Multidimensional pump-probe spectroscopy with entangled twin-photon states

Abstract

We show that entangled photons may be used in coherent multidimensional nonlinear spectroscopy to provide information on matter by scanning photon wave function parameters (entanglement time and delay of twin photons), rather than frequencies and time delays, as is commonly done with classical pulses. Signals are expressed and interpreted intuitively in terms of products of matter and field correlation functions using a diagrammatic close time path loop formalism which reveals the entangled quantum pathways of the fields and matter. The pump-probe signal measured when the pump and the probe are in a twin entangled state shows two-photon resonant contributions which scale linearly rather than quadratically with the incident beam intensity and reveal frequencies of off-resonant transitions. Two-dimensional spectrograms obtained by double Fourier transform of the signal with respect to the entanglement time and delay of the twins could provide detailed information on correlations among states and dynamical processes with high temporal resolution. The analogy with multidimensional time-domain optical techniques which use sequences of short classical pulses and pulse shaping algorithms is pointed out.

I. INTRODUCTION

Multidimensional techniques have long been used in NMR to study the structure and dynamics of complex molecules by measuring the response of spins to sequences of magnetic pulses [1]. Correlation plots obtained by varying several time delays spread the information in multiple dimensions thus improving the resolution and obtaining information unavailable from ordinary one-dimensional (1D) spectroscopy (e.g., absorption). Over the past 15 years these techniques have been extended to the femtosecond regime using infrared and visible pulses [2–4] and were applied to many types of systems and relaxation processes including vibrations in proteins, protein folding, hydrogen bonded liquids, and exciton dynamics in electronically excited photo-synthetic complexes [5]. Pulse shaping techniques make it possible to manipulate and control these signals [6–11]. All of these measurements employ classical fields and use multiple pulse delays or frequencies as control parameters.

In this paper we propose multidimensional signals which exploit the quantum nature of the field and are based on the manipulation of photon wave functions. These can now be engineered to create entangled photons which depend on multiple parameters that can be varied to generate two-dimensional (2D) correlation spectra. Entanglement is a purely quantum type of correlation among particles which exists even when they are spatially well separated [12,13]. Many applications to quantum computing, information and communication protocols, cryptography, and teleportation are based on the manipulation of entangled particles [14–16]. Nonlocal quantum correlations brought about by entanglement have been used to test the foundations of quantum mechanics by demonstrating violation of Bell's inequalities and resolving the Einstein-Podolsky-Rosen (EPR) paradox [17–19]. It is easier to maintain entanglement of photons than of massive particles. Entangled photons have been generated by a variety of schemes. These include atomic cascades [20–22], parametric down conversion (PDC) [23,24], or hyperparametric scattering (HPS) [25,26] in nonlinear crystals and, most recently, biexciton decay in semiconductors [27,28]. The photons may be entangled with respect to their polarization, momentum-position, energy-time, their orbital angular momentum, or number state. Photons may be hyperentangled when the entanglement is encoded in several degrees of freedom all mentioned above [29,30].

Entangled photons have been used recently in nonlinear spectroscopic applications, including two-photon absorption (TPA) [31,32,53], sum frequency generation (SFG) [33–35], two-photon-induced fluorescence (TPIF) [36–40], and two-photon-induced transparency (TPIT) [41]. The simplest signature of entanglement is that TPA signals scale linearly rather than quadratically with the pump intensity [42]. This reflects the fact that in these experiments, the entangled photons act as a single object. This effect has been predicted and demonstrated in molecular systems. As a consequence, power levels required for TPA excitation can be dramatically reduced by many orders of magnitude, becoming closer to those for single-photon excitation [36]. This could be used to minimize damage in two-photon microscopy. The spatial resolution can also be improved since TPA only takes place in regions where correlated photon pairs overlap in space and time [43,44].

In a recent study we calculated pump-probe (PP) signals using polarization entangled photons generated by type II PDC and postselected by the Mach-Zehnder interferometer. We showed that only the TPA pathways scale linearly with pump intensity and at low pump intensity dominate the other, stimulated Raman (SR) contributions, which scale quadratically. Thus the entangled photons enable quantum pathway selectivity and can help disentangle complex spectra with overlapping features [45]. Photon-entanglement-induced strong correlations of quantum pathways of matter have also been predicted for the difference frequency generation (DFG) technique [46].

Here we focus on a different type of entangled two-photon states known as “twin photons” [40,41,47,54]. The twins are generated by PDC of a single pump photon and correlated both in time and space. Their TPA which occurs within the entanglement time T₁₂ and area A₁₂ is strongly enhanced since it does not rely on the accidental simultaneous arrival of two uncorrelated photons. TPA with twin photons had also been used to detect off-resonant states. The sample becomes transparent for certain values of the entanglement time T₁₂, thereby revealing off-resonant state frequencies. Alternatively, one can fix the entanglement time and introduce a time delay |τ|≤T₁₂ between the twin photons. The Fourier transform of the TPA signal with respect to τ can also reveal the structure of the virtual states. Since the delay can be varied with 0.25 fs steps, the technique can monitor intermediate-state levels separated by ~eV and may therefore be used for atomic spectroscopy. Current theory is limited to the regime where the entangled time is shorter than the dephasing time [32]. We develop a microscopic theory of nonlinear optical signals with twin photons using close time path loop (CTPL) diagrams [48,49] and apply it to predict pump-probe signals in a three-level ladder |g〉, |e〉, |f〉 (Fig. 1). We focus on resonances in PP spectroscopy where the sum of the pump and probe frequencies is resonant with one of the molecular states |f〉 i.e., ω₁+ω₂=ω_fg. Difference frequency ω₁−ω₂ (stimulated Raman) type resonances are interesting as well, but will not be considered here. In Sec. II we show that the signals for general quantum optical fields are given by sums over products of matter and light quantum pathways. The pathways can be divided into two groups. The first, representing TPA, is expressed as a modulus square of a two-photon scattering amplitude $\left(T\mathrm{matrix}\right)~{\mid {\mathcal{T}}_{fg}(T_{12},\tau )\mid }^2$. It may be calculated as two-photon counting rate using Glauber's theory [23,39,50]. The second component [pump modulated probe (PMP)] represents a process where the pump acts as a “catalyst.” It modulates the probe absorption but returns to its initial state at the end. This group is given by the imaginary part of an amplitude product $~{\mathcal{T}}_{gf}^{\star }{\mathcal{T}}_{fg}^{\star }$ (not modulus square) and cannot be calculated as a photon counting rate. The TPA and PMP can be measured separately by fluorescence from the doubly excited |f〉 and the singly excited states |e〉, respectively. Each group has a characteristic resonances pattern. In Sec. III we show that the TPA and PMP contributions to the signal are identical when the fields are classical and the intermediate state |e〉 is off resonant. In Sec. IV we study the variation of the signal with the entanglement time T₁₂. We show that the TPA signal with twin photons scales linearly with the field intensity and is inversely proportional to the entanglement time. In Sec. V we introduce the time delay τ between the twins and present 2D correlation spectra obtained by a double Fourier transform with respect to the entanglement time T₁₂ and τ. Various groups of resonances can be clearly separated by these spectra which contain a rich pattern of cross peaks. In Sec. VI we discuss the connection with 2D signals obtained with shaped classical pulses. We show that two mutually delayed rectangularly shaped optical fields may be used to reproduce the TPA spectra obtained with twins. A general discussion is finally given in Sec. VII.

FIG. 1.

(Color online)(a) Pump-probe experiment with the twin photons. (b) One of the possible schemes for controlling the external time delay τ between the twins [40]. (c) The twin photon transition amplitude in Eq. (22) is nonzero in the diagonal strip of width $\sqrt{2}{T}_{12}$. The symmetry of photon interactions t₄↔t₂,t₃↔t₁ is broken when one of them is delayed. (d) The model molecular level scheme used in our simulations.

II. CTPL FORMALISM OF PUMP-PROBE SIGNALS

We consider an assembly of N three-level |g〉, |e〉, f〉 molecules [Fig. 1(d)] which are initially in the ground state |g〉. We focus on one component of a general PP signal obtained with two optical modes k₁ and k₂ where the sum of the two-photon frequencies coincides with one of the transitions ω₁+ω₂=ω_gf. The stimulated Raman part with ω₁−ω₂ resonances is neglected. The frequency domain (cw) PP signal with classical modes is defined as the change in the transmitted intensity of the probe with and without the pump [48]. We treat both beams on the same footing as shown in Fig. 1(a). One of the detectors (d₂) measures the change in intensity of the “probe” mode E₂ with and without the sample. The second detector (d₁) does the same for the “pump” mode E₁. We thus define two signals:

$$S_{\mathrm{PP}}^{d_2}({\omega }_2;{\omega }_1)=-\mathrm{Im}\frac{N}{T}{\int }_{-T}^{T}d{t}_4\langle E_2^{†}\left(t_4\right)P^{\left(3\right)}({\mathbf{k}}_2,t_4)\rangle ,$$ (1)

$$S_{\mathrm{PP}}^{d_1}({\omega }_1;{\omega }_2)=-\mathrm{Im}\frac{N}{T}{\int }_{-T}^{T}d{t}_4\langle E_1^{†}\left(t_4\right)P^{\left(3\right)}({\mathbf{k}}_1,t_4)\rangle .$$ (2)

Here P⁽³⁾(k, t) is the third-order polarization and 2T is the measurement time. The nonlinear polarization is induced by sequence of interactions with the optical fields. When the last interaction occurs with k₂ (k₁) mode it gives P⁽³⁾(k₂, t₄) [P⁽³⁾(k₁, t₄)].

We further define the following symmetric and asymmetric combinations of the two signals:

$$S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_1;{\omega }_2)=\frac{1}{2}[S_{\mathrm{PP}}^{d_1}({\omega }_1;{\omega }_2)+S_{\mathrm{PP}}^{d_2}({\omega }_2;{\omega }_1)],$$

$$S_{\mathrm{PP}}^{\mathrm{Asym}}({\omega }_1;{\omega }_2)=\frac{1}{2}[S_{\mathrm{PP}}^{d_1}({\omega }_2;{\omega }_1)-S_{\mathrm{PP}}^{d_2}({\omega }_1;{\omega }_2)].$$ (3)

In the rotating wave approximation the matter field coupling is given by

$$H_{\mathrm{int}}=V^{†}\left(t\right)E\left(t\right)+V\left(t\right)E^{†}\left(t\right).$$

Here the dipole operator is partitioned as V(t)+V^†(t), where

$$V\left(t\right)=\sum _j\sum _{k>j}{\mu }_{jk}{e}^{i{\omega }_{jg}t}\mid j\rangle \langle k\mid e^{-i{\omega }_{kg}t}$$ (4)

is the positive frequency (de-excitation) part; {|k〉} represents the eigenstates of the bare material system labeled according to increasing energy. V^† represents the excitation part of the dipole operator.

The field of mode j=1,2 is similarly partitioned as $E_j\left(t\right)+E_j^{†}\left(t\right)$, where the positive frequency component is

$$E_j\left(t\right)={\left(\frac{2\pi {\omega }_j}{\Omega }\right)}^{1∕2}{a}_j\exp (-i{\omega }_{j}t).$$ (5)

Here, a_j ($a_j^{†}$) is the annihilation (creation) boson operator for the j mode and Ω is the quantization volume.

Using the superoperator nonequilibrium Green's function (SNGF) formalism [48,51] and the loop diagrams of Fig. 2 we can express Eq. (1) in terms of four-point correlation function of the matter and the field and obtain

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{d_2}({\omega }_2;{\omega }_1)=& -\frac{2N}{T}{\int }_{-T}^{T}d{t}_4\mathrm{Im}{i}^3{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }d{t}_{3}d{t}_{2}d{t}_1[\theta (t_4-t_2)\theta (t_2-t_3)\theta (t_3-t_1)\langle V\left(t_1\right)V\left(t_3\right)V^{†}\left(t_2\right)V^{†}\left(t_4\right)\rangle \hfill \\ \hfill & \times \langle E_1^{†}\left(t_1\right)E_2^{†}\left(t_3\right)E_1\left(t_2\right)E_2\left(t_4\right)\rangle +\theta (t_4-t_2)\theta (t_2-t_1)\theta (t_1-t_3)\langle V\left(t_3\right)V\left(t_1\right)V^{†}\left(t_2\right)V^{†}\left(t_4\right)\rangle \langle E_2^{†}\left(t_3\right)E_1^{†}\left(t_1\right)E_1\left(t_2\right)E_2\left(t_4\right)\rangle \hfill \\ \hfill & +\theta (t_4-t_2)\theta (t_4-t_3)\theta (t_3-t_1)\langle V\left(t_1\right)V\left(t_3\right)V^{†}\left(t_4\right)V^{†}\left(t_2\right)\rangle \langle E_1^{†}\left(t_1\right)E_2^{†}\left(t_3\right)E_2\left(t_4\right)E_1\left(t_2\right)\rangle \hfill \\ \hfill & +\theta (t_4-t_2)\theta (t_4-t_1)\theta (t_1-t_3)\langle V\left(t_3\right)V\left(t_1\right)V^{†}\left(t_4\right)V^{†}\left(t_2\right)\rangle \langle E_2^{†}\left(t_3\right)E_1^{†}\left(t_1\right)E_2\left(t_4\right)E_1\left(t_2\right)\rangle ].\hfill \end{array}$$ (6)

Here the step functions θ(t−τ) maintain the time orderings of interactions along the loop (but not in real time). The interactions are time ordered within each branch of the diagram, but the two branches are not time ordered with respect to each other. By convention, the chronologically last interaction is an absorption and occurs at the left branch of the loop. The four terms in Eq. (6) correspond to diagrams i–iv of Fig. 2. $S_{\mathrm{PP}}^{d_1}({\omega }_1;{\omega }_2)$ is given by diagrams v–vii and can be obtained from $S_{\mathrm{PP}}^{d_2}({\omega }_1;{\omega }_2)$ [Eq. (6)] by interchanging {ω₁↔ω₂}. To proceed further we must calculate the optical field correlation functions by specifying its initial state.

FIG. 2.

CTPL diagrams for the contributions to the pump-probe signal which contain ω₁+ω₂=ω_fg resonances. By convention, the last interaction is an absorption and occurs at the left branch of the loop. Diagrams i–iv represent k₁ (pump) and k₂ (probe)[v–vii are k₁ (probe) and k₂ (pump)]. The top row diagrams represent pump modulated probe (PMP) where the system ends in the singly excited state |e〉〈e|. The bottom row diagrams represent two-photon absorption (TPA) where the matter ends in the doubly excited state |f〉〈f|.

III. PUMP-PROBE SPECTROSCOPY WITH CLASSICAL OPTICAL FIELDS

To set the stage and introduce the relevant building blocks of the signal we shall first consider conventional frequency domain PP performed with classical modes. In this case all optical field correlation functions in Eq. (6) may be factorized into the same product of the field intensities: 〈E^†E^†EE〉=|ε₁|²|ε₂|². The time arguments in the optical field correlation functions transformed to the loop times (the arguments of the step functions) provide the Fourier components of the material correlation functions and yield the signal

$$S_{\mathrm{PP}}^{d_1}({\omega }_1;{\omega }_2)=4N{\mid {\epsilon }_1\mid }^2{\mid {\epsilon }_2\mid }^2\mathrm{Im}{\chi }^{\left(3\right)}(-{\omega }_1;{\omega }_1,-{\omega }_2,{\omega }_2),$$

$$S_{\mathrm{PP}}^{d_2}({\omega }_2;{\omega }_1)=4N{\mid {\epsilon }_1\mid }^2{\mid {\epsilon }_2\mid }^2\mathrm{Im}{\chi }^{\left(3\right)}(-{\omega }_2;{\omega }_2,-{\omega }_1,{\omega }_1),$$

where the explicit CTPL expression for nonlinear susceptibility χ⁽³⁾ is given in Refs. [48,49]. For comparison with the entangled photon generated PP signal we recast the above expressions in terms of elements of the symmetric two-photon scattering T matrix,

$$\mathcal{T}({\omega }_1,{\omega }_2)=(V+V^{†})[G({\omega }_1+{\omega }_g)+G({\omega }_2+{\omega }_g)](V+V^{†}),$$ (7)

where G(ω) is the Fourier transform of the retarded Green's function operator G(τ)=θ(τ)exp(−iHτ),

$$G\left(\omega \right)=-i{\int }_{-\infty }^{\infty }d\tau G\left(\tau \right)e^{i\omega \tau },$$

where H is the matter Hamiltonian. We shall need the following matrix elements of the two-photon operator (two-photon transition amplitudes):

$$\langle f\mid \mathcal{T}({\omega }_1,{\omega }_2)\mid g\rangle ={\mathcal{T}}_{fg}({\omega }_1,{\omega }_2)={\mu }_{fe}^{\star }{I}_{eg}\left({\omega }_1\right){\mu }_{eg}^{\star }+{\mu }_{fe}^{\star }{I}_{eg}\left({\omega }_2\right){\mu }_{eg}^{\star },$$

$$\begin{array}{cc}\hfill \langle f\mid {\mathcal{T}}^{†}({\omega }_1,{\omega }_2)\mid g\rangle & ={\mathcal{T}}_{fg}^{\star }({\omega }_1,{\omega }_2)\hfill \\ \hfill & ={\mu }_{fe}^{\star }{I}_{eg}^{\star }\left({\omega }_1\right){\mu }_{eg}^{\star }+{\mu }_{fe}^{\star }{I}_{eg}^{\star }\left({\omega }_2\right){\mu }_{eg}^{\star },\hfill \end{array}$$

$$\langle g\mid \mathcal{T}({\omega }_1,{\omega }_2)\mid f\rangle ={\mathcal{T}}_{gf}({\omega }_1,{\omega }_2)={\mu }_{ge}{I}_{ge}\left({\omega }_1\right){\mu }_{ef}+{\mu }_{ge}{I}_{ge}\left({\omega }_2\right){\mu }_{ef},$$

$$\begin{array}{cc}\hfill \langle g\mid {\mathcal{T}}^{†}({\omega }_1,{\omega }_2)\mid f\rangle & ={\mathcal{T}}_{gf}^{\star }({\omega }_1,{\omega }_2)\hfill \\ \hfill & ={\mu }_{ge}{I}_{ge}^{\star }\left({\omega }_1\right){\mu }_{ef}+{\mu }_{ge}{I}_{ge}^{\star }\left({\omega }_2\right){\mu }_{ef}.\hfill \end{array}$$ (8)

Here we have introduced the auxiliary quantity

$$I_{eg}\left(\omega \right)=G_{ee}(\omega +{\omega }_g)=\frac{1}{\omega -{\omega }_{eg}+i\gamma },$$ (9)

where G_ee(τ)=θ(τ)exp(−iω_eτ− γτ), γ is a dephasing rate, and exp(iω_gτ) describes the free evolution of the molecule in the ground state |g〉.

${\mathcal{T}}_{fg}({\omega }_1,{\omega }_2)$ describes retarded TPA on the left branches of the loop diagrams in Fig. 2 where the loop time runs forward. ${\mathcal{T}}_{fg}^{\star }({\omega }_1,{\omega }_2)$ represents advanced TPA on the right branches where the loop time runs backward. Similarly, ${\mathcal{T}}_{gf}({\omega }_1,{\omega }_2)\left[{\mathcal{T}}_{gf}^{\star }({\omega }_1,{\omega }_2)\right]$ describes retarded (advanced) TP emission. The system evolution after the absorption of the two photons is described by the doubly excited state Green's function G_ff=θ(τ)exp(−iω_fτ) and we define

$$I_{fg}({\omega }_1+{\omega }_2)=G_{ff}({\omega }_1+{\omega }_2+{\omega }_g)=\frac{1}{{\omega }_1+{\omega }_2-{\omega }_{fg}+i\gamma }.$$ (10)

Using the loop diagram rules [48] and Eq. (9) we obtain

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{d_2}({\omega }_2;{\omega }_1)~& \mathrm{Im}{i}^6\sum _{e\prime e}{\mu }_{ge\prime }{\mu }_e^{\star }{\mu }_{e\prime f}{\mu }_{fe}^{\star }{I}_{fg}^{\star }({\omega }_1+{\omega }_2)\hfill \\ \hfill & \times [I_{e\prime g}^{\star }\left({\omega }_2\right)I_{eg}^{\star }\left({\omega }_1\right)+I_{e\prime g}^{\star }\left({\omega }_2\right)I_{eg}^{\star }\left({\omega }_2\right)\hfill \\ \hfill & +I_{e\prime g}\left({\omega }_1\right)I_{eg}^{\star }\left({\omega }_1\right)+I_{e\prime g}\left({\omega }_1\right)I_{eg}^{\star }\left({\omega }_2\right)].\hfill \end{array}$$

The four terms correspond to pathways i–iv in Fig. 2, respectively. The other signal $S_{\mathrm{PP}}^{d_1}({\omega }_1;{\omega }_2)$ can be obtained from the above equation by interchanging ω₁ and ω₂.

The symmetric signal (3) assumes a compact form in terms of the two-photon transition amplitudes (8),

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_1,{\omega }_2)=& S_{\mathrm{PMP}}({\omega }_1;{\omega }_2)+S_{\mathrm{TPA}}({\omega }_1;{\omega }_2)\hfill \\ \hfill \equiv & 2N{\mid {\epsilon }_1\mid }^2{\mid {\epsilon }_2\mid }^2\mathrm{Im}{I}_{fg}^{\star }({\omega }_1+{\omega }_2)\hfill \\ \hfill & \times [{\mathcal{T}}_{gf}^{\star }({\omega }_1,{\omega }_2){\mathcal{T}}_{fg}^{\star }({\omega }_1,{\omega }_2)+{\mid {\mathcal{T}}_{fg}({\omega }_1,{\omega }_2)\mid }^2].\hfill \end{array}$$ (11)

The two terms in the brackets represent the PMP and TPA groups of pathways, respectively, as shown in Fig. 2. To interpret these terms we recall that PP is accompanied by spontaneous light-emission process. Each partially time-ordered CTPL diagram may be dissected into a set of fully time-ordered Feynman diagrams, where chronologically the last interaction occurs at time t₄ [52]. In the TPA pathways the molecule ends up in the doubly excited state |f〉〈f| and two photons are absorbed. The fluorescence from this state can be detected and described by the TP counting [50] formalism $~{\mid {\mathcal{T}}_{fg}\mid }^2$. The fundamental difference between the single-photon absorption and the PMP is the involvement of the second photon. The state of the second mode at time t₄ is the same as its initial state, but it provides intermediate coherence between the ground and double excited states, i.e., catalyzes the absorption of the other mode. The PMP pathways are related to fluorescence from the intermediate states |e〉〈e|. Since only one of the photons is absorbed, their contribution is not described by the single-photon transition amplitude as in the photon counting formalism but by $~{\mathcal{T}}_{gf}^{\star }({\omega }_1,{\omega }_2){\mathcal{T}}_{fg}^{\star }({\omega }_1,{\omega }_2)$.

The asymmetric signal vanishes in the degenerate case (ω₁=ω₂) where the two detectors give the same PP signal. When the intermediate states |e〉 are off resonant one has $T_{fg}^{\star }({\omega }_1,{\omega }_2)=T_{fg}({\omega }_1,{\omega }_2)$. Equation (11) then assumes the two-photon counting form

$$S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_1,{\omega }_2)=4N{\mid {\epsilon }_1\mid }^2{\mid {\epsilon }_2\mid }^2{\mid {\mathcal{T}}_{gf}({\omega }_1,{\omega }_2)\mid }^2\delta ({\omega }_1+{\omega }_2-{\omega }_{fg}).$$

However, as will be shown in Sec. IV, these results do not hold for quantum optical fields, where the two photons are correlated. Additional terms coming from the PMP group of pathways contribute to all PP signals. As a result, spontaneous emission from the final |f〉 and the intermediate states |e〉 become comparable.

IV. PUMP-PROBE WITH TWIN PHOTONS

We now assume that modes k₁ and k₂ are initially prepared in a twin state [41] representing an entangled photon pair. This can be produced by spontaneous parametric down conversion in a birefringent second-order nonlinear crystal χ⁽²⁾. The photon pair is described by a nonseparable wave function even when the photons are physically separated [47],

$$\begin{array}{cc}\hfill \mid {\psi }^{\left(2\right)}\left(0\right)\rangle =& C\sum _{{\mathbf{k}}_1}\sum _{{\mathbf{k}}_2}{t}_I\exp \left(-i\frac{\Delta \omega t_I}{2}\right)\mathrm{sinc}\left(\frac{\Delta \omega t_I}{2}\right)\hfill \\ \hfill & \times \exp \left(-i\frac{{\left(\Delta \mathbf{k}\right)}_z{L}_z}{2}\right)\mathrm{sinc}\left(\frac{{\left(\Delta \mathbf{k}\right)}_z{L}_z}{2}\right)\mid {\mathbf{k}}_1,{\mathbf{k}}_2\rangle .\hfill \end{array}$$ (12)

The Fock state |k₁,k₂| has one photon in each mode k₁ and k₂; Δω=ω_p−ω₁−ω₂,Δk=k_p−k₁−k₂; t_I and L_z are the interaction time and length within the PDC crystal. The normalization factor $C={\chi }^{\left(2\right)}{\epsilon }_p∕\sqrt{A_{12}}$ is proportional to the nonlinearity of the PDC crystal, the pump electric field amplitude ε_p, and the entanglement area A₁₂ [24]. The detectors d₁ and d₂ can distinguish between the two modes by their polarization (vertical for k₁ and horizontal for k₂).

For the twin state each of the optical field correlation functions 〈E^†E^†EE〉 in Eq. (6) is factorized into a product of two field transition amplitudes [47],

$$\begin{array}{cc}\hfill & \langle 0,0\mid E^2\left(t_4\right)E^1\left(t_2\right)\mid {\psi }^{\left(2\right)}\left(0\right)\rangle \hfill \\ \hfill & =\theta (t_4-t_2)F(t_4,t_2,T_{12})+\theta (t_2-t_4)F(t_2,t_4,T_{12})\hfill \\ \hfill & \times \langle {\psi }^{\left(2\right)}\left(0\right)\mid E^{1†}\left(t_3\right)E^{2,†}\left(t_1\right)\mid 0,0\rangle \hfill \\ \hfill & =\theta (t_3-t_1)F^{\star }(t_3,t_1,T_{12})+\theta (t_1-t_3)F^{\star }(t_1,t_3,T_{12}),\hfill \end{array}$$ (13)

where

$$\begin{array}{cc}\hfill F(t_4,t_2,T_{12})=& \frac{2\pi {\chi }^{\left(2\right)}{\epsilon }_p}{\Omega }\sqrt{\frac{{\omega }_1{\omega }_2}{A_{12}{T}_{12}}}\hfill \\ \hfill & \times \exp [-i({\omega }_1{t}_2+{\omega }_2{t}_4)]\mathrm{rect}\left(\frac{t_4-t_2}{T_{12}}\right),\hfill \end{array}$$ (14)

rect(t) is the rectangular function equal to 1 for 0≤t≤1 and 0 otherwise, and T₁₂=(1/v₁−1/v₂)L_z is the entanglement time determined by the group velocities v₁(v₂) of the mode k₁(k₂) within the PDC crystal.

By substituting Eq. (13) into Eq. (6), we get

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{d_2}({\omega }_2;{\omega }_1,T_{12})=& -\frac{4N}{2T}{\int }_{-T}^{T}d{t}_4\mathrm{Im}{i}^3{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }d{t}_{3}d{t}_{2}d{t}_1\hfill \\ \hfill & \times [\theta (t_4-t_2)\theta (t_2-t_3)\theta (t_3-t_1)\langle V\left(t_1\right)V\left(t_3\right)V^{†}\left(t_2\right)V^{†}\left(t_4\right)\rangle F^{\star }(t_3,t_1,T_{12})F(t_4,t_2,T_{12})\hfill \\ \hfill & +\theta (t_4-t_2)\theta (t_2-t_1)\theta (t_1-t_3)\langle V\left(t_3\right)V\left(t_1\right)V^{†}\left(t_2\right)V^{†}\left(t_4\right)\rangle F^{\star }(t_1,t_3,T_{12})F(t_4,t_2,T_{12})\hfill \\ \hfill & +\theta (t_4-t_2)\theta (t_4-t_3)\theta (t_3-t_1)\langle V\left(t_1\right)V\left(t_3\right)V^{†}\left(t_4\right)V^{†}\left(t_2\right)\rangle F^{\star }(t_3,t_1,T_{12})F(t_2,t_4,T_{12})\hfill \\ \hfill & +\theta (t_4-t_2)\theta (t_4-t_1)\theta (t_1-t_3)\langle V\left(t_3\right)V\left(t_1\right)V^{†}\left(t_4\right)V^{†}\left(t_2\right)\rangle F^{\star }(t_1,t_3,T_{12})F(t_2,t_4,T_{12})].\hfill \end{array}$$ (15)

We now introduce the Fourier transform of the joined matter and entangled photon Green's functions,

$$I_{eg}(\omega ,T_{12})=-i{\int }_{-\infty }^{\infty }{G}_{ee}\left(\tau \right)\mathrm{rect}\left(\frac{\tau }{T_{12}}\right)\exp \left[i(\omega -{\omega }_g)\tau \right]d\tau =\frac{e^{i(\omega -{\omega }_{eg}+i\gamma )T_{12}}-1}{\omega -{\omega }_{eg}+i\gamma }.$$ (16)

By substituting Eqs. (4) and (13) into Eq. (15) and changing the time variables to the loop times, we can express the TPA signal in terms of the entangled photons and matter Green's functions as follows:

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{d_2}({\omega }_2;{\omega }_1,T_{12})=& \frac{16{\pi }^{2}N{\mid {\chi }^{\left(2\right)}\mid }^2{\mid {\epsilon }_p\mid }^2{\omega }_1{\omega }_2}{A_{12}{T}_{12}{\Omega }^2}\mathrm{Im}\sum _{f,e\prime ,e}{\mu }_{e\prime g}{\mu }_{eg}^{\star }{\mu }_{e\prime f}{\mu }_{ef}^{\star }{I}_{fg}^{\star }({\omega }_1+{\omega }_2)[I_{e\prime g}^{\star }({\omega }_2,T_{12})I_{eg}^{\star }({\omega }_1,T_{12})+I_{e\prime g}^{\star }({\omega }_2,T_{12})I_{eg}^{\star }({\omega }_2,T_{12})\hfill \\ \hfill & +I_{e\prime g}({\omega }_1,T_{12})I_{eg}^{\star }({\omega }_1,T_{12})+I_{e\prime g}({\omega }_1,T_{12})I_{eg}^{\star }({\omega }_2,T_{12})],\hfill \end{array}$$ (17)

where I_fg(ω₁+ω₂) is the material doubly excited state Green's function (10). The signal $S_{\mathrm{PP}}^{d_1}({\omega }_1;{\omega }_2,T_{12})$ can be obtained from Eq. (17) by interchanging ω₁ and ω₂.

Introducing the entanglement modified two-photon transition amplitude

$${\mathcal{T}}_{gf}({\omega }_1,{\omega }_2,T_{12})={\mu }_{ge}{\mu }_{ef}{I}_{eg}({\omega }_1,T_{12})+{\mu }_{ge}{\mu }_{ef}{I}_{eg}({\omega }_2,T_{12}),$$

$${\mathcal{T}}_{fg}({\omega }_1,{\omega }_2,T_{12})={\mu }_{ge}^{\star }{\mu }_{ef}^{\star }{I}_{eg}({\omega }_1,T_{12})+{\mu }_{ge}^{\star }{\mu }_{ef}^{\star }{I}_{eg}({\omega }_2,T_{12}),$$ (18)

we obtain the symmetric TPA signal induced by the twin photons,

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_1,{\omega }_2,T_{12})=& \frac{8{\pi }^{2}N{\mid {\chi }^{\left(2\right)}\mid }^2{\mid {\epsilon }_p\mid }^2{\omega }_1{\omega }_2}{A_{12}{T}_{12}{\Omega }^2}\hfill \\ \hfill & \times \mathrm{Im}{I}_{fg}^{\star }({\omega }_1+{\omega }_2)[{\mathcal{T}}_{gf}^{\star }({\omega }_1,{\omega }_2,T_{12})\hfill \\ \hfill & \times {\mathcal{T}}_{fg}^{\star }({\omega }_1,{\omega }_2,T_{12})+{\mid {\mathcal{T}}_{fg}({\omega }_1,{\omega }_2,T_{12})\mid }^2].\hfill \end{array}$$ (19)

As in Eq. (11), the two terms correspond to PMP and TPA pathways. It is interesting to note that the symmetric PP signal generated by twin photons (19) may not be described by the absolute square of the transition amplitude.

We have carried out simulations for the following model material system. The single excited state frequencies ω_eg={1.5, 1.6, 1.8} are off resonant (ω_eg−ω_p/2)T₁₂⪢γT₁₂. The doubly excited state at ω_fg=1 is resonant with the pump photon ω_gf =ω_p used to generate the degenerate twins ω₁=ω₂=ω_p/2. All dipole moments are real and identical.

We have varied the entanglement time T₁₂ from T_min=2¹²ΔT₁₂ to T_max=2T_min with the time stem ΔT₁₂=2π/[max(ω_eg+ω_e′g−ω_p)]=0.025. That is if the unit of energy is $\hslash {\omega }_p=1$ eV then Δτ=1.3 fs. This can be done by tilting the PDC crystal [36] which increases its size L_z.

In Fig. 3 we display the entanglement time spectra defined as follows:

$$S_{\mathrm{PP}}^{\nu }({\omega }_p∕2,{\omega }_T)=\mathcal{N}{\int }_{T_{\min }}^{T_{\max }}d{T}_{12}\exp \left(i{\omega }_{T_{12}}{T}_{12}\right)S_{\mathrm{PP}}^{\nu }({\omega }_p∕2,T_{12}),$$ (20)

where superscript ν can be {d1,d2,Sym,TPA,PMP} and $\mathcal{N}=1∕S_{\mathrm{TPA}}^{\mathrm{Sym}}({\omega }_p∕2,T_{12})$ is a normalization factor. The spectra show several groups of resonances. Group a correspond to the intraband resonances of the intermediate-state manifold 0≤ω_T12=ω_eg−ω_e′g≤0.3, group b corresponds to the 1≤ω_T12=ω_eg−ω_p/2≤1.3 resonances, and group c describes 2≤ω_T12=ω_eg+ω_p≤2.6 resonances. Note that the resonances of group b can be detected by a conventional pump probe with short well-separated classical pulses.

FIG. 3.

(Color online) The pump-probe spectra ${\log }_{10}\mid S_{\mathrm{TPA}}^{\nu }({\omega }_p∕2,{\omega }_T)\mid$ [Eq. (25)]. The signals recorded by the two detectors are identical S^d₁=S^d₂=S^Sym. Left (right) column correspond to large (small) dephasing rate, as shown.

All pathways (Fig. 2) which contribute to the signals interfere constructively. The left column in Fig. 3 uses a larger dephasing rate 2.68<T_12γ<5.36 than the right column 0.03<γT₁₂<0.05. The larger dephasing rate in the left column quenches the resonances in regions a and c. PMP pathways contribute to spectral regions b and c, while TPA pathways show resonances in regions i and ii. We note that for this technique S^d₁=S^d₂=S^Sym and the asymmetric signal S^Asym vanishes. In both columns the S^TPA signal has no resonances in region c whereas the S^PMP signal does not show resonances in region a.

V. PUMP-PROBE WITH MUTUALLY DELAYED TWIN PHOTONS

We next turn to a more general twin photon PP setup as proposed by Saleh et al. [32]. The twins remain degenerate ω₁=ω₂=ω_p/2 but we introduce a relative time delay 0<τ<T₁₂ between them by shifting the phase of the k₁ mode [12,40],

$$E_1\left(t\right)={\left(\frac{\pi {\omega }_p}{\Omega }\right)}^{1∕2}{a}_1\exp [-i{\omega }_p(t-\tau ∕2)∕2],$$

$$E_2\left(t\right)={\left(\frac{\pi {\omega }_p}{\Omega }\right)}^{1∕2}{a}_2\exp [-i{\omega }_p(t+\tau ∕2)∕2].$$ (21)

Using Eq. (21) we obtain the optical field transition amplitudes for diagrams i–iv,

$$\begin{array}{cc}\hfill & \langle 0,0\mid E_2\left(t_4\right)E_1\left(t_2\right)\mid {\psi }^{\left(2\right)}\left(0\right)\rangle \hfill \\ \hfill & =\theta (t_4-t_2)F(t_4,t_2,T_{12})+\theta (t_2-t_4)F(t_2,t_4,T_{12}),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \langle {\psi }^{\left(2\right)}\left(0\right)\mid E_1^{†}\left(t_3\right)E_2^{†}\left(t_1\right)\mid 0,0\rangle \hfill \\ \hfill & =\theta (t_3-t_1)F^{\star }(t_3,t_1,T_{12})+\theta (t_1-t_3)F^{\star }(t_1,t_3,T_{12}),\hfill \end{array}$$ (22)

where

$$F(t_4,t_2,T_{12})=\frac{\pi {\chi }^{\left(2\right)}{\epsilon }_p{\omega }_p}{\Omega \sqrt{A_{12}{T}_{12}}}\exp [-i{\omega }_p∕2(t_4+t_2)]\mathrm{rect}\left(\frac{t_4-t_2}{T_{12}-\tau }\right).$$

For diagrams v–viii one has to interchange t₄↔t₂,t₃↔t₁ in the above expression.

The signal in Eq. (15) now becomes

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{d_2}({\omega }_p∕2,T_{12},\tau )=& \frac{16{\pi }^{2}N{\mid {\chi }^{\left(2\right)}\mid }^2{\mid {\epsilon }_p\mid }^2{\omega }_p^2}{A_{12}{T}_{12}{\Omega }^2}\mathrm{Im}\sum _{f,e\prime ,e}{\mu }_{e\prime g}{\mu }_{eg}^{\star }{\mu }_{e\prime f}{\mu }_{ef}^{\star }{I}_{fg}^{\star }\left({\omega }_p\right)[I_{e\prime g}^{\star }({\omega }_p∕2,T_{12}-\tau )I_{eg}^{\star }({\omega }_p∕2,T_{12}-\tau )\hfill \\ \hfill & +I_{e\prime g}^{\star }({\omega }_p∕2,T_{12}-\tau )I_{eg}^{\star }({\omega }_p∕2,T_{12}+\tau )+I_{e\prime g}({\omega }_p∕2,T_{12}-\tau )I_{eg}^{\star }({\omega }_p∕2,T_{12}-\tau )\hfill \\ \hfill & +I_{e\prime g}({\omega }_p∕2,T_{12}-\tau )I_{eg}^{\star }({\omega }_p∕2,T_{12}+\tau )].\hfill \end{array}$$ (23)

The signal from the other detector is $S_{\mathrm{PP}}^{d_1}({\omega }_p∕2,T_{12},\tau )=S_{\mathrm{PP}}^{d_2}({\omega }_p∕2,T_{12},-\tau )$. Here $I_{eg}({\omega }_p∕2,T_{12}\mp \tau )$ describes the Fourier transform of the system (twin+matter) Green's function which depends on the delay τ between the entangled photons. This delay breaks the symmetry of the process making it dependent on the order in which the twins are absorbed or emitted even in the degenerate case [see Fig. 1(c)]. If mode k₁ precedes k₂, the system's evolution is described by the following Green's function: I_eg(ω,T₁₂−τ). When k₂ comes first we have I_eg(ω,T₁₂+τ). The broken symmetry implies that $S_{\mathrm{PP}}^{d_2}({\omega }_p∕2,T_{12},\tau )\ne S_{\mathrm{PP}}^{d_1}({\omega }_p,2∕T_{12},\tau )$.

Using the field transition amplitudes (22) the two-photon transition amplitude assumes the form

$$\begin{array}{cc}\hfill {\mathcal{T}}_{gf}({\omega }_p∕2,T_{12},\tau )=& \sum _e{\mu }_{ge}{\mu }_{ef}{I}_{eg}({\omega }_p∕2,T_{12}-\tau )\hfill \\ \hfill & +{\mu }_{ge}{\mu }_{ef}{I}_{eg}({\omega }_p∕2,T_{12}+\tau ),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{T}}_{fg}({\omega }_p∕2,T_{12},\tau )=& \sum _e{\mu }_{ge}^{\star }{\mu }_{ef}^{\star }{I}_{eg}({\omega }_p∕2,T_{12}-\tau )\hfill \\ \hfill & +{\mu }_{ge}^{\star }{\mu }_{ef}^{\star }{I}_{eg}({\omega }_p∕2,T_{12}+\tau ).\hfill \end{array}$$ (24)

The symmetrized signal (3) can be written in terms of the matter field transition amplitudes (24) as

$$S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_p∕2,T_{12},\tau )=\frac{1}{2}[S_{\mathrm{PP}}^{\mathrm{PMP}}({\omega }_p∕2,T_{12},\tau )+S_{\mathrm{TAP}}^{\mathrm{Sym}}({\omega }_p∕2,T_{12},\tau )],$$ (25)

where

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{\mathrm{PMP}}({\omega }_p∕2,T_{12},\tau )=& \frac{16{\pi }^{2}N{\mid {\chi }^{\left(2\right)}\mid }^2{\mid {\epsilon }_p\mid }^2{\omega }_p^2}{A_{12}{T}_{12}{\Omega }^2}\hfill \\ \hfill & \times \mathrm{Im}\sum _f{I}_{fg}^{\star }\left({\omega }_p\right){\mathcal{T}}_{fg}^{\star }({\omega }_p∕2,T_{12},\tau )\hfill \\ \hfill & \times {\mathcal{T}}_{gf}^{\star }({\omega }_p∕2,T_{12},\tau ),\hfill \end{array}$$ (26)

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{\mathrm{TPA}}({\omega }_p∕2,T_{12},\tau )=& \frac{16{\pi }^{2}N{\mid {\chi }^{\left(2\right)}\mid }^2{\mid {\epsilon }_p\mid }^2{\omega }_p^2}{A_{12}{T}_{12}{\Omega }^2}\hfill \\ \hfill & \times \sum _f{\mid {\mathcal{T}}_{fg}({\omega }_p∕2,T_{12},\tau )\mid }^2\delta ({\omega }_p-{\omega }_{fg}).\hfill \end{array}$$ (27)

Equation (27) is identical to the result of Ref. [32]. For the degenerate case ω₁=ω₂, the symmetry of the above signal implies the following symmetry with respect to the time delay $S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_p∕2,T_{12},\tau )=S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_p,2∕T_{12},-\tau )$.

We next turn to the antisymmetric signal [the difference between the detector d₁ and d₂ in Fig. 1(a)],

$$\begin{array}{cc}\hfill S_{\mathrm{PP}}^{\mathrm{Asym}}({\omega }_p∕2,T_{12},\tau )=& \frac{8{\pi }^{2}N{\mid {\chi }^{\left(2\right)}\mid }^2{\mid {\epsilon }_p\mid }^2{\omega }_p^2}{A_{12}{T}_{12}{\Omega }^2}\mathrm{Im}\sum _f{I}_{fg}^{\star }\left({\omega }_p\right)\hfill \\ \hfill & \times [{\mathcal{T}}_{gf}^{\star }({\omega }_p∕2,T_{12},\tau ){\mathcal{T}}_{gf}^{\prime \star }({\omega }_p∕2,T_{12},\tau )\hfill \\ \hfill & +{\mathcal{T}}_{gf}^{\prime }({\omega }_p∕2,T_{12},\tau ){\mathcal{T}}_{fg}^{\star }({\omega }_p∕2,T_{12},\tau )].\hfill \end{array}$$ (28)

Here we have introduced the antisymmetric two-photon transition amplitude

$$\begin{array}{cc}\hfill {\mathcal{T}}_{gf}^{\prime }({\omega }_p∕2,T_{12},\tau )=& \sum _e{\mu }_{ge}{\mu }_{ef}{I}_{eg}({\omega }_p∕2,T_{12}-\tau )\hfill \\ \hfill & -{\mu }_{ge}{\mu }_{ef}{I}_{eg}({\omega }_p∕2,T_{12}+\tau ).\hfill \end{array}$$ (29)

This signal vanishes in frequency-domain experiments with classical fields where the order in which the modes interact is not important,

$${\mathcal{T}}_{gf}^{\prime }({\omega }_1,{\omega }_2)={\mu }_{ge}{\mu }_{ef}{I}_{eg}({\omega }_1,{\omega }_2)-{\mu }_{ge}{\mu }_{ef}{I}_{eg}({\omega }_1,{\omega }_2)=0.$$

The asymmetric signal also vanishes at zero twin photon delay since ${\mathcal{T}}_{gf}^{\prime }({\omega }_p∕2,T_{12},0)=0$. Absorption of k₁ is always accompanied by absorption of k₂ and the same holds for emission. For the twin photons with delay τ the Green's functions do depend on the order of interaction and the asymmetric signal does not vanish. This signal is thus induced by the delay between twin photons.

In Fig. 4 we compare the individual detector readings (23), the symmetrized signal (25), and the TPA contribution (27) to the PP signal. First, we hold the entanglement time T₁₂ fixed and plot the Fourier transforms with respect to the time delay τ:

$$S_{\mathrm{PP}}^{\nu }({\omega }_p∕2,T_{12},{\omega }_{\tau })=\mathcal{N}{\int }_0^{T_{12}}d\tau \exp \left(i{\omega }_{\tau }\tau \right)S_{\mathrm{PP}}^{\nu }({\omega }_p∕2,T_{12},\tau ).$$ (30)

The entanglement time increases from zero with a step ΔτΔT₁₂ up to the value of the entanglement time T₁₂=T_min. As in Fig. 3 the resonances of the entanglement time PP spectra may be classified by the same three spectral groups a,b,c. Recall that the spectra with respect to the entanglement time associate the region a with PMP and region c with TPA. However their meanings for the spectra given by Eq. (30) is quite different. For PMP pathways region a resonances are produced by the pathways i, v, iii, and vii where emitted and absorbed photons follow the same chronological order; the region b resonances are given by the pathways ii, vi, iv, and viii where the chronological photon order for the emission is opposite to those of the absorption. For TPA the meaning of the regions of a and c is exactly opposite to those of PMP. This peculiar feature is a result of broken symmetry of the field transition amplitude. For small dephasing rate γT₁₂=0.03 (right column) PMP and TPA contribute to the same regions in all spectra and spectrally nonseparable by the pump-probe technique. At large dephasing rate γT₁₂=2.68 (left column) pathways v–viii do not contribute to regions a and c. PMP pathways i and ii contribute to the spectral regions b and c while TPA pathways iii and iv contribute to regions a and b. We note that the spectra obtained from the two detectors are different S^d₁≠S^d₂. Only the S^d₂ signal shows resonances in the nonclassical regions a and c. However, both S^Sym and S^Asym signals contain all resonances. The TPA signal misses the region c resonances and PMP misses resonances in region a; thus, one can separate PMP from TPA if focuses on the regions a and c only.

FIG. 4.

(Color online) The pump-probe spectra ${\log }_{10}\mid S_{\mathrm{TPA}}^{\nu }({\omega }_p∕2,T_{12},{\omega }_{\tau })\mid$ vs ω_τ and a fixed entanglement time T₁₂. The two columns differ by the dephasing rate, as indicated.

In Fig. 5 we display correlation 2D spectra between the entanglement and the delay times,

$$\begin{array}{cc}\hfill S_{\mathrm{TPA}}^{\nu }({\omega }_p∕2,{\omega }_{T_{12}},{\omega }_{\tau })=& \mathcal{N}{\int }_{T_{\min }}^{T_{\max }}d{T}_{12}\exp \left(i{\omega }_{T_{12}}{T}_{12}\right)\hfill \\ \hfill & \times {\int }_0^{T_{12}}d\tau \exp \left(i{\omega }_{\tau }\tau \right)S_{\mathrm{TPA}}^{\nu }({\omega }_p∕2,T_{12},\tau ).\hfill \end{array}$$ (31)

The resonances are classified into five regions in the (ω_T12,ω_τ) plane. The PMP pathways contribute to regions (b,b), (c,a), and (c,c); the TPA pathways create resonances at (a,a), (b,b), and (a,c). Therefore regions a,c and pathways c,a directly show the correlation between the PMP and TPA. We find that for both values of the dephasing rate the TPA and PMP signals miss the (c,c) and (a,a) resonances correspondingly. The correlation spectra from the two detectors are different. Unlike the delay spectra, detector d₁ reveals all spectral regions, while resonances (a,a) and (c,c) from detector d₂ are suppressed. Finally the off-diagonal, as well as (b,b), resonances are sensitive to the dephasing rate and vanish for large dephasing. The diagonal (a,a) and (c,c) resonances overlap as the dephasing rate is increased. In summary, the conventional frequency-domain PP signal when ω₁ and ω₂ are off resonant can be described as a two-photon counting process. The twin photons characterized by their frequencies, polarizations, and the entanglement time T₁₂ generate the symmetric PP signal [Eq. (19)] obtained from CTPL formalism. The signal consists of $\mathrm{TPA}~{\mid {\mathcal{T}}_{fg}\left(T_{12}\right)\mid }^2$ as well as $\mathrm{PMP}~{\mathcal{T}}_{gf}^{\star }\left(T_{12}\right){\mathcal{T}}_{fg}^{\star }\left(T_{12}\right)$ terms. The two terms may be observed separately by measuring two-photon-induced fluorescence from the doubly and singly excited states, respectively. Alternatively, they can be spectrally separated by focusing on certain regions in $S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_p∕2,{\omega }_{T_{12}})$. When the time delay τ is introduced the two detectors yield two different signals $S_{\mathrm{PP}}^{d_{1,2}}({\omega }_p∕2,{\omega }_{\tau })$ giving rise to the asymmetric signal $S_{\mathrm{PP}}^{\mathrm{Asym}}({\omega }_p∕2,{\omega }_{\tau })$ or spectra. For $S_{\mathrm{PP}}^{\mathrm{Sym}}({\omega }_p∕2,{\omega }_{\tau })$ the separation between PMP and TPA is possible only at large dephasing. The correlation between PMP and TPA pathways can be seen in the off-diagonal regions on the 2D correlation spectra $S_{\mathrm{PP}}^{\nu }({\omega }_p∕2,{\omega }_{T_{12}},{\omega }_{\tau })$. Spectral separation and correlation between PMP and TPA is important if the direct observation of the fluorescence from |e〉 or |f〉 is not possible due to selection rules or fast nonradiative relaxation of the |f〉 state.

FIG. 5.

(Color online) 2D correlation pump-probe spectra ${\log }_{10}\mid S_{\mathrm{TPA}}^{\nu }({\omega }_p∕2,{\omega }_{T_{12}},{\omega }_{\tau })\mid$ with entangled twin photons. The left and right columns represent different dephasing rates.

VI. PUMP-PROBE WITH RECTANGULARLY SHAPED CLASSICAL OPTICAL FIELDS

We now demonstrate how similar spectra can be obtained with shaped classical pulses. The key ingredients of the TPA signals are the optical field transition amplitudes, which can be shaped and manipulated. Let us assume that modes k₁ and k₂ are created from a rectangular (width T₁₂) pump beam with frequency ω_p by means of 50:50 beam splitter. Both modes are classical and initially in a coherent state |β〉. The beam-splitter-induced π/2 phase shift between the reflected k₁ and transmitted k₂ modes can be compensated by difference in their optical path from the splitter to the sample. We further introduce relative time delay τ in k₁ beam path. The optical field operators are

$$\begin{array}{cc}\hfill E_1\left(t\right)=& {\left(\frac{2\pi {\omega }_p}{\Omega }\right)}^{1∕2}{a}_1\exp [-i{\omega }_p(t-\tau ∕2)∕2]\hfill \\ \hfill & \times \mathrm{rect}\left(\frac{t-\tau ∕2-T_{12}∕2}{T_{12}}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E_2\left(t\right)=& {\left(\frac{2\pi {\omega }_p}{\Omega }\right)}^{1∕2}{a}_2\exp [-i{\omega }_p(t+\tau ∕2)∕2]\hfill \\ \hfill & \times \mathrm{rect}\left(\frac{t+\tau ∕2-T_{12}∕2}{T_{12}}\right).\hfill \end{array}$$ (32)

Making use of the following identity:

$$\mathrm{rect}\left(\frac{t_4-\tau ∕2-T_{12}∕2}{T_{12}}\right)\mathrm{rect}\left(\frac{t_2+\tau ∕2-T_{12}∕2}{T_{12}}\right)=\mathrm{rect}\left(\frac{\mid t_4-t_2+\tau \mid }{T_{12}}\right),$$

we obtain the following field transition amplitudes for diagrams i–iv:

$$\begin{array}{cc}\hfill & \langle {\beta }_1,{\beta }_2\mid E^2\left(t_4\right)E^1\left(t_2\right)\mid {\beta }_1,{\beta }_2\rangle \hfill \\ \hfill & =\theta (t_4-t_2)E(t_4,t_2,T_{12})+\theta (t_2-t_4)E(t_2,t_4,T_{12})\hfill \\ \hfill & =\frac{2\pi {\omega }_p}{\Omega }\exp [-i{\omega }_p∕2(t_4+t_2)]\hfill \\ \hfill & \times \left[\theta (t_4-t_2)\mathrm{rect}\left(\frac{t_4-t_2}{T_{12}-\tau }\right)+\theta (t_2-t_4)\mathrm{rect}\left(\frac{t_2-t_4}{T_{12}+\tau }\right)\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \langle {\beta }_1,{\beta }_2\mid E^{1†}\left(t_3\right)E^{2,†}\left(t_1\right)\mid {\beta }_1,{\beta }_2\rangle \hfill \\ \hfill & =\theta (t_3-t_1)E^{\star }(t_3,t_1,T_{12})+\theta (t_1-t_3)E^{\star }(t_1,t_3,T_{12})\hfill \\ \hfill & =\frac{2\pi {\omega }_p}{\Omega }\exp [i{\omega }_p∕2(t_3+t_1)]\hfill \\ \hfill & \times \left[\theta (t_3-t_1)\mathrm{rect}\left(\frac{t_3-t_1}{T_{12}-\tau }\right)+\theta (t_1-t_3)\mathrm{rect}\left(\frac{t_1-t_3}{T_{12}+\tau }\right)\right].\hfill \end{array}$$ (33)

Substituting Eq. (33) into Eq. (15) gives

$$\begin{array}{cc}\hfill S_{\mathrm{TPA}}^{\mathrm{Sym}}({\omega }_p∕2,T_{12},\tau )=& \frac{16{\pi }^{2}N{\omega }_p^2}{{\Omega }^{3}2}\mathrm{Im}{I}_{fg}^{\star }\left({\omega }_p\right)[{\mathcal{T}}_{fg}^{\star ,2}({\omega }_p∕2,T_{12},\tau )\hfill \\ \hfill & +{\mid {\mathcal{T}}_{fg}({\omega }_p,T_{12},\tau )\mid }^2]\hfill \end{array}$$ (34)

Equation (34) derived for shaped classical modes coincides with the entanglement-induced TPA signal [Eq. (25)] up to an overall scaling factor. Equation (25) scales linearly with the pump photon intensity, while the classical signal scales quadratically. The scaling factor is independent on the τ parameter. Therefore their Fourier spectra with respect to τ are identical.

Pump-probe spectroscopy performed with shaped classical pulses where the pump is split into two copropagating pulses separated by a delay τ gives information equivalent to four-wave mixing carried out with four pulses. 2D spectra can be obtained by, e.g., varying τ and the T₁₂. The twin delay τ plays a similar role but enters in a different way than with classical fields. The precise connection between the two approaches to multidimensional spectroscopy based controlling either the delays between classical pulses or the parameters of entangled photon wave function that determine their relative timing statistics will be an interesting topic for future investigations.

VII. CONCLUSIONS

In this paper we demonstrated how quantum degrees of freedom of the radiation field may be used as multidimensional spectroscopic probes of matter. Mutually delayed twin photons tuned to two-photon resonances ω₁+ω₂=ω_fg can provide direct information on off-resonant transitions. The twin photon state is a wave packet of two groups of modes that can be distinguished and detected separately by two detectors d₁ and d₂ by exploiting their different polarizations or propagation directions. The two detectors measure the intensity change in one of the beams with and without the sample. An intuitive microscopic formalism based on close time path loop (CTPL) diagrams is developed and employed to calculate the pump-probe signals generated with the entangled photons $S_{\mathrm{PP}}^{d_1}$ and $S_{\mathrm{PP}}^{d_2}$. Symmetric (asymmetric) signals are further defined as the sum (difference) of the detector readings $(S_{\mathrm{PP}}^{d_1}\pm S_{\mathrm{PP}}^{d_2})$. For general quantum optical fields the signal is recast in terms of quantum pathways of matter, each scaled with a corresponding four-point field correlation functions. Such signals may not be expressed in terms of ordinary causal response functions and susceptibilities as is the case for classical fields since they depend on correlated spontaneous fluctuations of field and matter. For classical fields the signal is given by a product of a susceptibility and field amplitude. In the entangled case the signal is given by a sum of such products. The symmetrized pump-probe signal can be divided into components which represent two groups of pathways $S_{\mathrm{PP}}^{\mathrm{PMP}}+S_{\mathrm{PP}}^{\mathrm{TPA}}$. TPA may be measured by fluorescence from the doubly excited states |f〉 and may be calculated as a modulus square of a two-photon counting amplitude. PMP on the other hand represents a process where one of the pump photon creates an intermediate coherence between the ground and double excited states and “catalyzes” the probe absorption, without changing its state at the end of the process. This component can be measured by fluorescence from the singly excited |e〉 state and is given by the imaginary part of a product of two-photon complex transition amplitudes (not modulus square). For classical off-resonant fields $S_{\mathrm{PP}}^{\mathrm{PMP}}=S_{\mathrm{PP}}^{\mathrm{TPA}}$, and the antisymmetric signal $S_{\mathrm{PP}}^{\mathrm{Asym}}$ vanishes. However it becomes finite with twin photons. The distinction between the pump-probe and TPA signals has not been clearly made in the literature, where the two terms are used interchangeably. 2D signals obtained by varying the delay time τ and the entanglement time T₁₂ reveal the off-resonant singly excited state frequencies and allow us to distinguish between various groups of resonances. Focusing on different regions of the spectra one can separate PMP and TPA contributions or measure the correlation between the two. This spectral resolution is especially important if direct measurement of the fluorescence from the |f〉 is impossible. We further showed that one can mimic the entangled-photon pump-probe signals by using two mutually delayed rectangularly shaped classical optical fields. The present formalism allows us to describe pulse shaping and photon entanglement within the same framework. This may allow the design of other types of time-domain spectroscopies which make use of the broad band width of entangled photons to probe relaxation processes in the resonantly excited states.