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# Multidimensional photon correlation spectroscopy of cavity polaritons

Contributed by Shaul Mukamel, December 27, 2017 (sent for review November 7, 2017; reviewed by Steven Cundiff and Angel Rubio)

## Significance

We propose a spectroscopic technique that can track the time-dependent state of a dressed molecule in an optical cavity (polariton) by measuring coincidence of two emitted photons. The proposed technique offers an independent control of the spectral and temporal resolution; single-photon detection allows for low-intensity measurements, which do not disturb the state of the cavity field, and time-dependent atom/cavity coupling provides a control tool. Tracking the evolution of the polariton states with time-dependent atom/cavity coupling should be of interest in photochemistry and photobiology and could improve fundamental understanding of many physical processes in strongly coupled atom/radiation states. Possible applications include chemical sensors and quantum information processing.

## Abstract

The strong coupling of atoms and molecules to radiation field modes in optical cavities creates dressed matter/field states known as polaritons with controllable dynamical and energy transfer properties. We propose a multidimensional optical spectroscopy technique for monitoring polariton dynamics. The response of a two-level atom to the time-dependent coupling to a single-cavity mode is monitored through time-and-frequency–resolved single-photon coincidence measurements of spontaneous emission. Polariton population and coherence dynamics and its variation with cavity photon number and controlled by gating parameters are predicted by solving the Jaynes–Cummings model.

Cavity quantum electrodynamics (QED) provides a powerful tool for studying quantum effects in matter (1⇓⇓⇓⇓–6). Due to strong coupling between electronic and nuclear degrees of freedom, molecular systems can undergo nonadiabatic dynamics, which is hard to detect. The nonadiabatic dynamics can be manipulated (7) when a molecule is coupled to a localized cavity mode. Earlier studies in atomic systems showed that the cavity photons can enhance cooperative signals, such as superradiance and subradiance (8, 9). The description of these phenomena is based on the joint photon–matter states known as polaritons (10, 11). Cavity polaritons have been applied to trapping and cooling of atoms (12) and prescribe a new recipe for cooling molecules (13, 14). Cavity effects can provide a tool to probe larger molecules (15, 16), where various many-body quantum effects play an important role. For instance, the polariton–polariton interaction strength can be directly probed in a high-quality microcavity (17). Strong coupling in cavity QED has been recently demonstrated for organic molecules (18⇓⇓–21) and photosynthetic light harvesting (22). Polaritons have been further investigated in chromophore aggregates (11) arising from electronic transitions (10) as well as from vibrational transitions (23), which in turn, allow the manipulation of chemical reaction rates and outcomes (24⇓⇓–27).

Cavity polariton dynamics can be investigated by nonlinear spectroscopy. IR and Raman spectroscopies have been recently used to show the enhancement of the spectra of vibrational polaritons in molecular aggregates (11, 28⇓⇓–31). More elaborate two-dimensional spectroscopic measurements have further provided experimental demonstration for multiexciton correlation effects (32). Coherent multidimensional spectroscopy can reveal correlations of matter dynamics during several time intervals controlled by sequences of short pulses to reveal material information (33, 34) by a coherent measurement of a signal optical field. Such correlation plots carry qualitatively higher levels of information than single-interval (1D) techniques. A recent theoretical study of vibrational polaritons using coherent 2D IR spectroscopy (35) has been reported.

In this paper, we propose to study polariton dynamics using a different class of incoherent multidimensional signals. Unlike coherent multidimensional techniques, which is based on carefully timed laser pulse sequences, incoherent techniques detect spontaneously emitted light, and the control knobs of such signals are based on single-photon gated detection. Time-and-frequency (TF)–gated

Analytical solutions for the time evolution of the population inversion for the time-dependent JC model can be derived for specific time profiles of the system–cavity coupling (41, 42, 45). The gated number of photons spontaneously emitted into noncavity modes and photon coincidence signals for this analytically solvable time-dependent JC model are given by two-point and four-point correlation functions of the dipole operator, respectively. We shall use a compact time ordered superoperator formalism in these calculations (46⇓⇓–49). These correlation functions and the corresponding Green’s functions that describe dynamics in the dressed field–matter (polariton) space will be calculated. The proposed signals provide a unique observation window for the population and coherence polariton dynamics.

The spontaneously emitted photons will be detected by a TF–resolved photon gating obtained by consecutive spectral and temporal gates with bandwidths

## Evolution of Polaritons in Time-Modulated Cavities

The two-level molecular system can be described by the Pauli matrices, which satisfy **1** *SI Text*). Our approach, however, is general and can treat an arbitrary time profile.

In addition to the strong coupling to the single-longitudinal cavity mode, the atom is weakly coupled to the vacuum modes described by *SI Text*. We use perturbation theory in

## Gated Photon Counting Signals

### Atomic Inversion.

We first present the atomic inversion (population difference of the two-level system) **S51**. Assuming that, initially, the atom is in the excited state (

### Gated Photon Counting Signals.

In the following, we denote a general *N*th-order correlation measurement as photon counting. Gated photon number and gated photon coincidence correspond to **S32**), we obtain (for the TF-resolved photon number signal)**S45** and S48 are given in *SI Text*.

## Results and Discussion

Using Eqs. **4** and **5**, we have simulated the photon number **2** is shown in Fig. 1 for the parameters given in *Materials and Methods*. Fig. 2 depicts the time–frequency dependence of the TF gated photon number. Consider first a fast variation *A* now shows a dominant peak at *B*. The side peaks signify the coherence origin of the photon coming from the superposition of the dressed atom–cavity states, which can only be observed during the time modulation of the coupling.

The photon coincidence signals are shown in Fig. 3 *A*–*D*. For rapid coupling variation corresponding to Fig. 2*A*, the coincidence counting depicted in Fig. 3*A* contains side peaks at *B*. At *C* is similar to Fig. 3*A*. Finally, for *D* shows a full grid of well-pronounced resonances with three prominent peaks for

We next compare the commonly used TF gated results of Figs. 2 *A* and *B* and 3 *A–D* with the simpler PS gating (50, 51), which uses the gating function *C* has a single dominant peak at *D*, the stationary PS gate might have higher spectral resolution, allowing us to visualize the strong side peaks at

Turning to coincidence counting, we note that the PS Eq. **5** has a low temporal and spectral resolution and misses some spectral features. For instance, the JC ladder of various peaks cannot be observed in Fig. 3 *E* and *G* for *F*, with much less contrast compared with the case of TF. Finally, for *H* vs. well-resolved peaks shown in Fig. 3*D* for TF. Therefore, the TF coincidence counting signal shows how the response to the time modulation of the coupling changes the configuration of the JC states participating in the photon emission for a given cavity photon number, which is missed by the PS. The resonances that manifest as cross-peaks in Fig. 3 *A*, *C*, and *D* mark the role of coherence between different dressed atom–cavity states that are governed by the coupling temporal profile, since the pair of photons can be generated from the superposition of two JC ladder states.

## Note on the Joint Temporal/Spectral Resolution

We identify two main differences between the PS and TF signals. In PS, both frequency and time resolutions are determined by a single parameter **S59** are independent, so that each field operator is gated separately. In contrast, the TF signal in Eq. **S60** depends on two gating parameters **S60**. To examine how these parameters determine the temporal and spectral resolutions of the signal, we performed an asymptotic expansion of Eq. **7**, such that, in zeroth order, one can approximate it at **S63**. This contains terms of the form (details are in *SI Text*) **S33**. To illustrate the role of the time gate, we can consider a high spectral resolution, so that

In summary, we have demonstrated how multidimensional photon counting can be used to reveal polariton dynamics in a cavity. The TF gated photon number and photon coincidence detection can capture subtle time-evolving features, such as dressed JC ladder polariton states and their correlations via cross-peaks in the 2D photon coincidence spectra. Note that, in the strong dissipation limit, the oscillations shown in Fig. 1 and contributing to the polariton dynamics would be significantly damped. This means that, for systems with the dynamics of the timescale similar to the cavity damping rate, one has to take into account the cavity leakage when evaluating the signals. Recent experiments in light harvesting molecules placed in a microcavity (22) observe large amounts of scattering into the microcavity system; however, the observed damping is not strong enough to destroy the strong coupling. The technique described in this paper can be useful for laser stabilization, where the cavity photon distribution and gain medium dynamics are monitored simultaneously. In this scenario, cavity photon statistics acts as an input into the polariton configuration captured by the photon counting signals, which consequently monitors the stability of the cavity radiation. The time-dependent coupling offers a versatile coherent control tool. Similarly to the pulse shaping technique used in ordinary spectroscopy, one can optimize the coupling temporal profile in a generic algorithm setup to optimize the control. To implement this control scheme, one has to match the coupling duration and profile to the timescale of the nuclear motion. In addition to the suppression of the dephasing and changing of the dynamical rates that are observed in the systems with the stationary cavity coupling, time-dependent coupling provides a unique control mechanism for tracking the polariton–polariton and other many-body correlation effects via optimizing the corresponding coherences visualized as the cross-peaks in 2D spectra in photon coincidence signals.

## Materials and Methods

The Hamiltonian [**1**] can be recast as **S2**, while **S3**. For the one-photon JC ladder, *SI Text*. The subspace in which **6** can be solved analytically for the sech function coupling **6** subject to initial conditions above is given by a simplified hypergeometric function

The matter correlation functions in Eqs. **4** and **5** are traced over the noncavity modes. Assuming initial wave function **S51**.

To get the photon counting signal, we have calculated two-point [**4**] and four-point [**5**] correlation functions:**7**], we have made use of Eqs. **8**–**10** to calculate the gated emission [**4**] and the coincidence counting [**5**]. Note that Eqs. **8**–**10** reduce to the simple Rabi oscillations **S57** and S58, respectively. Additional details are summarized in *SI Text*.

The simulations shown in Figs. 1–3 use typical parameters related to vibrational spectroscopy: vibrational frequency ^{−1}, coupling modulation ^{−1}, and centered at

## Acknowledgments

K.E.D. is supported by the Zijiang Endowed Young Scholar Fund and Overseas Expertise Introduction Project for Discipline Innovation (‘‘111 Project,’’ B12024). S.M. is supported by National Science Foundation Grant CHE-1663822 and Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy Award DE-FG02-04ER15571.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: dorfmank{at}lps.ecnu.edu.cn or smukamel{at}uci.edu.

Author contributions: K.E.D. and S.M. designed research; K.E.D. performed calculations and analyzed the data; and K.E.D. and S.M. wrote the paper.

Reviewers: S.C., University of Michigan; and A.R., Max Planck Institute for the Structure & Dynamics.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1719443115/-/DCSupplemental.

Published under the PNAS license.

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