PHYSICAL REVIEW RESEARCH 3, 013116 (2021)

Ultrastrong time-dependent light-matter interactions are gauge relative

Adam Stokes* and Ahsan Nazir † Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

(Received 12 August 2020; revised 2 November 2020; accepted 19 January 2021; published 8 February 2021)

Time-dependent light-matter interactions are a widespread means by which to describe controllable exper- imental operations. They can be viewed as an approximation in which a third system, the control system, is treated as external within the Hamiltonian. We demonstrate that this results in nonequivalence between gauges. We provide a physical example in which each different nonequivalent model coincides with a gauge-invariant description applied in a different experimental situation. The qualitative final-time predictions obtained from these models, including entanglement and photon number, depend on the gauge within which the time-dependent coupling assumption is made. This occurs whenever the interaction switching is sufficiently strong and nonadi- abatic even if the coupling vanishes at the preparation and measurement stages of the protocol, at which times the subsystems are unique and experimentally addressable.

DOI: 10.1103/PhysRevResearch.3.013116

I. INTRODUCTION even when interactions are present is also common, and this is equivalent to assuming an instantaneous switching of the Exploiting controlled light-matter coupling is important interaction (e.g., [44–47]). for quantum computation [1–5], quantum communication [6], The assumption of a time-dependent coupling, or of quantum metrology [7,8], and quantum simulation [9,10]. preparing and measuring particular noninteracting states, is In the search for scalable platforms operating at room highly nontrivial because light and matter quantum subsys- temperature, strong light-matter coupling has become of tems are defined differently by different gauges [48]. Within major interest through solid-state systems, such as semi- traditional weak-coupling regimes, focus has predominantly conductor quantum wells [11,12] and dots [13,14], through been placed on the Coulomb and multipolar gauges [47,49– two-dimensional [15] and organic [16] materials, and through 52]. The Pauli-Fierz representation, which attempts to isolate superconducting circuits [4,17–22]. the component of the electromagnetic field tied to the material Here, we consider the implications of the widespread and system, has also been used to calculate radiative corrections important practice of modeling controllable light-matter in- [53]. These previous studies have focused specifically on teractions by assuming that the coupling parameter η of the establishing gauge invariance of the S matrix [54–61], or model Hamiltonian H(η) depends on an external control else have considered the natural line-shape problem of spon- parameter [4,20,23–39]. This means that η varies in time: taneous emission in weak-coupling and Markovian regimes η → η(t ). For example, time-dependent couplings in cavity [44–47,50,62,63]. It is now well known that QED S-matrix QED can be used to realize a universal set of gates for quan- elements calculated using perturbation theory are independent tum computation [32] and ultrastrong ultrafast couplings are of the subsystem division at every order [55,60,61]. This result proposed to realize high-fidelity gates using superconducting is physically limited, however, because it is a direct conse- circuits [4]. Such models may also result from the rotation quence of the adiabatic switching condition definitive of the S of a model in which a subsystem is classically driven (e.g., matrix [55]. [26]). Time-dependent Hamiltonian components are of intrin- In scattering theory, virtual processes are allowed only sic importance in thermodynamics, where they are used to as intermediates within a “real process.” On the other hand, define work as a component of energy. They are also impor- beyond scattering theory, virtual effects are especially im- tant in optimal control theory, through models such as the portant in ultrastrong-coupling regimes and when dealing extended Rabi model [40–43]. The assumption of preparing with ultrafast interactions. Moreover, ultrastrong light-matter or measuring an eigenstate of the noninteracting Hamiltonian coupling is now a major field of study for both fundamen- tal and applied physics [26,64]. Likewise, increasingly fast interactions are becoming more and more prevalent and, in *[email protected] particular, may be advantageous in mitigating detrimental en- †[email protected] vironmental affects occurring over the course of a controlled process. Subcycle interaction switching was in fact achieved Published by the American Physical Society under the terms of the some time ago [18] and, more recently, sub-optical-cycle dy- Creative Commons Attribution 4.0 International license. Further namics have been achieved within the ultrastrong-coupling distribution of this work must maintain attribution to the author(s) regime by exploiting vacuum fluctuations rather than coherent and the published article’s title, journal citation, and DOI. driving [65].

2643-1564/2021/3(1)/013116(19) 013116-1 Published by the American Physical Society ADAM STOKES AND AHSAN NAZIR PHYSICAL REVIEW RESEARCH 3, 013116 (2021)

Here we consider controllable light-matter interactions, but vanish sufficiently rapidly at the boundaries of the integration we avoid the restrictive scattering-theoretic assumption of domain. Under a gauge transformation A → A − dχ, where χ adiabatic switching over infinite times, as is required for mod- is arbitrary, the Lagrangian is transformed to one that differs eling any platform that involves sufficiently fast and strong by a total time derivative, and which is therefore equivalent interaction switching. We also allow the gauge to be arbi- to L. trary. Since each gauge for H(η) provides a different physical In applications we often wish to control the interaction be- definition of the interacting “light” and “matter” quantum tween light and matter systems, such as atoms within a cavity. subsystems, the promotion η → η(t ) constitutes a different This control can only occur via a third system, such as a laser, physical assumption when applied in each different gauge. which we call the control subsystem. Often, the explicit inclu- Moreover, beyond scattering theory, different gauges gener- sion of this control subsystem via a fully quantum treatment is ally treat virtual processes very differently within sufficiently cumbersome or even intractable. In this case a simpler alterna- fast and strong interaction regimes. We show that as a result tive must be sought. The simplest approach is to promote the of this, in such regimes the final-time predictions obtained light-matter coupling parameter to a time-dependent function, from H(η(t )) are significantly different in different gauges. which then constitutes an implicit model for the control sub- This occurs even when the quantum subsystems are unique system. One could instead pursue an explicit description of the at the preparation and measurement stages at which times the control subsystem as classical and external, i.e., as possessing interaction vanishes. preprescribed dynamics. This external control approximation We show using the example of an atom moving through a could be implemented at either the Lagrangian level or the cavity that by including from the outset an explicit description Hamiltonian level, but it is not clear that these two approaches of the degrees of freedom that mediate the interaction, i.e., by will be equivalent. explicitly including the control system at the Lagrangian level, Even if one confines oneself to attempting to treat the one can obtain a more complete and gauge-invariant descrip- control subsystem as external, in many situations a tractable tion H˜ (t ). We demonstrate that each different model H(η(t )), model may still be unavailable. We therefore start here by each of which belongs to a different gauge, coincides with considering the simpler approach of using a time-dependent H˜ (t ) applied to a different microscopic arrangement of the light-matter coupling parameter, which is a widespread ap- overall system. Thus, when using the assumption of a time- proach within the literature. Our aim is to understand the dependent coupling η → η(t ), each gauge models a specific implications of this approach when dealing with very fast and microscopic arrangement. These findings place significant re- strong interactions. In due course, we will see within specific strictions on the validity of using this common method when examples how, if at all, this description differs from explicitly describing strong and fast interactions because determining modeling the control subsystem as external. which specific experimental arrangements a particular model Our first task is to define what is meant by an interac- H(η(t )) describes, requires a more complete theory, such as tion. The definition must be such that when the interaction H˜ (t ), which may be unavailable or intractable except in the vanishes, the theory reduces to two free theories. A natural simplest cases. approach to describing time-dependent interactions would be μ The paper is organized into five sections. In Sec. II we to modify the interaction Lagrangian density LI =−jμA provide theoretical background introducing time-dependent via the replacement LI → μ(t )LI , where μ(t ) is a time- interactions. In Sec. III we provide a simple toy model that dependent coupling function. However, this alone does not transparently demonstrates the implications of using a time- imply that the interaction vanishes when μ(t ) = 0, due to 0i dependent coupling parameter. In Sec. IV we consider a more Gauss’ law ∇ · E = ρ, where ρ = j0 and Ei = F . Instead, a realistic atom-cavity system which facilitates a comparison modified current may be defined as μ(t ) j. Whenever μ(t ) = between the time-dependent coupling method and more com- 0, one then recovers two independent and free theories with plete descriptions of the controlled light-matter interaction. In matter described by Lm, and the electromagnetic subsystem L = 2 − 2 / = Sec. V we consider the time-dependent coupling method when described by TEM (ET B ) 2 where E ET is trans- describing fast and strong interactions. Finally, we briefly verse and B = ∇ × A is the magnetic field. summarize our findings in Sec. VI. Solving ∇ · E = μ(t )ρ to obtain EL, and replacing jμ with μ(t ) jμ in Eq. (1) yields the Lagrangian  II. CONTROLLABLE ELECTROMAGNETIC μ(t )2 INTERACTIONS L = L + L + d3x ρφ m TEM 2 Coul A. Definition of interaction and external control approximation  − μ 3 ρ − · , Maxwell’s equations can be derived from the standard (t ) d x[ A0 J A] (2) QED Lagrangian i    where Ji = j , i = 1, 2, 3, and 1 L = − 3 μ + μν ,  Lm d x jμA Fμν F (1) ρ(x ) 4 φ (x) = d3x . (3) Coul 4π|x − x| where Lm and j are the free Lagrangian and the four-current of an arbitrary material system such that dj = 0, while F = dA This formulation accommodates an arbitrary time-dependent and A are the electromagnetic tensor and four-potential for interaction, arbitrary material and electromagnetic systems, an arbitrary electromagnetic system. All fields are assumed to and arbitrary choice of gauge.

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B. Nonequivalent Lagrangians −e with mass m bound in the potential Vext. Choosing the Coulomb gauge ∇ · A = 0 implies A = A . From Gauss’ law We now consider a gauge transformation Aμ → Aμ − ∂μχ, T ∇ · = μ ρ = μ φ which in terms of scalar and vector potentials reads as E (t ) we then obtain A0 (t ) Coul. Instead of the Coulomb gauge, we could choose the Poincaré gauge defined → − ∂ χ, → + ∇χ. 1 A0 A0 t A A (4) by x · A(x) = 0. We then obtain A(x) = AT(x) + ∇χ and A = μ t φ − ∂ χ 1 This transforms the Lagrangian L in Eq. (2)as 0 ( ) Coul t where   1 d 3 1 L → L + μ(t ) d x ρχ. (5) χ (x) =− dλ x · AT(λx). (6) dt 0 The right-hand side is equivalent to L if and only ifμ ˙ = 0. More generally, we can straightforwardly encode the choice Since the total electric charge is the conserved Noether of gauge in a real parameter α such that charge resulting from gauge symmetry, the nonequivalence μ α = + ∇χ α, α = μ φ − ∂ χ α, can be understood as a consequence of the fact that if ∂μ j = A AT A0 (t ) Coul t (7) ∂ μ ν = μ = 0, then ν (t ) j 0 if and only if ˙ 0. One is naturally α 1 1 where χ = αχ with χ given in Eq. (6). Note that AT led to seek a different modified current j˜, which includes α α  ν is gauge invariant [55], that is, A = A for all α and α , the external control μ(t ), but also satisfies ∂ν j˜ = 0. We T T because ∇χ α is necessarily longitudinal: ∇ × ∇χ α ≡ 0. perform this analysis in Appendix A. The construction of j˜ If we now apply the canonical procedure to derive the requires inverting the divergence operator, which introduces Hamiltonian from the Lagrangian in Eq. (2) we obtain an additional, equally significant, gauge arbitrariness into the formalism. Neither μ(t ) j nor j˜ results in a Lagrangian that 1 H α (t ) = [p + eμ(t )Aα (r)]2 + V + V (t ) provides invariant dynamics under a complete gauge trans- 2m ext self formation. Thus, introducing the single additional assumption  1    + 3 + μ α 2 + 2 , that the interaction is controllable has resulted in nonequiva- d x (t )PT B (8) lence between different gauges. 2 As we have noted, a more complete approach would where include an explicit model for the control subsystem. In  μ 2 Sec. IV A we will consider the example of an atom mov- (t ) 3 Vself (t ) = d x ρφCoul (9) ing in and out of a cavity, which is simple enough to be 2 amenable to such an approach. In this case, the control sub- is the infinite Coulomb self-energy of the charge, which is system is the atom’s center-of-mass motion. We will see that usually taken as renormalizing the bare mass and is then approximating the control subsystem as external at the Hamil- ignored, and where tonian level actually produces the same result as assuming  a time-dependent coupling parameter. Thus, the nonequiva- 1 α =− α λ δT − λ lence of models belonging to different gauges when using PT,i e d r j ij(x r) (10) the time-dependent coupling method can be understood as 0 the consequence of an approximation. The situation is anal- is the α-gauge transverse polarization. Interpreted as ogous to the effect of material energy-level truncation, which Schrödinger picture quantum operators the Hamiltonians of also produces nonequivalent models when applied in different different gauges are nonequivalent being unitarily related by a gauges [52,66–68]. However, while material level truncation generalized time-dependent Power-Zienau-Woolley transfor- is often straightforwardly avoidable, avoiding the approxima- mation as tion of a control system as being external may be much more α α =   , difficult. H (t ) Rαα (t )H (t )Rα α (t ) (11) In Sec. IV B we will see that if we instead approximate the where control subsystem as external at the Lagrangian level, then    the subsequent Hamiltonians belonging to different gauges  3 1 Rαα (t ) = exp i(α − α )μ(t ) d x P · A . (12) are equivalent. Furthermore, through comparison with this T T Lagrangian approach it is possible to determine whether or The nonequivalence of the Hamiltonians for distinct values not the simple time-dependent coupling method will be valid  of α follows from Eq. (11), which shows that H α (t ) = when applied within a given gauge. We find that the correct α   + ˙   gauge, if any, to employ when using the latter method, de- Rαα (t )H (t )Rα α (t ) iRαα (t )Rα α (t ), where the right-hand side is equivalent to H α (t ). pends on the microscopic details of the system. The procedure α is analogous to identifying a gauge that provides the most Equation (8)givesthe -gauge Hamiltonian with time- dependent coupling and no approximations have been made accurate material truncation by comparing the approximate μ nonequivalent models with a more complete gauge-invariant in its derivation, except the use of (t ) as a model for the theory [52,66–68]. control subsystem. Note that the canonical coordinate of the electromagnetic subsystem is the transverse vector potential AT, which is manifestly gauge invariant. The α-gauge vector C. Nonequivalent Hamiltonians potential Aα appearing in Eq. (8) is specified as a function To better understand the implications of the transforma- of AT given uniquely by Eqs. (6) and (7). In particular, these tion property (5) we consider the example of a point charge equations together with Eq. (10) imply that eAα (r) can be

013116-3 ADAM STOKES AND AHSAN NAZIR PHYSICAL REVIEW RESEARCH 3, 013116 (2021) written III. TOY MODEL  Time-dependent interactions between subsystems arise in α = + α∇ 3 1 · . eA (r) eAT(r) r d x PT AT (13) many and diverse areas of physics. Here, we consider a very simple light-matter model. This serves to clearly determine It is common to perform the electric dipole approximation the situations within which we can expect the gauge nonequiv- ≈ 1 ≈− δT (EDA) AT(r) AT(0) and PT,i(x) er j ij(x), which re- alence of models that result from assuming a time-dependent quires the resonant wavelengths to be long compared with coupling to become significant. We will see that it becomes the spatial extent of the material system set by Vext.Itisalso significant in the description of so-called virtual processes, common to neglect the infinite self-energy of the charge. One which typically become increasingly important with increas- then obtains the Hamiltonian ing coupling strength. 1 α = + μ − α 2 + H (t ) [p e (t )(1 )AT(0)] Vext A. Time-dependent Hamiltonian 2m  1  2  We suppose that a charge −e is confined in all spatial di- + d3x (x) + αμ(t )P1 (x) + B(x)2 . ε 2 T mensions except the direction of the polarization of a single (14) cavity mode, in which it is bound harmonically. The position operator is r = rε and the conjugate momentum is p = pε. The choice α = 0 provides the time-dependent version of the The material canonical commutation relation is [r, p] = i.The well-known “p · A” interaction of the Coulomb gauge, while field canonical commutation relation is, in the general case, the choice α = 1 likewise provides the time-dependent ver- given by − · sion of the well-known “ er ” interaction of the Poincaré ,  = δT −  . gauge. Both of these interaction forms are commonly found [Ai(x) j (x )] i ij(x x ) (16) within the literature. The Hamiltonians of different gauges Discretizing the modes within a cavity volume v, the fields continue to be nonequivalent and unitarily related as in can be expanded in terms of photon creation and annihilation Eq. (11) where now operators. Restricting the fields to a single mode kλ then gives   = − α − α μ · , † −ik·x ik·x Rαα (t ) exp [ i( )e (t )r AT(0)] (15) AT(x) = gε(a e + ae ), (17) which is simply the dipole approximation of Eq. (12). (x) = iωgε(a†e−ik·x − aeik·x ), (18) The canonical formalism explains why the nonequiva- α √ lence of the H (t ) occurs; in different gauges the theoretical where g = 1/ 2ωv, ω =|k|, ε ≡ εkλ is orthogonal to k, and † quantum subsystems are defined in terms of different gauge- a ≡ akλ with [a, a ] = 1. Equations (17) and (18) imply that invariant observables. In the α gauge the field canonical the cavity canonical operators now satisfy the commutation =− − α 1 momentum is ET PT. The Coulomb and multipo- relation lar gauges are special cases with =−ET and =−DT,  iεiε j  respectively, both of which are gauge-invariant observables. [AT,i(x), j (x )] = cos [k · (x − x )]. (19) Since the “light” and “matter” subsystems are defined using v the canonical operator sets {AT, } and {r, p}, they can also In the dipole-approximated Hamiltonian of Eq. (14)the only be specified relative to a choice of gauge. The interaction fields are evaluated at the dipolar position 0. The Hamil- being externally controllable constitutes a different physical tonian can therefore be expressed entirely in terms of the assumption when imposed on different physical subsystems cavity variables A = ε · AT(0) and = ε · (0). According that are defined relative to different gauges [48]. Thus, each to Eqs. (17) and (18), the commutator of these variables is H α (t ) describes a different experimental protocol, in which a [A, ] = i/v. Comparing this commutator, or the commutator different interaction is being controlled. This will be demon- in Eq. (19), with Eq. (16), reveals that the transverse delta strated directly by way of example in Sec. IV B. function is given within the single-mode approximation by  Presented with an experiment that we are asked to model 3 T d k 1 using a time-dependent coupling, we possess an infinity of δ (0) = ελ, ελ, −→ ε ε . (20) α ij π 3 i j i j nonequivalent models H (t ) which for each different value (2 ) λ v of α, we know to model a different experimental protocol. Without an argument to choose between the available models, It follows that the polarization self-energy term in the an ambiguity is encountered. Determining the correct model Hamiltonian becomes within the EDA and single-mode ap- may be difficult because as we shall see, the theoretical sub- proximations  systems differ between gauges only in their description of 2 2 1 3 2 e e 2 virtual processes. In weak-coupling regimes involving suffi- d x P (x) = rir jδij(0) −→ r . (21) 2 T 2 2v ciently adiabatic interactions the “ambiguity” described above is unproblematic because its consequences are usually negli- The dipole and single-mode approximations have no bearing gible in practical calculations. This is no longer the case in on gauge invariance or noninvariance because whether they sufficiently strong-coupling nonadiabatic regimes where, as is are performed or not, the Hamiltonians H α (t ) are equivalent if apparent in Eq. (23) below, α-dependent components of the and only ifμ ˙ = 0. The approximations are used here to enable interaction V α (t ) are not negligible. a simple and transparent example.

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√ Material bosonic ladder operators can be defined as b = because the two models possess different dependencies on the † 1/2mωm(mωmr + ip) with [b, b ] = 1. The Hamiltonian in underlying model parameters. Eq. (14) can now be written We remark that anharmonic material systems may also be α α = + α, considered. The independence of predictions for processes H H0 V (22) conserving the noninteracting part of a Hamiltonian is a com- † † pletely general result within scattering theory [48,55,56,60]. where H0 = ω(a a + 1/2) + ωm(b b + 1/2) and However, anharmonic matter does not generally admit a sim- 2 α η(t ) ω ple analytic treatment at the level of the model Hamiltonian. V = [(1 − α)2(a† + a)2 + δα2(b† + b)2] 4 An exception is when the material system is sufficiently an- − † † + † † harmonic that a two-level truncation can be made. In general, + iuα (t )(ab − a b) + iuα (t )(a b − ab) (23) this will break the gauge invariance of the theory and so must with δ = ω/ωm and be performed within a gauge in which the truncation is found eμ(t ) to be accurate for the properties of interest [52,66–68]. η(t ) = ημ(t ) = √ , (24) ω mv √ IV. EXAMPLE: UNIFORM MOTION THROUGH A CAVITY ± 1 uα (t ) = η(t )ωm δ[(1 − α) ∓ δα]. (25) 2 A. Time-dependent Hamiltonian The variation of μ(t ) could be interpreted as a model for B. Bare-energy conservation and α-independent predictions the motion of an external potential Vext, which moves in and In QED material systems are often interpreted as sur- out of contact with the electromagnetic fields. As we noted rounded by a cloud of virtual photons [53,55,69–72]. Two in Sec. II B, the system responsible for the potential can be examples of virtual processes are those described by the terms called the control system, which in a more complete descrip- † † + ab and a b having a coupling strength uα (t )inEq.(23), tion would be included explicitly via additional dynamical which are number nonconserving and so do not commute position and momentum variables R and K.Toshowhow α with H0. Inspection of Eq. (23) reveals directly that different the nonequivalence of the Hamiltonians H (t ) results from an subsystem divisions only differ in their description of virtual approximation, we consider a hydrogen atom consisting of a α processes. All number-nonconserving terms in Eq. (23)are charge +e with mass m2 at r2 and a charge −e with mass m1 at dependent, whereas the remaining number-conserving part is r1. The charge and current densities are ρ(x) = e[δ(x − r1) − α independent at resonance, which is precisely when this term δ(x − r2 )] and J(x) = e[r˙1δ(x − r1) − r˙2δ(x − r2 )]. Within conserves the bare energy, because the EDA i ω η(t )[H , (ab† − a†b)] = 0. (26) ρ(x) = er · ∇δ(x − R), (27) 2 m 0 Thus, despite the α dependence of the subsystems them- J(x) =−er˙δ(x − R) + eR˙ (r · ∇)δ(x − R), (28) selves, within the approximation of retaining only the = − interaction terms that conserve H ,allα dependence drops out where r r1 r2 is the relative position between the charges 0 = + / + of the theory. In this case, the bare vacuum coincides with the and R (m1r1 m2r2 ) (m1 m2 ) is the position of the cen- Hamiltonian ground state. This approximation is valid in the ter of mass. The second term on the right-hand side of Eq. (28) ∂ μ = traditional regime of weakly coupled nearly resonant systems, ensures the conservation of charge μ j 0, and also ensures a regime that can be understood as gauge nonrelativistic [48]. that the dipole’s interaction with the electric field induced Therein, one can pretend that the quantum subsystems,“light” by the atomic motion in the laboratory frame is properly and “matter,” are ostensibly unique, i.e., not gauge relative. In included. In particular, in the multipolar gauge this term cor- truth, this is not the case, and the pretense cannot be sustained rectly ensures the presence of the Röntgen interaction. The if the required approximation is not valid. Therefore, for ultra- current can be obtained from the nonrelativistic transforma- ρ = ρ =  + ˙ ρ strong and fast interactions, when assuming a time-dependent tion , J J R , which relates the (primed) atomic = coupling it must be determined which gauges describe which rest frame to the (unprimed) laboratory frame in which R ˙ ˙ = α experimental protocols and arrangements. In what follows, we Rt with R 0, up to a constant initial position. The -gauge verify that the correct gauge to use will generally depend on polarization field is the microscopic arrangement being considered. No one gauge α T P , (x) =−eαr jδ (x − R) (29) is universally correct. T i ij Ultrastrong and fast interactions are now of major impor- and the associated magnetization field Mα is such that ∇ × α = − ˙ α α = tance [26,64]. For such interactions a model corresponding to M JT PT. In the multipolar gauge 1 we obtain the α = 0 or 1, which are both commonly chosen gauges in light- expected multipolar expressions. In particular, ∇ × M1(x) = matter theory, will not generally produce even qualitatively −e∇ × [r × R˙ δ(x − R)]. accurate predictions if the underlying physics of the system Equations (27)–(29) can be used within the Lagrangian of is more correctly described by an interaction corresponding, Eq. (1) and the α-gauge Hamiltonian can be derived with + for example, to α ∼ αg = 1/(1 + δ), for which uα (t ) vanishes r, R, and AT as canonical coordinates. Details are given in identically. In fact, as we shall show, even conventional gauges Appendix B. If we approximate the center-of-mass position α = 0 and 1 generally give significantly different physical R as externally prescribed within the Hamiltonian, then we predictions when the coupling η(t ) is ultrastrong and ultrafast obtain a bipartite quantum system and the Hamiltonians of

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η = δ = 1 FIG. 2. 1and 2 . The average number of photons found α using H (t ) is plotted with time in units of the beam transit time tb = wc/ν assuming an initial state |0, 0 . The beam waist is wc = 20 μm, ωm is chosen in the microwave regime (energy ∼10 μeV), and ν = 10−3c where c is the speed of light. The final values are given where curves become straight, and are clearly different for different α.

atom to be in the xy plane such that zˆ · R(t ) = 0. The cavity FIG. 1. A cavity of length L supporting standing waves in the z canonical operators are given by direction and a Gaussian perpendicular mode profile with waist wc − ω 1 ∗ is depicted, along with a dipole er oscillating with frequency m. A (t, x) = √ [φ (x)a†(t ) + φ(x)a(t )], (32) = T ω At t 0 the cavity and dipole are noninteracting. The dipole follows 2 v a classical trajectory R(t ) through the cavity, entering the cavity at ω ∗ † t and exiting at t + τ. The Hamiltonian for this system is derived (t, x) = i [φ (x)a (t ) − φ(x)a(t )]. (33) 0 0 2v in Appendix C and can be realized using a time-dependent coupling φ(R(t )) =: μ(t )asinEq.(31). Letting φ(R(t )) = μ(t ), we see that the Hamiltonian in Eq. (31) is identical to that defined in Eq. (22), which was in turn obtained from Eq. (8). This shows that the nonequiv- different gauges become nonequivalent, being given by alence of the models Hα (t ) for a single atom obtained via the time-dependent coupling method can indeed be understood as α 1 2 H (t ) = [p + e(1 − α)AT(R(t))] + V (r) a consequence of approximating the center-of-mass motion 2m  (the control subsystem) as an external subsystem.   1 α Uniform motion of the dipole in and out of the cavity is + d3x [ + P (t )]2 + B2 , (30) 2 T described by a Gaussian function μ(t ). Significant α depen- dence of final predictions occurs when the interaction time α τ ∼ /ν ν = ˙ /ω where PT (t ) is explicitly time dependent due to its dependence wc ( R) is comparable to the cycle time 1 m.In on R. This expression clearly has the same structure as Eq. (8) the case of a microcavity with wc = 20 μm and ωm in the and its dipole approximation (14), which were obtained by microwave regime, this requires ν ∼ 10−3c, which is nonrel- assuming a time-dependent coupling. ativistic. We assume that the system is initially noninteracting To progress further, in Appendix C we specialize the above [μ(0) = 0] and starts in the ground state |0, 0 . The inter- Hamiltonian to the case of a Fabry-Perot Gaussian cavity action is switched on at time t0 > 0, and switched off at mode with mirrors orthogonal to the z direction, as depicted in t0 + τ>t0. Thus, at the preparation and measurement stages Fig. 1. Assuming as before that r = rε and p = pε, and that the definitions of the quantum subsystems are unique. In Fig. 2 the atomic potential energy V (r) is harmonic, the Hamiltonian the number of cavity photons is plotted as a function of time reads as with wcωm/ν ∼ 1, η = 1, and α = 0, 1,αg. The three gauges give different residual photon populations within the cavity 1 mω2 H α (t ) = [p + e(1 − α)A (R(t ))]2 + m r2 after the interaction has ceased, consistent with the sugges- 2m T 2 tion of the energy-time uncertainty relation. For longer and α2 2 slower interaction switching and weaker couplings than we 2 r 2 + e |φ(R(t ))| − eαr (R(t )) have shown all photon populations return to zero independent 2v of α. In contrast, when the interaction switching is on the 1 + ω a†a + , (31) order of a bare cycle and the coupling is sufficiently strong, 2 there is a significant probability that virtual photons created near the beginning of the switch-off are not reabsorbed before − 2+ 2 / 2 where φ(x) = eikze (x y ) wc is a Gaussian mode envelope, the atom has exited the cavity. They therefore detach and are with wc the Gaussian beam waist. We assume the path of the left behind, remaining inside the cavity. This cannot occur,

013116-6 ULTRASTRONG TIME-DEPENDENT LIGHT-MATTER … PHYSICAL REVIEW RESEARCH 3, 013116 (2021)   however, for values of α ∼ αg for which ground-state photons 2R × iωxˆ zˆ (a† − a) − (αδ − xˆ xˆ )(a† + a) . (37) are not explicit. i j 2 ij i j wc It is straightforward to verify that for α = 1 the right-hand B. Gauge-invariant class of approximate Hamiltonians side of this expression coincides with er · [R˙ × (∇ × AT(R))] In the case of a sufficiently simple moving bound-charge as required [cf. Eq. (36)]. Since the coefficient of a† − a on system, as considered above, a gauge-invariant set of Hamil- the right-hand side is α independent, there is no choice of α tonians can be derived. This can only be achieved by starting for which H α (t ) = H˜ α (t ) in general. However, if we make with an explicit model for the control system and requires the simplifying assumption that r = rε [giving H α (t )asin making the approximation that this control is external at the Eq. (31)], then we obtain Lagrangian level, rather than at the Hamiltonian level. The ˜ α − α =− μ α − 2 θ , procedure is not generally equivalent to assuming a time- H (t ) H (t ) e ˙ (t )rAT(0)[ cos ] (38) dependent coupling. where cos θ = ε · xˆ. Notice that if the switchingμ ˙ is suffi- In the case of the hydrogen atom within the EDA, we have ˜ α = α α , ciently slow, then H (t ) H (t ) independent of , whereas three subsystems with position coordinates r R, and AT.The ifμ ˙ is sufficiently fast then the predictions obtained from control subsystem with coordinate R can be treated as external different H α (t ) may become appreciably different. In contrast, at the Lagrangian level, such that it becomes a preprescribed the predictions obtained from H˜ α (t ) are always α independent dynamical vector R(t ) prior to the transition to the canonical (gauge invariant). The correct value of α to choose within formalism. Now only r and AT remain as dynamical variables. H α t H α t = H˜ α t α ˜ α ( ) is the value solving the equation ( ) ( ). This The resulting -gauge Hamiltonian, denoted H (t ), is given value depends on the orientation of the mode polarization and by dipole moment ε relative to the direction of motion xˆ. In other α α α H˜ (t ) =H (t ) + eR˙ · [(r · ∇)AT(R)] words, the correct value of to choose when employing the time-dependent coupling method, depends on the microscopic − α ˙ · ∇ · , e R [r AT(R)] (34) arrangement of the system (the microscopic context). α θ =±π/ ε where H (t )isgiveninEq.(30). Full details of the derivation Consider the arrangements 2( and xˆ orthogo- α ˜ 0 | = 0 are given in Appendix B. Unlike the Hamiltonians H (t ), as nal). From Eq. (38)wehaveH (t ) θ=±π/2 H (t ) whereas α ˜ α | = α α = Schrödinger picture operators the Hamiltonians H˜ (t ) satisfy H (t ) θ=±π/2 H (t )for 0. Therefore, to model these arrangements the correct value of α to choose when using ˜ α = ˜ 0 † + ˙ † α H (t ) R0αH (t )R0α iR0αR0α (35) H (t )isα = 0. Any other value will yield incorrect predic- tions as determined by comparison with the gauge-invariant such that Hamiltonians belonging to distinct gauges are α predictions provided by H˜ (t )|θ=π/2. Similarly, for the al- equivalent. It is instructive to consider the multipolar-gauge θ = ,π ε example (α = 1): ternative arrangements 0 ( and xˆ parallel) we have ˜ 1 | = 1 ˜ α | = α α =  H (t ) θ=0,π H (t ) whereas H (t ) θ=0,π H (t )for 2    α = 1 p 1 3 1 2 2 1. Therefore, 1 is the correct value to choose when H˜ (t ) = + V (r) + d x + P + B α 2m 2 T modeling these arrangements using H (t ). More generally, θ + · ˙ × . Eq. (38) shows that for modeling the arrangement a cor- er [R B(R)] (36) rect value of α to choose when using H α (t ) is a solution of The final term in this expression describes the Röntgen inter- α = cos2 θ. action in which the dipole experiences an effective electric We have demonstrated that the determination of when the field R˙ × B(R) due to the gross motion in the laboratory time-dependent coupling method will produce correct results frame [56,73,74]. Such an interaction also appears in the cannot be accomplished without recourse to a more complete complete Hamiltonian derived by keeping R as a dynamical description, which yields the constraint α = cos2 θ. Under α variable where it manifests via a nonmechanical canonical this constraint, H (t ) provides a gauge-invariant description α momentum K = MR˙ + er × B(R). The Röntgen interaction because it coincides with H˜ (t ) which provides a gauge- term is lost when R = R˙ t is prescribed as external within the invariant description by construction. Under the constraint complete Hamiltonian, which results in Eq. (30). that α = cos2 θ, the parameter α may be thought of as select- When using H˜ α (t ), any value of α can be chosen and the ing an experimental context rather than a choice of gauge. It final predictions will necessarily be α independent (gauge follows that it is not the case that the Coulomb gauge is always invariant). Let us therefore suppose that these predictions are correct when using a time-dependent coupling, contradicting “correct” (albeit approximate). It follows that the fixed values Ref. [67]. For example, within the system considered here, the of α for which H α (t ) = H˜ α (t ) are those allowed in order to Coulomb gauge will yield the correct description only when obtain “correct” results using H α (t ). Taking the fields given cos θ = 0. Subsequent time-dependent gauge transformation by Eqs. (32) and (33), we assume that R(t ) = (h − νt )xˆ, using R0α (t ) will of course then yield an equivalent descrip- implying uniform motion R˙ =−νxˆ from an initial position tion to H 0(t ) as noted in Ref. [67], but this equivalence class hxˆ outside of the cavity. Under these conditions, the difference of models is restricted to describing the experimental context H˜ α (t ) − H α (t ) is given by cos θ = 0. In Fig. 3 the average number of photons is plotted as a ˙ · · ∇ − α ˙ · ∇ · α eR [(r )AT(R)] e R [r AT(R)] function of time found using H˜ (t ). As expected, the num- eν ber differs between gauges when η = 0, due to the inherent = √ φ(R)εir j 2ωv relativity in the definition of the light quantum subsystem, but

013116-7 ADAM STOKES AND AHSAN NAZIR PHYSICAL REVIEW RESEARCH 3, 013116 (2021)

FIG. 3. All parameters are as in Fig. 2. The average number = | , = of photons is plotted with time in units of tb = wc/ν assuming an FIG. 4. Starting at t 0 in the ground state 0 0 of H0 α α † initial state |0, 0 . The dynamics is generated by H˜ (t ) and we have H (0), the average number of photons a a t in the cavity is plotted assumed cos θ = 0. As expected, during the interaction window the with time. The switch-on function μ(t ) is shown in Fig. 5 and is such average photon number differs between gauges. However, in contrast that the switch-on time is roughly 4/ωm. The remaining parameters η = δ = 1 ω to Fig. 2 all gauges predict the same final value. Because we have are 1and 2 ,and m is in the microwave regime. The final chosen cos θ = 0, the final value coincides with that predicted by values after the interaction has ceased are given where the curves H 0(t ). It is therefore identical to the final value of the α = 0curvein level off, and are clearly different for different α. Fig. 2. in contrast to the predictions obtained using H α (t )(cf.Fig.2), in Fig. 3 all gauges predict the same final value. Since we have This is a smoothed box function with a maximum of one t = t + τ/ μ t ≈ 1 τ chosen cos θ = 0, this final value coincides with that predicted at 0 2, such that ( 0 ) 2 , and is roughly the by H 0(t ) as given by the curve for α = 0inFig.2. Similarly, full-width at half-maximum. The parameter s controls the if cos θ = 1, the final value coincides with that predicted by smoothness of the switch-on. Through tuning of parameters H 1(t ), which is given by the α = 1 curve in Fig. 2. this general coupling function can take a variety of forms, Considering a uniform distribution of random orientations including close resemblance to a Gaussian, as occurs for θ, the average of Eq. (38)is uniform atomic motion through a Gaussian cavity. In what   follows, we determine the dependence on α of the final light α α 1 and matter properties that result from the dynamics generated E[H˜ (t ) − H (t )]θ =−eμ˙ (t )rA (0) α − . (39) α T 2 by H (t ) in nonadiabatic strong-coupling regimes. A naive example of time-dependent coupling comprises At resonance (δ = 1) the Jaynes-Cummings gauge αg = instantaneous switching of a constant interaction, but here the 1/(1 + δ) now gives the “correct” value, but the difference free evolution before and after the interaction window does |1/2 − αg| increases as the detuning moves away from reso- not alter the physical quantities of interest. Predictions for the nance. case of a constant interaction in the ground state |G of the full Hamiltonian H α are given in Appendix D. V. GENERAL TIME-DEPENDENT COUPLING A more realistic interaction switching is smooth, requiring finite time. We therefore use the general coupling function A tunable coupling function could be used to model any given in Eq. (40) within the simple Hamiltonian given in time-dependent interaction, such as those realized by address- Eq. (22). The dynamics of the system is found by numer- ing specific states of an atomic system [18], or laser-driven ically solving the closed set of differential equations for systems [53]. Switchable interactions are also commonly en- correlations of the form xy where x, y = a, a†, b, b†.For countered in superconducting circuits [20]. It is seldom the an initial Gaussian state these correlations suffice to com- case that the subsystem mediating a controllable interaction pletely characterize the final state [75]. We find that significant between two other subsystems will admit a straightforward α dependence of final predictions occurs if the interaction explicit model of the kind that we have been able to provide switching time is ultrafast, i.e., of the order of a bare cycle for the simple atom-cavity example considered above. It is ω−1,ω−1 m , and the coupling is sufficiently strong. For longer therefore important to understand more generally the extent switching times predictions from different gauges converge to which results obtained from the simple time-dependent as the interaction is switched off, such that no differences coupling method will apply only to a specific experimental remain by the end of the protocol. Figure 4 shows the average context. To this end, we consider the general coupling func- number of photons in the cavity as a function of time, when tion the switching time is roughly 4/ωm and the system starts in      | , = α st0 2 s τ the ground state 0 0 of H0 H (0). Again, both initially tanh sinh t − − t0 μ(t ) = 1 −  2  2  2 . (40) and finally there is no ambiguity in the definitions of the light s − s τ + − cosh 2 (t t0 ) cosh 2 ( t0 t ) and matter systems, which are uncoupled. Relevant subcycle,

013116-8 ULTRASTRONG TIME-DEPENDENT LIGHT-MATTER … PHYSICAL REVIEW RESEARCH 3, 013116 (2021)

= | , = FIG. 5. Starting at t 0 in the ground state 0 0 of H0 FIG. 6. η = 1andδ = 3 with τ, s,andωm as in Fig. 5. βc cor- α I α H (0), the mutual information ( ) at a final time t long after responds to room temperature while βm = 2βc. The final subsystem the interaction has been switched off is plotted as a function of α energy changes and net work are plotted with α. The net work and δ η for various combinations of and . The inset shows the coupling Ec are always positive, while Em becomes negative for certain envelope μ(t ) as a function of time. The interaction duration given α, implying that energy has left the initially cooler system and has τ = /ω ω by the difference in the dashed lines is 10 m with m chosen in entered the initially hotter system. This is due to the nonzero net = τ/ the optical range. The switch on occurs at roughly t0 2 and the work input. chosen value of s gives a switching time, represented by the arrow, of roughly 4/ωm.Theα dependence of I(α) varies significantly depending on the regime considered. I(α) is symmetric about the do not assume these temperatures are equal. If the subsequent light-matter interaction is relatively short on the order of ω−1 minimum of zero at αg = 1/(1 + δ)forallδ and η.Theα depen- m dence tends to be more pronounced further from resonance and for as in Figs. 5 and 6, and is also ultrastrong, then a clear separa- stronger coupling. tion of time and energy scales emerges. Weak environmental interactions can therefore be ignored over the timescales of interest. ultrastrong couplings have already been achieved in cavity Using the unitarity of the dynamics it is straightforward QED [18]. to show that changes in the energies of the subsystems de- Since the systems are initially uncoupled, it is natural to eq fined by Ex = tr[ρx (t )Hx] − tr[ρx Hx] with x = m, c,are assume that they are not correlated. Correlations may then bounded according to βmEm + βcEc  I  0[76,77]. If build up due to the subsequent interaction. Figure 5 shows the the interaction is also such that there is no net input of work, final mutual information I(α) at a time t after the interaction α i.e., H (t ) ≡ H0 ≡Em + Ec = 0, then we obtain has ceased. It exhibits a diversity of behaviors depending (β − β )E  0. Thus, without a net input of work, energy α η δ m c m on the values of , , and . As expected, the variations in cannot move counter to the initial temperature gradient. On the mutual information become increasingly pronounced as the other hand, if Em + Ec = 0, then by the end of the the coupling strength increases and one moves away from interaction the initially cooler system may have lost energy, δ = resonance 1. As noted in Sec. III, the weak-coupling with an accompanying increase in energy of the initially hot- resonance regime is gauge nonrelativistic, i.e., is such that ter system. Alternatively, both subsystems may simply gain all gauges will produce the same final predictions despite the energy due to the nonzero net work. The final energy that α α = α H (t ) being nonequivalent. As shown in Fig. 5,if g the has been exchanged between the systems after the protocol interaction does not generate any correlations for the values has finished is shown as a function of α in Fig. 6. It is clear η δ α of and chosen. Thus, this value of reproduces the that different qualitative thermodynamics can be realized by (ostensibly unique) result obtained within the weak-coupling varying only the parameter α, which controls the gauge. We regime even for ultrastrong coupling. The subsystems defined express once more that these qualitative differences in final relative to this gauge are those for which the ground state is properties occur even though the subsystems are uniquely much closer to the bare vacuum. defined at both the initial and final times. To exemplify the importance of our results, we show that due to the time dependence of the interaction even the qualita- VI. CONCLUSIONS tive predictions for energy exchange depend strongly on α.To this end, we consider a situation where the systems are not ini- We have studied the implications of gauge freedom for tially isolated from their environments. We therefore consider subsystem properties in QED when dealing with tunable eq an initial product state of two Gibbs states ρ(0) = ρm (βm ) ⊗ nonadiabatic, strong coupling. When the coupling is nonva- eq eq −β H ρc (βc ) where ρm (βm ) = e x x /tr(·), x = m, c with Hm = nishing, there are infinitely many nonequivalent definitions of † † ωm(b b + 1/2) and Hc = ω(a a + 1/2). These states result if the quantum subsystems. For strong enough coupling, “light” before their interaction the systems have separately weakly and “matter” subsystem properties like entanglement and coupled and equilibrated with Markovian environments at the photon number are significantly different for different subsys- β−1 β−1 corresponding temperatures m and c . For generality, we tem definitions. These differences persist in the case of final

013116-9 ADAM STOKES AND AHSAN NAZIR PHYSICAL REVIEW RESEARCH 3, 013116 (2021) subsystem predictions found when assuming a time- where  dependent coupling. The differences become increasingly    pronounced as the coupling switching increases in strength P(x) =− d3x g(x, x )ρ(x )(A3) and speed. We have shown directly that the time-dependent coupling in which ∇ · g(x, x ) = δ(x − x ). The polarization P satisfies assumption can be valid for fast and strong interactions, −∇ · P = ρ identically, but has arbitrary transverse com-  but only if one can identify and choose the correct gauge ponent because ∇ · gT(x, x ) ≡ 0. Defining P˜ = μ(t )P one within which to make the assumption when describing a given recovers the well-known continuity and polarization relations physical arrangement. This is necessary in order to obtain in terms of the modified quantities: even qualitatively reliable predictions. The correct choice of ∂ ˜ν = , −∇ · ˜ = ρ. gauge will generally depend on the specific microscopic ar- ν j 0 P ˜ (A4) rangement being considered. Its determination requires the The complete construction of the charge and current den- availability of, and comparison with, a more complete descrip- sities using auxiliary fields requires the introduction of the tion that explicitly includes the control degrees of freedom. magnetization M such that J = P˙ + ∇ × M. The modified This finding is of major importance for current technological magnetization required to give J˜ = P˜˙ + ∇ × M˜ is therefore applications including quantum communication, metrology, simply M˜ = μ(t )M. Since P and M are auxiliary fields for ρ simulation, and information processing, where the use of and J, they can be viewed as material analogs of the electro- time-dependent couplings is widespread and final subsystem magnetic potentials, which are auxiliary fields for E and B. properties are of central importance. Like the polarization P the definition of the magnetization M also possesses an arbitrary freedom. Indeed, the definitions ACKNOWLEDGMENTS of these auxiliary quantities in terms of the charge and current densities possess the same structure as the inhomogeneous This work was supported by the UK Engineering and Phys- Maxwell equations, but these equations are not supplemented / / ical Sciences Research Council, Grant No. EP N008154 1. by any homogeneous Maxwell-type equations. It follows that We thank Z. Blunden-Codd for useful discussions. any transformation of P and M that leaves the defining in- homogeneous equations invariant is permissible. Thus, j is APPENDIX A: INTRODUCTION OF A DIFFERENT invariant under a transformation by pseudomagnetic and pseu- MODIFIED CURRENT doelectric fields as

1. Charge conservation P → P + ∇ × U, M → M − ∇U0 − U˙ , (A5) Since the total electric charge is the conserved Noether- where U is an arbitrary pseudo-four-potential. The fields charge associated with gauge symmetry, the nonequivalence are in turn invariant under a gauge transformation Uμ → of the Lagrangians associated with different gauges can be Uμ − ∂μχ where χ is arbitrary. The modified current J˜ is ∂ μ ν = understood as a consequence of the fact that ν (t ) j not invariant under the transformation (A5), which results in μ = 0 if and only if ˙ 0. A second implication is that the J˜ → J˜ + μ˙ (t )∇ × U. ∇ · = μ ρ inhomogeneous Maxwell equations E (t ) (Gauss’ If we now replace the naive modified current μ(t )J that ˙ = ∇ × − μ law) and E B (t )J (Ampere’s law), cannot be appears in Eq. (2) of the main text with J˜ we obtain simultaneously satisfied. The method of modeling con-  trollable interactions between given subsystems using a L˜ = L + μ˙ (t ) d3x P · A. (A6) time-dependent coupling parameter is usually adopted at the Hamiltonian level, and is widespread. It is not our intention Since L˜ is not equivalent to L, it does not possess the same to advocate such an approach, but merely to understand its properties under a gauge transformation of the electromag- implications. The implications above follow from tracing back netic potentials, which gives the conventional approach to the Lagrangian level or to the  fundamental equations of motion. d L˜ → L˜ + d3x ρχ˜ (A7) Given the discussion above, one is naturally led to seek dt a different modified current j˜, which includes the external ν as desired. However, it can be seen immediately from Eq. (A6) control μ(t ), but also satisfies ∂ν j˜ = 0. Letting j˜0 ≡ ρ˜ = μρ and then considering ρ˜˙ reveals that the appropriate modified that unlike the original Lagrangian L, under the transfor- ˜ three-current J˜ must satisfy mation (A5) the Lagrangian L transforms to an equivalent Lagrangian if and only ifμ ˙ = 0. Thus, our construction ∇ · J˜ = μ(t )∇ · J − μ˙ (t )ρ. (A1) of j˜, P˜ , and L˜ has replaced one gauge noninvariance with another. The inhomogeneous Maxwell equations are now si- This clearly necessitates the addition of a nontrivial term to multaneously satisfied when written in terms of the modified the naive modified current μ(t )J. Solving Eq. (A1)forJ˜ quantities, but the modified current J˜ is not invariant under requires inverting the divergence operator, which introduces the transformation (A5) and therefore neither is Ampere’s law a new arbitrary element into the formalism. The solution can when written in terms of J˜. We stress that the freedom to be expressed as choose the transverse component of P is an important freedom within the theory, and is no less significant than the freedom ˜ = μ + μ , J (t )J ˙ (t )P (A2) to choose the potentials. Indeed, as we will see in Sec. A 2

013116-10 ULTRASTRONG TIME-DEPENDENT LIGHT-MATTER … PHYSICAL REVIEW RESEARCH 3, 013116 (2021) the freedom to choose PT is what gives rise to the well-known manifestly gauge-invariant interaction Lagrangian from the Poincaré-gauge dipolar interaction Hamiltonian −er · (0). outset. Such an interaction Lagrangian is given by Essentially the same result as Eq. (A6) above can be ob-     tained if instead of considering the current, one considers the  3 3 μ d L = d x[P · E + M · B] =− d x j Aμ + P · A , interaction Lagrangian. The standard interaction Lagrangian I dt μ density j Aμ is not gauge invariant, rather, under a gauge (A9) transformation Aμ → Aμ − ∂μχ it changes as    d which is clearly invariant under a gauge transformation. How- L =− 3 μ −→ − 3 μ + 3 ρχ. L I d xj Aμ d xj Aμ d x ever, under the transformation (A5) I transforms to an dt equivalent but different form as (A8)  Since the remaining Lagrangian components are manifestly   d L → L − d3x B · U, (A10) gauge invariant, according to Eq. (A8) the total Lagrangian I I dt changes under a gauge transformation by the addition of a total time derivative, meaning that the result is equivalent, but where we have used Faraday’s law B˙ =−∇ × E. The original μ μ not identical. This equivalence no longer holds if the inter- interaction j Aμ involves the physical matter fields j and the = μ L L action Lagrangian is LI (t ) (t ) I . The additional term that auxiliary electromagnetic fields Aμ, while the interaction I μ d 3 ρχ results from the gauge transformation is now (t ) dt d x , involves the physical electromagnetic fields and the auxiliary which is not a total time derivative. However, it is clear that matter fields. Thus, if we define a new time-dependent inter- this nonequivalence could be avoided if one starts with a action Lagrangian by

      3 μ d 3 L (t ) = μ(t )L =−μ(t ) d x j Aμ + P · A = μ(t ) d x[P · E + M · B], (A11) I I dt then we obtain a total Lagrangian that despite including the external control μ(t ), is invariant under a gauge transformation of the potentials. However, as is to be expected on the basis of the transformation property (A10), this comes at the price of producing a nonequivalent Lagrangian under the transformation (A5). Indeed, the interaction Lagrangian in Eq. (A11) is actually what ˜ ˜ L ˜  results from using the modified quantities j and P within I . To show this, we denote by L the total Lagrangian obtained by replacing in L, the current j and polarization P with their modified counterparts, and note that a quick calculation gives     μ 2  (t ) 3 3 μ d L˜ ≡ L + L + d x ρφ − d x j˜ Aμ + P˜ · A m TEM 2 Coul dt     μ 2 (t ) 3 3 μ d = L + L + d x ρφ − μ(t ) d x j Aμ + P · A . (A12) m TEM 2 Coul dt

It is now readily verified that gauge-fixing constraint that has the form    3    d d d x g(x , x) · A(x ) = 0, (A14) L˜  = L˜ − d3x P˜ · A = L − μ(t ) d3x P · A, dt dt (A13) where g must be the same choice of Green’s function as is made in Eq. (A3). Choosing the potentials  = + ∇χ, = μ φ − ∂ χ, where L˜ isgivenbyEq.(A6). The new Lagrangian L˜ is A AT A0 (t ) Coul t (A15) equivalent to L˜ which was the result we obtained by replacing where j and P with j˜ and P˜ in L. The only difference between L˜   3    and L˜ is that under a gauge transformation (4), L˜ transforms χ(x) = d x gT(x , x) · AT(x ) (A16) into an equivalent form, whereas L˜  is invariant. Whichever of these equivalent Lagrangians is considered, it is clear that un- means that Eq. (A14) is satisfied identically. The freedom like the original Lagrangian L, neither provides an equivalent to choose a gauge now reduces to the freedom to choose Lagrangian under the transformation (A5). Conversely L is gT, which uniquely specifies both the four-potential A and invariant under the transformation (A5), but does not provide the polarization P. Two special cases are given by the = ,  = an equivalent Lagrangian under a gauge transformation of the Coulomb gauge gT 0 and the Poincaré gauge gT,i(x x ) electromagnetic potentials. −  1 λδT − λ  x j 0 d ij(x x ). The invariance of L and of L˜  under gauge transformations requires that P is not altered by the gauge transformation, but 2. Gauge fixing this is no longer the case if both A and P are simultaneously →  In conventional approaches to nonrelativistic QED all determined by gT. A new choice of gT via gT gT will result gauge redundancies are eliminated simultaneously through a in a gauge transformation of both A and P. The latter will

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3 · 1 transform as in (A5) with the contribution d x PT possesses the well-known form −er · (0).  Ifμ ˙ = 0, then all forms of the Lagrangian are equivalent. ∇ × = 3   ,  − ,  ρ  . μ = U(x) d x [gT(x x ) gT(x x )] (x ) (A17) However, if ˙ 0 we have only been able to construct La- grangians that are at best partially invariant under a complete gauge transformation of both electromagnetic and mate- This (gauge) freedom in PT is central to quantum optics and rial potentials. If, in particular, we impose the constraint molecular electrodynamics because when transforming from (A14), which all standard nonrelativistic gauges satisfy, then the Coulomb to Poincaré (multipolar) gauges the additional d3x P · A = 0, implying that L, L˜, and L˜  all coincide. ∇ × = 1 contribution U PT provides the dominant interaction Moving afterwards to the canonical formalism results in 3 · 1 + 1 2 Hamiltonian d x [ PT (PT ) ]. This is the only nonvan- nonequivalent Hamiltonians as was shown in Sec. II C of the ishing interaction term in the dipole approximation, wherein main text.

APPENDIX B: FULL DESCRIPTION OF THE DIPOLE’S MOTION

We consider a two-charge system comprised of a charge e1 =−e with mass m1 at r1 and a charge e2 = e with mass m2 at r2. We introduce relative and center-of-mass coordinates as m r + m r r = r − r , R = 1 1 2 2 , (B1) 1 2 M where M = m1 + m2. We start with the standard Lagrangian    1 1 3 μ 1 μν L = m r˙ + m r˙ − d x j Aμ + F Fμν 2 1 1 2 2 2 4    1 1 1  = m r˙ + m r˙ − V (r − r ) + d3x ρ∂ χ α + J · A + E2 − B2 2 1 1 2 2 2 1 2 t 2 T    1 1 d 1  = mr˙ + MR˙ − V (r) + d3x J · A − Pα · A + E2 − B2 , (B2) 2 2 T dt T T 2 T where m = m1m2/M and V (r1 − r2 ) = V (r) is the intercharge Coulomb energy. The infinite Coulomb self-energies have been ignored. The remaining quantities are given by

ρ(x) = e[δ(x − r2 ) − δ(x − r1)], (B3)

J(x) = er˙2δ(x − r2 ) − er˙1δ(x − r1), (B4)  α =− 3 α ,  ρ  , PT (x) d gT(x x ) (x ) (B5)

α A = AT + ∇χ , (B6) where  1 α ,  =−α  − λδT − − λ  − , gT,i(x x ) (x R) j d ij[x R (x R)] (B7)  0 χ α = 3  α , ·  . (x) d x gT(x x) AT(x ) (B8)

α 1 α Here, gT is chosen such that gT gives the usual multipolar transverse polarization. Notice, however, that this means that gT depends on the center-of-mass position R. The electric dipole approximation (EDA) is obtained by retaining only the leading-order term in the multipole expansion of ρ about R, which for R˙ = 0 results in ρ(x) = er · ∇δ(x − R), (B9)

J(x) =−er˙δ(x − R) + eR˙ (r · ∇)δ(x − R), (B10)

1 =− δT − . PT,i(x) er j ij(x R) (B11) The second term in Eq. (B10) vanishes if and only if the atom is at rest in the laboratory frame. This term is vital for ensuring that the correct Röntgen interaction due to atomic motion is included. Substituting these expressions into Eqs. (B3)–(B6) and using the resulting expressions in the Lagrangian gives the Lagrangian within the EDA. This can be taken as the starting point

013116-12 ULTRASTRONG TIME-DEPENDENT LIGHT-MATTER … PHYSICAL REVIEW RESEARCH 3, 013116 (2021) for the canonical formalism with r, R, and AT as canonical coordinates. The momenta conjugates are denoted p, K, and , respectively. The resulting Hamiltonian is     1 1 1 2 H α = [K + e(r · ∇)A (R) − eα∇ r · A (R)]2 + [p + e(1 − α)A (R)]2 + V (r) + d3x + Pα + B2 2M T R T 2m T 2 T 0 = Rα0H R0α, (B12) where  α · 1 =− δT − , = − α 3 1 · = i er AT (R). PT,i(x) er j ij(x R) R0α exp i d x PT AT e (B13)

At this stage, the theory is completely gauge invariant because the H α are unitarily equivalent. The predictions for any physical observable are independent of the choice of gauge α. It is nevertheless the case that the quantum subsystems are defined differently in each different gauge. Subsystem properties like photon number and entanglement will generally depend on the definitions chosen, that is, they will depend on the choice of gauge relative to which the subsystems are defined.

1. Approximation of externally prescribed uniform gross motion in the Hamiltonian The approximation of an externally controlled coupling between the dipole and the field results from the assumption that the dynamical variable R(t ) = R˙ t (up to a constant initial position) is external and prescribed. This means that R˙ is also prescribed. With this assumption the Hamiltonian in Eq. (B12) becomes that of a bipartite quantum system, and reads as     1 1 1 2 H α (t ) = MR˙ 2 + [p + e(1 − α)A (R)]2 + V (r) + d3x + Pα + B2 , (B14) 2 2m T 2 T where now R and R˙ are given classical variables. Since the first kinetic term MR˙ 2/2 is not operator valued and for uniform motion is also constant in time, it can be ignored. Before approximating R(t ) as external, the Hamiltonians in Eq. (B12)were seen to be equivalent, but the H α (t )inEq.(B14) are not equivalent for different α, being related by α α H (t ) = Rαα (t )H (t )Rαα (t ), (B15)  where Rαα (t ) = exp[−ie(α − α )r · AT(R(t ))]. Equation (B15) shows that Hamiltonians associated with different gauges are not equivalent because α α H (t ) = Rαα (t )H (t )Rαα (t ) + iR˙αα (t )Rαα (t ), (B16) where the right-hand side of this inequality is equivalent to H α (t ). To obtain the Hamiltonian in Eq. (14) of the main text, which was obtained by assuming a time-dependent coupling μ(t ), one requires only that AT(R(t )) can be written AT(R(t )) = μ(t )AT(0). This is indeed the case in the example we consider in the main text and in Appendix C whereby an atom moves in and out of a Gaussian cavity beam for which AT(x) has the form εA(x), and we also assume that r = rε.

2. Approximation of externally prescribed uniform gross motion in the Lagrangian

If we approximate R = R˙ t as external at the Lagrangian level, then the remaining variables are r and AT.Theα-gauge Hamiltonian is  α 3 H˜ = p · r˙ + d x · AT − L        1 1 2 = [p + e(1 − α)A (R)]2 + V (r) + d3x + Pα + B2 + eR˙ · r · ∇ A (R) − eα(R˙ · ∇)r · A (R), (B17) 2m T 2 T T T where the constant kinetic energy MR˙ 2/2, which depends only on the external control, has been neglected. This is the Hamiltonian given in Eq. (34) of the main text. As Schrödinger picture operators these Hamiltonians are related by α α H˜ (t ) = Rαα (t )H˜ (t )Rαα (t ) + iR˙αα (t )Rαα (t ), (B18)  where, as before, Rαα (t ) = exp[ie(α − α )r · AT(R(t ))].

APPENDIX C: ATOM MOVING IN AND OUT OF A 1. Quantization of the free cavity FABRY-PEROT CAVITY We consider a Fabry-Perot cavity consisting of parallel In this Appendix we specialize the Hamiltonian derived mirrors in the xy plane separated by a distance L.Inthez above in Eq. (B14) to describe the interaction between a direction the electromagnetic field satisfies periodic boundary Fabry-Perot cavity and an oscillating dipole at an arbitrary conditions, with a Gaussian profile in the perpendicular direc- position within the cavity. tion xxˆ + yyˆ [78]. We restrict our attention to the fundamental

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Gaussian mode in the perpendicular direction. Although not independent. In obtaining Eq. (C7)wehaveused    necessary, for simplicity we also consider only the fundamen-  l+L − 2+ 2 / 2 tal standing wave mode in the z direction. It is straightforward d3x φ(x)2 = dze2ikz dxdye (x y ) wc = 0, to extend this model to the multimode case that includes l more standing-wave modes in the z direction. One could also (C8) consider additional Gauss-Hermite or Gauss-Laguerre modes where l is arbitrary such that l and l + L are the positions in the perpendicular direction. of the two cavity mirrors along the z axis. Equation (C8) In the present case the single-cavity mode is described by follows from the vanishing of the z integral due to the peri- a pure Gaussian beam propagating in the z direction such that odic boundary conditions in the z direction; k = nπ/L, n = classically the transverse vector potential is 0, 1, 2, 3,.... A(t, x) = ε Aau(x)e−iωt + c.c., (C1) Quantization is now straightforward via the replacement of the complex numbers a and a∗ with bosonic operators a and a† where ε is a transverse polarization in the xy plane and such that [a, a†] = 1. We thereby obtain the mode expansions u(x)e−iωt satisfies the paraxial scalar wave equation [78–80]. ε A(t, x) = √ [φ∗(x)a†(t ) + φ(x)a(t )], (C9) Anticipating the transition to the quantum theory we have ω A 2 v written the space and time-independent amplitude a as the ω product of a real normalization A and a complex number (t, x) = iε [φ∗(x)a†(t ) − φ(x)a(t )], (C10) a. We have also neglected a small nontransverse component 2v , = ˙ , ≡ − ω in the z direction [79,80]. We define (t x) A(t x) where a(t ) = ae i t in the free theory. All nonzero equal-time − , ET(t x) such that the cavity energy is canonical commutation relations are obtained from Eqs. (C9)   , † = 1   and (C10)using[a a ] 1; = 3 2 + 2 = 3 2, Hl d x [ET(x) B(x) ] d x (x) (C2)  εiε j ∗  ∗  2 [Ai(t, x), j (t, x )] = i [φ(x)φ (x ) + φ (x)φ(x )], 2v  (C11) where indicates that spatial integration is restricted to the ε ε = ∇ ×  i j ∗  ∗  cavity length L in the z direction, and B A.Wehave [Ai(t, x), A j (t, x )] = [φ(x)φ (x ) − φ (x)φ(x )], assumed that the magnetic and electric energies are the same 2ωv in the free theory. (C12)  2  To obtain an explicit expression for Hl that can be quan- [ i(t, x), j (t, x )] = ω [Ai(t, x), A j (t, x )]. (C13) tized, we consider the fundamental Gaussian mode solution − ω In particular, we have [A , ] = iε ε /v and [A , A ] = 0 = to the paraxial wave equation u(x)e i t such that [80] i j i j i j [ i, j] in agreement with Sec. III of the main text. w 2 2 2 2 2 u(x) = c e−(x +y )/w(z) eik(x +y )/2R(z)+iθ(z)+ikz (C3) The violation of relativistic causality implied by the nonva- w(z) nishing commutators of fields at spacelike separated events is a result of the approximations made, namely, the restriction with (0, 0, k) the wave vector such that k = ω and to a single radiation mode and the paraxial approximation.   1 z 2 The single-mode approximation eliminates the spatiotemporal z = kw2, w(z) = w 1 + , R 2 c c z structure necessary to elicit causality and has been discussed R in this context recently in Ref. [81]. These authors consider z2 z R(z) = z + R ,θ(z) =−arctan , (C4) the propagation direction only and show that by includ- z zR ing more standing-wave modes, consistency with relativistic causality is recovered. Here, our aim is to study the role of where w denotes the beam waist. For L  z we have c R the gauge parameter α in the light-matter interaction and for w(z) ≈ w , k(x2 + y2 )/2R(z) ≈ 0, and θ(z) ≈−π/2. In this c this purpose it suffices to restrict attention to the fundamental limit Eq. (C3) reduces to mode. As noted at the beginning of this section the single- − 2+ 2 / 2 u(x) ≈ φ(x) = eikze (x y ) wc , (C5) mode approximation is certainly not necessary and has been used here for simplicity. Without requiring any essentially where we have ignored a global phase e−iπ/2. We define the new theoretical machinery one can extend the present treat- cavity volume by ment in a straightforward manner to include more modes in  the transverse direction or in the z direction. Within the single- 1  πw2L v = d3x|φ(x)|2 = c (C6) mode treatment, which is adequate for the present purpose, the 2 2 canonical commutation relations (C11)–(C13) are necessary √ for the formal self-consistency of the framework developed. and choose the normalization A = 1/ 2ωv, such that substi- tution of Eq. (C1) into the right-hand side of Eq. (C2) yields 2. Cavity-dipole interaction ω ∗ ∗ v 2 2 2 Hl = (a a + aa ) = ( + ω A ), (C7) Using the above expressions for the field of a Gaussian cav- 2 2 ity mode the full Hamiltonian for atomic motion in and out of where A ≡ A(x = 0) and ≡ (x = 0). This cavity Hamil- the cavity is given by Eq. (B12). To preserve gauge invariance tonian is formally identical to the bare-cavity Hamiltonian we must also perform the single-mode approximation within of Sec. III, and in the free (noninteracting) theory it is α the material polarization. The appropriate approximation can

013116-14 ULTRASTRONG TIME-DEPENDENT LIGHT-MATTER … PHYSICAL REVIEW RESEARCH 3, 013116 (2021) be deduced by inspection of the linear polarization interaction component, which expressed in k space reads as   ω 3 † · = 3 ε · † d k PT(k) T(k) i d k λ [PT(k)aλ(k) 2 λ − † , PT(k)aλ(k)] (C14) where aλ(k) is the annihilation operator for a photon with momentum k and polarization λ, and ελ is the corresponding polarization unit vector, which is orthogonal to k. In writing Eq. (C14) we have used the Hermiticity of the real-space fields. Discretizing the modes in a volume v with periodic boundary conditions and restricting to a single mode kλ,the interaction becomes FIG. 7. The mutual information I(α,t ) is plotted with time in  units of tb = wc/ν assuming an initial state |0, 0 . The beam waist vω 3 † † † is wc = 20 μm, ωm is chosen in the microwave regime (energy d k P (k) · T(k) −→ i ε · [PT(k)a − P (k)a]. T 2 T ∼10 μeV) and ν = 10−3c where c is the speed of light. (C15) For the Gaussian cavity this must be equal to −er · (R) as μ(t ) = φ(R(t )). For uniform motion the coupling function − −ν 2/ 2 where (R) is given by Eq. (C10). It follows that the single- is μ(t ) = e (h t ) wc . mode approximation of PT(k) appropriate for the Gaussian In the case of uniform motion, the Gaussian coupling cavity is envelope incurs a relatively smooth switch-on. For a beam = μ e waist wc 20 m, with h substantially larger, so that the ε · Pα (k) =− (ε · r)φ∗(R). (C16) T v dipole starts well outside of the cavity, and for a dipole with microwave frequency ωm ∼ GHz, the gross dipolar speed The polarization self-energy term in the Hamiltonian is there- must be around ν = 10−3c in order that the interaction time fore τ ∼ wc/ν is comparable to the cycle time 1/ωm. The velocity   − 1 1 10 3c is not yet relativistic, but significantly larger than the d3x P (x)2 = d3k|P (k)|2 2 T 2 T velocities found in typical atomic beam experiments, which  are around three orders of magnitude smaller. In order to 1 3 ∗ ω /ν ∼ ν = d k (ελ · PT(k))(ελ · PT(k)) achieve wc m 1 with smaller either the cavity beam 2 λ waist must be further reduced, or slower dipolar oscilla- tions must be considered. However w ω /ν ∼ 1 is achieved, v ∗ c m −→ (ε · PT(k))(ε · PT(k)) significant differences occur in predictions associated with 2 different gauges within this regime, as shown in Fig. 7. e2 = r2|φ(R)|2, (C17) 2v APPENDIX D: GROUND STATE OF THE INTERACTING where r = ε · r. We can now specify all terms within the HAMILTONIAN, AND THE GROUND-STATE PHOTON complete Hamiltonian in Eq. (B12) for the case of a single- NUMBER AND MUTUAL INFORMATION mode Gaussian cavity. The Hamiltonians of different gauges are unitarily related and therefore equivalent. Assuming for A naive example of a time-dependent interaction comprises simplicity that r = rε and p = pε, when we approximate R(t ) instantaneous interaction switch-on and switch-off described μ = − − − + τ as external we obtain Eq. (31) given in the main text. The by the function (t ) u(t t0 ) u[t (t0 )] where u > + τ resulting Hamiltonians continue to be unitarily related, but are denotes the unit-step function. For final times t t0 the evolution of the system is composed of sequential evolu- no longer equivalent. Thus, it is the approximation of treat- α tions as U (0, t ) = U0(0, t0 )U (t0, t0 + τ )U0(t0 + τ,t ) where ing R(t ) as external that results in nonequivalence between α  − −  α  − −  , = i(t t )H , = i(t t )H0 gauges. U (t t ) e and U0(t t ) e . However, the free (uncoupled) evolution U0 does not alter either the os- cillator populations nor the final light-matter correlations. To 3. Mutual information find these observables one can set t0 = 0 and t = τ without Without loss of generality we can consider cavity mirrors loss of generality, which is equivalent to considering the full located at z =±L/2 centered at (0,0) in the xy plane. Any interacting system with a constant interaction μ(t ) = 1. In prescribed dipolar motion may now be considered. The sim- this case, it is more physically relevant to consider an initial plest case consists of uniform motion R˙ =−νxˆ starting from eigenstate of the full Hamiltonian H α rather than the free part rest at the point hxˆ, which yields the path R(t ) = xˆ(h − νt ). H0. Quite generally, paths satisfying zˆ · R(t ) = 0 for all t have the Of considerable interest are light-matter correlations in property that the Hamiltonian in Eq. (31)ofthemaintext the ground state |G of the full Hamiltonian H α. These are = I α = ρα + ρα with R R(t ) is identical to that in Eq. (23)ofthemain quantified by the mutual information G( ) S( m ) S( l ) text if the time-dependent coupling function therein is taken where S(ρ) =−trρ ln ρ and the reduced material and cavity

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ρα = α | | ρα = α | | states are defined by m trl G G and l trm G G , where respectively. The mutual information IG(α) is found to be 2 2 2 e 2 ωα = ω + (1 − α) . (D7) μα + 1 μα − 1 mv IG(α) = (μα + 1)ln − (μα − 1)ln , 2 2 The operators c, c† are related to a, a† by a local Bogoliubov (D1) Hα α transformation in l . The average number of -gauge renor- α = † where malized ground-state photons nc( ) c c G is    ω 2 e2 1 e2(α − α )2 ω2 1 2 g α μα = 1 + (α − αg) . (D2) nc(α) = ωg + + − . (D8) ωg mvωωm 4ωα mvωm,g ωg 2 α = α It is symmetric about the point g where it takes its Unlike the average in Eq. (D3) this average reaches a mini- minimum value of zero. mum of zero for α = αg. This can be understood by noting We also consider the average number of α-gauge photons that for this choice of α the Hamiltonian can be written in α = † na( ) a a G in the ground state. A straightforward calcu- number-conserving form as lation yields   2 α − α 2 ω2 g † 1 † 1 1 e ( g) 1 H = ωm,g d d + + ωg c c + na(α) = ωg + + − , (D3) 2 2 4ω mvωm,g ωg 2 ωω 1 ω ≡ ω + m † − † , where g αg and ie (d c dc ) (D9) mv ωm + ω ω e2 α = m ,ω2 = ω2 + α2. g ω + ω m,g m g (D4) where the renormalized material modes d are such that m mv 2 mω2 If one allows the definition of photon number to depend on p m,g 2 † 1 + r = ωm,g d d + . (D10) material parameters e and m, then the self-energy term 2m 2 2 e2(1 − α)2A2/2m = η2ω[(1 − α)2(a† + a)2](D5)The renormalized material modes d are connected to the bare material modes b via a local Bogoliubov transformation in can be absorbed into a redefinition of the local cavity energy g Hm. The ground state |G of the Hamiltonian is the vacuum as |0d , 0c annihilated by the operators d and c. Thus, n (α ) = v c g ˜ α = α + 2 − α 2 2/ = + ω2 2 0. It is important to note that unlike a full diagonalization Hl Hl e (1 ) A 2m ( αA ) 2 of the Hamiltonian, the partially diagonal form (D9) does † 1 not obscure the divisibility of the overall system into “light” = ωα c c + , (D6) 2 and “matter” subsystems. After a full diagonalization, the

FIG. 8. IG(α) is plotted as a function of α with δ = ω/ωm = FIG. 9. The ground-state photon number averages na(α)and 1 η = α α δ = ω/ω = 2 , for√ three values of the dimensionless coupling parameter nc ( ) are plotted as functions of with m 2. The strength e/(ω mv). The strength of the α dependence increases with increas- of the dependence on α increases with increasing η, as does the dif- ing η.Forallη the mutual information IG(α) is symmetric about ference between the two photon numbers na and nc. For sufficiently the minimum value of zero occurring at αg = 1/(1 + δ) for which weak coupling η  0.1, na and nc are indistinguishable within the the ground state |G is in fact separable (Appendix D). At resonance resolution of the plot. Both na and nc are minimum at α = αg.For δ = α = 1 I = I α 1wehave g 2 , implying G(0) G(1). Off-resonant values all couplings, nc is identically zero at g while na becomes nonzero of δ determine the shift of the minimum αg relative to the resonant for stronger coupling. For α → 1, na(α) → nc (α) because the self- 2 2 2 value; αg is shifted towards α = 1forδ<1, and towards α = 0for energy term e (1 − α) A /2m vanishes identically in the Poincaré δ>1. gauge α = 1.

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Hamiltonian can be written as the sum of two harmonic os- and then performing nothing but local operations within that cillator energies, but it is not possible to distinguish these gauge. harmonic oscillators such that one can be called “light” and Figure 8 shows significant variations in the mutual infor- the other “matter” in any meaningful way. This is because a mation IG(α), which become increasingly pronounced for completely diagonalizing transformation is necessarily nonlo- larger dimensionless coupling strengths η. Similarly, Fig. 9 cal with respect to the light-matter Hilbert space bipartition plots na(α) and nc(α) as a functions of α, showing that both of any gauge. On the other hand, the number-conserving form nonrenormalized and renormalized photon numbers vary sig- (D9) can be achieved by simply choosing a particular gauge nificantly with α.

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