# Signatures of indistinguishability in bosonic many-body dynamics

###### Abstract

We introduce a measure of the indistinguishability of bosonic, multimode many-particle configurations, and demonstrate its unambiguous relationship with the evolution of observables that probe two-particle interferences, even for finite inter-particle interaction strengths. In this latter case, the degree of indistinguishability also controls single-particle observables, due to the interaction-induced coupling of single-particle amplitudes.

Indistinguishability and interactions are the essential ingredients of the extraordinary diversity of physical phenomena displayed by many-particle quantum systems. In (thermodynamic or dynamical) equilibrium, these range from macroscopic quantum states of matter, as realized by Bose-Einstein condensates BEC or other, even more exotic phases ExoticPhases1 ; ExoticPhases2 , to the entanglement of distinct degrees of freedom Entanglement . The dynamical quantum manifestations of interactions and indistinguishability got into focus more recently, and offer an overwhelmingly rich phenomenology, from different types of localization to relaxation and transport phenomena Nandkishore2015 ; Altman2015 . Yet, little effort has actually been devoted to a systematic discrimination of indistinguishibility- versus interaction-induced dynamical features, even though the paradigm of Hong-Ou-Mandel (HOM) interference HOM:PRL87 ; MKaufman:Sci14 on a fair beam splitter is strongly indicative of non-trivial dynamical effects MTichy:PRA11 ; YSRa:PRCLE13 ; SHTan:PRL13 ; NSpagnolo:Nat13 ; SAgne:PRL17 ; AJMenssen:PRL17 exclusively due to the involved (non-interacting) particles’ indistinguishability. Only recently has the HOM line of thought been fully unfolded to the realm of truly many-particle dynamics, mostly in photonic settings MTichy:NJP12 ; SAaronson:ToC13 ; MTillmann:PRX15 ; VTamma:QIP16 ; VSShchesnovich:PRL16 ; JDUrbina:PRL16 ; MWalschaers:NJP16 ; ACrespi:NPho13 ; MABroome:Sci13 ; JBSpring:Sci13 ; Tillmann:NatComm13 ; JCarolan:NPho14 ; LLatmiral:NJP16 ; HWang:Nat17 , and some work andersson_quantum_1999 ; MTichy:PRA12 ; YLahini:PRA12 ; XQin:PRA14 ; PPreiss:Science15 ; WJMullin:PRA15 ; BGertjerenken:PL15 ; GDufour:arXiv17 did already touch upon the competition of interaction and indistinguishability in systems with few modes and/or particles.

It is the purpose of our present contribution to systematically expand this line of research, by exploring the impact specifically of the particles’ indistinguishability on the time evolution of interacting many-body systems, in the general framework of bosons which occupy a discrete set of coupled modes and belong to several mutually distinguishable species (comprising the experimentally readily accessible Bose-Hubbard model).
We first define a measure of the *degree of indistinguishability* (DOI) of many-body configurations which is, in contrast to other DOI measures HGuise:PRA14 ; VShchesnovich:PRA14 ; VSShchesnovich1:PRA15 ; MTichy:PRA15 ; MWalschaers:PRA16 introduced in the context of many-photon scattering, adapted to the study of systems evolving continuously in time from an arbitrary initial Fock state. Our measure is derived in the non-interacting case, where it has an intuitive interpretation in terms of two-particle interference and correlates directly with the *variance* of experimentally accessible single-particle observables (1POs), as demonstrated in Fig. 1. In the presence of interactions, we show that the DOI is even imprinted on the bare *expectation values* of 1POs, and still correlates with their fluctuations.

Let us consider a general many-particle system with a discrete set of mutually coupled external modes (e.g. photonic input and output modes coupled via a beam splitter array, or tunnel-coupled sites in an optical lattice), and with a discrete set of internal states, or ‘species’, (e.g. photon polarization
or hyperfine states of atoms). In this setting, we want to establish a quantitative measure of the DOI of a many-body configuration represented by a Fock state
where is the number of bosons of species in mode .
To investigate the direct consequences of (in)distinguishability, we assume that the Hamiltonian of the system is *species-blind*, in the sense that it neither resolves, nor modifies, the internal degree of freedom of the particles SM .

We first consider the non-interacting case, where the Hamiltonian takes the general form of a species-blind 1PO, , and the time-evolution of the bosonic operators is given by the matrix elements of the single-particle unitary evolution operator: . Under these conditions, many-particle interference is known to manifest itself only on the level of two-particle or higher order observables KMayer:PRA11 . Indeed, the expectation value of a general species-blind two-particle observable (2PO), , in a Fock state reads

(1) |

where is the total number of particles in mode . The amplitudes in the first line of Eq. (1) correspond to two-particle paths where one particle moves from mode to , is taken to by the observable, and from there moves back to , while the other particle follows the path

The above motivates the definition of a DOI measure for a many-boson Fock state as the sum over all the multiplicities of the crossed contributions in (1):

(2) |

where the normalization ensures values between , for maximally distinguishable states (when each particle is of a different species, or when all particles of the same species occupy the same mode), and , when all particles are mutually indistinguishable (i.e. only one species is present).

We find that our measure manifests itself most directly when the 2PO under consideration is the square of a species-blind 1PO, , as this ensures that the factors appear dressed by real and positive coefficients in Eq. (1). In particular, we consider fluctuations of on-site density operators ,

(3) |

with amplitudes . By averaging over time and subtracting the -independent contribution in Eq. (3), we define the level of fluctuation (LOF) of the 1PO in the Fock state ,

(4) |

where the overbar denotes time average, and correspond to in a state with the same total density distribution as but with () or () (i.e. in a fully distinguishable configuration, or in the state involving only one species, respectively ExpDOI ). Comparison of Eqs. (2) and (4) shows that, for a narrow distribution of the over , the measurement of the LOF directly gives access to the DOI. Specifically, we find that SM ,

(5) |

where and are, respectively, the standard deviation and the mean of the for all pairs .

It is instructive to study the behavior of our DOI measure in the special case of a two-mode system, such as a multi-component, species-blind, non-interacting Bose-Hubbard Hamiltonian (BHH) SM with sites.
In any two-mode system, only one coefficient, , contributes to , which therefore reproduces *exactly* the DOI measure .
For two bosonic species , and fixed total particle number , the configuration space of the system is determined by three parameters: the mode population imbalance, , and the species imbalances per site, and . The DOI measure then reads

(6) |

For , the space of non-equivalent Fock configurations is spanned by and , and is charted in Fig. 3 for . According to Eq. (6), having all particles of the same species [] corresponds to , whereas complete spatial separation of the two species [ implies . As shown in the top inset of Fig. 3, these two initial states seed, respectively, maximum and minimum values of the density fluctuation , as a direct consequence of the presence or absence of the two-particle crossed terms in Eq. (1) and Fig. 2(a). Furthermore, all states with have and yield the same fluctuation of 1POs over time if the bosons do not interact. Note that the DOI does not solely depend on the repartition of particles among species, but also [recall the structure of the crossed term in (1)] on how the species are initially distributed over the external modes. Indeed, can be written in terms of the inverse participation ratios (with respect to the external modes) of the density distributions of each species IPR . This interplay between external and internal degrees of freedom was discussed for the indistinguishability of two photons MCTichy:FdP13 ; PTurner:arXiv16 , but it has not been clearly resolved in previously introduced DOI measures HGuise:PRA14 ; VShchesnovich:PRA14 ; VSShchesnovich1:PRA15 ; MTichy:PRA15 ; MWalschaers:PRA16 .

Let us proceed to larger numbers of modes and species: We numerically demonstrate a remarkable - correlation in a species-blind BHH with sites and a total of non-interacting bosons, as shown in Fig. 1. We sample uniformly initial states out of the total available Fock space for each of the cases of , and distinct species. For each state , the fluctuation is calculated using Eq. (4)
and plotted versus the DOI value of , together with
the bound provided by Eq. (5).
We observe that the - correlation becomes even more pronounced for larger and/or DOIbound . These results demonstrate that our DOI measure is at the core of the time-evolution of 2POs in non-interacting systems [see Eq. (1)], and furthermore, that it can be characterized from the *variance*
of 1POs such as the on-site density of cold bosons in optical lattices, which are experimentally readily accessible.

We now expand our analysis to the interacting case, where, remarkably, the DOI is revealed
already in the *expectation value* of 1POs.
To see this, we complement the Hamiltonian by a species-blind, two-body interaction term .
For simplicity, we elaborate on the case of contact ‘on-mode’ interactions, ; our subsequent results, however, are valid for the most general . Solving the Heisenberg equation of motion
perturbatively in shows that, over time, any 1PO develops contributions in the form of second and higher order observables, whose importance is weighted by the interaction strength:
SM .
Here, is a 1PO corresponding to the non-interacting evolution, with an expectation value independent of the particles’ (in)distinguishability. In contrast,
is a 2PO, and its expectation value reads

(7) |

where is the amplitude of the ladder and crossed two-particle paths arising due to the interaction SM . These are illustrated in Fig. 2(b) for the one-mode density as 1PO.

By analogy with the result for 2POs in the non-interacting case [compare the structure of (7) to that of (1)], the DOI measure can be identified in the expectation value of 1POs in the interacting case, dressed by the amplitudes .
Interactions therefore imprint the DOI on the bare *expectation value* of 1POs. This is demonstrated in Fig. 4, where we show the expectation value for the two-species two-site BHH BJJBE1 ; BJJBE2 ; BJJBE3 ; BJJBE4 ; BJJBE5 ; BJJBE6 ; BJJBE7 ; MTichy:PRA12 ; BJJBE8 ; GDufour:arXiv17 subject to a tilt (to ensure a non-vanishing correction Usym ):

Within a regime of small , which depends on the system under consideration, the evolution of the on-site density is well described by Eq. (7). In particular, the initial slopes of the curves in Figs. 4(a2) and (b2) are uniquely determined by . For larger interaction strengths and/or times, higher order terms contribute to the expectation value of the observable, which additionally probe three-particle and higher processes [causing, e.g., states with the same to exhibit independent trajectories – see panels (a2) and (b2) of Fig. 4]. Nonetheless, the correlation between and persists beyond first order perturbation. This suggests that our measure of the DOI based on two-particle paths remains meaningful even in the presence of higher order processes.

Indeed, also the long-time signals , although more involved than in the non-interacting case, due to the appearance of extra frequencies (compare the top and bottom insets of Fig. 3), indicate that the time-averaged density fluctuation still correlates with the DOI of the initial state. This is demonstrated in Fig. 5, where we show as a function of SM .

For states with a homogeneous initial distribution of particles (first row of Fig. 5), one observes a striking correlation between and over the entire range of interaction strengths (also for – not shown in the figure). For states with a strongly imbalanced initial distribution of particles (second row of Fig. 5), this correlation also holds for weak interactions, but is lost for larger values of . Closer inspection of the system’s spectral structure shows that, in the regime of strong interactions, the dynamics is dominated by Fock states with the same interaction energy as the initial state, which, in the imbalanced case, include states with dissimilar density distributions [e.g. and in the double well]. The interaction-mediated higher-order processes connecting these states then contribute predominantly to , breaking the correlation to the DOI measure . A detailed characterization of this effect will be the subject of future work.

We conclude by generalizing our DOI measure to superpositions of Fock states , where each term has the same total density distribution but a different number of particles per species. The expectation value of a species-blind observable in such a state is additive, since by definition the observable cannot change the number of particles per species. Thus, we can additively generalize our DOI measure as . For the exemplary Hong-Ou-Mandel state , our measure coincides with Mandel’s indistinguishability parameter LMandel:OL91 . Using the additivity property of , the effects of indistinguishability in various generalizations of the Hong-Ou-Mandel setup (e.g. non-monotonicity in four-photon interference YSRa:PRCLE13 ; MTichy:PRA11 ) can easily be understood.

We have introduced a measure of the degree of indistinguishability (DOI) of a many particle quantum state, which is derived from the structure of two-particle transition amplitudes, and readily accessible by experimental monitoring of the fluctuations of one particle observables. Our measure incorporates the significance of internal as well as of external degrees of freedom for the DOI and for the associated many-particle interferences, and notably exploits the information encoded in the continuous dynamical many-particle evolution — inaccessible in many-paricle scattering scenarios. Our analysis also shows that interaction-induced interference reveals the DOI already in the expectation value of single particle observables, and that the DOI remains a meaningful concept in the presence of interactions. It appears suggestive that the characteristic dynamical features here described must have a structural counterpart in the underlying energy spectra and many particle eigenstates.

T.B. expresses gratitude to the German Research Foundation (IRTG 2079) for financial support and thanks Mattia Walschaers and Florian Meinert for helpful discussions. G.D. and A.B. acknowledge support by the EU Collaborative project QuProCS (Grant Agreement No. 641277). Furthermore, G.D. is thankful to the Alexander von Humboldt foundation. The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 40/467-1 FUGG.

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Supplemental Material

## I Multi-species Bose-Hubbard model

As a particular realization of the class of systems discussed in the manuscript, we consider a one-dimensional Bose-Hubbard Hamiltonian (BHH) for many bosons which may be mutually distinguishable by an internal degree of freedom . The BHH describes bosonic atoms restricted to the first energy band of an optical lattice, and it contains a nearest-neighbor hopping term with typical energy and a two-body on-site interaction of strength . In this case, the different species may correspond to different hyperfine atomic states.
The Hamiltonian of the system is chosen to be *species-blind*, i.e. it preserves the species type and acts on all bosons in the same way: All bosons have the same hopping energy independently of , as well as the same inter and intra-species interaction, . The measurement of a species-blind observable does not require to resolve the internal degree of freedom of the bosons in the measurement process. The species-blindness condition is also known in the literature as isospecificity MTichy:PRA12 .
The total Hamiltonian reads , where

(8) | ||||

(9) |

in terms of creation (annihilation) bosonic operators (), , . We consider Hamiltonian for a system comprising lattice sites in the presence of hard-wall boundary conditions.

### i.1 Dynamics in the non-interacting case

In the non-interacting case (), the dynamics of the system can be solved analytically. Heisenberg’s equations of motion for the bosonic operators read

(10) |

and therefore

(11) |

where are the matrix elements of the single-particle evolution operator in the single-particle Wannier basis.

For hard-wall boundary conditions, one has

(12) |

In order to asses the correlation between the indistinguishability measure [Eq. (2)] and the level of fluctuation (LOF) [Eq. (4)], we need to evaluate the time-averaged coefficients . The time average can be easily carried out analytically. However, the explicit evaluation of the resulting sums is rather involved. Nonetheless, the distribution of the values for and its dependence on the number of modes can be straightforwardly obtained numerically.

This is shown in Fig. 6. The mean values are independent of the mode considered. The standard deviation also shows a common trend with independently of . An exception occurs when the number of sites satisfy , i.e. if is the centre of the mirror symmetry of the system, when we observe a jump in roughly by a factor of two. Nevertheless, this isolated resonant increase does not change the global decay of with . As grows, the data is well fitted by the functions

(13) | ||||

(14) |

as demonstrated in Fig. 6. The ratio decreases with as

(15) |

approaching the minimum value as .

## Ii Estimation of the bounds for the correlation

In order to derive a bound for the deviation of the LOF of the density operator from the degree of indistinguishability (DOI) measure , we express both quantities in terms of weighted averages of non-diagonal elements of matrices defined on pairs of sites :

(16) |

The DOI and LOF can be respectively written as and , where

(17) |

and the product is performed entrywise: . We now express the difference between LOF and DOI in two different ways:

(18) |

Given that , we find, using successively both expressions of :

(19) | ||||

(20) |

where we have approximated the right hand side of the inequalities by assuming that is on the order of the unweighted standard deviation of the distribution of , while the weighted average is approximated by its unweighted counterpart . These approximations are valid for narrow enough distributions satisfying . The resulting estimation of the deviation of the LOF from the DOI is thus:

(21) |

A rigorous bound can be obtained by noting that and so that

(22) |

## Iii Diagrammatic representation of time-dependent observables

We give a diagrammatic interpretation of the time-dependent expectation values (1) and (7), both of which carry a signature of the DOI of the many-body configuration. We recall the most general form of species-blind single-particle observables (1POs) and two-particle observables (2POs):

(23) | ||||

(24) |

and their expectation values in the Fock state :

(25) | ||||

(26) |

We first consider a species blind 2PO, , of the form (24), with coefficients , evolving under a species-blind, non-interacting Hamiltonian . In the Heisenberg picture, is also a 2PO with matrix elements

(27) |

The coefficients are single-particle matrix elements of the evolution operator , as defined underneath Eq. (11). This expression is represented graphically in Fig. 7. The corresponding expectation value reads

(28) |

Diagrammatically, it is obtained by identifying each leg on the left of the diagram to one on the right and to a populated mode in . Taking and leads to the ladder term, with coefficient . For , one can also identify and , yielding a crossed term with coefficient .