The Hong-Ou-Mandel effect with atoms

A. M. Kaufman,1 M. C. Tichy,2 F. Mintert,3 A. M. Rey,1 and C. A. Regal11JILA, National Institute of Standards and Technology and University of Colorado,and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

2Department of Physics and Astronomy, University of Aarhus, DK–8000 Aarhus C, Denmark3Department of Physics, Imperial College, SW7 2AZ London, UK

Controlling light at the level of individual photons has led to advances in fields ranging fromquantum information and precision sensing to fundamental tests of quantum mechanics. A centraldevelopment that followed the advent of single photon sources was the observation of the Hong-Ou-Mandel (HOM) effect, a novel two-photon path interference phenomenon experienced by indistin-guishable photons. The effect is now a central technique in the field of quantum optics, harnessedfor a variety of applications such as diagnosing single photon sources and creating probabilisticentanglement in linear quantum computing. Recently, several distinct experiments using atomicsources have realized the requisite control to observe and exploit Hong-Ou-Mandel interference ofatoms. This article provides a summary of this phenomenon and discusses some of its implicationsfor atomic systems. Transitioning from the domain of photons to atoms opens new perspectives onfundamental concepts, such as the classification of entanglement of identical particles. It aids inthe design of novel probes of quantities such as entanglement entropy by combining well establishedtools of AMO physics — unity single-atom detection, tunable interactions, and scalability — withthe Hong-Ou-Mandel interference. Furthermore, it is now possible for established protocols in thephoton community, such as measurement-induced entanglement, to be employed in atomic exper-iments that possess deterministic single-particle production and detection. Hence, the realizationof the HOM effect with atoms represents a productive union of central ideas in quantum control ofatoms and photons.

Keywords: Hong-Ou-Mandel effect, Entanglement, Ultracold atoms, Quantum statistics, Many-particle in-terference, Entanglement entropy, Schmidt rank, Bose-Hubbard model

CONTENTS

I. Introduction 2

II. Experiments 3A. Photon HOM experiments and photon

sources 3B. Single-atom sources 3C. HOM effect with tunneling atoms 3D. HOM effect with twin-pair source 5

III. Explanation of the Hong-Ou-Mandel effect andmany-particle interference 5A. One particle: Single-particle interference 5B. Two-particle interference and the HOM

effect 6C. Many-particle interference beyond HOM:

More particles or more modes 7D. Bosonic bunching and fermionic

anti-bunching 8E. Interference versus statistics 8F. Loss of coherence and partial

distinguishability 81. Loss of coherence in the beamsplitter 92. Partial indistinguishability 9

IV. Entanglement and the Hong-Ou-Mandel effect 9A. Entangled states 10B. Symmetrization and separability 10C. Entanglement of identical particles 10

1. Complete set of properties (CSOP)analysis 11

2. Applying CSOP analysis to HOM 113. Second quantization and mode

entanglement 12D. Schmidt Rank analysis 12E. Creating particle entanglement through

measurement or interactions 13

V. Interaction in two and many-particleinterference 13A. Double well 14

1. Two atoms 142. Many atoms 14

B. Bose-Hubbard: N atoms in L lattice sites 151. U = 0 162. U →∞ 163. Finite U 164. Application of Schmidt rank to

Bose-Hubbard example 16

VI. Entanglement entropy 17

VII. Concluding Remarks 19

Acknowledgments 19

References 20

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I. INTRODUCTION

The Hong-Ou-Mandel (HOM) effect [1] is a two-particle interference phenomenon, wherein two indistin-guishable bosonic particles are interfered on a beamsplit-ter, and the particles always emerge on the same, butrandom output port (Fig. 1). In the original HOM ex-periment, two single photons were sent into each port of abalanced beamsplitter, and the population of the outputports was measured with single-photon detectors. Theexperimenters quantified the probability for joint detec-tion at the two detectors, that is, the likelihood that bothdetectors click within a small time window [1]. Classi-cally, one might compare the experiment to flipping twoindependent coins, where each coin represents a photonand the coin faces represent the output port on whichthe photon emerges. Quantum mechanics dictates thatphotons indistinguishable in all degrees of freedom be-sides the initial mode they occupy will always emerge onthe same output port, akin to the two independent coinsalways landing identically. Experimentally, the quantummechanical outcome leads to the “Mandel dip” in the jointdetection probability, which ideally extends to zero asthe photons are tuned to impinge on the beamsplitterat exactly the same time. The effect is traced to thebosonic nature of the photons, and a resulting destruc-tive interference of the paths that yield joint detection.If photons were fermionic, Pauli exclusion would preventthe particles from being in the same mode and lead to apreponderance of joint detections at the output ports.

a1

a2 b2

b1

(a) (b)

P (2, 0) =

P (1, 1) = +| |2p

T

pT

| |2pT

P (0, 2) = | |2p

Tip

R

ip

R

ip

R

ip

R

FIG. 1. Two-particle interference in the Hong-Ou-Mandel ex-periment. (a) Physical setup: Two identical particles impingeon the two input modes (a1, a2) of a beam-splitter and emergein two output modes (b1, b2). (b) Evaluation of event prob-abilities via Feynman paths. In the HOM effect the bosonicnature of photons results in destructive interference of theP(1,1) outcome.

The HOM effect plays an important role in quantumoptics and quantum information. It has become one ofthe textbook examples of an observed quantum inter-ference phenomenon that cannot be explained by semi-classical theory. Further, it is a standard techniquefor characterizing single-photon sources and is the keycomponent of linear optical quantum computation pro-tocols [2]. Generalization of the HOM effect to many-particle and many-mode interference leads to computa-tionally complex situations such as the boson sampling

problem [2, 3]. In recent years, a number of systemsbeyond optical photons have come to observe Hong-Ou-Mandel physics. Microwave-photons and phonons in iontraps have demonstrated the HOM effect [4, 5] and elec-trons have realized the fermionic counterpart to HOM [6].Further, as is a focus of this article, atoms can serveas “single particle sources” in analogy to “single photonsources” used in the HOM effect.

The challenge in realizing the HOM effect with atoms isplacing the particles in equivalent modes, defined by bothmotion and spin, and realizing effective atomic beam-splitters. In one paradigm, atoms are held in separatepotential wells, such as in an optical lattice or a double-well formed by focused laser beams known as opticaltweezers. Tunneling between the potential wells thenacts as an effective beamsplitter. Pairs of trapped atomsin the required pure single-particle states can be pre-pared independently, as for example in experiments inwhich individual 87Rb atoms were laser-cooled to theirmotional ground state [7]. Alternatively, the atoms canbe prepared collectively, as in optical lattice experimentsin which a Mott insulator is created via a phase tran-sition from a superfluid and observed with a quantum-gas microscope [8, 9]. A second paradigm uses freely-propagating atom sources. Experiments utilizing ultra-cold 4He atoms have created twin beams that can beinterfered with tunable temporal overlap, in analogy tothe original HOM photonic beamsplitter [10].

In addition to state preparation of individual atoms,the observation of the HOM effect with atoms requiresachieving negligible atom-atom interactions by appro-priately tuning the atomic scattering length or confine-ment. Manipulating non-interacting atoms not only al-lows the exploration of the HOM effect, but opens an en-tire paradigm for creating entangled quantum states thatuse measurement instead of interaction. These ideas havelong been explored in the context of linear optics with sin-gle photons. Such experiments feature a surprisingly richspace that can result from combining projective measure-ment with only linear elements such as beamsplitters andwaveplates [2]. On the other hand, trapped atoms offercontrollability that may enable explorations that are diffi-cult with photons – for example exploration of the role ofquantum statistics on distant atoms through interchangeof two particles [11].

We have multi-fold goals for this article. We would liketo provide a background on optical HOM experiments forthe atomic physics community; there are interesting par-allels between the HOM effect experiments that have nowbeen performed with atoms and those with photons. Fora rounded understanding of the HOM effect, it is usefulto realize that the HOM effect incorporates both coherentsuperposition of paths (modes) with single-photon inputsand the bosonic nature of photons [12], which demandsproper wavefunction symmetrization. As such, the HOMeffect is an excellent thought experiment for understand-ing entanglement in the presence of symmetrization. Onepotential misconception is to assume that symmetriza-

3

tion automatically implies entanglement due to the non-separability of the symmetrized wavefunction. We reviewone formalism that helps clarify the apparent contradic-tion, namely, the presence of non-separability but theabsence of the quantum fluctuations fundamental to en-tanglement [12–17].

Further, the HOM effect is only the simplest case ofricher types of many-particle interference, in which moreparticles and more modes participate. Those have beenthe subject of considerable studies in the optics commu-nity, both theoretically [18–20] and experimentally [21–27]. Atomic systems seem particularly well-suited to fu-ture studies of many-particle interference by combiningthe toolset required for two-particle quantum interference(the HOM effect) with the scalability and measurementtechniques endemic to cold atomic sources. While thetopic of fully general multipartite entanglement charac-terization is beyond the scope of this article, we presenta basic delineation between the kinds of entanglementpresent in two-particle quantum states of identical parti-cles.

Lastly, while photons in vacuum are non-interacting,atoms of course can interact, and their interactions canplay an important role in experiments. It is thus usefulto study how interactions affect many-particle interfer-ence, the range of entangled states that can be createdwith interactions, and how the HOM effect can be used tocharacterize the emergent many-body states. Already ex-periments with ultracold atoms in lattices have taken theconcept of HOM beyond two particles: Interfering copiesof a quantum state have been used to quantify entangle-ment entropy of an interacting gas [28, 29], and to studyquantum thermalization through entanglement [30].

II. EXPERIMENTS

A. Photon HOM experiments and photon sources

The HOM effect, and its name, come from an experi-ment with single photons created via parametric down-conversion in which a single pump photon is split into apair of lower-frequency photons [1]. The original exper-iment was titled “Measurement of sub-picosecond timeintervals between two photons via interference”. Thetechnique used quantum interference to determine if twophotons impinged simultaneously upon the beamsplittersand hence were indistinguishable in their arrival times.The photons used in HOM’s experiment were so-calledtwin pairs of entangled photons. If these pairs are per-mutation symmetric one sees the bosonic HOM-effect,but on the other hand if one uses bosons entangled inanother degree of freedom with anti-symmetric spatialwave-functions, they will leave in different ports as if theywere fermions [31, 32]. One of the most important as-pects of the the Hong-Ou-Mandel effect is the fact thatit is a quantum phenomenon that does not require theincident bosonic particles be correlated to begin with; in-

distinguishability is all that is required. The photons canoriginate completely independently and the HOM effectstill persists. Following the original HOM experiment,efforts were made to realize the effect with independentphoton sources. This was first explored in experiments ofRef. [33] and after it many subsequent experiments havebeen carried out [34–36]. For one particularly elucidatingexample, the HOM effect has been observed by creatingtwo independent photons via radiation decay from twoseparately trapped atoms [37].

B. Single-atom sources

Since the realization of Bose-Einstein condensa-tion [40], ultra-cold atom experiments have become auseful laboratory for the investigation of the role of quan-tum statistics and indistinguishability in the behaviorof many-particle systems. Atom optics phenomena suchas the atom laser [41, 42] and the Hanbury Brown andTwiss effect with gases of bosons were realized withina decade of Bose-Einstein condensates (BEC) [43–46].Recently increasing interest has turned to single-atomimaging and control. In optical lattices, tunneling dy-namics of individual atoms can be observed in quantum-gas microscopes [8, 47], and a variety of studies in whichboth interactions and quantum statistics play a role havebeen realized [48]. In this article we will discuss differentplatforms that have achieved the single-atom control andpure state preparation required for realizing the HOMeffect. Those include laser-cooled optical tweezer arrays,optical lattices, and twin-pair sources from BECs. Thesesystems and state preparation therein are summarizedin Fig. 2. In the following sections we discuss how theHOM effect has been observed in these different physicalplatforms.

C. HOM effect with tunneling atoms

An atomic beamsplitter, as required for the HOM ef-fect with atoms, can be realized by tunneling betweenclosely-spaced potential wells, such as the double-wellshown in Fig. 3(b), which forms the minimal setup forobserving the HOM effect in a bound system. Due tothe tunnel-coupling, the energy-eigenstates are coherentsuperpositions of the ground states of each potential well|L〉 and |R〉. An initially prepared atom in the leftwell evolves into a coherent superposition of the form|L〉 → cos(Jt) |L〉 + i sin(Jt) |R〉, where J is the tun-neling rate. Stopping the evolution at a specific timetHOM = π/4J yields an equal superposition associatedwith an effective 50 − 50 beamsplitter (R = T = 1/2).Position-resolved single-atom detection then plays therole of photon detection at output ports of the beam-splitter.

In Ref. [7], two optical tweezers were used to createa double-well potential in which a single atom was ini-

4

kx (klat)

kz

(kla

t)

ky (klat)-0.4 -0.2 0.0 0.2 0.4

0.5

1.0

1.5

-0.4 -0.2 0.0 0.2 0.4

0.00 0.10 0.20Averaged det. atom number

(b) (c)(a)Average det. atom number

FIG. 2. Experimental realization of single-atom sources analogous to single-photon sources of quantum optics. Some examplesof single-atom sources are: (a) Atoms in optical lattices. A microscope was used to individually observe atoms loaded from aBose-Einstein condensate into a two dimensional lattice and to image their quantum random walk trajectories [9]. (b) Atomsin optical tweezers. Individual neutral atoms were laser-cooled to their ground state [38] using alternating Raman sidebandtransitions and optical pumping (OP) between two hyperfine states. Optical tweezers were then used for real-time spatialpositioning of the atoms (right images) and individual atom imaging. (c) A twin-pair source of 4He. Twin-pairs were producedby a four-wave mixing process. Multichannel detection (left) enabled spatio-temporal imaging and to probe the two-dimensionalmomentum distribution of the atom pairs (right). Images from Ref. [39].

a1 a2b2 b1= =

(a) (b) Single atom images

J

(c)

45 c

m

a1 ab2=

a1 b2=a1 a2b2 b1= =

a1 a2b2 b1= =

z

y

time

z

a1

a2b2

b1

FIG. 3. Physical realizations of atomic beamsplitters for the HOM effect. (a) Free-space photonic beamsplitter with photonsimpinging on both input ports a1 and a2, and detected on two output ports b1 and b2. (b) Double-well potential with tunneling,as can be created with optical tweezers or lattices. Single atoms are mixed between the left and right wells by tunneling J , anddetected at the output by parity imaging on each site. (c) Propagating twin-pairs are generated from a BEC source, reflected,and then beam-split through Bragg processes [10].

tialized on each well. The experiment employed ground-state laser cooling in order to control all degrees of free-dom of two independently prepared 87Rb atoms [38, 49].The trapping parameters were set to realize a sufficientlysmall ratio of interaction to tunneling and the atomicspin used as a knob to tune distinguishability. In analogyto polarization of photons, a Mandel dip was observedvia rotation of the relative spin state [7]. The observa-tion of the HOM effect in the experiment demonstratedthe capability to create indistinguishable, and hence pureatomic quantum states in both motion and spin, andshowed that quantum statistics can strongly determinethe interference properties of laser cooled atoms individ-ually placed in their motional ground-state.

Experiments in optical lattices offer another way to ini-tialize single bosons for observing quantum interference

of single atoms. In fact, the use of optical lattices forthe observation of the HOM effect has long been consid-ered [50–52]. In recent successful efforts a Mott-insulatortransition [53] was used to create a uniformly-filled arrayof bosons. Via lattice manipulations and quantum-gasmicroscopy single-atoms were interfered, and the parti-cle number measured on individual sites after interfer-ence. The interference protocol consisted in decreasingthe lattice depth to allow particles to tunnel and thus toperform a random walk. In the regime where interactionswere made smaller than tunneling, interference phenom-ena arising from quantum statistics and closely-relatedto HOM were observed [9]. Quite recently, a general-ized form of HOM was employed to study entanglementin ground state and non-equilibrium Bose-Hubbard sys-tems [29, 30].

5

D. HOM effect with twin-pair source

Early ultracold-atom experiments probed density-density correlations in momentum distributions of ex-panding Bose or Fermi gases of alkali atoms and an-alyzed them using similar methods to typical photon-photon correlation experiments [54–56]. Experimentsusing metastable helium atoms, which provide uniquespatio-temporal resolution through microchannel platedetection, were particular fruitful in developing atom op-tics experiments. Using both bosonic and fermionic he-lium species [43, 45] these experiments observed the Han-bury Brown and Twiss (HBT) effect [57–60].

More recently it was demonstrated that metastable he-lium can be used also to create a direct source of atompairs for the HOM effect [10]. The experiment startedwith a Bose Einstein condensate of helium atoms in themetastable 1s2s3S1 internal state. A moving optical lat-tice created twin atom pairs in analogy to spontaneousfour-wave mixing of optical fields [Fig. 2(c)]. Using an-other optical lattice, the modes for each of the twin beamswere coherently superposed on an effective beam-splittervia Bragg diffraction [Fig. 3(c)]. The timing of this ef-fective beam-splitter process tuned the temporal overlap,and, hence, the atom distinguishability in analogy to thephysical configuration of the original photonic HOM ex-periment. In those experiments, as in the optical tweezerand lattice-based experiments, the presence of the HOMeffect required the realization of an effective beam-splitterand the preparation of indistinguishable bosonic atoms.Recent experiments have also expanded to two particlesand four modes [61].

III. EXPLANATION OF THEHONG-OU-MANDEL EFFECT ANDMANY-PARTICLE INTERFERENCE

One of the points we would like to make clear in our dis-cussion of the HOM effect is the relation both to the sta-tistical bosonic enhancement of multiply occupied single-particle states and the coherent superposition of paths.In this light, we will introduce the HOM effect by firstdiscussing single-particle interference, then two-particleinterference of the HOM effect, and then many-particleinterference.

The superposition principle in quantum mechanics ismost immediately illustrated using a single particle thatis prepared in a superposition of two distinct spatialstates, supported by different modes - for example, ina double-slit experiment or in a Mach-Zehnder interfer-ometer. Although the superposition principle is typi-cally discussed in the context of single-particle states,it applies equally well to states of two or more parti-cles: if |Ψi〉 ∼ a†i1 . . . a

†in|0〉 and |Ψj〉 ∼ a†j1 . . . a

†jn|0〉 are

two n-particle states, then so is any coherent superpo-sition of these two states. Just like coherent superpo-sitions of single particle states give rise to interference

that manifests itself in single particle observables, coher-ent superpositions of n-particle states give rise to inter-ference phenomena in n-particle space. The most imme-diate paradigm for an observable consequence of many-particle interference is the HOM effect [1], in which cor-relations in particle-like events are governed by wave-likeinterference in the high-dimensional many-particle space.Many-particle interference comes hand-in-hand with theparticular particle statistics; typically, bosons interfere ina different way than fermions do.

A. One particle: Single-particle interference

To develop some intuition, let us first establish somebasic ideas and concepts of single particle interference ina Mach-Zehnder-interferometer and subsequently moveto two-particle interference. Initially, a particle in theoptical mode a1 is prepared in the state

a†1 |0〉 , (1)

where |0〉 denotes the vacuum and a†1 is a particle cre-ation operator. The particle impinges on a beam-splitter,which induces the time-evolution

a†1 → U a†1U−1 =

√T b†2 + i

√Rb†1, (2)

where R and T are the reflection and transmission coeffi-cients of the beam-splitter connecting input modes a1, a2with output modes b1, b2. The phase shift acquired uponreflection is chosen to be π.

Due to the unitarity of the time-evolution U inducedby the optical element, reflectivity R and transmittivityT satisfy R + T = 1. The beam-splitter thus generatesa coherent superposition of a particle in the upper and aparticle in the lower arm,(√

T b†2 + i√Rb†1

)|0〉 , (3)

where we exploited the invariance of the vacuum underparticle-number-preserving time-evolutions, U |0〉 ∝ |0〉.

Subsequently, the particle in the lower arm passes aphase shifter, such that it acquires a phase φ relative tothe upper arm,

b†2 → eiφb†2, (4)

and the state becomes(eiφ√T b†2 + i

√Rb†1

)|0〉 (5)

At a second beam-splitter, the two arms are joinedagain, and the time-evolution

b†1 →√T c†2 + i

√Rc†1 , (6)

b†2 →√T c†1 + i

√Rc†2 , (7)

(similarly to Eq. 2) eventually leads to the final state((−R+ Teiφ

)c†1 + i

(√RT +

√RTeiφ

)c†2

)|0〉 (8)

6

P (1, 0) = +| |2

P (0, 1) = +| |2

(a) (b)

a1

a2 b2

b1c1

c2

pT

pT

pT

pTi

pR i

pR

ip

R ip

R

FIG. 4. Single-particle interference in the Mach-Zehnder in-terferometer. (a) Setup of two beam-splitters and mode la-bels. A relative phase φ is acquired in the lower arm, leadingto interference fringes in the probability to find the particle inthe upper output arm c1. (b) The probabilities for the eventscan be understood as arising from the coherent sum of single-particle path amplitudes. They reflect the acquired phase φ.

from which we can read off the probabilities P (1, 0) andP (0, 1) to find the particle in the upper (first) and in thelower (second) arm respectively.

These probabilities

P (1, 0) = R2 + T 2 − 2RT cos(φ) (9)P (0, 1) = 4RT cos2(φ/2) (10)

depend on the relative phase φ, acquired between the twoarms, since the amplitude for the pertinent event is thesum of the two complex path amplitudes.

That is, two physically distinguishable, alternativepathways (the particle travels through the lower mode,the particle travels through the upper mode) are coher-ently super-imposed, leading to interference fringes as afunction of φ. As a result, the probability to observe agranular particle-like event is governed by the wave-likeinterference of the two pathways.

B. Two-particle interference and the HOM effect

The HOM effect can be understood as an interferenceeffect, but is inherently not a single particle effect as theinterfering paths describe more than one particle. Indeed,the HOM effect can not be understood in terms of single-particle interference.

Instead of a single particle, we now have to considertwo identical particles in the two input modes of a beam-splitter:

|ΨHOM,in〉 = a†1a†2 |0〉 (11)

and detectors placed at the output modes of the beam-splitter as depicted in Fig. 1(a).

Because the particles do not interact, they propagateindependently, and the beamsplitter is characterized bysingle particle dynamics similar to Eq. 2. Since the HOMsetup leaves no room for an additional phase shifter asin the Mach Zehnder setup, we will leave the phase thata particle acquires upon reflection a free parameter. The

action of the beamsplitter thus reads

a†1 →√T b†2 + ieiϕ

√Rb†1 , (12)

a†2 →√T b†1 + ie−iϕ

√Rb†2 . (13)

The relative phase of ϕ can be chosen at will, but unitar-ity prevents the two relative phases in the relations fora†1 and a†2 to be chosen independently.

With this, the final state of the particles after passingthe beamsplitter reads(i√RT

[eiϕ(b†1

)2+ e−iϕ

(b†2

)2]+ T b†2b

†1 −Rb†1b†2

)|0〉 .

Up to here, the discussion applies to both fermions andbosons, but in identifying probabilities P (2, 0), P (0, 2)to find both particles in the upper or lower arm, or theprobability P (1, 1) to find one particle in each arm, oneneeds to distinguish between the two types of particles.

In the case of bosons, the final state reads

i

(√RT

[eiϕ(b†1

)2+ e−iϕ

(b†2

)2]+ (T −R)b†1b

†2

)|0〉 ,

due to the relation b†1b†2 = b†2b

†1. The state remains nor-

malized to unity thanks to the properties of bosonic cre-ation operations, which will be discussed in more detailbelow.

Focusing on the final state with one particle per outputmode, b†1b

†2 |0〉, one can – just like in the Mach-Zehnder in-

terferometer – identify two physically distinct paths thatconnect the initial and the final states: Either both par-ticles are reflected and feed the component −Rb†1b†2, orboth particles are transmitted and contribute to T b†2b

†1.

Interestingly, the phase relation between the quantumamplitudes is fixed to −1 = eiπ, independently of thephase shift that a single particle acquires upon reflec-tion. Unitarity requires that the phase shift between areflected and a transmitted particle is fixed. Moreover,any phase applied in any input mode only contributesas a global phase. Hence, the setup leaves no room toacquire a variable relative phase.

In the case of fermions, all contributions of doubly oc-

cupied states vanish since(b†1

)2=(b†2

)2= 0, and the

final state reads

−(T +R)b†1b†2 |0〉 = −b†1b†2 |0〉 (14)

because of the relation b†1b†2 = −b†2b†1. Independent of

transitivity and reflectivity of the beamsplitter, one willthus always find one particle per mode.

The interference between the two discussed pathways isthus constructive in the case of fermions and destructivein the case of bosons. Perfect destructive interference forbosons is achieved for R = T = 1/2, which is the casethat gives rise to the HOM effect [1].

It should also be stressed that these interference effectsare manifest in the probability to find two particles in thesame mode or to find one particle in each mode. These

7

probabilities are reflected by two-particle observables, i.e.observables made of two creation and two annihilationoperators, namely b†1b

†2b1b2,

(b†1

)2 (b1

)2,(b†2

)2 (b2

)2.

Proper single-particle observables (containing only onecreation and one annihilation operator) like the expec-tation value for the number of particles in one mode,i.e. b†1b1 and b†2b2, yield expectation values of unity forboth bosons and fermions, independently of the proper-ties of the beamsplitter, and they show no signature ofinterference.

C. Many-particle interference beyond HOM: Moreparticles or more modes

In its simplicity, the two-particle, two-mode HOM ef-fect is a paradigm for the quantum behavior of a fewindistinguishable particles as it reduces the essence ofquantum statistics to a microscopic setting. Ultimately,this microscopic behavior is driven by the same cause asthe macroscopic behavior of liquids, solids and gases. Byincreasing the number of particles and modes involvedin the experiment [21], however, one encounters varioussurprising aspects, due to the interplay of many-bodycomplexity and many-body coherence.

The behavior of many identical bosons falling onto thetwo input modes of a balanced beamsplitter exhibits,both granular interference on the level of individual par-ticles, as well as coarse-grained structures that can bedescribed via a semiclassical approach [62, 63], in whichthe number of particles is represented by a macroscopicwavefunction without well-defined phase. As a promi-nent example for a granular interference effect, considerthe same number of particles impinging onto a beam-splitter, i.e. an initial state of the form(

a†1

)N (a†2

)N|0〉 . (15)

The time-evolution is directly inherited as in Eq. 13. As adirect generalization of the suppression of the transition(1, 1)→ (1, 1), the probability to find an odd number ofparticles in either output mode vanishes [63–65], whichcan be shown for the general case by symmetry considera-tions. In one experiment studying multi-particle interfer-ence, the suppression of the transition (2, 2)→ (3, 1) wasaccompanied with a narrowing of the temporal width ofthe interference due to multi-particle effects [66]. On topof this fine-grained interference pattern, a coarse-grainedstructure appears: When many particles impinge on bothmodes of the beamsplitter, the probability to find a bal-anced number in the output modes is suppressed, andunbalanced arrangements are enhanced [63]. The inter-play of fine-grained and coarse-grained structures can beseen in Fig. 5.

When increasing the number of modes as well as thenumber of particles, the number of possibilities to arrangethe particles in the modes explodes combinatorially. On

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a2 b2

b1

FIG. 5. Two-mode many-particle scattering example. Ex-perimental setup on the left and calculation on the right.For many distinguishable particles that impinge on a beam-splitter, we expect to find a binomial distribution (green dia-monds). For N = 4 (red circles) and N = 22 (blue squares)bosons that fall onto both input modes of a beamsplitter, theresulting interference pattern exhibits granular even-odd-likepatterns, as well as a coarse-grained overall structure, whichis reproduced well by a semi-classical approach (black solidline). The abscissa shows the fraction of particles encounteredin one output mode, the ordinate is the respective probabil-ity, which was normalized such that it matches the continuousprobability distribution of the semi-classical approach. Figuretaken from Ref. [12].

the other hand, the freedom in choosing a unitary scatter-ing matrix A that governs the many-mode time-evolution

bn →∑m=1

Anmam, (16)

in analogy to Eq. 13 allows the formulation of very differ-ent experiments [67]. While general statistical tendencies– bosons prefer bunched arrangements over spread-outconfigurations while the Pauli principle for fermions al-lows only such spread-out configurations [19] – prevailin all configurations, the very computation of individualtransition probabilities becomes prohibitively difficult forbosons. Indeed, the transition probability becomes thepermanent of the respective scattering matrix [68], forwhich the computational complexity is well-established.As a consequence, even the mere simulation of manybosons that scatter off a random setup is a computation-ally hard problem, dubbed boson-sampling [3], which hasattracted wide interest, fueled also by its experimentalfeasibility for few photons [69–72].

Whereas an unstructured random scattering setupdoes not allow systematic physical statements on the be-havior of the particles in the scattering process, struc-tured setups with symmetries allow the formulation ofstrong rules for the suppression of output events: If asymmetry is present in the input arrangement of bosonsand the setup is symmetry-preserving, the output ar-rangements need to fulfill the symmetry as well. Thisprinciple was applied to the Fourier-matrix [18, 19, 73,74], as well as to scattering matrices with other kinds ofsymmetries [20, 75]. The experimental implementationswith up to five photons [22, 76] may also provide one wayto verify the functionality of an alleged boson-sampling

8

computer [77].

D. Bosonic bunching and fermionic anti-bunching

We now turn to understanding the two-particle andmany-particle interference in a number of contexts. Re-turning to Fig. 1(b), note that independent of the parti-cle species, a given final state with both particles in onemode (either the first or the second one) is fed by exactlyone many-particle path. One particle must be reflected,the other one is necessarily transmitted, and no interfer-ence of different two-particle paths takes place. Never-theless, the probability to find both particles in one out-put mode does not coincide with RT as one might haveexpected for an event in which one particle is transmit-ted and one is reflected. This result would hold true fordistinguishable particles, but the probability for bosonsis enhanced by a bunching factor 2, whereas coincidenceevents are completely suppressed for fermions. The ori-gin of this behavior is the state-space-structure of bosonsand fermions. The Pauli principle for fermions strictlyprohibits the multiple occupation of any single-particlestate, in any basis,

N > 1⇒ (b†)N |0〉 = 0, (17)

while the bunching tendency of bosons is reflected by theover-normalization of multiply populated states,(

b†1

)2|0〉 =

√2 |2〉 . (18)

Probing the occupation of the final Fock-state |2, 0〉 or|0, 2〉, allows us to witness the fermionic suppresion andbosonic preponderance of double-occupied states – butnot the coherent superposition of several many-particlepaths.

The suppression for fermions is by no means specificto the two-particle HOM setup, and applies to many-particle interference. Indeed, the probability to find sev-eral fermions in the same single-particle state is fully sup-pressed in every thinkable setup – the Pauli principle isuniversal [78, 79]. Similarly, also the bosonic enhance-ment of a bunching event in which all particles end inthe very same mode fulfills [12, 80, 81]

PB = PDn!∏lj=1 rj !

, (19)

where PB is the probability for n bosons prepared withoccupations r1, . . . , rl to end in the same output mode,and PD is the analogous probability for distinguishableparticles. The enhancement factor n!/

∏j rj ! is indepen-

dent of any phases acquired in the system and reflectsmerely the bosonic state-space-structure.

E. Interference versus statistics

Often, Dirac’s famous dictum [82], “Each photon theninterferes only with itself. Interference between different

photons never occurs.” is said to be overcome in view ofthe HOM effect. However, it remains fully valid in thepresent context: The first particle does not interfere withthe second – the two particles propagate independentlyand reach each output mode with a definite probability,independently of the presence of the other one. Never-theless, coherent superpositions of different two-particlestates results in interference, but this interference is onlyapparent in two-particle observables as discussed in thespecific case of the HOM effect above. When probing anysingle-particle observable, e.g. the average particle num-ber in each output mode, there is no difference betweenbosons, fermions or distinguishable particles

We have treated the two types of events – doubly oc-cupied bunching events (2,0) and coincident events (1,1)– separately. Indeed, we witness two rather indepen-dent phenomena in one setup: The statistical bosonic en-hancement or fermionic suppression of multiply occupiedsingle-particle states and the coherent superposition ofmany-particle paths. Of course, due to the conservationof probability, an enhancement of (1,1) needs to be ac-companied by a suppression of (2,0), and vice versa, andit may seem suspicious to view bosonic/fermionic sta-tistical behavior and many-particle interference as twoseparate physical phenomena. Admittedly, these phe-nomena have a common origin: the indistinguishabilityof particles of the same species. The dependence of the(2,0)-event on the (1,1)-probability is, however, not byany means universal, it is unavoidable in the basic two-particle HOM effect due to the small size of the setup:When increasing the number of modes [83, 84] and/orthe number of particles [63], statistical effects (bosonicbunching and fermionic suppression of multiply occupiedevents) and many-particle interference (governing coinci-dent events) become quite independent phenomena. Inparticular, bosons and fermions can exhibit very similarinterference patterns in single (or few) particle observ-ables. Many-particle interference on the other hand ismuch more sensitive to phase variations and a meticu-lously prepared initial state than statistical bosonic orfermionic effects [19].

F. Loss of coherence and partial distinguishability

Interference effects can generally be easily be washedout by noise or decoherence. It is illustrative to under-stand the effect of loss of coherence at the beamsplitteron the HOM effect. To provide an illustrative calcula-tion, we consider here mainly the case of two-particleinterference. However, as just mentioned many-particleinterference will exhibit different sensitivities.

Further, because the presently discussed interferenceeffects are based on particle indistinguishability, they arealso easily affected by imperfect state preparations thatmake different particles distinguishable by some physicalproperty.

9

1. Loss of coherence in the beamsplitter

The beamsplitter with the relation defined in Eq. 2induces a perfect phase relation between the two outgoingstates; that is, the splitting is perfectly coherent.

In order to discuss reduction of phase coherence, it isinstructive to consider a beamsplitter following Eq. 13that describes a coherent operation for any value of ϕ.Ideally ϕ is a constant, and in fact to utilize the HOMeffect as a quantum resource ϕ must be controlled, butin real experiments ϕ may fluctuate. In this section, wewill briefly discuss the impact of such fluctuations onthe HOM effect and entanglement properties of the finalstate. For further discussion of the mode entanglementassociated with the HOM effect see Sec. IVC, where wewill assume a non-fluctuating phase can be verified.

The state resulting from the beamsplitter Eq. 13 witha given phase ϕ in an HOM experiment reads

(T −R) |1, 1〉+ i√

2RT (eiϕ |2, 0〉+ e−iϕ |0, 2〉). (20)

with |1, 1〉 = b†1b†2 |0〉, |2, 0〉 = b†1b

†1 |0〉 and |0, 2〉 =

b†2b†2 |0〉. For a balanced beamsplitter with R = T = 1/2

one will thus always observe perfect bunching indepen-dently of the value of ϕ.

In the case of a shot-to-shot-fluctuating phase de-scribed in terms of a distribution µ(ϕ), the resultingmixed state (i.e. density matrix) averaged over the phasefluctuation reads

%α =

2RT i√

2RTQα 2RTα2

−i√

2RTQα∗ Q2 −i√

2RTQα

2RT (α∗)2 i√

2RTQα∗ 2RT

(21)in the basis |2, 0〉, |1, 1〉, |0, 2〉 with

α =

∫dµ(ϕ)e−iϕ , (22)

and the short hand notation Q = T − R. Because theprobabilities for the observation of mode occupations isindependent of the value of ϕ, they are also not af-fected by phase fluctuations. That is, a perfect HOMdip can be observed even for maximal phase fluctuationsfor which α = 0. These phase fluctuations however havea strong impact on the entanglement properties of thefinal state [14, 85]. Similarly, many-particle, many-modeinterference will see detrimental effects from phase fluc-tuations.

2. Partial indistinguishability

If both particles impinging on the beamsplitter overlapperfectly, one observes the HOM effect discussed thus far,but if both particles propagate through the beamsplitterone after the other, no such effect exists. Similarly foratoms distinguished in another degree of freedom suchas motion or spin, interference will not occur. To discuss

the range of cases in between, we can consider a situa-tion in which the spatial wave functions of both particlesdo not overlap perfectly. To this end, we can assumethe particle in mode 1 to be prepared in the state a†1 |0〉,and the particle in mode 2 to be prepared in the state(cos θ a†2 + sin θ a†2) |0〉, where a†2 |0〉 is a state that yieldsno overlap of the two particles in the beamsplitter atall. Varying θ from 0 to π/2 thus permits to continu-ously scan from perfectly indistinguishable particles tocompletely distinguishable particles.

It is instructive to first consider the case in which thetwo particles do not overlap at all, that is, the classicalcombinatorial case. From the relation

a†2a†1 → Tb†2b

†1 −Rb†2b†1 + i

√RT (eiϕb†1b

†1 + e−iϕb†2b

†2)

one directly reads off the probability RT to find bothparticles in the first or second output mode respectively,and the probability T 2 + R2 to find one particle in eachoutput mode.

In the case of a general input state (cos θ a†2 +

sin θ a†2) |0〉 the corresponding probabilities for finite over-lap read for bosons

PB(2, 0) = PB(0, 2) = RT (1 + cos2 θ) (23)PB(1, 1) = (R− T )2 + 2RT sin2 θ (24)

and for fermions

PF (2, 0) = PF (0, 2) = sin2 θ RT (25)PF (1, 1) = cos2 θ + sin2 θ (R2 + T 2) (26)

= R+ T − 2 sin2 θ RT. (27)

Hence, the overlap of the wavefunctions can be read offin the event probabilities. This makes the HOM effect adirect quantitative probe for indistinguishability.

IV. ENTANGLEMENT AND THEHONG-OU-MANDEL EFFECT

So far, we have examined how interference emergeswhen indistinguishable particles are superposed amongtwo or many modes, emphasizing the role of multi-particle path interference. We showed that a multipar-ticle observable cannot be inferred from the expectationof single-particle interference, but rather depends on thequantum statistics of the particles (Sec. III B). Whenmultiparticle observables display correlations that cannotbe predicted by single-particle observables, often quan-tum entanglement plays an important role. However, onemust be careful when considering entanglement of iden-tical particles because in this circumstance the typicalentanglement criterion of non-separability can be mis-leading.

In this section, we will explore the entanglementpresent in the context of the HOM effect and its gen-eralizations. The HOM effect involves multiple particles,but also multiple modes, or paths. We will introduce

10

concepts we refer to as mode entanglement and parti-cle entanglement [14], which illustrate the different waysnon-separability are manifest in relevant quantum states.In particular, we reserve the term particle entanglementfor a situation in which, due to entanglement, a physi-cal reality cannot be assigned to a particle individually,and a salient point of our discussion is that no particleentanglement is present in any single or multi-particleHOM effect with spinless non-interacting particles. Wewill first generally discuss entanglement and related con-cepts, then, we use these concepts to define entangle-ment for identical particles in a way that assures thepresence of correlated fluctuations associated with theessence of entanglement. Along the way we compare andcontrast to prototypical Bell states in the particle’s spinor polarization degree of freedom. Bell state creation inatomic physics is mainly associated with entangling in-teractions, but, as we will describe, generalizations of theHOM effect also enable Bell state creation through mea-surement, effectively transferring entanglement betweenexternal (mode) and internal degrees of freedom.

A. Entangled states

As stated, our goal for this section is to discuss thevarying roles of entanglement in the HOM effect, and wedo so by first addressing entanglement generally. Entan-glement gives rise to correlations of measurement resultsthat can not be accounted for in terms of classical prob-abilities. A prototypical example of an entangled stateis a Bell state. It consists of two distinguishable parti-cles, which we will assume have spatial positions L andR, with internal states |↑〉 and |↓〉 that could for exam-ple correspond to the alignment of an internal degree offreedom along a quantizing direction. In this basis, theBell state |Φ+〉 can be written,

|Φ+〉 =1√2

(|↑〉L |↑〉R + |↓〉L |↓〉R) . (28)

Local measurements in the basis |↑〉 , |↓〉 have maxi-mally undetermined outcomes, i.e. the probabilities toproject on |↑〉 and |↓〉 are both 50%. Despite the max-imal fluctuations in single-particle observables, once thestate of one of the particles has been measured, the out-come of a subsequent measurement on the second par-ticle is determined. One will always observe that theparticles have the same spin state, and hence two-pointobservables (i.e. correlations) are not fluctuating. Cru-cially, this is not limited to measurements in the bases|↑〉L , |↓〉L and |↑〉R , |↓〉R. For any basis in L (i.e. anyunitary rotation of the basis vectors), there is a corre-sponding basis in R such that measurement results areperfectly correlated, while results of each measurementalone remain completely undetermined. While classicallysuch correlations could be produced in a particular basis,it is only through quantum effects that correlations canpersist in multiple bases.

Entangled states are often formally expressed as a statethat is not separable. A separable pure state is one whichcan be written as,

|ψ〉 = |ψ〉L ⊗ |ψ〉R (29)

of subsystem states |ψ〉L and |ψ〉R. Any pure state thatcannot be written in this way, such as the state Eq. 28with respect the spin-1/2 subsystems at L and R, is usu-ally considered entangled. As we will discuss in whatfollows, identical particles can complicate this somewhatsimple diagnostic formula.

B. Symmetrization and separability

Sec. IVA treats the particles as distinguishable. Theexistence of two degrees of freedom (e.g. spatial positionand spin) allows reference to the particle at position Land the one at R, each with its own spin degree of free-dom. As such, the quantum statistics of the particlesdo not have physical consequences. By contrast, in theHOM effect the bosonic particles have a single degree offreedom (spatial position, i.e. mode), and the quantumstatistics are essential.

By writing out the quantum states involved in theHOM effect, the complications that arise from identicalparticles are apparent. Adopting a first quantized nota-tion, we will refer to the particles in the two-particle stateas i and ii. In general, if we have a bosonic state with aparticle in state |α〉 and the other |β〉 (with 〈α |β〉 = 0),then the symmetrized state is written,

|ψsym〉 =1√2

(|α〉i |β〉ii + |β〉i |α〉ii) , (30)

which is manifestly symmetric under exchange of the par-ticle labels i and ii. Although this describes the simplescenario of two particles in orthogonal quantum states, itexhibits non-separability with respect to the particle la-bels i and ii. This raises the question of whether the non-separability constitutes entanglement. The goal of thefollowing sections is to bring further clarity to this point,namely, how the non-separability intrinsic to all multi-particle state of identical particles should be squared withentanglement.

C. Entanglement of identical particles

To analyze the presence of entanglement in Eq. 30, wefirst appeal to the more basic notion of entanglement de-veloped in Sec. IVA. The state |Φ+〉 exhibits correlatedfluctuations in the spin basis associated with spatially-separated particles. The state |ψsym〉, on the other hand,does not exhibit any fluctuations: an observer alwaysmeasures a particle in state |α〉 and another particle instate |β〉. In the state |Φ+〉, the subscripts representing

11

position and the ket arguments representing spin corre-spond to the eigenstates of observable operators. By con-trast, the particle label subscripts i and ii in |ψsym〉 arenot physical as they do not correspond to the eigenstateof any observable. Hence, although the state |ψsym〉 isnon-separable, it lacks the physical properties associatedwith the state |Φ+〉. This suggests separability alone isan insufficient metric for identifying entanglement whereidentical particles are concerned.

1. Complete set of properties (CSOP) analysis

There is a framework that clarifies the separability co-nundrum, which is described in Refs. [12–14]; we closelyfollow the treatment of these references here. To moti-vate the idea behind the framework, one must relax theconstraint of separability for non-entangled states, butretain the more fundamental quantum-correlated fluctu-ations that underlie entanglement.

The framework we summarize is rooted in the followingidea. When two particles are non-entangled, one can as-cribe a “complete set of properties", i.e. physical reality,to each of the particles, irrespective of which particle it isthat carries each of these sets. For the state in Eq. 30, oneof the particles is always in state |α〉 and the other in |β〉.Even though formally the unphysical particle label asso-ciated with the states (|α〉 or |β〉) fluctuates, there are nomeasurable fluctuations in any physical observable, and,as such, no entanglement between the particles.

Reference [13] provides a formalism for identifyingwhether a particle within a two-particle quantum statehas a complete set of properties (CSOP). The formal-ism relies on the construction of the symmetric operatorEP , with unity expectation value when one of the par-ticles in a symmetrized two-particle state has a CSOP.To develop the formalism, consider the single-particleHilbert space spanned by the orthonormal set xj. If aBose-symmetrized state |ψsym〉 consists of particles witha CSOP, then there is a projector in the single-particleHilbert space P = |xj〉 〈xj |, satisfying [12, 13],

1 = 〈ψsym| EP |ψsym〉 , (31)

with,

EP = P ⊗ (I− P ) + (I− P )⊗ P, (32)

where the Kronecker product ⊗ is taken with respect tothe single particle Hilbert spaces associated with eachparticle. When these equations are satisfied, one of theparticles has a CSOP associated to the state |xj〉, and thecorrelated fluctuations intrinsic to entanglement cannotbe present. The other particle is always in a state |x〉(in general, a superposition of states in the orthonor-mal set excluding |xj〉) orthogonal to |xj〉, satisfying1 = 〈x| I−P |x〉 = 〈x|∑l 6=j |xl〉 〈xl|x〉. However, becausetwo bosons can share the same quantum-state, and yield0 = 〈ψbos| EP |ψbos〉, we must provide the additional pre-scription that a non-entangled state either satisfies Eq. 32

or describes two bosons occupying the same quantumstate. If neither is true, the particles are entangled. Thecriteria can be summarized as follows: two bosons arenot entangled when either they share the same quantumstate, or are the symmetrized product of two orthogonalquantum states [12–14]. While the projection operatorEP exists for the exemplary state |ψsym〉 of Eq. 30, thereis provably no such operator for |Φ+〉 and so it exhibits“particle entanglement”. Lastly, we show in Sec. IVDthat the prescription here in terms of the symmetric op-erator EP can be equivalently cast in terms of the Schmidtrank, which also provides a useful tool for understandingthe effect of interactions.

2. Applying CSOP analysis to HOM

We now apply the CSOP formalism to elucidate theentanglement in the input and output state in the HOMexperiment. We will consider two kinds of entanglement– that we refer to as mode and particle entanglement– and whether each is manifest in the HOM effect. Todo so, we will go between a first quantized and secondquantized notation where appropriate to the discussion.

We start with first quantization in order to apply thetreatment of Sec. IVB to the HOM effect. We considertwo spinless bosons that initially occupy distinct spatialmodes. These modes could be a propagating free-spacemode of a photon or atom, or the bound state of a har-monic potential. We will adopt the latter scenario of adouble well with bound states (summarized in Sec. II C)for the following discussion. We will call the bound statemodes |L〉 and |R〉. The initial two-particle state is,

|S〉 =1√2

(|L〉i |R〉ii + |R〉i |L〉ii) . (33)

We can now ask whether the particles are entangled ac-cording to the metric established in Section IVB. We seekthe projector P that satisfies Eqs. 31 and 32. Trivially,if we take P = |R〉 〈R|, then 〈S| EP |S〉 = 1. The inputstate |S〉 describes particles that are not entangled, be-cause it lacks any measurable quantum fluctuations thatmight support correlations [13].

We now consider the output state after HOM interfer-ence. In a double well, a tunnel-coupling between twobound states realizes a beam-splitter interaction, induc-ing the single-particle transformations of the form (irre-spective of the particle label),

|L〉 , |R〉 → 1√2

(|L〉+ i |R〉) , 1√2

(|R〉+ i |L〉),(34)

12

which when implemented on |S〉 yields,

|S〉 → 1√2

(|S〉 − |S〉+ i |L〉i |L〉ii + i |R〉i |R〉ii)

=i√2

(|L〉i |L〉ii + |R〉i |R〉ii)

≡ |+〉 , (35)

where in the last line we have introduced a notationfor the ouput state, |+〉. We now analyze |+〉 for thepresence of particle entanglement. Again, from the or-thonormal space |L〉 , |R〉 we seek the pure-state pro-jector P = |p〉 〈p| that satisfies Eqs. 31 and 32. Im-portantly, |p〉 = 1√

2(|L〉+ i |R〉) is sufficient, leading to

projectors in Ep that correspond exactly to the beam-splitter transformed states. As in |ψsym〉 in Eq. 30,the beamsplitter results in |α〉 = 1√

2(|L〉+ i |R〉) and

|β〉 = 1√2

(|R〉+ i |L〉). One of the particles is always inthe spatial superposition given by |α〉, and the other inthe superposition given by |β〉, and the correspondingsymmetrized two-particle state is |+〉. According to theprescription from Section IVB the existence of P impliesthat there is not particle entanglement in the state |+〉.Hence, neither the input or output state in the HOMeffect requires particle entanglement.

Lastly, we stress that the usage of a double well inthe above example does not restrict the generality of theargument. In particular, the double well has the featurethat the input modes and output modes are the same (seeFig. 3). However, had we redefined the output modesin the above to be distinct from the input modes, thefinal quantum state would still lack particle entanglementaccording to the CSOP prescription.

3. Second quantization and mode entanglement

While the first-quantized picture demonstrates an ab-sence of particle entanglement in the state generated inthe HOM experiment, a second-quantized treatment em-phasizes the presence of mode entanglement [86]. As de-scribed in Sec. III, we are interested in the initial state oftwo particles occupying two distinct input modes, withassociated creation operators a†1 and a

†2. As in Eq. 11, the

initial state of two modes each with a single excitationis |ΨHOM,in〉 = a†1a

†2 |0〉 = |1, 1〉. This state is physically

equivalent to the state |S〉 from the first quantizationpicture. On the other hand, the output state after a bal-anced beamsplitter is |ΨHOM,out〉 = 1

2 ((b†1)2+(b†2)2) |0〉 =1√2(|2, 0〉 + |0, 2〉), which is equivalent to the state |+〉.

Note if a single photon were to impinge on one port of thebeamsplitter, the output state 1√

2(|1, 0〉+|0, 1〉) would be

similarly mode entangled.In this second-quantized notation, we now ask whether

there is separability between the two physically ob-servable mode degrees of freedom and the Fock statequantum number within each mode. The initial state

|ΨHOM,in〉 is a product state with respect to these degreesof freedom, while the output state is not. Therefore, theinitial state lacks mode entanglement, while the outputstate acquires it due to the application of the beamsplit-ter. We also stress that there is no logical inconsistencyin applying separability in this context, because we areasking about separability between physically observabledegrees of freedom, whereas the separability in questionwithin a generic symmetrized state (Eq. 30) is related tothe unphysical particle labels.

D. Schmidt Rank analysis

As shown in Ref. [13], the CSOP prescription can belinked to the Schmidt rank, which quantifies entangle-ment between the sub-systems of a quantum state. TheSchmidt rank provides an intuitive partition between howmode and particle entangled states are generated. Quitesensibly, mode-entangled states can only result whenthere are interactions between the modes, and particle-entangled states can only result when there are interac-tions between the particles (or projective measurement).

The Schmidt rank treatment discussed here pertainsstrictly to bosons, but can be modified to the case offermions. We consider the case of two bosons, 1 and2, represented in a basis of orthonormal single-particlestates |xj〉. We use a first quantized picture in or-der to emphasize the presence of symmetrization and theemergence of particle entangled states. In general, anyBose-symmetrized two-particle state can be written infirst quantization as,

|ψ2p〉 =∑q,j

vq,j |xq〉i |xj〉ii , (36)

where vq,j is the amplitude associated with particle 1being in |xq〉 while particle 2 is in |xj〉. This is the tradi-tional representation, for example, for a pair of particlesoccupying two orthogonal quantum states. Because theseare bosons, symmetrization implies vq,j = vj,q. The ma-trix v (with elements vq,j) can be diagonalized [13], suchthat we can rewrite the two-particle wave function in thediagonal basis |xj〉,

|ψ2p〉 =∑k

vk |xk〉1 |xk〉2 , (37)

where vk is the amplitude associated with both particlesoccupying the same state |xk〉. To discriminate the pres-ence of particle entanglement, we introduce the “Schmidtrank”, which is the number of terms in the summationwith non-zero vk. The particles are entangled when theSchmidt rank exceeds two [13, 14]. When the Schmidtrank is two, the state is the symmetrized product of twoquantum states (not necessarily orthogonal); when theSchmidt rank is one, the particles are in the same exactquantum state.

13

The Schmidt rank treatment can be straightforwardlyapplied to the HOM effect. The input state |S〉 hasSchmidt rank of two, with |x±〉 = 1√

2(|L〉 ± |R〉). This

rank is preserved by the unitary transformation achievedby the beamsplitter to produce |+〉, whose first-quantizeddescription is in the format of Eq. 61 (i.e. already ex-pressed in the diagonal basis). Both the input and out-put two-particle states are the symmetrized product oforthogonal single-particle quantum states. Hence, thereis no particle entanglement in the HOM effect. This isto be expected; if one starts with a pair of non-particle-entangled bosons in two orthogonal quantum states, thenthe action of a single-particle Hamiltonian (i.e. no inter-actions) will leave the state in that form, because theunitary Schrodinger evolution preserves the orthogonal-ity of the evolving states. In other words, the two-particlestate remains a symmetrized product of two orthogonalquantum states, and the Schmidt rank of two is invariantwith time.

E. Creating particle entanglement throughmeasurement or interactions

When measurement or interactions are at play, it ispossible to produce a two-particle quantum state whoseSchmidt rank exceeds two, and hence, contains particleentanglement. For example, consider the state |Φx〉 be-low, which is related to |Φ+〉 by basis rotation. Thisstate is routinely produced with ions via their Coulombinteraction [87] and single-particle spin rotations. In firstquantization, for bosons this state is written,

|Φx〉 =1

2(|L〉i |R〉ii + |R〉i |L〉ii) (|↑〉i |↓〉ii + |↓〉i |↑〉ii) .

(38)In the basis |xj〉 = |L, ↓〉 , |L, ↑〉 , |R, ↓〉 , |R, ↑〉, theassociated matrix v is of the form,

v =

0 0 0 1/20 0 1/2 00 1/2 0 0

1/2 0 0 0

, (39)

When this matrix is diagonalized to yield v, the diagonalelements of v represent vk associated with Eq. 61. Upondiagonalizing, we have that,

v =

−1/2 0 0 00 −1/2 0 00 0 1/2 00 0 0 1/2

, (40)

and so, vk = −1/2,−1/2, 1/2, 1/2. The states(which are the eigenvectors of v) attached to these co-efficients are |xk〉 = 1√

2− |L, ↓〉 + |R, ↑〉 ,− |L, ↑〉 +

|R, ↓〉 , |L, ↓〉+ |R, ↑〉 , |L, ↑〉+ |R, ↓〉. Plugging these co-efficients and states into Eq. 61, one finds |Φx〉 repre-sented in first quantization. Because this expansion is

unique [13] and there are four terms, the Schmidt rankis four, implying |Φx〉 is particle entangled.

Even when interactions are absent, it is possible tocreate probabilistic particle entanglement through a co-operation of quantum statistics and measurement [88].This can occur when distinguishable particles – eitherphotons via their polarization or atoms via their spin –are placed in an HOM interferometer. The role of mea-surement is to post-select on a particular symmetrizationmanifold of a distinguishable state, which allows for se-lection of a triplet (|Φ+〉) or singlet (|Φ−〉) in the dis-tinguishing degree of freedom, both of which describeparticle-entangled states. For example, consider a pairof bosonic atoms in opposing spin states undergoing adouble-well beam-splitter interaction. Because the par-ticles are distinguishable, they do not undergo HOM in-terference; half the time they will reside in the same welland half the time they will not. Due to the symmetriza-tion of the quantum state, post-selecting on those exper-iments in which the atoms end up in different wells inturn post-selects on the spin state of the atoms beingin a singlet. Hence, measurement allows the probabilis-tic synthesis of a particle-entangled state. Indeed, thisbasic principle underlies linear quantum computer archi-tectures with photons, whose lack of interaction does notpermit the same kind of entangling gates seen with ionsor neutral atoms.

Finally, even in the absence of a spin degree-of-freedom, it is possible to generate particle entanglement.In Sec. V we undertake a detailed analysis of interac-tions of spinless bosons, i.e. only particles with only aspatial degree of freedom, and we give an example of howthe Schmidt rank can characterize particle entanglementeven in this case (Sec. VB4).

V. INTERACTION IN TWO ANDMANY-PARTICLE INTERFERENCE

Photon experiments naturally achieve a noninteract-ing system where the dominant effect of quantum statis-tics is clear. However, in ultracold atomic systems in-teratomic interaction can easily be relevant. The inter-play between quantum statistics and interactions can behighly non-trivial and give rise to novel phenomena ab-sent in photonic systems, and of course is a natural partof condensed-matter systems. In this section we will fo-cus on this interplay. We start by considering the sim-plest possible system of two particles in a double well.Next we discuss the situation of multiple atoms in a dou-ble well and finally treat the most interesting case consist-ing of many atoms in many wells. We consider both thecase of bosonic and fermionic particles. When analyzingthe non-interacting case we of course arrive at analogousresults to those considered for multi-particle interferencewith photons in Sec. III C, but it is interesting to see thisappear out of formalisms more typical to interacting coldatoms, such as the Bose-Hubbard model.

14

A. Double well

1. Two atoms

From a physical standpoint, the main role of inter-actions is to introduce energy shifts that depend onthe number of particles per mode (site). These shiftscan then suppress the HOM dip observed with non-interacting particles. This effect can be cleanly seen inthe case of a double well potential with two bosonic par-ticles.

The Hamiltonian for the interacting atoms is given by

HDW = −J(a†1a2 + a†2a1) +

2∑j=1

U

2nj (nj − 1) , (41)

Here U is the interaction energy cost of having two parti-cles in the same lattice site and nj = a†j aj is the numberoperator. For two atoms the Hilbert space is spanned bythree states |2, 0〉, |0, 2〉, and |1, 1〉.

A good starting point to study the HOM effect is thestate |±〉 = |2,0〉±|0,2〉√

2. Due to the reflection symmetry of

the Hamiltonian only the |+〉 state couples to the |1, 1〉state. The |−〉 is decoupled from the rest. The dynamicsin the subspace spanned by the states |1, 1〉 , |+〉 reducesto

HDW =

(0 −2J−2J U

)(42)

Consequently if one prepares the system in the |ψ(0)〉 =|1, 1〉 state at time t = 0, the probability of remaining inthat state after some time t, P1,1 = |〈1, 1 |ψ(t)〉 |2 is givenby

P1,1(t) =

(1− 16J2

Ω2sin2[Ωt/2]

)(43)

with Ω2 = 16J2 + U2. From this expression one con-cludes that P1,1(t) ≥ U2/(16J2 + U2). The minimumvalue min(P1,1(t)) = 0 is only reached for noninteractingparticles, U = 0, at the HOM dip when tHOM = π/(4J).Any finite U introduces an energy shift between singlyand doubly occupied states. This shift introduces a netdetuning that brings the system away from the resonantcondition required for perfect destructive interference. Inthe strong interacting limit U J then P1,1(t) → 1.The absence of the HOM dip reflects the suppression oftunneling due to a large onsite repulsion. Two identicalfermions will also exhibit P1,1(t) = 1, due to the Pauliblockade. In fact, 1D bosons in the U/J 1 limit, andwith nearest-neighbor tunneling, exhibit the phenomenaknown as fermionization [89]. In this case, any observablethat can be written in terms of site occupation (number)operators maps to the one computed if the particles wereinstead non-interacting fermions. Nevertheless, stronglyinteracting bosons still obey bosonic statistics.

Fermionic particles with an internal spin degree of free-dom σ =↑, ↓ that allows the population of sites with twoparticles in the same mode can exhibit similar behav-ior to bosonic particles. For the relevant case where oneof the atoms is ↑ and the other ↓, the Hilbert space con-sists of the following states: |↑↓, 0〉,|0, ↑↓〉,|↑, ↓〉 and |↓, ↑〉.Here |↑, ↓〉 and |↓, ↑〉 are states with one atom per well inopposite spin configuration and |↑↓, 0〉,|0, ↑↓〉 states withtwo atoms in one well and zero in the other. The spins inthe doubly occupied states must be in a singlet spin con-figuration due to fermionic statistics. The Hamiltonianis now

HFDW = −J∑σ

(cσ†1 cσ2 + cσ†2 cσ1 ) + Un↑j n↓j , (44)

with cσi fermionic annihilation operators acting on a par-ticle at site i = 1, 2 and spin σ. A good basis to use is theone spanned by the states: |±〉 = (|↑↓, 0〉 ± |0, ↑↓〉)/

√2,

|S〉 = (|↑, ↓〉 − |↓, ↑〉)/√

2, and |T 〉 = (|↑, ↓〉+ |↓, ↑〉)/√

2.Because tunneling preserves the spin of the particles,

and the system exhibits reflection symmetry, the states|T 〉 and |−〉 can not tunnel and are decoupled from otherstates. They have energy 0 and U respectively. Thestates |S〉 and |+〉 on the other hand are coupled by tun-neling and obey exactly the same equations of motionsas those described by Eq. 42. The identical behavior be-tween these states and the corresponding bosonic statescan be easily understood by noticing that two fermionsin a spin singlet state, have the same symmetric spatialwave function as two identical bosons.

Now we proceed to discuss the case of many-atomsand many modes. When the atoms are non-interactingtheir behavior maps to the one we have already brieflydescribed in Sec. III C. Below, however, we will use aslightly different but complementary approach to de-scribe the physics. Not only can it help the reader togain a broader perspective, but can also facilitate theunderstanding of interference in the presence of interac-tions.

2. Many atoms

Let’s now consider the case of many bosonic particles,N , in a double well. To connect to the N = 2 case let usfirst understand the non-interacting limit. For that letus remind ourselves of the Hadamard Lemma,

e−iHDWta†jeiHDWt =

∞∑m=0

(−it)mm!

[HDW, a†j ]m, (45)

where

[X,Y ]m = [X, [X,Y ]m−1], [X,Y ]0 = Y (46)

Furthermore, we have

[a, (a†)N ] = N(a†)N−1 (47)[aN , a†

]= NaN−1 (48)

15

Using those properties for U = 0 one finds

[H, a†m]1 = Ja†3−m (49)

e−iHta†jeiHt =

∞∑m=0

(−itJ)m

m!a†2−(j+1 mod 2) (50)

= cos(tJ)a†j − i sin(tJ)a†3−j (51)

where we clearly see oscillations between the two wells.By applying the time-evolution for a finite time t =π/(4J), we obtain again a balanced beam-splitter in thedouble well.

To understand the role of interactions in the case ofmany bosonic particles, N , in a double well one canuse the Schwinger bosons representation. This natu-rally emerges by noting that the Hilbert space is spannedby the Fock states |n1, N − n1〉, with 0 ≥ n1 ≥ N ,which correspond to having n1 particles in one well andN − n1 in the other. Those states can be mapped to aspin S = N/2 particle with magnetic quantum numberSz = n1−N/2. Using this mapping the Hamiltonian canbe written as

HDW = −2J(Sx) + US2z , (52)

where Sx = (a†1a2 + a†2a1)/2, Sy = (a†1a2 − a†2a1)/(2i)

and Sz = (a†1a1 − a†2a2)/2. In Eq. 52 we have omittedconstant terms proportional to (n1 + n2)2 and (n1 + n2),since the Hamiltonian conserves particle number . ThisHamiltonian is the so called Lipkin-Meshkov-Glick Model[90–95]. From Eq. 52, it is clear that tunneling generatesrotations of the collective Bloch vector along the x direc-tion.

In the context of the HOM effect, the relevant ob-servable is the probability to populate a state with n1atoms in one of the wells and N − n1 in the other,Pn1,N−n1

(t) = |〈n1, N − n1|ψ(t)〉|2. When evaluated attHOM , Pn1,N−n1

(tHOM ) is the probability for a particu-lar fraction to be encountered in one of the two outputs ofthe beamsplitter as discussed in Sec. III C and displayedin Fig. 5. This probability can be computed from the mo-ments of Snz (t). Because 〈Snz (t)〉 =

∑Nm=0(A)nmPm,N−m

with Anm = (m − N/2)n, the later equation can be in-verted and we obtain

Pm,N−m(t) =

N∑n=0

(A−1)mn〈Snz (t)〉, (53)

where A−1 is the inverse of the matrix A. For the non-interacting system Sz(t) = cos(2Jt)Sz(0)+sin(2Jt)Sy(0),and therefore

〈Snz (t)〉 =⟨(

cos(2Jt)Sz(0) + sin(2Jt)Sy(0))n⟩

(54)

One clearly sees that 〈Snz (t)〉 exhibits interference fringeswith periodicity τ = π/J in agrement with Eq. 51. More-over at tHOM , 〈Snz (tHOM )〉 = 〈Sny (0)〉. If the initial state

Ê Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ

Ê

ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê‡ ‡ ‡ ‡ ‡ ‡

‡‡‡ ‡

‡‡‡

‡ ‡‡‡‡ ‡ ‡ ‡ ‡ ‡

Ï

Ï

Ï

Ï

Ï

ÏÏÏÏÏÏÏÏÏÏÏÏÏ

Ï

Ï

Ï

Ï

Ï

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

Fraction of particles in the left well

Prob

abili

ty

U/J=0 U/J=1

U/J=10

FIG. 6. Two-mode many-particle scattering example, includ-ing interactions between the particles. This plot is to be com-pared to Fig. 5, which uses the same parameters, but studiesnon-interacting particles. The interactions lead to a break-down of the suppression odd-particle events, a reduction ineven outcomes compared to the non-interacting case, and nar-row the distribution so that the initial input state is preservedin the output.

is an eigenstate of Sz(0) then all odd moments of Sy(0)vanish and only the even ones remain finite. If at t = 0,PN/2,N/2(0) = 1 at tHOM PN/2,N/2(tHOM ) = 0 reflect-ing destructive interference. For example for the N = 2case discussed before, P1,1(tHOM ) = 1 − 〈S2

z (tHOM )〉 =

1 − 〈S2y(0)〉 = 0, consistent with the expectations. In

Fig.6 we show P11,11(tHOM ) for the non-interacting sys-tem which is the double well analog of the beamsplittercase shown in Fig. 5.

As in the N = 2 case, a finite U introduces an energycost to move one particle from one well to the other dis-rupting the perfect destructive interference exhibited bynon-interacting particles, i.e. PN/2,N/2(tHOM ) > 0. Inthe large U limit the particle motion is suppressed andlimU→∞ Pm,N−m(t)→ Pm,N−m(0) (see Fig. 6) similar tothe N = 2 case. From this analysis we see that althoughthe many-particle system exhibits a more complex be-havior, one can identify similar features seen in the twoparticle case.

B. Bose-Hubbard: N atoms in L lattice sites

Let’s now consider the situation where N atoms areprepared in L sites of a deep optical lattice where to anexcellent approximation only nearest-neighbour tunnel-ing is relevant. We assume that interactions are localand experienced when two or more atoms occupy thesame lattice site. The Hamiltonian describing this situa-tion is the Bose-Hubbard model given by

HBH = −J∑〈i,j〉

(a†i aj) +U

2

N∑j=1

nj (nj − 1) , (55)

Here 〈i, j〉 means nearest-neighbor lattice sites.

16

The Bose-Hubbard model dynamics shares many com-mon features to the ones described in the double wellsetup. To illustrate the similarities we will first considertwo limiting cases, U = 0 and U → ∞. We will restrictthe analysis to a 1D lattice.

1. U = 0

In the non-interacting limit, the Hamiltonian reducesto HBH = −J∑〈i,j〉(a†i aj) and can be exactly diago-nalized by the unitary transformation (assuming peri-odic boundary conditions) bq = 1√

L

∑Lj=1 e

iqj aj . Hereq = 2π

L nq is the quasimomentum nq = 0, . . . , L − 1. Us-ing it, one can compute the evolution of the operator

a†n(t) = e−iHBHta†neiHBHt =

L∑m=0

Anm(t)a†m(0) (56)

here Anm(t) = im−nJm−n(2Jt), with Jn(x) Bessel func-tions of the first kind. Note that in contrast to thedouble-well case the dynamics is no longer periodic withJ . Instead the particle exhibits a quantum random walkas a function of time. Using the fact that for x n2

Jn(x) ∼√

2πx cos(x − π/4 − nπ/2), one can see that

transport of the particle is ballistic, because the proba-bility density of a particle initially localized at site n is〈nm(t)〉 = 〈a†m(t)am(t)〉 = |Jm−n(2Jt)|2, which expandslinearly in time.

In the case of N identical bosons initially lo-calized at sites n1, . . . , nN, the evolution of theequal time normal ordered correlator Pm1,...,mN

(t) =〈a†m1

(t)a†m2(t) · · · a†mN

(t)am1(t)am2

(t) · · · amN(t)〉 is given

by the permanent (see also Sec. III C) of the followingmatrix

Pm1,...,mN(t) = |Perm(A)|2

A =

An1m1

(t) An1m2

(t) . . . An1mN

(t)An2m1

(t) An2m2

(t) . . . An2mN

(t)· · · ·

AnNm1

(t) AnNm2

(t) · AnNmN

(t)

(57)

.This observable displays the key ingredients required

for the HOM: (1) quantum statistics, which manifestsin the requirement to compute the permanent of sin-gle particle functions, and, (2) coherent superpositionof paths, which arise because of the nonzero contribu-tion of the interference terms when the square of thepermanent is computed. To explicitly observe the con-tribution from (1) and (2), let’s consider the simplecase of two particles initially located at sites L/2 andL/2 + 1. The probability of finding the particle at thesame initial location is given by Pn1=L/2,n2=L/2+1(t) =

|An1n1

(t)An2n2

(t)+An2n1

(t)An1n2

(t)|2 = |J 20 (2Jt)−J 2

1 (2Jt)|2 ∼sin[4tJ ]2/(π2(tJ)2). At tHOM , PL/2,L/2+1(tHOM ) ∼ 0,which can be interpreted as the equivalent destructive

interference effect that gives rise to the HOM dip forL = 2, as well as a consequence of symmetrization (en-capsulated in the permanent) and a non-zero interferenceterm An1

n1(t)An2

n2(t)An2

n1(t)An1

n2(t). A similar picture holds

for many particles.

2. U → ∞

In the strongly interacting limit, the repulsion betweenthe particles forbids the population of sites with morethan one particle and in one dimension, the local observ-ables of the bosonic system resemble the ones of non-interacting fermions. In this limit, instead of the perma-nent one needs to compute a determinant

Pm1,...,mN(t) = |det (A)|2 (58)

If one analyzes the two particle case, as we did above,one obtainsPn1=L/2,n2=L/2+1(t) = |An1

n1(t)An2

n2(t) −

An1n2

(t)An2n1

(t)|2 = |J 20 (2Jt) + J 2

1 (2Jt)|2 ∼ 1/(π2(tJ)2).At tHOM , a constructive interference arises due to thedifferent sign of the term −An1

n1(t)An2

n2(t)An2

n1(t)An1

n2(t) in

striking contrast to the destructive one found for theU = 0 limit.

3. Finite U

For the intermediate interacting regime the dynamicsis more complex because finite U introduces additionalterms in Eq. 56 of the form

e−iHBHta†neiHBHt =

∑m

Anm(t)a†m

+U∑m

Mm(a†m)2am + etc. (59)

The above equation implies that the set of states that canbe produced with both the tunneling and interactions islarger than what can be produced with just tunneling[96]. During the dynamics tunneling and interactionscouple the modes and the particles, respectively [9] andthus mode and particle entangled states are not mutuallyexclusive. The tunneling produces the number fluctua-tions among the modes (in this case, these are the latticesites), but the interactions allow for states that are trulyparticle entangled. The states exhibit number correla-tions inaccessible to systems exhibiting only tunneling.To further elucidate this notion, in the next section wediscuss a specific example of a particle-entangled state ofspinless, indistinguishable bosons.

4. Application of Schmidt rank to Bose-Hubbard example

In Sec. IVD, we showed how the Schmidt rank indi-cates entanglement for the canonical example of a Bell-

17

State of distinguishable particles. The dynamics of theBose-Hubbard model illustrate how particle-entangledstates of spinless bosons — that is, indistinguishableparticles with only a spatial degree of freedom — canemerge as well. For example, in the recent publication ofRef. [9], a pair of strongly-interacting particles undergoesa quantum walk under the influence of the Bose-HubbardHamiltonian, with J U . In this limit the two particlesform a bound state which tunnels as a single object withan effective tunneling rate J2/U . Starting from the ini-tial state |ψ(0)〉 = (a†0)2 |0〉, the particles remain largelybound and undergo a quantum walk with an effectivetunneling rate of 2J2/U . In time, the state evolves as,

|ψ(t)〉 =∑n

inJn(4J2

Ut)

(a†n)2√2|0〉 , (60)

where n is the site index, Jn is the Bessel function ofthe first kind, and t is time. For times where morethan two terms in the summation are non-vanishing, thestate exhibits particle entanglement. This can be seen byrewriting |ψ(t)〉 in the first quantization decomposition ofEq. 61,

|ψ(t)fq〉 =∑n

inJn(4J2

Ut) |n〉1 |n〉2 , (61)

where |n〉 is the localized single-particle wavefunction ofsite n. This summation will exhibit more than two termswhenever the doublon is delocalized over more than twosites, which occurs quickly after a few effective tunnelingtimes of (J2/U)−1. By contrast, in the absence of inter-actions, two particles would independently tunnel leadingto a symmetrized wavefunction of Schmidt rank equal to1.

VI. ENTANGLEMENT ENTROPY

In addition to the challenges of classifying and gener-ating quantum entanglement, its quantification presentsyet another experimental obstacle. While HOM interfer-ence can be exploited to generate varying types of entan-glement, generalized HOM interference can be harnessedto detect entanglement in many-body states, as we dis-cuss in the following.

One way of mathematically quantifying entanglementis through the entanglement entropy. It quantifies thedegree of non-separability across a cut that partitionsthe system into subsystems. For example, for a chain ofspins, one might ask how the spins in the right half of thechain are entangled with those in the left half. Or, for aBose-Hubbard chain, we might consider mode entangle-ment between the right and the left half of the system.In general, we can ask whether a state is separable byintroducing a cut through the system, thereby definingtwo subsystems each with an associated complete basisin the left and right Hilbert spaces (HL andHR) (Fig. 7).

For any bipartite state |ψ〉, there is a unique decom-position — the Schmidt decomposition — such that,

|ψ〉 =∑j

cj |χLj 〉 |χR

j 〉 (62)

where for the different terms in the sum we have〈χLi |χL

j 〉 = 〈χRi |χR

j 〉 = δij , and |χL,R〉 ∈ HL,R. Thestate |ψ〉 is non-separable whenever there is more thanone non-vanishing term in this sum; otherwise, the stateis a product state with respect to the partitioning. En-tanglement entropy is linked to the degree of mixing ofthe reduced density matrix of just one of the subsystems,namely,

ρS =∑j

|cj |2 |χSj 〉 〈χS

j | (63)

where S = L, R. Whenever |ψ〉 is entangled, ρS is a mixedstate because there is more than one term in the sum inEq. 63. When a subsystem state is mixed and the fullsystem state is pure, the full system state is entangled.Hence, we refer to the loss of local purity, or, equivalently,the generation of local entropy, as entanglement entropy.It can be quantified with the von-Neumann entanglemententropy SvN = Tr(ρLlog(ρL)) = Tr(ρRlog(ρR)) or theRényi entropies Sn = log(Tr(ρnL)) = log(Tr(ρnR)). Then = 2 Rényi entropy is the logarithm of the purity of thesubsystem density matrix.

By definition, entanglement is invariant under coher-ent dynamics within the individual subsystems (local uni-taries) [97]; that is, a state UL⊗UR |ψ〉, with unitaries ULand UR on the left and right subsystem, carries the sameamount of entanglement as the state |ψ〉. The experi-mental assessment of entanglement following a setup withboth left and right subsystems as input of a beam split-ter [98, 99] will not necessarily assign the same amountof entanglement to |ψ〉 and to UL ⊗ UR |ψ〉, but thereare functions of quantum states that are invariant underall local unitaries. The norm is a particularly simple ex-ample, but it provides no information on entanglementproperties. All other invariant functions are necessarilynon-linear in |ψ〉 and 〈ψ| (or the density matrix % in thecase of mixed states), and the simplest non-linear func-tion is bi-linear. Because all measurements on quantummechanical systems yield expectation values 〈ψ|A |ψ〉 ofan observable A, i.e. a linear function, non-linear func-tions can be measured only with extra effort.

A way to realize the measurement of bilinear functionsis to work with two versions, also called twins, of a quan-tum system simultaneously. The two twins of the lefthalf are described in terms of the Hilbert space HL⊗HL

rather than the Hilbert space HL of a single twin. Anal-ogously, the two twins of the right half are described interms of the Hilbert space HR ⊗ HR. The central ideais to prepare the two twin systems, each consisting of aleft and a right twin, in the same fashion [100, 101]. Thestate of both twins together is then given by |ψ〉1⊗ |ψ〉2,where |ψ〉 is the state of an individual twin, and the in-dex ‘1’ and ‘2’ labels the two twins. Expectation values of

18

such twin states provide a characterization and quantifi-cation of entanglement, which has the desired invarianceproperties [102–106].

The degree of mixing of reduced density matrices %L,Rthat quantifies the entanglement of the underlying purestate [107–110] can be expressed as

tr%2S = tr ΠS (%S ⊗ %S) (64)

in terms of two identical, reduced density matrices %S,where ΠS is the permutation operator defined via therelation ΠS |i〉S1

⊗ |j〉S2= |j〉S1

⊗ |i〉S2, where |i/j〉S1

are subsystem states of the first twin of subsystem Sand |i/j〉S2

are states of the second twin. Consequently,proper entanglement measures for pure states |ψ〉 can beconstructed based on expectation values of the permuta-tion operators ΠL and ΠR with respect to the twin-state|ψ〉 ⊗ |ψ〉 [111, 112].

In the case of mixed states, the local purities alone cannot assess entanglement properties, because for examplea direct product of mixed single sub-system states wouldbe consistent with low local purities. The entanglementof mixed states can, however, be estimated if in additionto local purities also the purity of the entire state is ac-cessible [113, 114]. The resulting quantitative estimateis particularly good for weakly mixed states [115]; and itremains valid under imperfect preparation of the copies[116] as long as they remain uncorrelated [117].

Such assessments of entanglement in terms of puritieswere implemented for photons [118–120] but current ex-periments with atomic gases permit to implement analogmeasurements for substantially larger systems [121–123].

The realization of the required measurement of permu-tation symmetry for the twin-states of each subsystemcan indeed be based on a beam splitter with the action

a†S1→ (a†S1

+ a†S2)/√

2

a†S2→ (a†S1

− a†S2)/√

2 ,(65)

where a†S1and a†S2

are the creation operators for the twotwin-subsystems under consideration. Its action on aFock state

|ij〉S = |i〉S1⊗ |j〉S2

=1√i!j!

(a†S1)i(a†S2

)j |0〉 (66)

reads

|ij〉S →(a†S1

+ a†S2)i(a†S1

− a†S2)j

2i+j√i!j!

|0〉 = |χij〉S . (67)

Because the application of the permutation Π amountsto the exchange of a†S1

and a†S2, from Eq. (67) one can

directly read off the relation

ΠS |χij〉S = (−1)j |χij〉S . (68)

That is, the beamsplitter maps states with an even num-ber of particles in the second twin-subsystem onto statesthe are symmetric under Π, whereas states with an odd

number of particles in the second twin-subsystem aremapped onto anti-symmetric states. Because the beam-splitter operation is self-inverse, also any symmetric stateis mapped onto a state with an even number of particlesin the second twin-subsystem, and any anti-symmetricstate is mapped onto a state with an odd particle num-ber. Consequently, local purities can be measured withthe help of the beamsplitter in Eq. (65) and a measure-ment of particle number that can be realized with stateof the art optical read-out methods [124, 125].

For the measurement of the global purity of an N -bodysystem one merely needs to notice that the global permu-tation is just the tensor product of all local permutations.That is, the ability to measure all local purities is suffi-cient to measure the purity of the N -body states, andof the purities for all M -body (M < N) reduced densitymatrices [123, 126].

In a recent experiment [29] with 87Rb atoms in an op-tical lattice entanglement for systems of up to four atoms(per twin) was observed. The system can be modeled wellwith a Bose-Hubbard Hamiltonian (Eq.(55)). It inducesthe beam splitter relation defined in Eq. (2), that differsfrom Eq. (65) in terms of the phase shift that a particleacquires when tunneling. The difference between bothbeam splitters could be compensated through appropri-ate phase shifts exp(±iπ2 a

†S2aS2

) before and after thebeam splitter. The measurement of the particle numberis un-affected by any phase shift after the second beamsplitter, so that no phase shift after the beam splitterneeds to be applied. If, however, the necessary phase shiftbefore the beam-splitter is not applied, a measurementof particle number in the second twin systems, results inthe measurement of tr% exp(iπ2 a

†S2aS2)% exp(−iπ2 a

†S2aS2)

rather than tr%2. Due to the Cauchy-Schwarz inequal-ity (tr%1%2)2 ≤ tr%21 tr%22, this still results in a reliablelower bound of the purity. In addition, for Fock states,the phase shift before the beam splitter results in a globalphase without observable consequences, so that the beamsplitter Eq. (2) including phase shift results in the correctassessment of purity for all experimentally investigatedstates.

In the regime of strong interactions U J , the groundstate is a Fock state with no mode entanglement, butwith growing J/U , the atoms get increasingly delocalizedand mode entanglement builds up. Rather high puritiesfor single-site and two-site reduced density matrices wereobserved for U/J & 10, and, in the regime 10 & U/J & 1a rapid decrease in purity was observed, consistent withthe theoretical expectation. The purity of the four-atomstate showed no significant change in purity during thevariation of U/J , which underlines that the changes inlocal purities are indeed a result of mode entanglement,and not merely of an increase of entropy that could resultfrom experimental imperfections.

In the superfluid regime J U , one expects that thedynamics following the preparation of a Fock state wouldresult in periodic oscillations of mode entanglement. Fora system with two atoms and two sites, the dynamics fol-

19

Entangled stateProduct state

TracePure TraceMixed

(a) (b)

even particle # in Output 2

even particle # in Output 1 odd particle #

odd particle #

2 identical, N-particlebosonic states

FIG. 7. Entanglement entropy. (a) A quantum state can be partitioned into subsystems L and R, that represent subspacesof the full Hilbert space in which the quantum state lives. When the quantum state is pure and non-entangled, it can bewritten as a separable product of pure states in L and R. When it is pure and entangled, the full quantum state cannot bewritten this way, and in either subspace the state is mixed, i.e. entropic. (b) Through the use of beamsplitter interactionsand number-resolved counting, it is possible to measure local and global quantum state purities. In particular, the countingstatistics (the parity) after the application of a beamsplitter is sensitive to quantum state overlap of interfered states, fromwhich purities can be deduced. This allows for the measurement of entanglement entropy. Adapted from Ref. [29].

lowing the initial state |1, 1〉, would result in a balancedsuperposition of all three accessible Fock states |0, 2〉,|1, 1〉 and |2, 0〉 (at Jt = arctan(

√2)/(2π)) ∼ 0.68, i.e. the

state with highest mode entanglement achievable in thissystem. At Jt = π/(4) one would expect a balancedsuperposition of the two states |0, 2〉 and |2, 0〉 whichhas less mode entanglement. Subsequently the systemevolves again into a maximally entangled state, followedby the separable initial state [29].

VII. CONCLUDING REMARKS

Increasingly, ultracold atom experiments have turnedto single-atom imaging and manipulation to gain a newlevel of understanding. With this perspective, a varietyof experiments akin to those with single photons haveappeared, and the Hong-Ou-Mandel effect with atoms isexemplary of such newly-enabled experiments. A goalof this article was to discuss fundamental concepts thatconsequently arise – such as many-particle interferenceand entanglement of identical particles – which until nowwere not prominent players in ultracold atom studies.Bringing these concepts to the fore also emphasizes thenew opportunities that arise with ultracold atoms, due toefficient detection, the ability to introduce interactions,and by being massive. Applying these capabilities toexperimental directions inspired by photonic protocolshave, and will, continue to lead to new directions.

Future experiments may harness many-particle inter-ference of atoms in a variety of ways. As addressed above,the incorporation of quantum beamsplitters into coldatom technology has allowed measurement of entangle-ment entropy through many-particle interference, yield-ing an intersection of ideas in quantum optics, atomicphysics, and condensed matter physics. Many-particle

interference is also fundamental to the boson samplingproblem, for which cold atom technologies could augmentthe accessible system sizes and complexity. Furthermore,photons are routinely entangled in polarization or fre-quency degrees of freedom by utilizing measurement andquantum statistics; the entanglement results from mul-tiparticle interference, and circumvents the challenge ofproducing entanglement with non-interacting particles.

From a theoretical standpoint, the subtleties surround-ing entanglement of identical particles raises several ques-tions. This article addressed the case of entanglementbetween two particles, relying on the CSOP prescription.As just noted, there are many future pursuits that en-tail large samples of atoms; delineating the various formsof entanglement in such cases will be of significant im-port, and are not within the scope of the CSOP criterion.Relatedly, experiments are naturally imperfect and cre-ate only approximately pure quantum states, that is, thestates are mixed: theoretically defining particle entangle-ment in mixed quantum states of identical particles willbe similarly important, both for classifying and devisingmethods for quantifying entanglement.

ACKNOWLEDGMENTS

We thank M. Foss-Feig, M. Greiner, M. Holland, B.J. Lester, and Yiheng Lin for conversations. We ac-knowledge funding from the NSF under grant numberPHYS 1734006, the David and Lucile Packard Founda-tion, the Office of Naval Research under award numberN00014-17-1-2245, AFOSR FA9550-13-1-0086, AFOSR-MURI Advanced Quantum Materials, NIST and DARPAW911NF-16-1-0576 through ARO.

20

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