Theory of coherent control with quantum light

We develop a coherent control theory for multimode quantum light. It allows us to examine a fundamental problem in quantum optics: what is the optimal pulse form to drive a two-photon-transition? In formulating the question as a coherent control problem, we show that—and quantify how much—the strong frequency quantum correlations of entangled photons enhance the transition compared to shaped classical pulses. In ensembles of collectively driven two-level systems, such enhancement requires nonvanishing interactions.


Introduction
The nonlinear interaction between faint light and matter on a single atom/molecule and few-photon level is of great fundamental and practical interest: while Gedanken experiments involving single photons and single quantum emitters have recently come within reach of experimental verification [1][2][3], similar applications in single molecule spectroscopy may unravel the quantum dynamics of photo-sensitive materials beyond the ensemble average [4], and promise to elucidate, for instance, the role of fluctuations in energy transport [5,6]. Similarly, few-photon experiments in nonlinear media are able to create an effective interaction between photon pairs, and are employed to construct all-optical transistors [7][8][9][10][11][12]. Clearly, the feeble probe and signal fields in such experiments pose a formidable challenge: in order to detect the typically very weak nonlinear effects at small photon numbers (on the order of one), one seeks to, either, optimize the nonlinearity of the optical medium, or to manipulate the light fields: the former route includes the cavity-enhanced coupling of light to the medium [13][14][15], or the enhancement of the medium's nonlinearity by additional strong driving fields [7,8], large dipoles in highly excited Rydberg states [16,17], or molecular design of target molecules [18,19]. In the latter route, strong focussing of the light beams, i.e. the choice of a suitable geometry, was exploited to detect the nonlinear response on a single molecule level [1][2][3].
Here we present an alternative approach to enhance nonlinear interactions specifically in the highly relevant regime of weak intensities, by employing ideas from coherent control [20][21][22]. Whereas coherent control originally employs strong classical pulses to manipulate the interference between different excitation pathways, recent advances in the manipulation of the two-photon wavefunction of time-frequency entangled photons [23][24][25][26] open the possibility to apply coherent control techniques to interactions on the single-photon level. We will demonstrate that quantum-mechanical entanglement between photons can enhance the efficiency of the photons in carrying out a given task beyond the classically achievable limit (i.e. induced by optimally shaped pulses). Incidentally, our results challenge commonly held beliefs in coherent control, according to which the quantum nature of light is detrimental [22], and open a new avenue for exploiting quantum correlations in the control of molecular processes.
Specifically, we examine a prototypical example of nonlinear light-matter interaction-the resonant twophoton absorption in a three-level system driven by either photon pairs [27] or famished 'classical' coherent light pulses [28]. The three-level system represents the simplest theoretical model to feature a two-photon transition, and will allow us to extract the impact of quantum correlations in the most transparent way. Using a variational approach, we derive optimal pulse forms for classical and quantum two-photon states, in order to excite the target state f at a given time t. Our analytical results demonstrate that the optimal pulse form is, in general, given by an entangled two-photon state. Only if the matter system consists of an ensemble of noninteracting two-level atoms, classical light can perform as well as quantum light.
Note, in contrast, that it is well known that the two-photon absorption rate scales linearly with the pump intensity creating the entangled photons, while it scales quadratically for classical pulses [29]. Yet this scaling behavior, which originates in the creation process, cannot per se be attributed to entanglement per se. Indeed, with the right choice of parameters, downconversion can produce separable two-photon states [30], which nevertheless scale linearly in the pump intensity. Let us also mention that similar, yet different, scenarios as compared to the present one have been considered earlier in the literature: Dayan [31] theoretically analyzed offresonant two-photon absorption (i.e. with the intermediate state far off-resonant) of multimode quantum light, and concluded that 'excluding the linear intensity dependence of the coherent signal, all the other effects considered in this paper are completely described within the classical framework', and 'can be created by appropriately shaping classical pulses'. Our results show that this claim does not hold in the resonant case. Finally, let us stress that our present study concerns the quantum correlations between photons (as controlled by shaping their spectral features), rather than quantum fluctuations within predefined modes, which also have been studied elsewhere, and which also report an enhancement due to squeezed light with the same mean photon number as a coherent reference state [32][33][34].

Two-photon absorption
We consider a system of three electronic states with a ground state g, intermediate states e, and target state f as depicted in figure 1(b). The system is illuminated by two (possibly quantum) light fields, described by field operators ŵ ( ) E 1,2 . It is initialized in the ground state, and our goal consists in maximizing the population in state f at time t by controlling the joint quantum state ψ of the two light fields. To this end, we consider the functional The first term in equation (1), á ñ ŷ ( ) p t f , denotes the probability of the system to be in the target state f at time t. The expectation value is taken with respect to the fields' quantum state ψ, which drives the two-photon transition. The second term in equation (1) introduces a Lagrangian multiplier λ to enforce the constraint that the product of the mean photon numbers be N 2 photons, with the photon number operator given by . This peculiar normalization separates the well-known linear scaling of the two-photon absorption rate [29] of entangled photons from the impact of quantum correlations: by construction, a separable two-photon state is normalized just like a classical pulse, so any changes in the excitation probability ( ) p t f must stem from quantum correlations.
Since we focus on the interaction with weak fields, we obtain ( ) p t f perturbatively, where, at second order in the perturbation, the transition amplitude between g and f reads [35] ò ò   describing the matter response to the absorption of photons with frequencies w a and w b . m ge and m ef are the dipole moments connecting adjacent electronic states, g e and g f are the inverse lifetimes of the respective electronic levels, and E 0 the field normalization [36]. The summation over e includes several possible intermediate states. Our subsequent analysis is restricted to the simplest case of only one single state, but all the results may be straightforwardly generalized to a manifold of e-states. Each excitation of the matter is accompanied by a photon annihilation at the corresponding frequency, rendering equation (2) a field operator acting on the joint space of the two fields 1 and 2.
We further note that, in writing equation (3), we allow for pulses extending infinitely long in time, which simplifies our subsequent discussions. The (more involved) general response for finite interaction times will be given elsewhere. Equation (3) is symmetric with respect to the exchange w w « a b , which means that we assume each field couples to both transitions. In a situation where, for instance, field 1 only drives the g-e transition, and field 2 the e-f transition, we would have to eliminate the second summand in equation (3) to obtain the correct result.
To find the solution of our control problem, we require the first variation of the functional(1) to vanish, d = J 0, and obtain Fredholm integral equations (see appendix A) which define the optimal pulse forms. This approach may be carried out for arbitrary sample systems. In the present case, when there is a unique target state f, the integral equationsmay be solved analytically when changing to an appropriate basis of orthogonal functions: as we shall see, this basis is found in the Schmidt decomposition of the matter response function [37] å w w where y { } k and f { } k are orthonormal basis sets, and the r k are real weights, which may be chosen positive.

Classical versus quantum light
We first consider the excitation by multimode coherent states with low mean photon number. Since the field correlation function in á ñ y ( ) p t f is normally ordered [38], we may replace the field operators in equation (2) by the field amplitudes [36] such that-for this two-photon process-the light states behave like classical light fields with amplitudes . This justifies why, in the following, we shall refer to them as weak, 'classical' light pulses. The optimal frequency distributions A 1 and A 2 , equations (A.6) and (A.7), are given by the eigenfunctions pertaining to the largest singular value r 1 in equation (4) Using the orthonormality of the Schmidt basis y { } k and f { } k , we may readily carry out the frequency integrations in equation (2), and the maximal classical probability to excite the target state f at time t then reads While this maximal probability is time-independent, the ideal solutions A 1 and A 2 depend on the final time t through the time dependence of T t in equation (4). As we show in the appendix A.1, a mixture of classical states (i.e. classical correlations) cannot enhance the probability any further. The simple factorization of the four-point field correlation function in equation (2) into products of field amplitudes, as implied by equations (5) and (6), is no longer possible when describing general quantum states of the field. In particular, when dealing with a general two-photon state, we need to replace by the so-called 'two-photon wavefunction' [29,37] which describes the transition amplitude from the initial state yñ | to the vacuum ñ |0 , upon absorption of photons at frequencies w a and w b , respectively. The variation of f quantum (see appendix A.2) yields the optimal twophoton state where  is the normalization constant. The state may be visualized as the time-reversed field emitted by the system in a two-photon decay  f g, similar to the two-level atoms studied in [39][40][41]. The ideal excitation probability induced by equation (9) To compare classical and quantum results directly, we divide the classical result, equation (7) by the squared mean photon numberN 2 . Equations (7) and (10) then demonstrate that, in principle, entangled photons may be tuned such that they drive resonant two-photon absorption processes more effectively than classical laser pulses. Due to the joint wavefunction, the phase of each pair of frequency components w w ( ) , a b in equation (9) is tuned such that the corresponding transition amplitudes-given by equation (3)-sum up constructively. With classical light(6), the phases of w a and w b are determined independently, thus limiting the transition probability.
As a specific example, let us consider excitation along the states   S P D 5 5 5 . This particular level scheme was already used in [28] to study the influence of the spectral envelope of laser pulses on the two-photon absorption, and for a similar three-state ladder in rubidium in [27] to investigate the absorption of entangled photon pairs. We obtain the rather remarkable ratio  figure 2: the enhancement factor G ( ) E forms a convex function of its arguments Δ and δ, which takes its unique minimum value E=1 (i.e. no enhancement compared to excitation by classical pulses) at d D = = 0.
To understand this minimum, we note that we can rewrite the two-photon response function for d D = = 0 as a product of Lorentzian response functions,  - These Lorentzian factors can be interpreted as single-photon response functions for the transition  g e, for which the transition amplitude in first-order perturbation theory reads [35] Here, the amplitude w ( ) A may describe, both, a classical field amplitude or a single-photon spectral envelope. Hence, at d D = = 0 the two-photon response function factorizes into the product of single-photon response functions, and no correlations between the first and the second absorption event can induce any quantum advantage (compare equation (9)). Such a situation occurs, for instance, when the sample system consists of noninteracting two-level atoms, for which entangled two-photon absorption was investigated in [43][44][45]: it was conjectured in [43] that entanglement may enhance two-photon absorption. Our present analysis shows, however, that entanglement cannot enhance the efficiency of the two-photon absorption process in such systems beyond the classical limit: if we consider two two-level atoms with transition frequencies w e and w ¢ e , and each photon interacts with only one of the two-level atoms, we can modify equation (12) to write The two-photon response again factorizes, and there can be no enhancement beyond the classical limit due to photonic entanglement. By comparing optimal pulse forms rather than specific models of light in [43][44][45], our analysis unambiguously settles this debate.
Since the above discussion also establishes that entangled photons indeed can enhance the two-photon transition probability when either D ¹ 0 or d ¹ 0, we next investigate the optimal pulse forms in detail. Figure 3 depicts the optimal classical field amplitudes (panel (a)) and two-photon wavefunction f w w ( ) , a b (panel (b)) at the minimum in figure 2, i.e. when the two-photon transition amplitude factorizes as in equation (12). Clearly, the two pulse forms are identical, as was clear from our theoretical analysis. Since in our present study equation i e e , i.e. in the time domain by a 'rising exponential' shape. This behavior was to be expected from previous results on single-photon excitation of a two-level atom [39][40][41], where a 'rising exponential' temporal shape was shown to couple most efficiently to the matter system. In the case of noninteracting two-level atoms, it is quite intuitive that this pulse shape is still optimal for two-photon excitations, when the two absorption events are not correlated.
Real part of the optimal two-photon state (8) for the same matter system as panel (a). This is the parameter regime with enhancement factor E=1 (see figure 2).
The situation changes substantially when the two-photon state does enhance the transition probability: in figure 4, we depict the case where the second transition energy is larger than the first one, with g D = 5 e , and the target state lives longer than the intermediate state, g g = 2 f e (as was also the case in our rubidium example above): the optimal classical pulses (panel (a)) show a broad structure in the area w w w w w w . While, at first glance, these optimal pulses appear very different from the harmonic case before, this structure can still be understood by the interference of three Lorentzians: in addition to the harmonic Lorentzian of figure 3, one more Lorentzian contributes for, each, the nonvanishing Δ and δ.
In contrast, the optimal entangled wavefunction (panel (b)) forms a narrow structure along the antidiagonal. Its width along the diagonal becomes a lot narrower than in figure 3(a) . This is only possible in the quantum regime, since the absorption of the first photon affects the wavefunction of the other photon, thereby influencing the second absorption event [46]. The two-photon wavefunction is a clear indicator of strong frequency anticorrelations in the pure two-photon state considered here [37,47,48], and it is these correlations that are responsible for the 140% enhancement  The broad bandwidth of the classical pulses is reflected in a steep rise of both the intermediate and target state populations near t=0. The intermediate state e decays more rapidly than the target state f, such that the optimal classical pulse has to strongly excite this state shortly before the population in f shall be maximized.
In contrast, the populations induced by the absorption of entangled photons rise much slower. Since frequency-anticorrelated photons arrive in pairs [21], the time the system spends in the intermediate state is minimized. It further means that at any point in time there is a finite probability to find the system in the intermediate state. This driving causes a perfectly symmetric progression of the target state population ( ) p t f which rises with the same speed it subsequently decays again. This could be anticipated from the optimal state(9) which is given by the complex conjugate (hence, the time-reverse) of the matter response (3). The inability of the classical pulses to reproduce this symmetric time evolution lies at the heart of the quantum advantage discussed here.

Summary and outlook
We investigated optimal pulse forms to efficiently drive a resonant two-photon transition with weak fields. Our study represents-to the best of our knowledge-the first theoretical investigation of coherent control theory in the presence of quantum correlations. Whereas previous studies on coherent control with quantum light focus on the manipulation of interference between excitation pathways [22], where the quantum properties of light may in fact be detrimental, our study shows that photonic entanglement potentially renders the two-photon absorption process significantly more efficient than classical pulses-except for two-photon absorption in noninteracting systems. Such shaping of the two-photon wavefunction provides a useful application of entangled photons as a spectroscopic probe of photosensitive samples where large photon fluxes are to be avoided [50][51][52].
Our introduction of the Schmidt decomposition-a concept from quantum information theory-to analyze the material response function could have applications well beyond the simple systems considered here, as it offers a new perspective on the nonlinear response of (possibly complex) quantum systems. The Schmidt coefficients-very much like in the case of photonic entanglement, where they determine the effective dimension of the Hilbert space available for quantum information purposes [37]-contain information about couplings between resonances in the quantum system. The current approach extracts this information from the optical response of the system, and could thus provide a tool to analyze entanglement with optical measurements, for instance, the interaction-induced entanglement of excitons in complex systems [53].
Future work could investigate the impact of quantum correlations beyond the perturbative limit to analyze experiments such as those carried out by the Sandoghdar [1][2][3], or the Leuchs groups [54,55], and beyond the two-photon limit when incoherent contributions need to be taken into account [31]. Furthermore, we have not considered off-resonant two-photon transitions in this manuscript. We believe however that our formalism could be expanded to include off-resonant intermediate states by imposing additional constraints in the functional (1).
The variation of the Lagrange multiplier, , similarly yields the normalization condition ò ò Using the Schmidt decomposition(4) of the matter response function, it becomes apparent that are solutions of the integral equations. Hence, the solutions pertaining to the largest Schmidt eigenvalue r 1 maximize the classical two-photon absorption within this set of solutions. Strictly speaking, we do not need to assume identical amplitudes in each beam. Any pair 0, 1 also solves the integral equations. However, in the low-intensity regime considered here this reshuffling of amplitude between the two beams does not add any new physics to the problem, so we restrict our attention to solutions(A.6) and (A.7). where we have used the triangle inequality in the first estimation. To proceed, we invoke the Hölder inequality [56] å å å Similarly, an incoherent mixture of different eigenfunctions y k and f k with weightsp k cannot enhance the maximal transition probability (7)

A.2. Photon pairs
In case the initial state of the light field ψ is given by a two-photon state, the field enters in equation (2) with its two-photon wavefunction, f w w (see equation (8)). The functional(1) is then given adjacent grid points. The eigenvectors pertaining to larger eigenvalues r k 2 rapidly start to oscillate with the frequency, which may necessitate an enhanced frequency resolution. Note further that due to the long tails of the resulting Lorentzian pulse shapes very large frequency ranges need to be sampled to obtain a convergent result.
We used the optimal quantum state(9) as a reference: according to our previous analysis, the optimal population in the target state f induced by quantum light has to coincide with the numerical result for r 1  We note that, to obtain the largest eigenvalue of equation (B.1), an increase of the step size beyond g 5 e only changes the result by on the order of 10 −6 for a grid size of g 100 f , and even less for g 200 f . Instead, to enhance the numerical precision, it is necessary to sample a very large area around the central frequencies, in order to cover the very large tails of the Lorentzian distribution to a sufficient extent.