Hong-Ou-Mandel interferometry and spectroscopy using entangled photons

Optical interferometry has been a long-standing setup for characterization of quantum states of light. Both the linear and the nonlinear interferences can provide information about the light statistics an underlying detail of the light-matter interactions. Here we demonstrate how interferometric detection of nonlinear spectroscopic signals may be used to improve the measurement accuracy of matter susceptibilities. Light-matter interactions change the photon statistics of quantum light, which are encoded in the field correlation functions. Application is made to the Hong-Ou-Mandel two-photon interferometer that reveals entanglement-enhanced resolution that can be achieved with existing optical technology.


Introduction
Quantum states of light provide an exciting platform for observing and controlling matter beyond what is possible classically [1][2][3][4]. Quantum states are very sensitive to the external environment which makes them useful probes of matter. Quantum features of light have long been used in metrology and quantum information while lately there has been a growing activity in utilizing them in spectroscopic applications. Interferometry offers a robust detection schemes of quantum light. In this paper we present a novel spectroscopy based on the Hong-Ou-Mandel (HOM) [5] two-photon interferometric setup. Observables measured by the interference of two waves depend on two times separated by a delay which can be controlled by the propagation path difference of the mixed waves. The unified picture of second-order and fourth-order interferences in a single interferometer have been demonstrated in [6]. Previous works attempted either to utilize HOM-like measurements to address properties of the beam splitters or the individual emitters [7] or utilize four-wave mixing for the quantum light generation [8] and characterization [9].
Interferometric signals can be recast in terms of moments of the field operators posterior to the interaction with the matter, thereby revealing its statistics. The signal-field operator, (see Methods and Supplementary Note 1) is given by [10] ̂( +) ( , ) = −̂( , ), (1) Where is the signal mode frequency and ̂ is the matter dipole operator. The interferometric setup naturally gives rise to two characteristic timescales and respective length scales. First, the response interval in which light-matter interaction occurs. It is determined by the pulse envelope, the spatial dimension of the sample and the response time. Second is the pulse relative delay interval determined by the interference region governed by the interferometer dimensions. Here, we consider the field-matter interaction region to be localized compared to the spatial dimensions of the interferometer. We further consider a sequence of ultrafast coherent excitation pulses -which are classical for all practical purposes -followed by an interaction with the quantum state of light. The interferometer operation mode is shown in Fig. 1: when the pulse interval and the maximal response interval of the sample cτ R , are smaller than the free propagation distance between the sample and the interferencedetection location ( * ≪ ) where * ≔ max{ , } -the measured response functions are classical. The response in this regime is highly localized temporally and immediately after the pulse interacts with the sample, the matter degrees of freedom can be traced out. The excitation and deexcitation period is dominated by the duration of the narrowband envelope of the quantum field given by combined with the pulse delay interval . The maximal duration of this interval is defined by * which is smaller than a few hundred micrometers even for a picosecond pulse which is well within the narrowband region. For example, for a transform limited Gaussian pulse with central wavelength = 1064 and pulse duration Δ = 2 which occupies ≈ 600 . Finally, the interferometer length scale denoted specifies the free propagation distance between the incident beams, the beam-splitter and the detector.
Typical interferometer length is in the order of few centimeters which justifies the separation of timescales -considering the interaction interval to be localized around the sample compared with . Moreover, the Rayleigh distance is typically a few meters in this setup, thus one can consider the propagation as unidirectional for all practical purposes. For a femtosecond pulse In the present work we combined the interferometric detection (HOM) with wave mixing that involves both classical and quantum light beams to address more complex nonlinear optical processes and the corresponding components of the nonclassical response function. We investigate how the quantum state of light and its statistics are modified by interaction with matter. In particular, we address the following two issues of the quantum nonlinear interferometric spectroscopy. The first issue is regarding the nature of the change of the quantum state and its statistics. The second point investigates the details of the matter information that can be deduced from the change in the statistics of the field. These questions are explored by using an interferometric setup traditionally used to study quantum states of light and now applied for investigation of the matter degrees of freedom via extraction of the matter response functions. We therefore focus on accuracy enhancement of such responses and their deviations from classical susceptibilities.

Results and Discussion
The proposed experiment combines several conventionally used optical techniques such as fourwave mixing, beam splitting, and Hong-Ou-Mandel (HOM) interferometer, and three-photon absorption spectroscopy. In the following sections we present each technique independently highlighting the main principle and the underlying theoretical model that will be used to describe each part of the setup. We finally combine both techniques in the setup shown in Fig. 2a and discuss the resulting HOM spectroscopic measurements. In the experimental setup the three classical light beams generated are combined with the quantum light pulse produced by the parametric down conversion (independently from the classical three pulses). The corresponding level -scheme is shown in Fig. 2b. The main goal of the proposed measurement is to use interferometric (HOM-like) detection to investigate the (3) nonlinear susceptibility that combines an absorption of the three classical fields and the transmission of one quantum field shown in Fig. 2c. The corresponding Feynman diagram and HOM signal are shown in Fig. 2d and e, respectively. Unlike the Kerr process which requires high intensity laser pulses to produce third-order nonlinear response, here we deal with resonant absorption of each of the light beams participating in the four-wave mixing. This nonlinear process is the main focus of our study.
The general third order nonlinear optical process generates various signals that are well studied in where is the wavevector of the entangled photon containing the 2 3 phase matching directions [11]. Note that The Hong-Ou-Mandel interferometric signal. We next turn to the HOM two-photon interferometer in the presence of matter. The electric field is transformed by the relatively displaced beam-splitter (BS) depicted in Fig. 4 according to where the linear phase results in the ± relative time delay, corresponding to the ± displacement of the BS. √ and √ are the transmission and reflection coefficients. We focus on the photon coincidence signal depicted in Fig. 1 given by a joint probability to detect one photon in and one photon in separated by delay given by where is a normalization factor. We employ the superoperator notation, = and = , the superoperator ± represents an anti/commutator ± = ± . Note that the superoperator time ordering Ƭ , which is an operator in Liouville space is different from the standard Glauber's normally ordered operators [19]. The plus-minus and the left-right superoperators are linked by a linear transformation. Below we focus on a narrowband pump. Extension to a broadband pump is outlined in Supplementary Note 2.
The narrowband HOM spectrometer. In their seminal paper, HOM have used the narrowband wavefunction (see Eq. (16) and Methods). Following this procedure where in path ′ ′ (top branch of the interferometer) a sample composed of many molecules is placed, and the signal is given by the four-wave mixing setup depicted in Fig. 3.
We focus on the * ≪ regime, where the spatial extent of the photon wavepacket after the interaction is small compared to the dimensions of the interferometer. Calculating the coincidence count according to Eq. (5) using Eqs. (2) and (15) we obtain, Here the convoluted response is given by the functional ( ) = ( ) * (3) 3 are the frequencies of the three classical waves. The pre-factor containing the central frequency and the setup geometry is given by 0 = −1/2 ( ħ 0 2 ) 2 | (∆ )| 2 . In the absence of matter, ( ) = ( ) and we recover the HOM interference [5] signal. In that case the extra phase factor that appears in the second term in Eq (6) can be shifted at the frequency integration by → + /2 . When the material sample is added, a reference frequency is set. This can be compensated by equivalently translated matter response. When the coincidence counting is not temporally gated, we obtain the signal by integration over , where 0 = 0 ( 2 + 2 ) ∫ | ( )| 2 , and = 0 . For large BS displacement , the overlap term -the second term in the R.H.S. of Eq. (7) -vanishes due to diminishing correlations of the relatively shifted response. It assumes a Wigner-function form in the { , } space. As is reduced, the overlap term increases, introducing the hallmark dip in the HOM interference pattern.
A material sample added in one of the pathways affects the overlap term. Matter information is revealed in Eq. (7) by the variation of the HOM dip with the convoluted response ( ) Wignerfunction as illustrated in Fig. 2e. Note, that the response function ( ) is calculated using three classical fields followed by a single photon field in the last interaction. While it is not unusual that quantum enhanced performance is dramatically eroded by the loss of a single photon, this is not the case here. Several noise sources can be considered such as losses associated with single photon sources, non phase matched contributions, and imperfect transmission and detection efficiency.
First, the proposed setup in the single photon regime allows overcoming the noise because of the photon correlation measurement. The classical incoming fields contain a large number of photons and are thus insensitive to losses compared to other classical technique. When the single photon contribution has losses, the signal vanishes due to violation of the phase-matching due to momentum conservation. Second, the improved performance is attributed to the non-classical correlations between the photon pair, not from their Fock-state characteristics. Contributions to such losses can originate from non-phase matched signals, like spontaneous emission adding vacuum fluctuations to the transmitted beams. Third, losses occurring after the mixing can be modeled by a beam splitter with an empty port [27]. Recent interferometry. Experiments in the four-wave mixing setup performed in a multiphoton regime [28] indicate the reduction of quantum correlations is proportional to the square of the transmission ratios of the light beam intensities that characterize such losses. Single photon experiments have substantially lower transmission ratios and are therefore robust against such losses. Finally, it has been shown that similar biphoton spectroscopy measurements are robust against the external noise at the detection stage such as background thermal radiation [29], even under the signal-to-noise ratio reaches 1/30. (3) . We now turn to the setup depicted in Fig. 2a. The. molecule is modelled by a four level system { , 1 , 2 , } with transition energies 1 = 1 − = 3 , 2 1 = 2 , 2 = 1 . It undergoes three interactions with classical light pulses with controlled delays, the fourth-interaction is taken to be one photon produced by a parametric down-converted shown in Fig. 2a. The signal with phase matching at s = a + b + c can be generated in the four electronic state system. One can surely select another phase matching direction, out of the eight possible ones. Each direction contains a different type of material information governed by the set of pathways containing in the corresponding (3) nonlinear susceptibility. The third order nonlinear susceptibility is calculated using perturbative field-matter interactions arranged in the diagram in Fig. 2d. Following the general approach outlined in chapter six of Ref. [12] we obtain:

Interferometric detection of
where = + + implies energy conservation and are transition dipole moments.
The nonlinear susceptibility contains one-, two-, and three-photon resonances determined by the transition frequencies and dephasing rates (linewidth) , , = , 1 , 2 , . Interaction of quantum systems changes their state in the course of light-matter interaction at a single-photon regime [23][24][25]. Each interaction enhances the correlation, and the system becomes more inseparable. This may be employed in novel quantum spectroscopic setups, which extract matter information from optical probes. While single-photon states can be easily described in the photon number (Fock) basis, they are less suitable for probing phase-shifts due to number-phase uncertainty [13,14]. Multiphoton (entangled) states provide a richer playground for improving the temporal resolution imprinted by matter on the optical probe and is the focus of our study.
Multimode squeezed states [15] may be useful since the number-phase uncertainty can be further tuned in order to reach a desired joint frequency-time resolution.

Methods
Optical signals description. We start with the joint light-matter Hamiltonian, ̂=̂+̂+̂, (9) where , represent the matter and electromagnetic field respectively, and ̂ is their coupling.
The electric field operator is partitioned into positive and negative frequency components,

Additional information
Supplementary information is available for this paper at     After integration we finally obtain the field operator, One can further neglect the vacuum background contributions ̂( −∞).

Supplementary Note 2. * ≫ -generalized susceptibilities
Nonlinear spectroscopy with quantum states of light can be roughly divided into two levels.
On the first level which was considered so far, the matter degree of freedom is characterized by a set of causal response functions (susceptibilities) which are the result of consecutive interactions followed by a single measurement [4,11]. In this level the observable is classical, and it is measured with a unique probe that in some cases allows higher accuracy due to quantum enhancements [12]. where the (reduced) single photon density matrix after tracing the ′ ′ photon out is given by, The  [4,11] with one important difference, the time difference between measured interactions is entirely controlled by the experimentalist. In contrast, spontaneous signals require integration over the interaction time.
This signal scrambles the time ordering and gives rise to novel quantum observables, thus providing novel matter observables denoted as generalized susceptibilities. These go beyond the present study.

Supplementary Note 3. The broadband HOM spectrometer
When a broadband pump is used, performing Schmidt decomposition to the time-energy entangled photons is particularly useful.

Schmidt decomposition of broadband pump
In the case of an ultrashort pump, it is convenient to present the photon-pair using the Schmidt decomposition, whereby the amplitude takes the form [13,14], The single-photon amplitudes obey the coupled integral equations [9], where the kernels are given by tracing over the counter photon, be used below.

The broadband HOM signal
We now examine the ultrafast variant of this experiment. Using Eq.
Suppose one is interested in a pulse envelope given by ( ) which is not achievable experimentally. Since the weights of each mode is known, a simple reweighting can achieve a desired pulse envelope. Multiplying Eq. (S13) from the left and right by ( ′) and integrating w.r.t. ′ yields ( ) = ∑ ( ) where = ∫ ′ * ( ′ ) ( ′ ) . Summing the postmeasurement results mode by mode with the corresponding weights results in the desired ( ). One example of such resummation is by assuming the weights = * ( ) which can be used to scan the signal using a tunable narrowband profile with variable frequency .
When infinite number of modes are available, the pulse envelope converges to delta distribution ( ) → ( − ).

Supplementary Note 4. Mach-Zehnder Interferometric Spectroscopy
We now consider the setup of Fig. S1 in which the incoming photons (not necessarily entangled in this scenario) first interfere, here both photons interact with the matter. We assume that only the second BS in Fig. S1 in translated by ± , posterior to the interaction with the matter. It is straightforward to generalize the resulting expressions to include additional shift in the first BS. We monitor photon counting difference as a function of the shift parameter . The input-output relation for the field operators after the first and second BS is given with a single-and double prime notations respectively,  where √2 ( ) =̂′ (−) ( )̂( ) +̂′ (+) ( )̂ †( ) and √2̂′ (+) ( ) =̂( +) ( ) −̂( +) ( ) is the combined field that is coupled to the matter. Eq(S15) bears strong resemblance to the one introduced in [11] for = 0. However, there are two fundamental differences. First, there is a phase difference between the field in the definition of the signal and the one in the coupling Hamiltonian. Second, Eq. (S15) depends on the additional controlled time difference .
In this setup, it would be useful to employ either a multimode squeezed state or entangled pair initial state of the probe. Both are two-photon states and provide a rich optimization playground for quantum enhancement of metrology applications. Eq. (S15) essentially measures phase difference between the two incoming beams. The number-phase uncertainty may be exploited to demonstrate the measurement accuracy as a function of controlled variables such as the squeezing parameters or entanglement time.