Nonlinear optical signals and spectroscopy with quantum light

Conventional nonlinear spectroscopy uses classical light to detect matter properties through the variation of its response with frequencies or time delays. Quantum light opens up new avenues for spectroscopy by utilizing parameters of the quantum state of light as novel control knobs and through the variation of photon statistics by coupling to matter. An intuitive diagrammatic approach is presented for calculating ultrafast spectroscopy signals induced by quantum light, focusing on applications involving entangled photons with nonclassical bandwidth properties—known as “time-energy entanglement.” Nonlinear optical signals induced by quantized light fields are expressed using time-ordered multipoint correlation functions of superoperators in the joint field plus matter phase space. These are distinct from Glauber’s photon counting formalism which uses normally ordered products of ordinary operators in the field space. One notable advantage for spectroscopy applications is that entangled-photon pairs are not subjected to the classical Fourier limitations on the joint temporal and spectral resolution. After a brief survey of properties of entangled-photon pairs relevant to their spectroscopic applications, different optical signals, and photon counting setups are discussed and illustrated for simple multilevel model systems.

Figure 1

(a) Linear entangled TPA rate and quadratic nonlinear random TPA rate in a porphyrin dendrimer at different entanglement times. The inset shows the dominant effect of the entangled TPA at low input flux of correlated photons. From [146]. (b) Two-photon absorption $5S_{1/2}\to 4D_{5/2,3/2}$ in atomic rubidium vs the time delay between the two photons (left panel) and vs the pump frequency (right panel). From [50].

Figure 2

(a) The three-level scheme which we consider in most of the review consists of a unique ground state $|g⟩$, and single excited state manifold $|e⟩$ and doubly excited manifold $|f⟩$. The vertical arrows indicate dipole-allowed transitions. (b) The single loop diagram representing the evolution of the wave function. On the left-hand side, the vertical line represents the ket and on the right-hand side it corresponds to the bra. The wave function in this closed time path loop diagram propagates forward in a loop from the bottom branch of the diagram along the ket branch to the top of the diagram. It then evolves backward in time from the top to the bottom of the right branch (bra). This forward and backward propagation is similar to the Keldysh contour diagram rules ([124]). Horizontal arrows represent field-matter interactions. (c) The corresponding three ladder diagrams for the evolution of the density matrix. Here the density matrix evolves forward in time upward from the bottom to the top of the diagram. In order to achieve a doubly excited-state population starting from the ground state the density matrix has to undergo four interactions represented by the absorption of two photons represented by four inwardly directed arrows. The three diagrams represent different time orderings of ket vs bra interactions. Together with their mirror reflected diagrams (interchanging left and right sides of the diagram) the total six diagrams are lumped together in the single loop diagram (b).

Figure 3

Absolute value of the two-photon correlation function (63) of entangled-photon pairs with (a) strong frequency anticorrelations ${\sigma }_p=0.6/T_2$ [entanglement entropy (40) $E=1.9$], (b) very weak correlations ($E=0.018$), ${\sigma }_p=3.5/T_2$, and (c) strong positive frequency correlations ${\sigma }_p=50/T_2$ ($E=1.7$).

Figure 4

(a) The two-photon correlation function $h_{12}(\omega ,\omega )$, Eq. (63), plotted vs the frequency $\omega$ in units of $T_2$, and for mean photon numbers (with increasing, dashed) $\overline{n}\sim 0.1$, 1, 10, and 100. (b) The same for the autocorrelation function $g_1(\omega ,\omega )$, Eq. (64).

Figure 5

(a) Experimental layout of the entangled-photon shaper: A computer-controlled spatial light modulator (SLM) is used to manipulate the spectral phase of the entangled photons. The photon pairs are detected in the inverse process of the PDC—sum-frequency generation (SFG). The SFG photons are subsequently counted in a single-photon counting module. In order to demonstrate two-photon interference oscillations, a Mach-Zehnder interferometer is placed between the last prism and the SFG crystal. (b) The SFG counts (circles) and the calculated second-order correlation function (line) of the unperturbed wave function as a function of the signal-idler delay. (c) The same for the shaped wave function. From [199].

Figure 6

(a) Scheme of two noninteracting two-level atoms $A$ and $B$ and corresponding many-body state diagram where ground state $g$ corresponds to both atoms in the ground state, $a$—atom $A$ is excited and $B$ is in the ground state, $b$—atom $B$ is in the excited state and atom $A$ is in the ground state, and $ab$—both atoms are in the excited state. The arrows directed upward represent two-photon excitations. (b) Relevant set of diagrams corresponding to atom $A$ being in the excited state to $\sim |{\mu }_A{|}^2|{\mu }_B{|}^2$ in field-matter interactions. $\alpha$ and $\beta$ run over $a$ and $b$ to account for all possible permutations in the excitation pathways.

Figure 7

The temporal profiles of two photons emitted by a cascade source illustrate time-frequency entanglement: the red curve represents the marginal probability $P({\tau }_{\alpha })$ Eq. (85), and the blue curve corresponds to the conditional probability $P({\tau }_{\beta }|{\tau }_{\alpha })$ Eq. (84). The intrinsic time ordering of the photons, $\alpha$ first, followed by $\beta$, suppresses the excitation pathway where $\beta$ is absorbed first, followed by $\alpha$, inducing joint two-atom excitation.

Figure 8

Level scheme of the multilevel model employed in this review: two singly excited states $e_1$ and $e_2$ with energies 11 000 and $11500{\mathrm{cm}}^{-1}$ are coupled to doubly excited states $f_1$ and $f_2$, as indicated. Furthermore, $e_2$ decays to $e_1$ within $1/k\simeq 30\mathrm{fs}$.

Figure 9

(a) The density matrix for doubly excited states ${\varrho }_{f{f}^{\prime }}(t)=T_{f^{\prime }g}^{*}(t)T_{fg}(t)$, Eq. (100), prepared by the absorption of entangled photons with pump frequency ${\omega }_p=22160{\mathrm{cm}}^{-1}$. (b) The same, with ${\omega }_p=24200{\mathrm{cm}}^{-1}$. From [239].

Figure 10

Variation of the double-excited-state populations $p_f(t)$, Eqs. (102)–(104), with the pump frequency ${\omega }_p$ after excitation by entangled photons with strong frequency anticorrelations (solid, red), by laser pulses matching the pump bandwidth ${\sigma }_p$ (dashed, blue), and laser pulses matching the photon bandwidths $\sim 1/T$ (dot-dashed, green). The total $f$ population is normalized to unity at each frequency, i.e., $\sum _f{p}_f(t)=1$.

Figure 11

(a) Relative contribution of the coherent two-photon contribution $h_{12}$, Eq. (63), to the full time-integrated population of $f_2$ plotted vs the mean photon number $\overline{n}$, Eq. (59). The dashed line shows strong entanglement with entanglement time $T=10\mathrm{fs}$, and the solid line shows weaker entanglement with $T=100\mathrm{fs}$. (b) The same for an intermediate state moved to ${\omega }_{e_2}=18500{\mathrm{cm}}^{-1}$.

Figure 12

Entangled virtual-state spectroscopy: the TPIF signal vs the Fourier transform of an additional time delay between the two photons (see text) in hydrogen. From [230].

Figure 13

(a) TPIF action spectrum $S_{\mathrm{TPIF}}(\mathrm{\Gamma })$, Eq. (114), induced by entangled-photon pairs with the same parameters as in Fig. 10. The fluorescence created by the state $f_1$ is shown separately as a dashed line, and the signal from $f_2$ as a dot-dashed line. (b) TPIF action spectrum induced by laser pulses with bandwidth ${\sigma }_0=100{\mathrm{cm}}^{-1}$.

Figure 14

Two-photon absorption (TPIF in our notation, see Sec. 3b1) in a semiconductor quantum well: excited-state resonances within a continuum of delocalized states may be enhanced with frequency anticorrelations (black) or suppressed with positive frequency correlations (red). From [229].

Figure 15

(a) Experimental setup for the LOP protocol: By sending each beam through a Franson interferometer, the photon excitation may occur via either a delayed or an “early” photon. (b) Top panel: Two-dimensional fluorescence spectrum induced by classical light. Bottom panel: The same spectrum induced by entangled photons. From [215].

Figure 16

(a) Pulse sequence for LOP (top panel) and LAP (bottom panel) scanning protocols. (b) Loop diagram for the TPA process with indicated loop delays for the nonspecified phase cycling that depends on chronological time ordering between pulses 1 and 4. (c) Ladder diagrams for the TPA signal with selected phase-cycling components corresponding to the double-quantum-coherence term ${\mathbf{k}}_{\mathrm{I}}$: $-{\varphi }_d-{\varphi }_c+{\varphi }_b+{\varphi }_a$, rephasing ${\mathbf{k}}_{\mathrm{II}}$: $-{\varphi }_4+{\varphi }_2+{\varphi }_1-{\varphi }_3$, and nonrephasing ${\mathbf{k}}_{\mathrm{III}}$: $-{\varphi }_4+{\varphi }_2-{\varphi }_3+{\varphi }_1$. Both loop $s_j$ and ladder $t_j$ delays, $j=1$, 2, and 3 are indicated. The transformation between two is different for each diagram. The time translation invariance implies ${\omega }_1+{\omega }_2-{\omega }_3-{\omega }_4=0$.

Figure 17

Left column: $S_{\mathrm{LOP}}({\mathrm{\Omega }}_1,{\tau }_2=0,{\mathrm{\Omega }}_3)$ for the molecular dimer model of [215] calculated using classical light for (a) rephasing ${\mathbf{k}}_{\mathrm{II}}$ Eq. (123); (b) nonrephasing ${\mathbf{k}}_{\mathrm{III}}$ Eq. (124); and (c) the sum of the two ${\mathbf{k}}_{\mathrm{II}}+{\mathbf{k}}_{\mathrm{III}}$. Right column: The same for $S_{\mathrm{LAP}}({\tilde{\mathrm{\Omega }}}_1,t_2=0,{\tilde{\mathrm{\Omega }}}_3)$ Eqs. (119) and (120). Note that for the sums of rephasing and nonrephasing components for the LAP signal we flipped the rephasing component to obtain absorptive peaks as is typically done in the standard treatment of photon echo signals. The difference between the two columns stems from the display protocol and is unrelated to entanglement.

Figure 18

Left column: LOP signal $S_{{\mathbf{k}}_{\mathrm{II}}+{\mathbf{k}}_{\mathrm{III}}}({\mathrm{\Omega }}_1,{\tau }_2=0,{\mathrm{\Omega }}_3)$ for a molecular trimer with classical light Eqs. (123) and (124) (a), entangled light (125) with $T_e=100\mathrm{fs}$ (b). Right column: LAP signal $S_{{\mathbf{k}}_{\mathrm{II}}}({\tilde{\mathrm{\Omega }}}_1,t_2=0,{\tilde{\mathrm{\Omega }}}_3)$ using classical light Eq. (119) (c), entangled light with $T_e=100\mathrm{fs}$, Eq. (128) (d). Intraband dephasing ${\gamma }_{e{e}^{\prime }}=1\mathrm{meV}$. All other parameters are given in Sec. 5 of [65].

Figure 19

Diagrams representing the third-order contributions to Eq. (131).

Figure 20

(a) 2D signal $S_{\mathrm{DQC}}^{(\mathrm{LAP})}({\tilde{\mathrm{\Omega }}}_1,{\tilde{\mathrm{\Omega }}}_3)$ Eq. (162) (absolute value), showing correlation plots with different pump bandwidths $\sigma$ as indicated. The bottom panel is multiplied by 6. (b) Same as (a) but for the 2D signal $S_{\mathrm{DQC}}^{(\mathrm{LAP})}({\tilde{\mathrm{\Omega }}}_1,{\tilde{\mathrm{\Omega }}}_2)$ Eq. (163). Parameters for simulations are given by [219].

Figure 21

(a) Schematic of time-and-frequency-resolved photon coincidence measurement. (b) Loop diagram for the bare signal Eq. (e15) in a gated measurement. (c) Loop diagram for correlated two photon measurement Eq. (e26). Dashed lines represent the dynamics of the system driven by the field modes. ${\tau }_i$ and ${\tau }_s$ can be either positive or negative.

Figure 22

Linear absorption experiment of [119] with coincidence detection. (a) The entangled-photon pairs in beams $E_s$ and $E_r$ are split on a beam splitter. $E_s$ is employed as a probe transmitted through the sample, while $E_r$ is detected in coincidence.The absorption spectrum of the $E{r}^{3+}$ ion in a yttrium aluminum garnet (YAG) crystal around 650 nm obtained by single-photon counting (blue filled squares, left $Y$ axis) and coincidence counting (red open squares, right $Y$ axis) at various values of signal-to-noise ratios in the channel with the sample: (b) $1/2$ and (c) $1/30$.

Figure 23

(a) The IPP setup: the entangled-photon pair in beams $E_s$ and $E_r$ are split on a beam splitter. A classical, actinic, ultrafast pulse ${\mathcal{E}}_a$ excites the sample, and $E_s$ is employed as a probe in a pump-probe measurement, while $E_r$ is detected in coincidence. (b) The level scheme for the two-state jump (TSJ) model considered in this work. The $g-e$ transition is far off resonant from the spectral range of the entangled-photon wave packet, which couples only to the $e-f$ transition. (c) Diagram representing the pump-probe measurement. From [241].

Figure 24

(a) TPC signals with ${\sigma }_p=2000{\mathrm{cm}}^{-1}$ and $T=90\mathrm{fs}$, ${\omega }_r=10400{\mathrm{cm}}^{-1}$ (blue, dashed) and $11400{\mathrm{cm}}^{-1}$ (red, dot-dashed) as well as the classical pump-probe signal (black, solid) with $\sigma =1000{\mathrm{cm}}^{-1}$ are plotted vs the dispersed frequency $\omega$ with a time delay set to $t_0=3\mathrm{fs}$. (b) Same for $t_0=30$, (c) 60, and (d) 90 fs. (e)–(h) Same as (a)–(d), but with classical bandwidth $\sigma =100{\mathrm{cm}}^{-1}$. The classical signal is normalized, such that its maximum value at $t_0=0$ is equal to 1. Similarly, the TPC signals are normalized to the maximum value of the signal with ${\omega }_r=11400{\mathrm{cm}}^{-1}$ at $t_0=0$. From [241].

Figure 25

Top row: (a) FSRS level scheme for the tunneling model, (b) pulse configuration, and (c) loop diagrams (for diagram rules see Appendix pp1). (d), (e) The same as (b) and (c) but for IFSRS. The pairs of indices (0,1), etc., indicate the number of photons registered by detectors $s$ and $r$ in each photon counting signal $(N_s,N_r)$

Figure 26

(a) Time-frequency Wigner spectrogram for classical light, (b) the same as (a) but for an entangled twin state given by Eq. (205). The insets depict a 2D projection. (c) A window function for FSRS ${\mathcal{E}}_s^{*}(\omega ){\mathcal{E}}_s(\omega +i{\gamma }_a)$ (black), and IFSRS ${\mathrm{\Phi }}^{*}(\omega ,{\overline{\omega }}_r)\mathrm{\Phi }(\omega +i{\gamma }_a,{\overline{\omega }}_r)$ different values of $T_1$. (d) Spectrum of the eigenvalues ${\lambda }_n$ in the Schmidt decomposition (34) for entangled state with amplitude (205). Parameters for the simulations are listed in [66].

Figure 27

(a) Absorption for a time evolving vibrational mode $i$ vs $\omega -{\omega }_p$ for slow tunneling rate $k=18{\mathrm{cm}}^{-1}$ and narrow dephasing ${\gamma }_a=9{\mathrm{cm}}^{-1}$, (b) same as (a) but for fast tunneling rate $k=53{\mathrm{cm}}^{-1}$ and broad dephasing ${\gamma }_a=43{\mathrm{cm}}^{-1}$. (c), (d) The same as (a), (b) but for the classical FSRS signal. (e), (f) The same as (a), (b) but for $S_{\mathrm{IFSRS}}^{(1,1)}$. (g), (h) The same as (a), (b) but for $S_{\mathrm{IFSRS}}^{(2,1)}$ vs ${\overline{\omega }}_s-{\omega }_p$. Parameters for the simulations are listed in [66].

Figure 28

Left column: $S_{\mathrm{IFSRS}}^{(1,1)}$ signal vs ${\overline{\omega }}_s-{\omega }_p$ for (a) entangled state (205), (b) correlated, and (c) uncorrelated separable states. (d)–(f) The same as (a)–(c) but for the $S_{\mathrm{IFSRS}}^{(2,1)}$ signal. Parameters for the simulations are listed in [66].

Figure 29

Three-wave process involuting classical (solid line) and quantum (dashed line) modes. Black lines represent incoming fields, and red lines correspond to generated light.

Figure 30

Heterodyne-detected DFG: (a) phase-matching condition ${\mathbf{k}}_3={\mathbf{k}}_1-{\mathbf{k}}_2$, (b) molecular level scheme, and (c) loop diagrams.

Figure 31

Heterodyne-detected SFG: (a) phase-matching condition ${\mathbf{k}}_3={\mathbf{k}}_1+{\mathbf{k}}_2$, (b) level scheme, and (c) loop diagrams.

Figure 32

Three-wave process with two classical and one quantum mode: (a) phase-matching condition, (b) molecular level scheme, and (c) loop for the incoherent TPIF (C1) and coherent homodyne SFG (C2).

Figure 33

Three wave processes involuting two quantum and one classical mode: (a) phase-matching condition, (b) molecular level scheme, and (c) loop diagrams for the incoherent TPEF (C1) and coherent type-I PDC (C2).

Figure 34

Type-II PDC: (a) phase matching, (b) molecular level scheme, and (c) CTPL diagrams for the signal from ${\mathbf{k}}_3$ (C1) and ${\mathbf{k}}_2$ (C2) modes polarized along the $x$ direction.

Figure 35

(a) Schematic of the PDC experiment and (b) the three-level model system used in our simulations. (c) One out of four loop diagrams [the remaining three diagrams are presented by 62] for the PCC rate of signal and idler photons generated in type-I PDC [Eq. (265)]. The left and right diagrams represent a pair of molecules. Blue (red) arrows represent field-matter interaction with the sample (detectors). There are four possible permutations ($s/i$ and $s^{\prime }/i^{\prime }$) which leads to four terms when Eq. (267) is substituted into Eq. (266). (d) Absolute value of semiclassical susceptibility $|{\chi }_{+--}^{(2)}\mathbf{(}-({\omega }_p-{\omega }_i),-{\omega }_i,{\omega }_p\mathbf{)}|$ (arb. units) Eq. (268), and (e) the susceptibility $|{\chi }_{LL-}^{(2)}\mathbf{(}-({\omega }_p-{\omega }_i),-{\omega }_i,{\omega }_p\mathbf{)}|$ Eq. (267) vs pump ${\omega }_p$ and idler frequency ${\omega }_i$. We used the standard potassium titanyl phosphate (KTP) crystal parameters outlined in the text. (f) Real and (h) imaginary parts and (j) absolute value of ${\chi }_{+--}^{(2)}$ (red lines) and ${\chi }_{LL-}^{(2)}$ (blue lines) for off-resonant pump ${\omega }_p-{\omega }_{gf}=10{\gamma }_{gf}$. (g), (i), and (k) The same as (f), (h), and (j) but for resonant pump ${\omega }_p\simeq {\omega }_{gf}$.

Figure 36

Left column: Two-dimensional PCC rate (log scale in arb. units) calculated using quantum theory Eq. (175) assuming a single monochromatic pump with frequency ${\omega }_p$. (a) Idler detector resonant with intermediate level ${\overline{\omega }}_i={\omega }_{eg}=282\mathrm{THz}$, while (b) ${\omega }_{eg}-{\overline{\omega }}_i=2\mathrm{GHz}$. Right column: The same as the left but for a pump made out of two monochromatic beams with frequencies ${\omega }_p-{\omega }_p^{\prime }=2\times {10}^{-6}{\omega }_p$.

Figure 37

Top row: (a) Absorption (276) and (b) fluorescence (277) line shapes vs displaced frequency $\mathrm{\Delta }\overline{\omega }=\omega -\frac{1}{2}({\omega }_a+{\omega }_b)$. Frequency dispersed time-resolved fluorescence (284) displayed as a snapshot spectra for molecule (c) ($a$) and (d) ($b$). Bottom row: (e) PCC for different transition energies of molecules excited at ${\omega }_p={\omega }_b^0+{\lambda }_b$, (f) PCC for different values of the SD time scale ${\mathrm{\Lambda }}_a$ and ${\mathrm{\Lambda }}_b$ and fixed linewidth ${\mathrm{\Gamma }}_a$, ${\mathrm{\Gamma }}_b$ according to Eq. (275), PCC for different frequency gate bandwidths at (g) fixed time gate bandwidth ${\sigma }_T=100\mathrm{MHz}$ and for (h) different time gate bandwidths at fixed frequency gate bandwidth ${\sigma }_{\omega }=100\mathrm{MHz}$. Molecules have distinct SD time scales ${\mathrm{\Lambda }}_a=15\mathrm{MHz}$, ${\mathrm{\Lambda }}_b=1\mathrm{MHz}$, and ${\omega }_b^0-{\omega }_a^0=1\mathrm{MHz}$.

Figure 38

Time-and-frequency-resolved measurement of PCC with spectral diffusion. (a) Schematic of the PCC experiment with two indistinguishable source molecules, and (b) the two-level model of the molecule with SD used in our simulations. (c) Loop diagrams for the PCC rate of emitted photons from two molecules (for diagram rules see Appendix pp1). The left and right branches of each diagram represent interactions with ket and bra of the density matrix, respectively. Field-matter interactions with the pump pulses $p_1$ and $p_2$ (blue), spontaneously emitted $s$, $s^{\prime }$, $r$, and $r^{\prime }$ photons (red) and detectors (brown).

Figure 39

Diagram construction in a quantum electrodynamics (QED) formulation: The vertical blue lines indicate the evolution of the matter density matrix. An arrow pointing toward (away from) it corresponds to a matter excitation $V^{†}(\tau )$ [deexcitation $V(\tau )$], which is accompanied by a photon annihilation $E(\tau )$ [creation $E^{†}(\tau )$]. In the Liouville space formulation, interactions on the left side represent left superoperators (1), and interactions on the right side represent right superoperators (2).

Figure 40

Loop diagram construction in a QED formulation: Using the diagram rules given in the text, the diagram translates into the Heisenberg picture expression (top) or the Schrödinger picture expression (bottom). The corresponding field correlation functions are reordered as in the loop diagram.

Figure 41

Ladder diagram construction in a QED formulation: Using the diagram rules given in the text, the diagram translates into the Heisenberg picture expression (top) or the Schrödinger picture expression (bottom). The corresponding field correlation functions are reordered as in the loop diagram.