Correlations of indistinguishable particles in non-Hermitian lattices

A novel approach to investigate the dynamics of indistinguishable particles in non-Hermitian lattice systems is presented, allowing an efficient calculation of quantum correlations between these particles in the presence of losses. Particular attention is paid to quasi-parity-time-symmetric systems, for which we numerically analyze two-particle quantum random walks for a variety of input states. Our results show how in some scenarios coherence is lost, inducing classical random walks, while in others the characteristic signatures of bosonic and fermionic exchange symmetry prevail.


Appendix A. Quantum Langevin equation for coupled systems
In the following, we deduce the quantum Langevin equation for our systems from a general approach by principally following similar considerations as in [1]. Regard the total Hamiltonian of the overall configuration under the assumption of weak coupling between reservoir and lossfree system with strengths g m,n m , H total =Ĥ 0 +Ĥ reservoir +Ĥ interaction whereβ n m denotes the propagation constants of the reservoir modes and β m and C j,m are defined in the main text. Thus, we find the Heisenberg equations for the system field operatorsâ † q and the reservoir field operatorsb † r q as coupled differential equations: r q (z),Ĥ total ] = iβ r qb † r q (z) + ig q,r qâ † q (z) . where the second term on the rhs acts as a noise operatorf † q = i r q g * q,r qb † r q (0)e iβ rq z . For the last term on the rhs we replace the summation by an integral r q → V (2π ) 3 d 3k r q → dβ r q D(β r q ) with the spectral density of states D(β): With Fermi's golden rule decay rates γ q = 2π D(β q )|g q (β q )| 2 and the introduction ofĤ system = H 0 + iÂ loss =Ĥ 0 + ih M m=1 γ m 2â † mâ m one obtains the quantum Langevin equation: With the linear propagation operator U q, j (z) = (e iĈ z ) q, j the formal solution of (A.6) iŝ Appendix B. Calculation of the particle number correlation Here we present an extensive derivation of the particle number correlation and prove that this quantity does not depend on the noise operators. Calculating the particle number correlations by using (A.7) ) and four ( (16) q,r ) noise operators. In the following we will show exemplarily the contribution of each group by an extended calculation for one member. For the sake of clarity we slip the z-argument of the propagation operator elements U j,k . Any other dependence will be written explicitly. Taking into account that the noise operators are Markovian and the average value of the noise equals zero as well as the zero temperature approximation (ZTA) for the reservoir, we get: . . , 11: . . , 15: × ι q ,κ r ,λ r ,π q g * q,ι q g * r,κ r g r,λ r g q, e iβ ιq ζ e iβ κr ζ e −iβ λr ζ e −iβ πq ζ = 0.
= 0 because in the ZTA the average particle number for each reservoir mode vanishes.
As all terms with α > 1 do not contribute it holds for the particle number correlation: which is independent of any noise operators. Abstract. A novel approach to investigate the dynamics of indistinguishable particles in non-Hermitian lattice systems is presented, allowing an efficient calculation of quantum correlations between these particles in the presence of losses. Particular attention is paid to quasi-parity-time-symmetric systems, for which we numerically analyze two-particle quantum random walks for a variety of input states. Our results show how in some scenarios coherence is lost, inducing classical random walks, while in others the characteristic signatures of bosonic and fermionic exchange symmetry prevail.

Introduction
Understanding quantum random walks of indistinguishable particles and their mutual correlation allows deep insight into the very nature of quantum mechanical processes in discretized space. In particular waveguiding lattices provide an exceptional tool for the study of such phenomena [1,2]. To date, the correlations of identical particles have been explored in various systems, such as lattices with a transverse potential gradient [3] or disorder [4], Glauber-Fock lattices [5], two-dimensional lattices [6] and lattices with periodic boundaries [7]. However, an intrinsic property of all these quantum systems is their Hermiticity.
Recently, parity-time(PT )-symmetric systems attracted much attention throughout the physics community. First proposed in 1998 [8], such Hamiltonians are not necessarily Hermitian, but still can exhibit purely real eigenvalue spectra. Since this concept was introduced in optics by anti-symmetric gain-loss distributions [9], numerous works were devoted to exploring optical PT -symmetric phenomena [10][11][12][13][14][15], culminating in the first experimental demonstration of PTsymmetry in physics [16,17]. However, all of the reported effects have a purely classical origin, leaving the analysis of a true quantum evolution in PT -symmetric lattices an open question. Guo et al [12] showed that systems with anti-symmetric modulated loss distribution behave equivalently to those with gain-loss structure due to a transformation into an exponentially damped co-moving frame. For example a periodic gain-loss structure with equal gain and loss is equivalent to a structure with periodic no-loss-loss distribution where the loss is twice as before. We call such structures 'quasi-PT -symmetric'. However, due to noise from vacuum fluctuations we are limited to transformations that result in vanishing net gain for the analysis of quantum evolution in non-Hermitian lattices. Nevertheless, despite this limitation, quasi-PTsymmetric quantum systems give rise to fascinating dynamics. In this paper, we provide the first theoretical analysis of quantum random walks of identical particles in quasi-PT -symmetric lattices. We further demonstrate how such dissipative quantum systems can be described by a set of coupled equations that allows the efficient simulation of extended configurations and investigate the correlation behavior of indistinguishable separable and entangled particles. We show how the symmetry of the lattice leads to independence or even decoherence of the quantum walkers for certain input configurations, while preserving strong correlations for others.

Theory
The general approach to describe a dissipative quantum systems is to weakly couple the loss-free system with a reservoir. Thus, the total Hamiltonian of the overall configuration is represented by the sumĤ total =Ĥ 0 +Ĥ reservoir +Ĥ interaction (1) with the Hamiltonian of the subsystem without lossesĤ 0 . The evolution of such an open system can be analyzed by, e.g. the Lindblad formalism [18] which evaluates the evolution of the system's full density matrix subjected to the interaction with the reservoir. Hence, the numerical effort in solving the evolution equations grows strongly nonlinearly with system size. In a lattice with M sites the overhead is O(M 4 ), rendering even moderate system sizes computationally intractable. In our work, we follow a different approach, which scales more favorably.
We consider a lattice consisting of M coupled sites with an arbitrary transverse loss profile. The non-Hermitian Hamiltonian of the dissipative system (lattice with no back coupling from the reservoir) with the creation (annihilation) operatorâ † q (â q ) for mode q, is defined aŝ H system =Ĥ 0 + iÂ loss with the Hermitian Hamiltonian of the loss-free system and the Hermitian Operator describing the losŝ Here β m are the propagation constants in each lattice site and energy eigenvalues ofĤ 0 , respectively, γ m are the loss coefficients and C j,m are the coupling coefficients between sites j and m that form a matrixĈ. Analogously to [19,20] for weak coupling between system and reservoir, one can derive the quantum Langevin equation as the corresponding equation of motion (see appendix A) withC q, j = (β j + i γ j 2 )δ j,q + C q, j , z denoting the evolution coordinate of the system and the noise operatorsf † q which ensure the commutator (anti-commutator) relations [â j ,â † k ] = δ j,k ({â j ,â † k } = δ j,k ) (with the Kronecker delta δ j,k ) for bosons (fermions). Formal integration yieldŝ where in analogy to [1,21] U q, j (z) = (e iĈ z ) q, j are the matrix elements of the linear z-propagation operatorÛ . The average occupation number n q (z) in mode q is It is typically assumed that the noise operators are Markovian (no memory effect) and the average value of the noise is equal to zero. This yields n (1) q = n (2) q = 0. Following [19], one obtains n (3) q ∼ n R with the particle density of the reservoir n R = exp hω kT − 1 −1 in equilibrium at frequency ω and temperature T . In the zero temperature approximation (ZTA) this contribution vanishes 5 . Therefore, we find the remarkable result that n q (z) = M j,k=1 U q, j (z)U * q,k (z) â † j (0)â k (0) is independent of noise becauseâ † q (0) is not related to the noise operators. Thus, a numerical treatment of the average occupation number of any excitation-bosonic, fermionic and irrespective of the number of launched particles-in a lattice with loss can be performed by neglecting the noise operators in (4), i.e. the standard Heisenberg equation for the non-Hermitian HamiltonianĤ system : When only two indistinguishable particles evolve in the lattice, one can analyze their mutual particle number correlation q,r (z), which characterizes the probability to find one particle at lattice site q and the other one at site r . We find (see appendix B) which is also independent of the noise term and only determined by the z-propagation operator U , i.e. by the coupling dynamics. Hence, both the evolution of the average particle number and the particle number correlation can be computed using the simple equation (7). Note that our approach reduces the overhead to O(M 2 ), as compared to O(M 4 ) for the Lindblad method, enabling the numerical analysis of much larger systems without any additional assumptions. Furthermore, we emphasize that this concept is applicable to any non-Hermitian lattice with arbitrary loss profile, irrespective of its physical manifestation, e.g., spin chains, electrons in crystalline solids, Bose-or Fermi-Hubbard systems, molecular chains or photonic lattices.

Numerical investigations
Let us now consider two identical bosons (b)/fermions (f) being launched into the lattice at sites m and n (m = n). These separable input states |ψ (b/f) =â † m (0)â † n (0) |0 (with the vacuum state |0 ) yield the particle number correlations [1,4] ( where '+' corresponds to bosons and '−' to fermions. In contrast, injecting two bosons simultaneously into either site m or n results in a symmetric (s) or anti-symmetric (a) pathentangled input state |ψ (s/a) For all considered input states one finds n q (z) = U q,m 2 + U q,n 2 .
To answer the open question of a true quantum evolution in loss-modulated quasi-PTsymmetric systems, we consider a lattice where only even numbered sites exhibit loss (see figure 1(a)). Furthermore, the coupling shall be restricted to adjacent sites such that C j,m = C δ j,m±1 . In order to investigate the impact of such a loss structure on the mutual correlation of bosons, we consider an array of evanescently coupled optical waveguides [22] with identical propagation constants β m = β. Thus, the lattice can be described in a frame co-propagating with β so that rapid phase oscillations can be separated from the transverse dynamics (â † q =â † q e −iβz ). Fermions, on the other hand, can be investigated directly in cold atoms trapped in optical lattices [23], where the spatial coordinate z is replaced by time, or simulated in waveguides lattices [24,25]. We performed extensive numerical simulations for an array of M = 50 sites, a length of 15 cm, a coupling constant C = 1 cm −1 and alternating loss with γ q = 0 cm −1 for odd and γ q = 4 cm −1 for even q. In an envisioned experiment, this can be realized by a controlled variation of the fabrication parameters of the waveguides [26]. We consider three different scenarios: (i) two particles are injected into two adjacent channels m = 25 (lossless) and n = 26 (lossy) ( figure 1(b)); (ii) two particles are injected into next-nearest sites m = 24 and n = 26 (both lossy) (figure 1(c)) and (iii) lossless next-nearest neighbors m = 25 and n = 27 ( figure 1(d)). All other possible input states result in similar dynamics as long as the particles do not reach the boundary, depending on whether an even or an odd number of lattice sites is in between the input channels. A striking feature of these quasi-PT -symmetric lattices is that they give rise to a diffusive wave evolution despite being invariant along z. This means the variance of the wave packet spreads linearly with the propagation which can only be achieved by longitudinal disorder, i. e., propagation constants randomly varying along z (corresponding to a temporal fluctuation in the quantum mechanical context) [27] in the Hermitian system. Figure 2 shows the correlation functions (b/f) q,r for separable bosonic/fermionic particles as well as for the symmetric/anti-symmetric path-entangled bosons (s/a) q,r after a quantum Output distribution and evolution of the average particle number n q for two adjacent inputs (b) and next neighbor injection in lossy (c) or loss-free (d) channels for coupling constant of C = 1 cm −1 and loss coefficient γ q = 4 cm −1 for even q. The output distributions are normalized to their maximum value respectively. random walk with an evolution distance of z = 15 cm, for the three input conditions described above. As a first intuitive result we mention that in all observed cases the average particle number distribution n q almost vanishes in the lossy channels (see figure 1) and the particle number correlation q,r approaches significant values only in the loss-free sites (see figure 2). Comparing these results with the loss-free case ( q = 0, ∀ q) one can clearly observe circumstances where the shapes of the correlations are changed drastically due to loss whereas being basically preserved in others.
The correlation function of bosons launched into adjacent waveguides (figure 2(a)) maintains the characteristic bimodal shape of the Hermitian loss-free case [1], with the peaks being slightly broadened and closer to the excited sites due to the diffusive propagation. In contrast, for next-nearest neighbor excitation of lossy channels the bosons are no longer correlated and thus evolve as independent but coherent wave packets through the structure as evident from the equal height of the four peaks ( figure 2(b)). The situation changes completely when both bosons are launched into loss-free next-nearest channels. In this case, the nonclassical correlation of a coherent random walker is destroyed and the classical broad Gaussianshaped correlation of two incoherent diffusive particles is formed around the excitation center ( figure 2(c)).
For fermions, we find that launching two separable particles into adjacent lattice sites again preserves the main features of the anti-bunching correlation pattern of the Hermitian system, (see figure 2(d)). However, fermions injected in next-nearest channels show a behavior that is very different from that in the Hermitian system, where the correlation pattern shows a square shape indicating that both fermions will travel with high probability far from the excited channels and will always be found in different guides (due to the Pauli exclusion principle). In contrast, in our system, the correlation function (f) q,r changes significantly. When both input sites exhibit loss, we find a complex correlation pattern that shows signatures of anti-bunching as well as distinct correlation ridges parallel to the bunching diagonal (figure 2(e)). Note that the row sum of the correlation function r q,r does not equal the average particle number n q (see figure 3) as in a Hermitian system [1,2]. While the former quantity accounts for probabilities of two particle events, the latter also includes scenarios where one particle is lost.
Nevertheless, also in the quasi-PT -symmetric system the fermions always occupy different sites. When the input channels are loss-free, a distinct anti-bunching behavior of the correlation function is observed (figure 2(f)), very similar to the case where two neighboring modes are excited. These transitions strikingly demonstrate how the loss profile affects the correlation patterns of the evolving particles.
In the case of the symmetric and anti-symmetric input states one finds similarly a strong dependence of the correlations on the input configuration. For adjacent site excitation (s/a) q,r shows how the presence of losses can lead to full decoherence as heralded from the single Gaussian peak in figures 2(g)-(j). On the other hand for next-nearest neighbor excitation the particle number correlations explicitly show the random walkers' behavior of indistinguishable bosons (figure 2(l)) as well as the behavior of distinguishable but coherent walkers (figure 2(h)) and fully classical incoherent walkers (figure 2(i)) and additional an intricate shape exhibiting bunching as well as anti-bunching properties (figure 2(k)).

Conclusion
Summarized, we have introduced a new method to efficiently investigate the quantum random walks of identical particles in lattice structures with arbitrary loss profile. By virtue of this method, we analyzed the correlations of separable bosonic and fermionic input states as well as entangled symmetric and anti-symmetric input states in a quasi-PT -symmetric lattice structure. We found that, depending on the position of the excited sites, the quantum correlations of a corresponding homogeneous loss-free system may either prevail, alter significantly or get destroyed forming the correlation of distinguishable or even classical random walkers.
whereβ n denotes the propagation constants of the reservoir modes and β m and C j,m are defined in the main text. Thus, we find the Heisenberg equations for the system field operatorsâ † q and the reservoir field operatorsb † r as coupled differential equations where the second term on the rhs acts as a noise operatorf † q = i r g * q,rb † r (0) e iβ r z . For the last term on the rhs we replace one summation by an integral r → V (2π ) 3 d 3k r → dβ r D(β r ) with the spectral density of states D(β) One obtains with decay rates γ q = 2π D(β q )|g q,r (β q )| 2 from Fermi's golden rule and the introduction ofĤ system =Ĥ 0 + iÂ loss =Ĥ 0 + ih M m=1 γ m 2â † mâ m the quantum Langevin equation whereC q, j = (β j + i γ j 2 )δ j,q + C q, j . With the linear propagation operator U q, j (z) = (e iĈ z ) q, j the formal solution of (4) iŝ