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Exact N-Point Function Mapping Between Pairs of Experiments With Exact N -point function mapping between pairs of experiments with Markovian open quantum systems Etienne Wamba1;2;3;4, and Axel Pelster4 1 Faculty of Engineering and Technology, University of Buea, P.O. Box 63 Buea, Cameroon, 2 African Institute for Mathematical Sciences, P.O. Box 608 Limbe, Cameroon, 3 International Center for Theoretical Physics, 34151 Trieste, Italy, 4 Fachbereich Physik and State Research Center OPTIMAS, Technische Universit¨atKaiserslautern, 67663 Kaiserslautern, Germany E-mail: [email protected], [email protected] September 11, 2020 Abstract We formulate an exact spacetime mapping between the N -point correlation functions of two different experiments with open quantum gases. Our formalism extends a quantum-field mapping result for closed systems [Phys. Rev. A 94, 043628 (2016)] to the general case of open quantum systems with Markovian property. For this, we consider an open many-body system consisting of a D-dimensional quantum gas of bosons or fermions that interacts with a bath under Born-Markov approximation and evolves according to a Lindblad master equation in a regime of loss or gain. Invoking the independence of expectation values on pictures of quantum mechanics and using the quantum fields that describe the gas dynamics, we derive the Heisenberg evolution of any arbitrary N -point function of the system in the regime when the Lindblad generators feature a loss or a gain. Our quantum field mapping for closed quantum systems is rewritten in the Schr¨odingerpicture and then extended to open quantum systems by relating onto each other two different evolutions of the N -point functions of the open quantum system. As a concrete example of the mapping, we consider the mean-field dynamics of a simple dissipative quantum system that consists of a one-dimensional Bose-Einstein condensate being locally bombarded by a dissipating beam of electrons in both cases when the beam amplitude or the waist is steady and modulated. 1 Introduction and thermal states, and the spectrum of quasi-particles [1]. Correlations of order one or two have been, however, the most frequently measured correlations. Recently, there has The concrete and complete description of the dynamical be- been immense progress in recipes for measuring higher-order havior of any physical system can be given by correlations [1]. correlations [10] to the ultimate goals of non-destructively ex- For quantum systems with many degrees of freedom, the com- ploring quantum systems [3, 11] and achieving quantum-field plete information can be obtained from quantum field theory tomography [12]. In order to explore the quantum coher- using multitime-multipoint correlation functions. Using such ence properties of massive particles, it has been possible to functions, it is possible to directly express any observable of observe higher-order correlations up to the sixth order [13]. the system [2, 3, 1]. First-order (one-body) and second-order The knowledge of correlations finds applications mostly in (two-body) correlations have been investigated for trapped quantum metrology [12, 14], quantum information [14, 15], cold gases by using involved interferometry in atom chip [4] and quantum simulations [16, 17] where they allow reading, or free expansion [5]. In quasi-one dimensional trapped cold verifying and characterizing quantum systems. arXiv:2009.04270v2 [cond-mat.quant-gas] 10 Sep 2020 gases, quantum fluctuations are enhanced, and first-order cor- relations are important in describing the coherence properties Relevant many-body quantum systems, both for closed and of the system, especially in the quasi-condensate regime when open cases, can be designed and controlled with an unprece- the system turns smoothly from a decoherent system to a dented precision due to recent technical development of quan- finite-size Bose-Einstein condensate [5]. Higher-order corre- tum gas experiments [18]. In open systems [19, 20, 21], in lations and the way they decompose to lower orders are very particular, localized dissipation provides new possibilities to important in many-body systems, since they provide useful in- engineer robust many-body quantum states and allows us to formation on the interactions, the structure and the complex- study fundamental quantum effects like quantum Zeno dy- ity of the system [6, 7, 8] including, for instance, decoherence namics [22, 23] and non-equilibrium dynamics in an ultracold and thermalization [9], the discrimination between the ground quantum gas [24, 25]. Despite those achievements, however, 1 numerous regimes of many-body quantum systems are still 2.1 Revisiting the exact quantum-field mapping challenging both theoretically and experimentally. We consider a D-dimensional quantum gas which can be a Motivated by those theoretical and experimental efforts for mixture of several species of type k with possibly different achieving and controlling open quantum systems, we here masses M in any arbitrary state. The gas can be a single- or study the dynamics of a class of open quantum systems k multi-component bosonic or fermionic system, or even a mix- through an exact N -point function mapping between differ- ture of bosons and fermions. All observables of the gas may ent experimental situations. We aim at finding how the data be expressed in terms of the time-evolving second-quantized of a pair of experiments can be mapped onto each other. field operator ^ (r; t), which destroys a particle, and of its One of the experiments may be interesting but challenging k Hermitian conjugate field operator ^y(r; t), which correspond- enough to be achieved, for instance, due to technical limi- l tations. Our current contribution is restricted to equal-time ingly creates a particle of type k, at position r and time t. multipoint correlation functions which does not capture au- We require the field operators to satisfy the canonical (anti- tocorrelations, i.e., correlation functions for different times. )commutation relations But a lot of information can be obtained from quantum field h ^ ^y 0 i D 0 k(r; t); l (r ; t) = δkl δ (r − r ) ; (1) theory using those simpler correlators. Moreover, using those ± correlators, for instance, one would not have to worry about h i the effects of wave function collapse from the first measure- where A;^ B^ = A^B^ ± B^A^, such that the theory can be ± ment changing the results of the second measurement. The equally applicable to fermions (anticommutation) and bosons work is organized as follows. In Sec. 2, starting from previous (commutation) [26]. There has been tremendous experimen- results on the quantum-field mapping in the Heisenberg pic- tal progress in creating and monitoring various kinds of dy- ture, we establish a corresponding result in the Schr¨odinger namics of ultracold quantum gases with a large range of two- picture by expressing a pair of many-body wave functions particle interaction, U (r; r0; t), and one-particle interac- in terms of second quantized fields, and then relating them klmn tion, Vk(r; t), imposed by external trapping fields. Without onto each other. In Sec. 3, we present the spacetime mapping loss of generality, we will drop the species type subscript index of a pair of N -point functions that would represent two dif- in what follows. In general, experimental measurements in a ferent experiments with lossy many-body quantum systems. quantum gas can be expressed in terms of an N -point corre- The quantum-field mapping for closed systems formulated in lation function. The N -point or N -body correlation function Ref. [26] is rewritten in terms of wave functions, and then is a general expectation value of the field operators that de- further generalized to open systems described by the Lindblad scribe the quantum gas properties. As from here, we omit the equation with loss or gain channels through the N -point cor- species indices to simplify the notations. Our theory is gen- relation functions. Such an equation has proven to be useful in eral, but we restrict ourselves to spatial correlations, and thus describing the dynamics of Markovian open quantum systems, the equal time N -point function can be written as [26, 27, 13]: see for instance [22, 23]. Sec. 4 deals with a concrete example that illustrates the mapping between two experimental situa- *2 N 3 2 N 3+ tions where dissipation is created on a typical bosonic system 0 Y ^y 0 Y ^ F (R; R ; t) = 4 rj0 ; t 5 4 (rj; t)5 ; (2) by a Gaussian beam of electrons. Numerical computations j=1 j=1 are performed with the mean-field Gross-Pitaevskii equation 0 0 0 to quantify our findings. The work ends in section 5 which is where R = (r1; :::; rN ) and R = (r1; :::; rN ). For notational devoted to the conclusion and summary of our results. simplicity, we introduced the subscript index j0 ≡ N + 1 − j. It is worth noting that the order of operators in the N -point function is quite rigid for fermions due to the anticommuta- 2 Schr¨odinger-pictureversion of the tivity of annihilation/creation operators. For bosons, setting j0 = j would still be a correct notation. Let us mention in exact quantum-field mapping for passing that the N -point function F (R; R0; t) is sometimes closed systems denoted G(N ) (R; R0). We consider a quantum gas that evolves under two different Spacetime mappings have long been used, essentially as coor- experimental conditions, but in a related way, for some par- dinate transformations to put theoretical problems into forms ticular pairs of interparticle interactions (U; U~) and trapping where they can easily be solved. A brief survey of solution- potentials (V; V~ ). Suppose both evolutions are described by a oriented mappings was provided in Ref. [26]. Using spacetime pair of different sets of quantum fields ( ^; Ψ),^ and their conju- transformations for mapping two different parameter regimes gates ( ^y; Ψ^ y).
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