Multidimensional pump-probe spectroscopy with entangled twin-photon states

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.

Figure 1

(Color online) (a) Pump-probe experiment with the twin photons. (b) One of the possible schemes for controlling the external time delay $\tau$ 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 $t_4\leftrightarrow t_2,t_3\leftrightarrow t_1$ is broken when one of them is delayed. (d) The model molecular level scheme used in our simulations.

Figure 2

CTPL diagrams for the contributions to the pump-probe signal which contain ${\omega }_1+{\omega }_2={\omega }_{fg}$ resonances. By convention, the last interaction is an absorption and occurs at the left branch of the loop. Diagrams i–iv represent ${\mathbf{k}}_1$ (pump) and ${\mathbf{k}}_2$ (probe) [v–vii are ${\mathbf{k}}_1$ (probe) and ${\mathbf{k}}_2$ (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 $|f⟩⟨f|$.

Figure 3

(Color online) The pump-probe spectra ${\text{log}}_{10}\left|S_{\text{TPA}}^{\nu }\left({\omega }_p/2,{\omega }_T\right)\right|$ [Eq. (25)]. The signals recorded by the two detectors are identical $S^{d_1}=S^{d_2}=S^{\text{Sym}}$. Left (right) column correspond to large (small) dephasing rate, as shown.

Figure 4

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

Figure 5

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