, in this case |μ〉=|ν〉=|0〉 and \(\langle {0}|\rho ^{(in)}_{B}|{0}\rangle =\langle {0|0}\rangle =1\). On the other hand, T|0〉=|0〉. So that \(\langle {f\,0}|{\mathcal{S}}|{i\,0}\rangle =\langle {\mathrm {T} i\,0}|{\mathcal{S}}|{\mathrm {T} f\,0}\rangle \), which implies P(|i〉→|f〉)=P(|Tf〉→|Ti〉).
Let us go back to Eq. (79) and define the probability \(p(\nu)\equiv \left \langle \nu \right |\rho ^{(in)}_{B}\left | \nu \right \rangle \) to be in the state |ν〉, and the probability of transition from |ν〉 to \(|{\mu }\rangle P_{if}(\mu \rightarrow \nu)=|\langle {\mu }|{\mathcal{S}}_{if}|{\nu }\rangle |^{2}\) where \({\mathcal{S}}_{if}=\langle {f}|{\mathcal{S}}|{i}\rangle \) is a scattering operator acting only on B. The question about the reciprocity of A can be formulated in the following way:
This happens if
When the initial density matrix of B is T-invariant, then p(Tμ)=p(μ) and the condition for A being reciprocal reduces to a detailed balance condition: the total probability for B to go to |μ〉 is equal to total probability to quit |μ〉.
In this Section, we address reciprocity for the multiple scattering of light from atoms. To make it easier, we model the atom by one single closed and isolated dipole transition. This dipole transition relates a ground state of total angular momentum F_{g} and an excited state of total angular momentum F_{e}. The transition from one state to the other is achieved by absorbing or emitting one photon. The frequency of this atomic transition is ω_{0} and the width of the excited state is Γ.
The hamiltonian for the system {atom + photon } is [34]:
where \(\hat {P}_{e}\) is the projector onto the excited state, d is the dipole operator associated with the atomic transition and E the electric field [28]:
As \(\mathrm {T} a^{\dagger }_{{\boldsymbol k}\boldsymbol \epsilon }\mathrm {T}^{\dagger }=a^{\dagger }_{-{\boldsymbol k} \boldsymbol \epsilon ^{*}}\) and \(\phantom {\dot {i}\!}\mathrm {T} a_{{\boldsymbol {k}}\boldsymbol {\epsilon }}\mathrm {T}^{\dagger }=a_{-{\boldsymbol {k}} \boldsymbol {\epsilon }^{*}}\), it is easy to check that H is T-invariant. Thus the interaction between one photon and one atom, and in particular a scattering event, is time-reversal invariant. This implies that it is also reciprocal.
Another question is whether reciprocity holds for the photons when the atoms are not observed. Let us consider the simple case when the atoms are uniformly distributed on the Zeeman sub-levels of their ground state, that is when \(\rho _{at}\propto \unicode{x1D7D9}\). This situation occurs for example in the scattering of light from atoms in a magneto-optical trap. We are then in the situation mentioned in Section 5.5.2, where the full system is decomposed into two subsystems, one of which is observed (here the photon) and the other is prepared in a statistical mixture (the atoms). In this case, the scattering remains reciprocal. This is confirmed by the calculation of the probability for an incident photon |kε〉 to be scattered into |k^{′}ε^{′}〉 [35]:
depending only on the polarization states (the w_{i}’s depend only on F_{g} and F_{e}). We find effectively P(kε→k^{′}ε^{′})=P(−k^{′}ε^{′}^{∗}→−kε^{∗}).
Coherent backscattering (CBS) is an interference effect involving scattering paths which visit the same scatterers in reverse order [36]. The amplitudes associated with each reversed path may be equal if reciprocity holds, leading to a maximally constructive interference. It turns out that the interference is non negligible only inside a small cone around the backscattering direction. Coherent backscattering of light from atoms is obtained when shining light at an atomic cloud and averaging over the external and internal degrees of freedom of the atoms, see Fig. 1 for an experimental implementation [37].
We consider the transition between the initial state |kε,m_{1}…m_{N}〉, which describes the incoming photon |kε〉 and N atoms prepared in the Zeeman substates |m_{1}…m_{N}〉, and the final state |k^{′}ε^{′},m1′…mN′〉. Along both reversed paths, the atoms experience the same transition |m_{1}…m_{N}〉→|m1′…mN′〉.
Using the shorthand {m} and {m^{′}} to represent the collection of spin states m_{1},(•••),m_{N} and \(m^{\prime }_{1}, (\cdot \cdot \cdot), m'_{N}\), we deduce from Eq. (75) that reciprocity relates the amplitudes \(\langle {{\boldsymbol {k}}'\boldsymbol {\epsilon ^{\prime }},\{m'\} }|{\mathcal{T}}|{{\boldsymbol {k}}\boldsymbol {\epsilon },\{m\} }\rangle \) and \(\langle {-{\boldsymbol k}\boldsymbol\epsilon^{*},\{-m\}\mid\mathcal{T}\mid-{\boldsymbol k}'\boldsymbol\epsilon'^{*},\{-m'\}}\rangle\) through:
Sufficient conditions for the amplitudes of the reversed paths to be equal are k^{′}=−k (backscattering), ε^{′}^{∗}=ε and \(\phantom {\dot {i}\!}m_{i}'=-m_{i}, \forall i\). The first two conditions can be easily fulfilled, but the third generally does not hold, unless F_{g}=0 that is to say \(\phantom {\dot {i}\!}m_{i}'=m_{i}=0\).
Thus, if the ground state is non-degenerate, the amplitudes of the reversed paths can be related by reciprocity and be equal. Otherwise, generic reversed scattering paths are not related by reciprocity and there is no reason for their amplitudes to be equal. The coherent backscattering effect can be lower when the atomic ground state is degenerate.
It has been shown by Müller [35] and Jonckheere [38] that the amplitudes which interfere to produce the coherent backscattering effect are not reciprocal, unless \(\phantom {\dot {i}\!}\boldsymbol {\epsilon ^{\prime }}=\boldsymbol {\epsilon }^{*}\) and w_{2}=w_{3}. This latter condition is fulfilled only if the atomic ground state is not degenerate F_{g}=0, or when F_{g}=F_{e}→∞. Experimentally, the reciprocity of reversed paths results in an enhancement factor 2 of the backscattered intensity. It has been observed for ^{88}Sr atoms which have a non-degenerate ground state [39], see Fig. 2, and for classical scatterers [40] (Rayleigh scattering is mathematically equivalent to scattering from atoms with non-degenerate ground state). In marked contrast, the CBS signal obtained with atoms having a ground state level with non-zero spin achieves very low contrasts, see Fig. 3 [37, 41–43].
In the presence of an external magnetic field \({\boldsymbol B}_{0}=B_{0}\hat {\boldsymbol e}_{z}\), the Zeeman sub-levels of the ground and excited states are split (Zeeman effect). The quantization axis is chosen parallel to B_{0}.
If one considers the global system {atom + photon + external field }, the system is reciprocal. However, reciprocity is broken if we only consider the sub-system {atom + photon }, keeping B_{0} unchanged:
If the scattering of one photon from one atom is non reciprocal, it is all the more so for the multiple scattering of one photon by N atoms. Moreover, in the dilute atomic gas regime, the photon experiences Faraday rotation during its propagation between two successive scatterers. We have seen in Section 2.6.2 that this breaks reciprocity too. Thus, the sub-system { light + atoms } is non-reciprocal in the presence of an external magnetic field. We shall show however that the sub-system { light } alone can be reciprocal under well chosen circumstances.
In the presence of a very large external magnetic field, μ_{B}B_{0}≫Γ where μ_{B} is the Bohr magneton, the Zeeman structure of the ground and excited states of the atoms is split. In such a situation, it is possible to shine at the atomic cloud a light resonant with an extremal transition, for example |F_{g},m=F_{g}〉→|F_{g}+1,F_{g}+1〉. The interest of such a configuration is that the transition is closed and isolated. All other transitions are out of resonance (Zeeman splitting) and cannot be excited. Moreover, they cannot be fed by the radiative cascade from the level |F_{g}+1,F_{g}+1〉. We thus have achieved an effective two-level atomic system.
As the chosen extremal transition can only be excited by a left circular polarization \(\hat {\boldsymbol e}_{+}\) (with respect to the B_{0}-axis), the transition matrix is proportional to the projector \(|{\boldsymbol {r},F_{g},\hat {\boldsymbol {e}}_{+}}\rangle \langle {\boldsymbol {r},F_{g},\hat {\boldsymbol {e}}_{+}}|\): The matrix elements are
Reciprocity holds for the photons if
that is to say:
Neither the reciprocity of the global system {atom + photon + external field }, nor the violation of reciprocity by the subsystem {atom + photon } does give us an indication if this equality generally holds or not. Nevertheless, we notice that it can be rewritten
It holds if either ε^{′}^{∗}=ε or k^{′}∥k⊥B.
In the case of coherent backscattering, the condition for reciprocity is the same as for single scattering events. This is due to the fact that after the first scattering, only the polarization \(\hat {\boldsymbol {e}}_{+}\) can propagate. Using the transversality of light and the fact that k^{′} is parallel to k in the backscattering direction, we deduce that the interfering paths are reciprocal if either ε^{′}^{∗}=ε or k⊥B.
This shows that although the subsystem {light + atoms } is neither time-reversal invariant nor reciprocal, the subsystem {light} can be reciprocal under well chosen conditions [44, 45], see Fig. 4.
In this pedagogical paper, we have introduced the time-reversal operator and its fundamental properties and we have detailed the transformation properties of physical systems under its action, both in classical electromagnetism and in quantum mechanics. We have also introduced the important concept of reciprocity and discussed its link and differences with time invariance. In particular, we have shown that time-reversal invariance always implies reciprocity. However, the converse does not generally hold: reciprocity does not imply time-reversal invariance as illustrated by systems with absorption. These systems obey reciprocity (“If you can see me, I can see you”) but do not satisfy time-reversal invariance (absorption becomes gain). By the same token, the contraposition always holds true: systems breaking reciprocity break time-reversal invariance. And here again, the converse does not generally hold: systems breaking time-reversal invariance can still fulfil reciprocity (as exemplified by absorption) or not (as exemplified by the Faraday effect).
In the framework of quantum mechanics, we have shown when a subsystem of a composite system is reciprocal. We have illustrated these notions with the experimental observation and measurement of the coherent backscattering effect with cold atoms shone by laser light. To us, the most striking result that we report is the restoration of reciprocity in the case of coherent backscattering of light from atoms in an external magnetic field : the system {light + atoms } is neither time-reversal invariant nor reciprocal (because of the external magnetic field), but the subsystem {light} becomes reciprocal when selecting a single atomic transition in the limit of a large magnetic field.
The properties of linear operators, discussed at length in the literature on quantum mechanics, are usually expressed in the bra-ket formalism of Dirac. We shall see that this formalism is not well suited for anti-linear operators. The scalar product of two vectors |ψ〉 and |ϕ〉 of the Hilbert space will thus be noted (|ψ〉,|ϕ〉).
The property of anti-linearity for A is expressed by:
It is clear from this definition that the product of a linear operator with an anti-linear operator is anti-linear, and the product of two anti-linear operators is linear. More generally, a product involving p linear operators and q anti-linear operators is linear if q is even, and anti-linear if q is odd. We also notice that A does not commute with a complex scalar.
A acts also on the dual element 〈ν| of a vector |ν〉:
Here, the parenthesis are necessary: the product of a scalar, a linear operator and a anti-linear operator is not associative.
The right and left matrix elements of A are
From the properties Eq. 91 and Eq. 92, we deduce that \(A^{(r)}_{\mu \nu }\) and \(A^{(l)}_{\mu \nu }\) are related by an anti-linear function. As they are numbers, this anti-linear function is of the form f(z)=λz^{∗}. Moreover, nothing distinguishes an anti-linear operator from a linear operator when only real numbers are involved. So λ=1 and the right and left elements of A are complex conjugated:
This shows that the Dirac notation 〈μ|A|ν〉 is not adapted to anti-linear operators: one has to specify if A acts on the left or on the right.
The right matrix elements of A can be expressed as scalar products:
We define the adjoint operator A^{†} of A by expressing also the left matrix elements with scalar products:
This is equivalent to saying that the dual operator 〈A^{†}μ| associated to the vector A^{†}|μ〉 is equal to (〈μ|A). Therefore, A and A^{†} are related by the important relation:
A^{†} has also the usual properties
The anti-linear operator A is said to be anti-unitary if \(AA^{\dagger }=A^{\dagger } A=\unicode{x1D7D9}\). It is then easy to deduce
A consequence of this is the transformation law of a linear operator B under the anti-unitary operator A:
so that B transforms into ABA^{†} under the action of A. Notice that this operator is indeed linear, as a product containing two anti-linear operators. There is no ambiguity on which side it acts, and we thus use the Dirac notation.
Let {|ϕ_{a}〉} be a basis of the Hilbert space. It defines a representation of the system. We define the complex conjugation operator \({\mathcal{C}}_{\phi }\) associate with this representation by:
It is easy to see that \(\mathcal{C}_{\phi }^{2}=\unicode{x1D7D9}\) and \(\mathcal{C}^{\dagger }_{\phi }=\mathcal{C}_{\phi }\). So \(\mathcal{C}_{\phi }\) is anti-unitary. \({\mathcal{C}}_{\phi }\) depends on the chosen representation because of the phases of the basis vectors.
It is possible to relate without restriction the matrix element O_{ba}=〈b|O|a〉 of a linear operator O between two states |a〉 and |b〉 to the matrix element of the operator TO^{†}T^{†} between the time-reversed states |Tb〉 and |Ta〉:
We thus have the equation
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Olivier Sigwarth wishes to thank the Centre for Quantum Technologies, Singapore, for its kind hospitality.
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The authors have contributed equally to the work. The authors read and approved the final manuscript.
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