1 Light emission from strongly driven many -body systems Andrea Pizzi12 Alexey Gorlach3 Nicholas Rivera14 Andreas Nunnenkamp5 and Ido Kaminer3

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Light emission from strongly driven many-body systems

Andrea Pizzi1,2†, Alexey Gorlach3†, Nicholas Rivera1,4, Andreas Nunnenkamp5, and Ido Kaminer3

1Department of Physics, Harvard University, Cambridge 02138, Massachusetts, USA

2Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom

3Solid State Institute, Technion–Israel Institute of Technology, Haifa 32000, Israel

4Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge 02139, Massachusetts, USA

5Faculty of Physics, University of Vienna, Boltzmanngasse 5,1090 Vienna, Austria

Corresponding authors: I.K. (kaminer@technion.ac.il) and N.R. (nrivera@mit.edu)

† equal contribution

Strongly driven systems of emitters offer an attractive source of light over broad spectral

ranges up to the X-ray region. A key limitation of these systems is that the light they emit is

for the most part classical. We challenge this paradigm by building a quantum-optical theory

of strongly driven many-body systems, showing that the presence of correlations among the

emitters creates emission of nonclassical many-photon states of light. We consider the

example of high-harmonic generation (HHG), by which a strongly driven system emits

photons at integer multiples of the drive frequency. In the conventional case of uncorrelated

emitters, the harmonics are in an almost perfectly multi-mode coherent state lacking any

correlation between harmonics. By contrast, a correlation of the emitters prior to the strong

drive is converted onto nonclassical features of the output light, including doubly-peaked

photon statistics, ring-shaped Wigner functions, and quantum correlations between

harmonics. We propose schemes for implementing these concepts – creating the correlations

between emitters via an interaction between them or their joint interaction with the

background electromagnetic field (as in superradiance). By tuning the time at which these

processes are interrupted by the strong drive, one can control the amount of correlations

between the emitters, and correspondingly the deviation of the emitted light from a classical

state. Our work paves the way towards the engineering of novel many-photon states of light

over a broadband spectrum of frequencies, and suggests HHG as a diagnostic tool for

characterizing correlations in many-body systems with attosecond temporal resolution.

Introduction

The creation and control of many-photon quantum states of light are important problems

with applications across the natural sciences. Realizations of squeezed quantum light states open

new avenues in spectroscopy and metrology, providing novel information on samples [1] and

enabling highly sensitive measurements beyond classical noise limits (e.g. in the detection of

gravitational waves [2,3]). At the same time, encoding quantum information on the quantum state

of light facilitates applications in quantum computing, simulation, and communication [4].

Several pioneering investigations have demonstrated a range of many-photon quantum states of

light, such as squeezed light [2,3,5–7], bright squeezed vacuum [8–11], displaced Fock

states [12], Schrodinger kitten [13,14] and cat states [15,16], subtracted squeezed states [17],

and others [18]. Many of the established techniques for generating quantum light at optical

frequencies rely on materials with a nonlinear optical response. Such nonlinear materials can be

typically described using a “perturbative” nonlinear response, where the induced polarization is

for example quadratic or cubic in the applied electric field.

At the other extreme of nonlinear optics are “non-perturbative” or “strong-field” effects

like high-harmonic generation (HHG), in which a very intense optical pulse creates radiation at

very high frequencies, even beyond hundredfold the frequency of the drive [19,20]. As such, HHG

is an attractive source of ultra-short pulses of high-frequency light. The potential of HHG for the

generation of nonclassical high-frequency light has, however, remained largely unexpressed. In

fact, many past semi-classical approaches [21–23] and more recent fully quantized ones [14,24]

have established that the output harmonics in HHG are in an almost precisely coherent (thus,

classical) state (apart from the notable exception of post-selected cat states in the driving

frequency [14]). Nonetheless, these works all focus on the scenario of uncorrelated emitters,

leaving open important questions about many-body aspects underlying HHG. In particular, to what

extent do correlations between the emitters affect the state of light created in the HHG process?

In this work, we develop a quantum-optical theory of light emission by strongly driven

many-body systems. We use this theory to show that many-body correlations in the emitters can

render the output radiation strongly nonclassical (Figure 1). To demonstrate this concept, we show

that HHG from a correlated many-body state of emitters features exotic photonic states, for

instance, super-Poissonian and doubly peaked photon number statistics, ring-shaped Wigner

functions, and strong correlations between harmonics. These features strongly contrast with

conventional HHG from uncorrelated emitters, in which the output harmonics are described by

almost-perfectly Poissonian photon statistics, Gaussian Wigner function, and uncorrelated

harmonics.

Indeed, the quantum state of the emitted light can be shaped by creating different

correlations among the emitters. We show this general idea by investigating two concrete scenarios,

one in which correlations among the emitters are induced through collective superradiant emission,

and one in which they arise from dipole-dipole type interactions. Our study makes the first step

towards the creation of bright high-frequency light with engineered quantum properties. Applying

this concept in reverse, characterizing the quantum photonic state of the emitted light will enable

to infer the many-body correlations of the material with high-temporal-resolution.

Figure 1. Quantum theory of light emission by strongly driven many-body atomic systems.

(a) HHG can be understood as a single-particle, strong-field three-step process: (i) an intense drive

laser tears off an electron from the atom, (ii) the electron is accelerated by the electric field, (iii)

the electron recombines with the atom, converting its energy into an energetic photon at higher

harmonics. The spectrum features are peaks at the odd harmonics, a characteristic plateau and a

cutoff. (b) The many-body correlations among the atoms can arise from spontaneous collective

emission (superradiance) or interatomic interactions. (c) Our theory marries the description of

many-body effects with that of strong-field physics, giving access to unique quantum properties

of the emitted light.

Results

We develop a quantum-optical theory that describes the interaction among an intense

driving field, a quantum-correlated (many-body) atomic system, and the quantized radiation

emitted from it. For a given initial atomic condition, our theory produces a full portrait of the

emitted quantum light, including the Wigner function, photon number statistics, and correlation

between different harmonics. First, we consider the standard scenario in which the atoms are

initially in their ground state and show that the emission resulting from the strong drive is in this

case essentially coherent: the Wigner function and photon statistics of the harmonics are almost

Gaussian and Poissonian, respectively, and the different harmonics are uncorrelated. Second, we

investigate the effect of strong atomic correlations on the output radiation, showing that, when the

emitters are initially in a correlated (thus, non-separable) many-body state, the output radiation

becomes nonclassical, featuring non-Gaussian Wigner functions, non-Poissonian photon statistics,

and correlated harmonic pairs. Third, we propose two experimentally relevant schemes, exploiting

inter-atomic interactions and superradiance, respectively, to induce correlations in the atomic

initial condition, resulting in controllably nonclassical output radiation. An expanded and detailed

version of the following theory is presented in the Supplementary Information (SI). In the

following, Hartree atomic units (, and ) are used unless otherwise

specified.

1 Quantum theory of strongly driven many-body systems

We consider  microscopic emitters (henceforth referred to as ‘atoms’ for convenience)

interacting with a strong driving field. The Hamiltonian describing the system reads





















, (1)

where  and 

 are the position and dipole moment of the -th atom, respectively, 





 is the free-field Hamiltonian, with 

 and  the creation and annihilation

operators of a photon with wavevector  and polarization , respectively, and 



 is the single

particle Hamiltonian describing the outermost electron of the th atom. The eigenvalue problem





 is solved via standard space discretization procedures, resulting in a discrete

single-particle spectrum composed of hundreds of levels, most of which in the continuum, 

 (see Supplementary Section S1).

The atomic system is driven with a laser whose field we take as a multimode coherent state



, where  is the parameter of a coherent state with frequency .

Using a unitary transformation (generated by a displacement operator) [24–28], the electric field



 can be separated into a classical part 

 and a quantum

part 

 , where  

,  is the polarization vector of the

mode , and  is the volume. The term 

 represents quantum fluctuations of the electric field.

We are interested in describing emission in vacuum, for which  and the modes are

continuous. Under the action of the displacement operator, the initial photonic state changes from

 to the vacuum with zero photons , and the photonic state only describes the radiation

on top of the drive laser. Assuming the electric field 

 to be polarized the x direction, the light-

matter coupling term in Eq. (1) can be rewritten as 





. In its single-particle version with

 atoms, the above model is known to capture the main features of HHG, including the

characteristic plateau and cutoff of the emission spectrum [21,22], see Figure 1b. Here, we wish

to investigate the effects that many-body atomic correlations for  have on the emitted output

light. Due to the exponential size (in ) of the Hilbert space, such a many-body problem is in

general a formidable one. We make it tractable by considering the simplest yet far from trivial

scenario, one in which the spatial arrangement of the atoms is negligible. In this case, the state of

the system can remain symmetric, meaning invariant under any permutation of the atoms, thus

making the atoms effectively indistinguishable. While the assumption of indistinguishable atoms

is physically justified in some cases, e.g., when the atoms are in a small volume with respect to the

involved interaction ranges and wavelengths, it can qualitatively describe multiple many-body

phenomena even when not fully justified, similarly to what happens for instance for

superradiance [29,30], spin squeezing [31], and countless equilibrium phase transitions [32,33].

Indeed, this assumption opens the way to much analytical progress, which brings the ultimate

numerical simulation of the system into reach and greatly facilitates physical intuition on the core

involved physics.

The first key step is to note that, thanks to the atoms' permutational symmetry above, and

according to a procedure analogous to second quantization, the state of the atomic system can be

expressed in terms of atomic Fock states  with  the number of atoms

in the -th single particle level of 

 (Supplementary Section S2). In terms of standard

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1Lightemissionfromstronglydrivenmany-bodysystemsAndreaPizzi1,2†,AlexeyGorlach3†,NicholasRivera1,4,AndreasNunnenkamp5,andIdoKaminer31DepartmentofPhysics,HarvardUniversity,Cambridge02138,Massachusetts,USA2CavendishLaboratory,UniversityofCambridge,CambridgeCB30HE,UnitedKingdom3SolidStateInstitute,Techn...

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1 Light emission from strongly driven many -body systems Andrea Pizzi12 Alexey Gorlach3 Nicholas Rivera14 Andreas Nunnenkamp5 and Ido Kaminer3

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