Entanglement induced interactions in binary mixtures

# Entanglement induced interactions in binary mixtures

## Abstract

We establish a conceptual framework for the identification and the characterization of induced interactions in binary mixtures and reveal their intricate relation to entanglement between the components or species of the mixture. Exploiting an expansion in terms of the strength of the entanglement among the two species, enables us to deduce an effective single-species description. In this way, we naturally incorporate the mutual feedback of the species and obtain induced interactions for both species which are effectively present among the particles of same type. Importantly, our approach incorporates few-body and inhomogeneous systems extending the scope of induced interactions where two particles interact via a bosonic bath-type environment. Employing the example of a one-dimensional spin-polarized ultracold Bose-Fermi mixture, we obtain induced Bose-Bose and Fermi-Fermi interactions with short-range attraction and long-range repulsion. With this, we show how beyond species mean-field physics visible in the two-body correlation functions can be understood via the induced interactions.

Introduction.– The experimental progress in ultracold atomic systems has made them an ideal platform for the investigation of quantum many-body physics due to the extraordinary controllability of the trapping geometries and the interactions (1). After the realization of Bose-Einstein condensation (BEC) (2) and degenerate fermionic gases (3), increasing interests also raised in the studies of atomic mixtures, covering Bose-Bose (4); (5); (6), Bose-Fermi (7); (8); (9); (10); (11); (12) and Fermi-Fermi mixtures (13); (14); (15). For such mixtures, the induced interaction (16), which is acting effectively between the particles of one species due to the presence of the second species is of particular interest. For example in a Bose-Fermi mixture, the latter can alter the Fermi-Fermi interaction (17) or the Bose-Bose interaction (18), both becoming evident in the excitation spectrum. Originating from particle and condensed-matter physics, it has become a well-established concept that the exchange of bosonic (quasi-) particles leads to induced interactions. For a crystalline solid, two electrons may acquire an attractive interaction by the exchange of phonons, according to the famous Fröhlich Hamiltonian (19). In addition to the induced electron-electron interaction, the latter includes also a periodic potential for the electrons. Such an approach significantly simplified the way of studying mixtures while, at the same time, it successfully describes the in-depth physics including, e.g., the celebrated BCS theory for superconductivity (20). Note, however, that the physics of the Fröhlich Hamiltonian focuses on the so-called system-bath regime where the feedback of the electrons on the phonons is usually neglected.

Beside the efforts on macroscopic ensembles, recent experiments (21); (22); (23); (24); (25) also focus on few-body physics revealing for example the formation of the Fermi sea (21), the fermionic pairing with an attractive interaction (22), as well as the fermionization of two distinguishable fermions (25), which pave the way for future experiments on few-body mixtures. Here, however, the notion of an infinitely extended bath-type environment of non-interacting modes ceases to be valid.

The focus of this work is to establish a conceptual framework for induced interactions. In particular, we show how the entanglement among the species creates the induced interactions and derive a general expression for the effective single-species Hamiltonians including the induced interactions. Our approach relies on an expansion in the weak-entanglement regime being a necessary condition for the validity of the effective single-species description. Importantly, the analysis of the induced interaction is derived from the eigenstates, which in general can be obtained by any ab-initio numerical method, making it applicable for both the ground state and the excited states of the mixture. Moreover, our approach naturally incorporates few-body and inhomogeneous systems extending the scope of induced interactions where two particles interact via a bosonic bath-type environment. By using a one-dimensional spin-polarized ultracold Bose-Fermi mixture as an exemplary system, we illustrate how induced interactions can give qualitative insights into a many-body state. We compute the ground-state of the latter system on a species mean-field level and via the ab-initio multi-layer multi-configuration time-dependent Hartree method for mixtures (ML-MCTDHX) (26); (27), which accounts for all correlations. We show how, e.g., two-body correlations, not captured in the species mean-field, can be understood via the induced interactions and that a reduction to an effective single-species description is qualitatively valid.

Theoretical approach.– The generic Hamiltonian for a binary mixture is given by , where denotes the Hamiltonian for species and , respectively. represents the interaction between the two species. For the moment, we restrict to be a local two-body interaction. Employing a Schmidt decomposition (28), an exact eigenstate of the mixture can be uniquely written in the form

 |Ψ⟩=∞∑i=1√λi |ψAi⟩|ψBi⟩, (1)

where are the Schmidt numbers with , which are real positive numbers and obey the constraint originating from the normalization of the wavefunction . The Schmidt numbers directly reveal the strength of the entanglement, note that for the case while all , the mixture is non-entangled (28). The denotes the -th Schmidt-orbital for species with . In addition, all Schmidt-orbitals form an orthonormal basis. It is worthwhile to note that the species mean-field (SMF) approximation assumes the total wavefunction to be of a simple product form, i.e. , as aforementioned, it excludes entanglement among the two species. For a mixture described by , such a product ansatz implies that the mutual impact of the species is merely an additional (induced) potential (29). It should be pointed out, albeit the simplifications, the SMF approximation is already beyond the Hartree-Fock mean-field approximation, since the later simply assigns a single permanent (Slater determinant) to bosonic (fermionic) species, while keeping the total wavefunction as (30).

Projecting with the -th Schmidt-orbital and multiplying with the from the left onto the time-independent Schrödinger equation for the mixture, we obtain

 ∞∑i=1√λq√λi⟨ψ¯σq|^H|ψ¯σi⟩|ψσi⟩=μq|ψσq⟩, (2)

with , for , and , with being the eigenenergy of the state . Moreover, due to the orthogonality of the Schmidt-orbitals, we have , with representing the transition amplitude between the Schmidt-orbital products and (26). Note that, as long as the Hamiltonian of the mixture posseses the time-reversal symmetry, the eigenfunction can always be chosen real (31), and since all the Schmidt numbers are real as well (28), the are real numbers, i.e. .

With this knowledge, we rewrite Eq. (2) as

 ⎡⎣λ1H¯σ11|ψσ1⟩+∑i≠1√λ1√λiH¯σ1i|ψσi⟩⎤⎦=μ1|ψσ1⟩, (3) |ψσq⟩=[μq−λqH¯σqq]−1⎡⎣∑j≠q√λj√λqH¯σqj|ψσj⟩⎤⎦ (q>1), (4)

with . Substituting Eq. (4) into Eq. (3) yields the expression

 ⎡⎣λ1H¯σ11|ψσ1⟩+∑i≠1∑j≠i√λ1λi√λj% H¯σ1i[μi−λiH¯σii]−1H¯σij|ψσj⟩⎤⎦=μ1|ψσ1⟩. (5)

So far the derivation is completely general and does not include any approximations. However, now we would like to focus on the situation where the two species are weakly entangled. The weak entanglement regime is defined via

 √λ1≈1   and   √λi≠1≪1, (6)

i.e. the first Schmidt-orbital carries the dominant weight, which explains our focus on in Eqs. (3) and (5). Importantly, this first Schmidt-orbital also contains the leading contributions to the properties of the many-body state such as energies, densities and correlation functions. This can be seen through the decomposition of the reduced n-body density matrix for species ,

 ρσn =trNσ−n[tr¯σ|Ψ⟩⟨Ψ|]=∑iλi trNσ−n|ψσi⟩⟨ψσi| ≈trNσ−n|ψσ1⟩⟨ψσ1|, (7)

here , denote tracing out the species and tracing out particles of species , respectively. Besides, we employed the conditions (6) in the last step, assuming all the terms of order are negligible. Equipped with this knowledge, we perform a Taylor expansion of Eq. (5). This is achieved by assuming all to be of order (or smaller) with and neglecting terms of O(). As our main result, we obtain the effective Hamiltonian for species , which is given by

 ^Hσeff=⎡⎣H¯σ11+∑i≠1√λiH¯σ1iH¯σi1t1i⎤⎦, (8)

with the associated effective Schrödinger equation

 ^Hσeff|ψσeff⟩=Eeff|ψσeff⟩. (9)

Equation (9) not only introduces a significant simplification in the study of mixtures but also allows for profound insights as we will see below. Here some considerations are in order: (i) The effective state can be understood as an approximation to in Eq. (5), which contains the dominant physics in the spirit of Eq. (7). (ii) It is worth to note that, the effective Hamiltonian , however, depends on the many-body state of the coupled system. Since is a general eigenstate of the mixture, the deduction of the effective Hamiltonian is applicable for both the ground-state and the excited-states of the mixture. (iii) If the mixture is non-entangled, the exact eigenfunction is of a product form, i.e. for all , therefore, only the first term in the effective Hamiltonian (8) is present which corresponds to the SMF case. Since is a local two-body interaction, the effective Hamiltonian becomes with being an additional (induced) potential. We will refer to as the SMF induced potential in the following discussions. Mathematically, it is the partial trace with respect to the species over the inter-species interaction (29). (iv) For the weak-entanglement regime, we obtain the physics beyond the SMF approximation. Since for , the first term on the right-hand side of Eq. (8) is reminiscent of the above SMF effective Hamiltonian. The second term which solely originates from the inter-species entanglement contains, beside additional potential term (see below), the induced interaction . Note that, the induced interactions always exist in both species (see also below). Importantly, this induced interaction is mediated via the Schmidt-orbitals from species . Moreover, it is a series with monotonously decreasing pre-factors , which suggest the possibility to truncate the sum over , since the contributions from Schmidt-orbitals with large can be neglected (29). (v) The last important observation is that for the case , the so-called system-bath regime, the induced interaction in the bath-species becomes negligible. This is because is proportional to , while scales with , leading to an induced interaction for species directly proportions to . Hence, for , there is negligible induced interaction among the particles, the species thereby becomes an ideal bath-type system. Meanwhile, by changing perspectives, the induced interaction in the species becomes increasingly important.

Induced interactions in a Bose-Fermi mixture.– Let us now elucidate the induced interactions and induced potentials for a mixture which is not a typical example of the applications of induced interactions: we discuss the few-body ensemble of a 1D spin-polarized ultracold Bose-Fermi mixture. The model Hamiltonian is , where

 ^Hσ =∫dx ^ψ†σ(x)hσ(x)^ψσ(x), (σ=b,f) ^Hbf =gbf∫dx ^ψ†f(x)^ψ†b(x)^ψb(x)^ψf(x), (10)

with being the single-particle Hamiltonian with harmonic confinement and containing a contact interaction (32); (33) between the two species with interaction strength . To demonstrate the physics originating from the inter-species entanglement, we focus on the case of inter-species interaction only with and explore a mixture with two fermions and two bosons . Note that, our approach is also valid when Bose-Bose and Fermi-Fermi interactions are taken into account. Moreover, this interaction strength is far beyond the perturbative regime with the interaction energy being comparable to the total kinetic energy . For such a Bose-Fermi mixture, the associated effective Hamiltonian for species is

 ^Hσeff=^Hσ+^Vσind+^Hσind, (11)

with

 ^Vσind =∫dx[Vσ1(x)+Vσno(x)]^ψ†σ(x)^ψσ(x), (12) ^Hσind =12∫dx1dx2 Hσind(x1,x2)^ψ†σ(x1)^ψ†σ(x2)^ψσ(x2)^ψσ(x1), (13)

representing the induced potential and induced interaction, respectively (see also below). In the following discussions, both the induced interactions and the induced potentials are obtained by ab-initio ML-MXTDHX simulations (26). In addition, we also compare to the results of exact diagonalization (ED) simulations, showing a qualitative agreement (29).

The induced potential consists of two terms in which the first one

 Vσ1(x)=gbfγ¯σ11(x) (14)

can be viewed as the SMF contribution. Here is the reduced one-body transition matrix element for species (26). From Eq. (14), we see that is proportional to both inter-species interaction strength and , the contribution made by the first Schmidt-orbital to the reduced one-body density for species . As discussed above, in the weak-entanglement regime we have , i.e. highly resembles the SMF induced potential [cf. Fig. 1(a,b), blue dashed and red solid lines]. The second term is given by

 Vσno(x)=gbf∑i≠1√λi~t1i[γ¯σ1i(x)γ¯σi1(x)+2β¯σ1iγ¯σi1(x)], (15)

with and , in which results from normal-ordering of in Eq. (8) while stems from cross-terms such as . It is fair to treat as a correction to , due to the pre-factors , rendering the magnitude of small in comparison to . In order to elaborate on the net confinement that a particle feels, we introduce the effective potentials as , as shown in Fig. 1(a,b) (black solid lines). We observe that, due to the presence of induced potentials, the effective potentials for both species deviate significantly from the original harmonic confinement forming either a double-well pattern (fermionic) or a tighter confinement (bosonic). In addition, the SMF effective potentials are presented as well [cf. Fig. 1(a,b), blue dashed lines].

Now we turn to the induced interaction, which reads

 Hσind(x1,x2)=gbf∑i≠12√λi~t1iγ¯σ1i(x1)γ¯σi1(x2). (16)

Unlike the Fröhlich Hamiltonian where the induced interaction is a second order perturbation term with respect to , here, it derives from the first order expansion with respect to the square root of the Schmidt numbers , therefore, it does not restrict the interaction strength to be small. In Fig. 2(b,e), we present the induced interactions among the fermions and the bosons, respectively. Importantly, the computed induced interaction preserves the particle exchange symmetry for identical particles as well as the parity symmetry of the original Hamiltonian [see Eq. (10)]. Furthermore, we notice that, unlike the contact Bose-Fermi interaction, the induced interaction is long-ranged and becomes, depending on the relative coordinate , attractive for small particle distances, repulsive for increasing , and vanishes at large [cf. Fig. 2(c,f)]. Apart from the dependence on the relative coordinate , the induced interaction also varies as a function of the mean position of the particles within the trap , since our system is not translationally invariant. These novel features differ from the situation in homogenous systems, where only the relative coordinate is involved and is usually attractive (17); (18); (30). In addition, albeit the similar spatial patterns, the induced interactions for the bosons and the fermions show noticeable differences with respect to their strengths and ranges. We observe that the fermionic induced interaction has almost a twice as large maximal value as the bosonic induced interaction, while its range is smaller.

The induced interaction can be an essential quantity for understanding in-depth the physics of the mixture. In particular, it can explain the behaviour of the pair-correlation function (34), which is given by

 gσ2(x1,x2)=ρσ2(x1,x2)ρσ1(x1)ρσ1(x2), (17)

with and being the reduced two- and one-body density, respectively. Physically, indicates the correlation between two particles, i.e. it is a measure for the probability of finding one particle at while the second is at . Through the division by the one-body densities, the function excludes the impact of the inhomogeneous density distribution and thereby directly reveals the spatial two-particle correlations induced by the interaction. In Fig. 2(a,d) we present the function obtained by the ab-initio ML-MCTDHX simulations. We find that, due to the inter-species entanglement, the bosonic pair-correlation function deviates significantly from the result of the SMF approximation given by for all and . Furthermore, based on the profile of the bosonic induced interaction [cf. Fig. 2(e)], we can conclude that the attractive part of the interaction increases the probability of finding two bosons next to each other and, hence, results in near the diagonal line , while the repulsive part suppresses bosonic bunching and leads to at larger relative distance [cf. Fig. 2(d)]. In contrast, the induced interaction among the fermions has minor impact due to the Pauli-exclusion principle leading to a similar to the SMF approximation [cf. Fig. 2(a) and Fig. 3(a,c)].

In order to test the applicability of our effective Hamiltonian, we compute the pair-correlation functions for both species using only the effective single-species Hamiltonian (11) and present them in the Fig. 3. Compared to the exact function, we find qualitative agreement between both methods suggesting that our effective Hamiltonian can evidently explain the important physics.

Conclusions.– We have introduced a new approach to derive the induced interactions in a binary mixture made of two species for the weak-entanglement regime. A theoretical framework leading to an effective single-species Hamiltonian has been presented. This Hamiltonian contains the induced interaction as well as an induced potential. Importantly, the induced interaction directly originates from the physics beyond the species mean-field and unveils the relationship to the inter-species entanglement. As an example, we employ a 1D spin-polarized ultracold Bose-Fermi mixture to explore our scheme. The profile of the induced interactions has been derived for both species yielding an induced interaction containing a short-range attraction and a long-range repulsion which is position dependent due to the inhomogeneity of the system. Finally, in order to test the applicability of our effective Hamiltonian, we have compared the pair-correlation functions of ML-MCTDHX simulations with simulations of the effective Hamiltonian and find them in good agreement. Our approach opens up the way to study induced interactions in binary mixtures in particular for few-body and inhomogeneous systems. Thereby, it is of specific interest how the functional form of the induced interaction depends on systems properties such as the particle statistics, the spatial dimension, the particle number, or the underlying interspecies coupling. Apart from being readily generalizable to the multi-species mixtures via a suitable bipartition, our approach should be extendable to time-dependent entangled states in the future. This generalization would enable us to compare to the commonly employed time-dependent perturbation theory picture.

###### Acknowledgements.
Acknowledgments.– The authors acknowledge fruitful discussions with Sven Krönke, Kevin Keiler and Maxim Pyzh. Moreover, the authors thank Richard Schmidt for a detailed feedback on the manuscript. This work has been financially supported by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB 925 âLight induced dynamics and control of correlated quantum systemsâ and by the excellence cluster “The Hamburg Centre for Ultrafast Imaging-Structure, Dynamics and Control of Matter at the Atomic Scale”.

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