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 Schrödingerpicture 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 correlations 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 ψˆ (r, t), which destroys a particle, and of its One of the experiments may be interesting but challenging k Hermitian conjugate field operator ψˆ†(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 ˆ ˆ† 0 i D 0 ψk(r, t), ψl (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,ˆ Bˆ = AˆBˆ ± BÂˆ, 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 Schrödinger 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- * N   N + tions where dissipation is created on a typical bosonic system 0 Y ˆ† 0 Y ˆ F (R, R , t) =  ψ rj0 , t   ψ (rj, t) , (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 Schrödinger-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,ˆ Ψ),ˆ and their conju- transformations for mapping two different parameter regimes gates (ψˆ†, Ψˆ †). The evolutions with {ψ,ˆ V, U} and {Ψˆ , V,˜ U˜} of an experiment is fundamental for physics. But to our thus satisfy the same Heisenberg equation as well as canon- knowledge this non-traditional use of mappings was formally ical (anti-)commutation relations. The Heisenberg-picture proposed only recently as an exact quantum-field mapping for quantum-field operators are mapped as follows [26]: studying challenging dynamical regimes of isolated quantum gases [26]. In what follows, taking advantage of such a map- − iM λ˙ r2 D/2 Ψ(ˆ r, t) = e 2~ λ λ ψˆ (λr, τ) , (3) ping for closed systems [26], we formulate a corresponding Schrödinger-pictureversion of the mapping that is valid for where λ = λ(t) is the free parameter of the problem. It fullfills both bosonic and fermionic systems. the two conditions λ(0) = 1 and λ˙ (0) = 0, where λ˙ (t) ≡ dλ/dt

2 2 ˆ ˆ ˆ and dτ/dt = λ with τ = τ(t), so that the fields in both exper- field operator at time t, we have ψ(r, t) = ψt(r) ≡ ψH. iments coincide at initial time, meaning that the correspond- For the Schrödinger-picturefield operator at all times (cor- ing experiments are supposed to be run with almost identical responding to the Heisenberg-picture field operator at time ˆ ˆ ˆ ˆ initial state. This can be achieved practically by preparing t = 0), we have ψ(r) = ψ(r, 0) = ψ0(r, 0) ≡ ψS. For the a state, and then adiabatically splitting it into two. For an Schrödinger-picturestate at time t, we have |ψti = |ψiS. interparticle interaction with homogeneity degree s (equal to And for the Heisenberg-picture state at all times (correspond- D for contact interaction), the potentials map as ing to Schrödinger-picturestate at time t = 0), we have |ψ0i = |ψi ≡ |ψiH. ˜ 0 2 0 2−s 0 U(r, r , t) = λ U(λr, λr , τ(t)) = λ U(r, r , t), Equation (7) above relates both second-quantized field op- 1 ˆ ˆ V˜ (r, t) = λ2 V (λr, τ(t)) + Mr2λΩˆ 2λ , (4) erators ψ(r) and ψ(r, t) in Schrödinger and Heisenberg pic- 2 tures, respectively, providing a way of writing the wave function and the N -point function (2) using non time-evolving where we use the time differential operator Ωˆ = λ−2∂/∂t ≡ field operators. Let |0i be the (position-space) vacuum ∂/∂τ. Using the quantum-field mapping above, we obtain state ket, |ψti the state ket of the system at time t, and that the quantum gas experiment described by the N -point ψ (r1, ··· , rN , t) the position-space wave function for the gas function (2) can be mimicked by an equivalent experiment with N particles, where rj is the position of the jth parti- described by a rescaled N -point function F˜, such that cle in D-dimensional space. Then the wave function can be obtained by projecting the state ket on the position space: * N   N + ˜ 0 Y ˆ † 0 Y ˆ F (R, R , t) =  Ψ rj0 , t   Ψ(rj, t) . (5) ψ (r1, ··· , rN , t) ≡ hr1, ··· , rN |ψti j=1 j=1 D N E Y ˆ In this expression, we used again j0 ≡ N + 1 − j. We find = 0 ψ(rj) ψt . (8) that the N -point functions F˜ and F of the two experimental j=1 situations are related onto each other through the mapping identity The many-body wave function ψ describes all the particles at a time and satisfies the following normalization condition N − iM λ˙ P (r2−r02) 0 N D 2~ λ i i 0 F˜ (R, R , t) = λ e i=1 F (λR, λR , τ) . (6) Z N Y D ∗ d rj ψ ψ = N. (9) Here, the scaling factor is not only a function of the dimension j=1 D, but also a function of the number N . Unlike for second-quantized field operators, the number of 2.2 Expressing the wave function in terms of particles N explicitly appears in the many-body wave function second-quantized fields as well as in the many-body Hamiltonian, see below. Using the unitary transformation (7), we can transfer the The dynamics of a quantum system can be mathematically time dependence from the state to the field operator and formulated in the two equivalent and most important repre- rewrite the many-body wave function (8) into the following sentations of quantum mechanics, which are the Schrödinger form and Heisenberg pictures. A natural question that emerges is how the exact mapping reads in the Schrödingerpicture, D N E Y ˆ which is the originally approved and most popular representa- ψ (r1, ··· , rN , t) = 0 ψ(rj, t) ψ . (10) tion. Unlike the Heisenberg picture, the Schrödingerpicture j=1 regards the quantum-field operators associated to all physical quantities of a quantum-mechanical system to be fundamen- The many-body wave function is expressed in terms of evolv- tally fixed in time. Meanwhile quantum systems are repre- ing quantum fields. This opens a possibility for using the sented by time evolving wave functions or state kets. The quantum-field mapping to construct the mapping of many- Schrödinger-picturequantum fields can readily be expressed body wave functions. in terms of Heisenberg-picture quantum fields by introduc- 0 ˆ † ˆ ing a unitary operation Uˆ(t, t ) such that ψH = Uˆ ψS Uˆ. The subscripts S and H refer to Schrödingerand Heisenberg rep- 2.3 Mapping identity for many-body wave resentations, respectively. In what follows, we will omit the functions subscripts for notational simplicity, but Heisenberg variables will still be identifiable through an explicit time t in the ar- Consider two potential experiments A and B performed with guments or subscripts. Thus we can write the same Heisenberg (stationary) state and different Hamilto- nians. Each of both evolutions with {ψ, V, U} and {Ψ, V,˜ U˜} ˆ ˆ† ˆ ˆ ˆ ψ(r, t) = U ψ(r) U, |ψti = U|ψi. (7) satisfies the following Schrödingerequation:

For the sake of clarity, we recall the following textbook for- ∂ i |ψ i = H|ψ i, (11) mulations which are equivalent: For the Heisenberg-picture ~∂t t t

3 where the SchrödingerHamiltonian for the D-dimensional the Lindblad equation. Then invoking the Heisenberg evo- quantum gas with N particles can be written as lution and the picture-independence of expectation values, which makes equivalent all pictures of quantum mechanics, N X 2 we present a mapping identity for relating onto each other two H = − ~ ∇2 + V (r , t) rj j 2Mj different evolutions of N -point functions. Note that the cor- j=1 responding calculations are performed for both bosonic and N N X X fermionic systems. + U(rk, rj, t). (12) j=1 k>j,k=1 3.1 Necessary tools in Schrödingerpicture The pairs of potentials (V,U) and (V,˜ U˜) can be related onto each other according to Eq. (4). It should be noted that Obtaining the mapping of N -point functions requires the use the time t in the Hamiltonian represents an explicit time- of second quantized fields in the Schrödingerpicture into the dependence, and not the result of an evolution of the Hamil- appropriate Lindblad equation. Those ingredients are pre- tonian operator under a unitary transformation. sented in this section. As for the wave function, it is a key tool of the Schrödinger picture and thus basically defined in terms of second quan- 3.1.1 The N -point function in terms of second quantized tized fields in Schrödinger picture. Using Eq. (10) the many- fields in the Schrödingerpicture body wave functions at time t corresponding to the two ex- In order to get the N -point function in terms of Schrödinger- periments B and A, however, can be written as functions of picture quantum-field operators, we can start from the def- second quantized fields. We get inition of N -point functions (2), and then apply the trans- N formation (7) and its conjugate. Since N -point functions are D E Y ˆ expectation values, they remain the same irrespective of the Ψ(r1, ··· , rN , t) = 0 Ψ(rj, t) Ψ , j=1 quantum-mechanical representation used to describe them. N For that, we can write D Y E ψ (λr , ··· , λr , τ) = 0 ψˆ(λr , τ) ψ . (13) 1 N j     j=1 * N N + 0 Y ˆ† 0 Y ˆ F (R, R , t) =  ψ rj0   ψ (rj) , (15) Since the exact space-time mapping is known for evolving j=1 j=1 quantum-field operators, it becomes clear that both wave functions can be mapped onto each other. That can be done where j0 ≡ N +1−j, and F now denotes the N -point function by combining together Eqs. (3) and (13). We obtain that in the Schrödingerpicture. In contrast to (2), the time de- the position-space wave functions in the two potential exper- pendence in the equation above is actually only implicit. It is iments will be exactly related through the following identity effectively redeemed, however, when the non-evolving operators are applied onto evolving state kets or wave functions. In N −i M λ˙ P r2 ND 2 λ j both the Heisenberg and Schrödingerpictures, we have kept ~ j=1 Ψ(r1, ··· , rN ; t) = λ 2 e the same notation for the N -point function. × ψ (λr1, ··· , λrN ; τ(t)) . (14) In the Schrödingerpicture, since operators associated to physical quantities are non time evolving, the evolution of ob- Similarly to the quantum-field mapping, the exact wave func- servables is given by the equations of motion for the state kets tion mapping consists of a non trivial time-dependent trans- onto which operators can be applied. The time dependence formation of space and a multiplication of many-body wave of all observables is thus determined by the time dependence functions by a Gaussian phase factor. Here that factor explic- of the state kets or of the density matrix, and a master equa- itly depends on the number of particles and their positions at tion in the Lindblad form may describe the dynamics of the a given time. dissipative quantum system.

3 Mapping of different evolutions of 3.1.2 The Lindblad master equation N -point functions in an open system Quantum gases are in general well isolated from the environment or bath. However, particle losses and heating can occur, The result above is obtained for closed quantum systems. for instance, due to background gas collisions, and photon Here we aim at generalizing the identities to dissipative sys- scattering and trap shaking, respectively. We restrict our- tems described by the Lindblad master equation by mapping selves to the case of systems that satisfy the Markovian (or the N -point functions of their evolutions. We start by pre- Born-Markov) approximation. Then the connection between senting the useful ingredients in Schrödingerpicture. Notably, the system and the bath is so weak that the bath forgets any we rewrite the N -point function using second quantized fields information received from the system much faster than the in the Schrödingerpicture and present the appropriate Lind- evolution we want to follow. We then expect a clean separa- blad equation. Afterwards we derive the Schrödinger-picture tion between the typical correlation time of the system’s fluc- evolution of an N -point function of a quantum gas that obeys tuations and the time scale of the evolution described. In that

4 case, the evolution of the density matrix and the full quan- quantum systems, dissipation can interplay with interparti- tum fields of the system should be based on a master equation cle interactions, which affects the coherence and the density in Lindblad form [29]. The Lindblad formulation is interest- fluctuations of the system [39]. ing because not only it is less difficult to solve compared to We consider a more general system where dissipation is non-Markovian equations, but also it may allow describing continuous and time-dependent but with only a single kind of the behavior of both bosonic [30, 31] and fermionic [32] sys- dissipative process. The Lindbladian may be written as tems. Moreover, it turns out to give an acceptably accurate Z description of recent experimental efforts to control dissipa- Lˆ[ˆρ] = − dDr Qˆ†Qˆρˆ +ρ ˆQˆ†Qˆ − 2QˆρˆQˆ† , (19) tive quantum systems of bosons [22] or fermions [23], see [33] for an overview. We use such a master equation as the theo- where the jump operators can be space- and time-dependent. retical model governing the dynamics of the class of quantum Their specific form depends on the kind of coupling between systems studied in this paper. In its standard form in the the system and its environment. Choosing the jump operators ˆ ˆ p Schrödingerpicture, the Lindblad equation reads [34]: to feature a pure loss, i.e., Q = ψ(r) γ(r, t)/2, we obtain Z γ(r, t) ∂ρˆ h i Lˆ[ˆρ] = − dDr ψˆ†ψˆρˆ +ρ ˆψˆ†ψˆ − 2ψˆρψˆ † . (20) i~ = H,ˆ ρˆ + i~ Lˆ[ˆρ], (16) 2 ∂t − The jump operator used in this Lindbladian is a general an- where Hˆ in the von Neumann term is the Hamiltonian of the nihilation operator applicable to both bosons and fermions. system expressed in terms of second quantized fields in the The dynamics of a system in a regime where the only dis- Schrödingerpicture as sipation channel is a gain, can readily be reproduced from the current work by taking the jump operator to be Qˆ = Z 2 D † ~ 2 ˆ† p Hˆ = d r ψˆ (r) − ∇ + V (r, t) ψˆ(r) ψ (r) γ+(r, t)/2 where γ+ is the gain rate. A regime with 2M (17) both gain and loss channels can also be readily deduced by 1 Z simply summing the corresponding Lindbladians to get the + dDrdDr0 ψˆ†(r)ψˆ†(r0)U (r, r0, t) ψˆ(r0)ψˆ(r). 2 full Lindbladian. The standard Lindblad master equation (16), which is rel- Even though Eq. (16) cannot describe general open systems evant to the Schrödingerpicture, is expressed in terms of the since it is an approximate quantum master equation, it suit- non evolving second-quantized field operators. The Lindblad ably describes the outcomes of quantum gas experiments as evolution based on time evolving second-quantized field op- pointed out above. As it can be seen, the quantum fields operators, which are convenient for the Heisenberg picture, will erators are not time-dependent, only the potentials have an be discussed below. explicit time-dependence, i.e., their time change is not due to the action of a quantum mechanical operator. The Lindbla- dian term for lossy systems may read 3.2 Lindblad evolution of N -point functions The time dependence of expectation values is completely car- D2−1 X ˆ ˆ† 1 h ˆ† ˆ i ried out by the density matrix in the Schrödinger picture. The L[ˆρ] = γj QjρˆQj − ρ,ˆ QjQj . (18) 2 + equation of motion of the expectation value of any arbitrary j=1 operator Oˆ is given by the evolution rule D denotes the dimension of the system’s Hilbert space and ρ ∂hOiˆ ∂ρˆ = Tr Oˆ . (21) represents the density matrix of the reduced (open) system. ∂t ∂t Furthermore, Qˆj are Lindblad operators (or quantum jump operators) which represent arbitrary linear operators taken Note that the explicit time dependence of Schrödinger’sfield in the Hilbert space of the system, each acting as an inde- operators is here ignored. As one can readily see, the N - ˆ ˆ pendent incoherent process. Those Lindblad generators may point function is expressed in the form hOi ≡ F, with O ˆ be taken to be time-dependent, which allows a more general being defined in terms of the set of operators {ψi} and their description of open systems that encloses the systems sub- adjoints at different points. Thus we can write jected to an external time-dependent field. In the case of ∂F 1 h i a one-dimensional leaky lattice, for instance, the correlated = Oˆ, Hˆ + Tr OˆLˆρˆ . (22) ∂t i − two- and three-body particle losses are expressed in terms of ~ dissipation [35, 36, 38, 37] and γj denote the respective loss Using the Schrödinger-picture form of the N -point func- ˆ† ˆ tion (15) and the Lindblad equation (16) into the rule (22), or coupling rates at sites j. Meanwhile Qj and Qj play the roles of the particle creation and annihilation operators for exploiting the mathematical properties of the trace operator bosons, respectively. Apart from losses, the Lindblad equa- leads to the equation sought for. Let us provide the detailed tion may also include gains in some terms of the sum. The derivation of the equation. balanced loss and gain can strongly affect the properties of For simplification purpose, let us rewrite the N -point func- many-body systems. It has been shown that the balanced loss tion in terms of a general operator and gain between lattice sites lead to the existence of station-  N  N ary states and the phase shift of pulses between two lattice ˆ ˆ0† ˆ0† ˆ ˆ Y ˆ0† Y ˆ O = ψN ...ψ1 ψ1...ψN =  ψj0  ψj. (23) sites in many-particle systems [38]. In bosonic many-body j=1 j=1

5 Notice that the index j0 ≡ N + 1 − j is used. F in this case is by using in a smart way the following (anti)commutator iden- a single time multipoint correlation function. The outline of tity the derivation of the time evolution equation of the N -point ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ function can be given as follows. We expand separately the [A, BC]− = [A, B]±C ∓ B[A, C]±, (26) two terms in the right-hand side of Eq. (22), considering the hermiticity of the Hamiltonian, i.e., Hˆ † = Hˆ . where + denotes the anticommutator and − the commutator. Then using the canonical commutation and anticommutation rules for bosons and fermions, respectively, we can evaluate 3.2.1 Dissipation term the two key commutators in the right-hand side of Eq. (25). From Eq. (20), we can write the dissipation term of Eq.(22) We obtain as N " ˆ # 00 X 00 0 ∂O Z γ(r00, t) [ψˆ , Oˆ] = (−1)j δD r00 − r , Tr OˆLˆρˆ = − dDr00 j ˆ0† j=1 ∂ψj 2 (24) (27) 00 00 00 N " # † 00 † 00 00 † ˆ × Tr Oˆψˆ ψˆ ρˆ + Oˆρˆψˆ ψˆ − 2Oˆψˆ ρˆψˆ . 00 X 00 ∂O [ψˆ †, Oˆ] = (−1)1+j δD (r00 − r ) . j ˆ j=1 ∂ψj Using the cyclicity (invariance under cyclic permutation) and additivity properties of the trace operator, we express the dis- We have j00 = 0 and j00 = N − j for bosons and fermions, re- sipation term in terms of two useful commutators as follows: spectively. The factor with −1 is reminiscent of anticommu- Z 00 tativity of fermionic ﬁelds since a negative sign would appear D 00 γ(r , t) ˆ00† ˆ00 Tr OˆLˆρˆ = − d r Tr ψ [ψ , Oˆ]−ρˆ each time two annihilation/creation operators are swapped in 2 the general operator Oˆ for a fermion. As a consequence, we 00 (25) Z γ(r , t) 00 + dDr00 Tr [ψˆ †, Oˆ] ψˆ00ρˆ . have 2 − 0 ∂Oˆ ∂Oˆ 00 ψˆ † = ψˆ = (−1)j Oˆ. (28) Remember that all possible pairwise combinations of the N j ˆ0† ˆ j ˆ N ∂ψj ∂ψj annihilation operators in the set {ψj}j=1 commute (anticom- mute) for bosons (fermions), which is also the case for the set 0 A subsequent substitution of Eqs. (27) ˆ † N of corresponding creation operators {ψj }j=1. In the following and (28) into Eq. (25) yields derivation we deal with bosons and fermions simultaneously

N " ˆ ! ˆ !# N 0 N 1 X 00 0 ∂O ∂O X γ(rj, t) D E X γ(rj, t) D E Tr OˆLˆρˆ = (−1)1+j γ(r0 , t)Tr ψˆ † ρˆ + γ(r , t)Tr ψˆ ρˆ = − Oˆ − Oˆ . 2 j j ˆ0† j ˆ j 2 2 j=1 ∂ψj ∂ψj j=1 j=1 (29)

−→ ←−0 0 Thus the dissipation term for the N -point function yields The energy operators h j ≡ h(rj) and h j ≡ h(rj) apply only ˆ ˆ0† rightwise on ψj and leftwise on ψj , respectively. The function N 0 h(rj) is given by the expression X γ(rj, t) + γ(rj, t) D E i Tr OˆLˆρˆ = − i Oˆ . (30) ~ ~ 2 j=1

3.2.2 The von Neumann term 2 ~ 2 In order to expand the Hamiltonian term of Eq.(22), let Aˆn = h(r) = − ∇ + V (r, t) 0 r ˆ0† ˆ † ˆ ˆ ˆ ˆ 2M (32) ψn ...ψ1 ψ1...ψn, with n = 1, ..., N , which means AN = O. Z h i + dDr00 ψˆ†(r00)U(r00, r, t)ψˆ(r00), Then we can express the commutator Aˆn, Hˆ in terms of − h i next commutator Aˆn−1, Hˆ as −

h ˆ ˆ i h ˆ0† ˆ ˆ ˆ i An, H = ψn An−1ψn, H − − ˆ0† h ˆ ˆ i ˆ which is nothing but a second-quantized many-particle Hamil- =ψn An−1, H ψn (31) − tonian. Eq. (38) provides the evolution of N -point functions 0 ←−0 −→ in a dissipative quantum system described by the Lindblad + ψˆ † − h Aˆ + Aˆ h ψˆ . n n n−1 n−1 n n equation. Let us iterate the formula (31) down to n = 1. For

6 illustration, the ﬁrst two iterates are obtain the following expression for the commutator

h i 0 h i Aˆ , Hˆ =ψˆ † Aˆ , Hˆ ψˆ N −1 " N −j ! n n n−1 n h i 0 −→ − − ˆ ˆ X Y ˆ † ˆ O, H = ψN +1−i Aj h j+1 0 ←−0 −→ − ˆ † ˆ ˆ ˆ j=0 i=1 +ψn − h nAn−1 + An−1 h n ψn   (35) h i 0 h i N # ˆ ˆ ˆ † ˆ ˆ ˆ ←−0 Y An−1, H =ψn−1 An−2, H ψn−1 ˆ ˆ − − − h j+1Aj ×  ψi . ˆ0† ←−0 ˆ ˆ −→ ˆ i=j+1 +ψn−1 − h n−1An−2 + An−2 h n−1 ψn−1, (33) For the validity of the above equation, the following conven- tions are assumed: and the last two iterates are 0 N h i 0 h i 0 ˆ ˆ ˆ † ˆ ˆ ˆ Y † Y A2, H =ψ2 A1, H ψ2 Aˆ0 = , Ψˆ = , and Ψˆ i = . (36) − − I N +1−i I I i=1 i=N +1 ˆ0† ←−0 ˆ ˆ −→ ˆ +ψ2 − h 2A1 + A1 h 2 ψ2 (34) Let us mention in passing that we can alternatively derive h ˆ ˆ i ˆ0† ←−0 −→ ˆ A1, H =ψ1 − h 1 + h 1 ψ1. that relation through a substitution of Eqs. (17), (26) and − h i (27) into the commutator Oˆ, Hˆ . ˆ − We introduce the quantity A0 = I (identity). Using backward substitution in the above set of equations up to n = N we

3.2.3 The evolution equation Putting the dissipation and Hamiltonian terms (30) and (35) together, we ﬁnally get the evolution equation   N −1 * N −j ! N + N 0 ∂F X Y 0 −→ ←−0 Y X γ(ri, t) + γ(r , t) i = ψˆ † Aˆ h − h Aˆ ψˆ − i F i . (37) ~ ∂t N +1−i j j+1 j+1 j  i ~ 2 j=0 i=1 i=j+1 i=1

The N -point function function F is here evaluated using Schr¨odinger’snon-evolving ﬁeld operators. We can rewrite the above equation as

N −1 * N ! j !  N  + ∂F X Y 0 Y −→ Y i = ψˆ † ψˆ h ψˆ ~ ∂t N +1−i i j+1  i j=0 i=1 i=1 i=j+1 (38) N −1 * N −j ! j ! N ! + N 0 X Y 0 ←−0 Y 0 Y X γ(ri, t) + γ(r , t) − ψˆ † h ψˆ † ψˆ − i F i , N +1−i j+1 i i ~ 2 j=0 i=1 i=1 i=1 i=1

ˆ ˆ where the field operators are explicitly given by ψj ≡ ψ(rj) operators, i.e., in the Heisenberg picture. In order to take 0 ˆ † ˆ† 0 advantage of the mapping, we thus have to write the equa- and ψj ≡ ψ (rj). The evolution of N -point functions can reproduce the evolution of basic quantities such as the den- tion of motion of the Schrödinger N -point function into the sity, the atom number, and higher-order correlations, thereby Heisenberg-picture form. One would think that it suffices to giving some concrete interpretation for the N -point function. replace the density operator by an evolving field operator in Eq. (38) represents an equation of motion for various correla- order to write the Lindblad equation into the Heisenberg pic- tions of fermionic systems. The same equation describes the ture. Such a solution is quite naive for at least two reasons. behavior of bosonic systems. Such an equation therefore uni- The density matrix is not a standard quantum-mechanical fies the dynamics of bosons and fermions as far as correlation operator, but a projective operator built out of state kets. functions are concerned. Moreover the resulting equation fails to evolve the product of two operators properly, which is one fundamental way of testing the equations of motions of quantum mechanical op- 3.3 Mapping identity for two different erators. Thus the mapped system would be physically not evolutions of N -point functions realisable and consequently irrelevant. In order to resolve the discrepancy, a Langevin force may be introduced into the While the N -point function above evolves in the Schrödinger problem, leading to the Heisenberg-Langevin equation. The picture, the useful exact mapping for closed many-body sys- price to pay when using such a counterpart of Lindblad equa- tems is expressed in terms of time-evolving quantum-field

7 tion is to deal with the stochasticity of the system, see the are unchanged irrespective of the quantum mechanical books [40, 41]. A cheaper and more experiment-oriented so- picture and using again the unitary transformation (7), lution, however, consists in considering only the evolution of the equation of motion of N -point functions (38) can expectation values. The N -point function encloses expecta- be rewritten using the time-evolving quantum ﬁelds as tion values of various measurable physical quantities. Considering that expectation values of given operators

N −1 * N ! j ! N + ∂F 1 X Y 0 Y −→ Y = ψˆ † (t) ψˆ (t) h (t) ψˆ (t) ∂t i N +1−i i j+1 i ~ j=0 i=1 i=1 i=j+1 (39) N −1 * N −j ! j ! N + N 0 1 X Y 0 ←−0 Y 0 Y X γ(ri, t) + γ(r , t) − ψˆ † (t) h (t) ψˆ †(t) ψˆ (t) − F i , i N +1−i j+1 i i 2 ~ j=0 i=1 i=1 i=1 i=1

0 −→ ˆ ˆ ˆ † ˆ† 0 where ψj(t) ≡ ψ(rj, t), ψj (t) ≡ ψ (rj, t), h j(t) ≡ h(rj, t), the same equation (39) derived from the Lindblad master ←− and h 0 (t) ≡ h(r0 , t), with equation (16). Therefore, for a given experiment A with a j j dissipative quantum gas described by the set {ψ, U, V, γ, F}, 2 an equivalent but different experiment B described by the ~ 2 h(r, t) = − ∇ + V (r, t) set {Ψ, U,˜ V,˜ γ,˜ F}˜ will exist, such that both experiments can 2M r Z (40) be exactly mapped onto each other. The mapping identities + dDr00 ψˆ†(r00, t)U(r00, r, t)ψˆ(r00, t). are given by Eqs. (3), (4), (6), and (42). This makes our exact space-time mapping relevant for both non-dissipative The difference between Eqs. (38) and (39) is that the opera- and dissipative quantum systems. The N -point functions can tors in the latter are expressed in the Heisenberg picture, so capture any observables, including the N -body non-local cor- that we can have h(r, t)ψˆ(r, t) ≡ i~ ∂ψˆ(r, t)/∂t in the absence relation functions [42, 43], and the hydrodynamic fields [44] of dissipation, which is simply the Heisenberg equation. Like- which are the mean density hnî = hψˆ†(r)ψˆ(r)i, the field cor- 0 ˆ† 0 ˆ wise, the operator h(r, t) in Eq. (40) is the Heisenberg picture relations G1(r − r ) = hψ (r )ψ(r)i, the density correlations equivalent of h(r) in Eq. (32). Like their Schrödingerpicture C(r − r0) = hnˆ(r0)ˆn(r)i − hnî2, and the density-density corre- −→ ←−0 lation G (r − r0) = h(ˆn(r0) − hnî)(ˆn(r) − hnî)i/hnî [11, 27]. counterparts, the operators h j(t) and h j(t) still apply only 2 ˆ ˆ0† on ψj(t) and ψj (t), respectively, in the direction indicated by the arrow. 4 A concrete example of mapping Eq. (39) may describe the time evolution of expectation values of various physical quantities in a given experiment, say between two experimental situations A. Using the mapping (3), we can show that the corresponding expectation values in another experiment, B, still obey a In this section, we describe two possibly achievable experi- similar equation with potentials and dissipation rate different ments with bosons, say A and B. By using numerical experi- but appropriately related to those in A. A straightforward ments we show how a spacetime mapping can be used to mim- calculation leads to the Lindblad evolution of the mapped ick the evolution of correlations in B from the direct evolu- N -point function (6). Such an evolution is governed by an tion of correlations measured in A. More clearly, the wording equation similar to Eq. (39), with the substitutions ’experiments’ we use does not actually mean achieved experiments, since we have performed no experimental works. It ˆ ˆ ˆ0† ˆ † 0 simply denotes a realistic situation that can be experimentally ψj → Ψ(λrj, τ), ψj → Ψ (λrj, τ), 2 explored or reproduced in a quantum gas laboratory. Data for h → h˜(λr , τ), h˜(r, t) = − ~ ∇2 + V˜ (r, t) the mapping are here given by the condensate wave function, j j 2M r (41) Z which is the simplest example we could take for illustrating + dDr00 Ψˆ †(r00, t)U˜(r00, r, t)Ψ(ˆ r00, t). our results. Such a wave function is generated through numerical computations of the imaginary Gross-Pitaevskii (iGP) More importantly, the corresponding mapping identity for N - equation. While the iGP equation does not directly stem from point functions is the same as in Eq. (6), but in addition a Eq. (38), we show in the Appendix how it can be derived di- mapping identity for local dissipation rates has to be taken rectly in a similar way. There we also show how its derivation into account, which reads can follow from Eq. (38).

2 γ(ri, t) → γ˜(ri, t) = λ(t) γ(λri, τ(t)). (42) 4.1 Description of the experimental situations Thus the N -point functions F and F˜ which are related We consider two potential experiments with simple lossy in accordance with the quantum-ﬁeld mapping, both satisfy quantum systems made of a harmonically trapped and elon-

8 gated ultracold gas of bosons with repulsive interaction onto The function φ depends on time t and space x, and represents which is acting a highly controlled dissipation. In experiment the condensate wave function. It corresponds to the field op- A, the gas with constant interparticle interaction is confined erator ψˆ, i.e., hψî ≡ φ. The coefficient g(t) is the strength of 2 2 in a harmonic trapping potential VA = MωAr /2 and the interaction between the gas particles. The condensate is har- amplitude of local dissipation rate is modulated in time. We monically trapped such that V (x) = ω2x2/2. The prefactor achieve a corresponding experiment B with trapping poten- γ(x, t) is the local dissipation rate given by [22, 48]: tial V = Mω2 r2/2 and modulated interparticle interaction B B 2 − x strength by choosing γ(x, t) = σ(t)e 2w(t)2 , (46)

1 2 λ(t) = p , (43) where σ ∝ I/w , with I(t) being the electron current and a cos(2ωt) + b w(t) the waist of the beam which are both tunable indepen- where a = (1 − η2)/2 and b = (1 + η2)/2, and we take η to dently. A similar dissipation rate produced by a cylindrically be the ratio of trapping frequencies in both experiments, η = focused laser beam was used to create a single Gaussian defect in the study of dissipative transport of Bose-Einstein conden- ωA/ωB, and ω ≡ ωB. The free parameter satisfies λ(0) = 1 and λ˙ (0) = 0, and shows how to modulate the dissipation and sates [51]. In what follows, we will numerically solve the Gross- interaction strengths in experiments A and B. The time tA when the data are measured in experiment A can be mapped Pitaevskii equation above for a Bose-Einstein condensate with imaginary potential to examine the dynamics of the system onto the time tB when the corresponding data are obtained in experiment B, such that for the two experiments A and B. In the numerical experiment A, the interaction strength gA(t) = g0 and the beam Z tB 2 waist wA(t) = w0 are constant. The amplitude of the loss rate tA = λ(t) dt. (44) 2 0 is modulated in time in such a way that σA(t) = σ0/λ(t) . In the numerical experiment B, the waist of the beam as well If we take for instance t = 20π, we obtain t ≈ 41.89. Even B A as the interaction strength are modulated in time such that though the present setting is not very general, as it does not w (t) = w /λ(t), and g (t) = g λ(t). The amplitude of the allow to independently tune the strength and modulation fre- B 0 B 0 loss rate σ (t) = σ is constant. Thus the amplitude and quency of the interparticle interaction, the scenario we de- B 0 waist of the beam is differently modulated in both experi- scribe here is experimentally achievable. For a contact inter- ments. With the above choice of parameters, experiments A action (s = D), the modulation of interparticle interactions and B may be described by the quantum fields ψˆ and Ψ,ˆ re- can be realized with a time-dependent magnetic field via Fesh- spectively, and then can be related onto each other through bach resonances [45, 46]. Experimentally it has been possible our mapping. We perform the numerical analysis based on to engineer tunable dissipations in quantum systems using a the split-operator method including the fast Fourier trans- technique called scanning electron microscopy. The working form. The initial state will be prepared by relaxation from a principle of this electron microscopy for quantum gases con- condensate background within the Thomas-Fermi regime. sists in scanning a focused electron beam over the atom cloud. The electron beam is shot onto the condensate, and the colli- sion of electrons with the trapped atom cloud locally causes 4.2 Mean-field mapping of the experiments the ionization of single atoms turning them onto untrapped We consider the case where the beam is narrow, such that the ions that escape from the condensate [22]. The fleeing ions average waist is less than a quarter of the initial condensate can then be subsequently detected by an ion detector [47]. radius. Only the particles located around the trap center The electron beam creates a fully controllable and environ- are then kicked out of the condensate. We display the results mentally induced imaginary potential acting on the ultracold from direct computations of the Gross-Pitaevskii equation for quantum gas. both experiments, and then concretely show how the result of Theoretically, an ultracold quantum gas with a source or one experiment can be recovered from the other by using the sink of particles is best described by the Lindblad master mapping. equation. In the mean-field regime, however, such a system may be well described by a Gross-Pitaevskii equation 4.2.1 Direct evolutions of the system in both experiments with an imaginary potential where contact interaction is assumed [22]. One of the reasons the Gross-Pitaevskii equa- In order to depict the evolutions of the system, we may com- tion is preferred to the Lindblad equation is easier numeri- pute the time evolution of any dynamical variables like the cal treatment since the Gross-Pitaevskii equation is in gen- condensate radius, the density, or the volume integrated local eral much more tractable in the limit of large atom num- ν-body correlation function defined as [27, 42, 52, 53] bers where solving more accurate equations is hard and time- Z consuming. Without loss of generality, we consider the system g(ν)(t) ∝ |φ(x, t)|2(ν+1)dx, (47) to be one-dimensional. In the dimensionless form, the Gross- Pitaevskii equation with imaginary potential can be written where ν = 0, 1, 2, 3 correspond to the density, and local one-, as follows [48, 49, 50]: two- and three-body correlation functions, respectively. 2 2 ∂φ 1 ∂2φ In Fig. 1, we portray the densities |φA(x, t)| and |φB(x, t)| i = − + V (x)φ + g(t)|φ|2φ + iγ(x, t)ψ. (45) ∂t 2 ∂x2 for both experiments A and B, respectively. In experiment

9 A, the only time-dependent parameter is the amplitude of the loss rate, and thus only excitations with small amplitude form on top of the cloud as shown in Fig. 1(a). In experiment B both the beam waist and the interparticle interaction strength are nontrivially changing with time in a periodic way. The condensate strongly breathes as its radius and maximum density vary periodically in time, see Fig. 1(b). It is clear that experiments A and B are very diﬀerent, but as we will show in what follows, they can be exactly mapped onto each other using our mapping.

4.2.2 Using the mapping to mimick the evolution of B from the direct evolution of A

Suppose the data from experiment A as portrayed in Fig. 1(a) are described by the field operator ψˆ or the condensate wave ∗ function φA. We reconsider those data for times t ∈ [0, t ] where t∗ ≈ 41.9, and we display the density, the condensate radius, and the correlators g(1) and g(2) in Figs. 2(a), (c), (e), respectively. Using these data, we can deduce the corresponding data of experiment B as portrayed in Fig. 1(b) without running it. For this, we introduce the data in the mapping (4), and obtain the data described by the field operator Ψˆ or condensate wave function φB (data labelled in the plots by mapped A), which we display in Figs. 2(b), (d), (f) comparatively with data from Fig. 1(b) for times t ∈ [0, 20π]. Figure 2: (Color online) Exactly mapping the data in the experimental We readily realize that the density evolution in Fig. 2(b) is situation B from the data in the experimental situation A. exactly identical to the one in Fig. 1(b). Moreover, there The left panels (a), (c) and (e) show the evolution of the denis a perfect overlap between the condensate radii and local sity, condensate radius and local correlations in experiment A for runs up to time t∗, respectively. The right panels (b), correlations in Figs. 2(d), (f). Thus the data mapped A ob- (d), (f) depict a comparison between the data from the direct tained by applying the mapping to A are nothing but the numerical experiment B and the result mapped A obtained data directly obtained from experiment B. Hence our map- by mapping the data from the direct numerical experiment ping clearly allows to exactly mimick the evolution of B from A, through the density, condensate radius and local correlations, respectively. the direct evolution of A. Conversely, it is possible to get the data in Fig. 1(a) from those in Fig. 1(b). In that case, we have to inverse the relation (4) in such a way that the field 4.2.3 Special cases operator ψˆ is expressed as a function of the field operator Ψ.ˆ The same mapping holds for any possible correlators or The mapping may take a particularly significant meaning in experimental data that can be measured in both experiments. some particular regimes.

Very large beam waist. We consider a regime when the electron beam is large enough to hit the condensate anywhere at a time. Because the waist is very large (we take e.g., −3 w0 = 15.63, keeping σ0 = 7.970 × 10 ), the local dissipation rate reduces to γ(x, t) ≈ σ(t) which is space-independent. Then the mapping transforms a time periodic modulation of the dissipation rate γ(t) onto a time periodic modulation of the interaction strength g(t) between the particles of the dissipative quantum gas. Dissipation from an electron beam can then be used to control the interaction strength between the Figure 1: (Color online) Density evolution computed directly using the particles of a quantum gas. It is true that controlling the in- Gross-Pitaevskii equation (45) for times up to t = 20π in experimental situations A (a) and B (b). The dashed line terparticle interaction in many types of quantum gas species is ∗ in panel (a) depicts the time tA ≈ 41.9 ≡ t which may be usually done via magnetic, optical, and magneto-optical Fes- mapped onto tB = 20π according to Eq. (44). The values of hbach resonances. The speciﬁc type of Feshbach resonance is simulation parameters used are ωA = 1.5, ωB = 1.0, g0 = chosen based on the type of particle, and is consequently de- 10.0, σ = 3.985 × 10−2, and w = 0.1563. 0 0 pendent on the resonance width, which turns out to be very narrow for some species. Therefore, Feshbach resonance tech-

10 nique may not be efficient in some regimes, for instance, due be used to control the interaction between particles in open to the narrowness of the resonance width or a faster depletion quantum many-body systems. Our result is a general theo- of the quantum gas sample through recombination processes. retical model that can be useful for mimicking quantum gas In such regimes, our method can be appropriate. experiments with dissipation and for benchmarking relevant theoretical approximations, especially in challenging regimes of dissipation that make the system hardly tractable. In our Weak interaction regime. In the limit when g0 tends to future works, it may be worth trying to further generalize our zero, the quantum gas is close to an ideal gas and the ef- mapping to explicitly describe autocorrelation functions and fect of dissipation can be studied alone. The mapping then possibly non-Markovian open quantum systems. transforms a time periodic modulation of the dissipation rate amplitude σ(t) onto a time periodic modulation of the beam waist w(t). In this case, the mapping helps matching the Acknowledgements effect on the condensate of changing the electron current for fixed beam waist and the effect of changing the beam waist at Part of this work was done during a research visit of EW at the constant electron current. This regime can also well describe Technical University Kaiserslautern, Germany. EW thanks the behavior of polaritons and photon condensates where the his host Prof. Dr. James R. Anglin for that opportunity and interparticle interaction is very weak in general [54, 55]. It for deep and fruitful discussions thereof. EW acknowledges should be noted that the mapping is still valid even when financial support from the Alexander von Humboldt Founda- mean field breaks down, since it is orginally formulated in tion, and from the Simons Associateship joint programme of terms of quantum fields. the Abdus Salam International Center for Theoretical Physics and of the Simons Foundation. The work is also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research 5 Conclusion Foundation) under the project number 277625399 - TRR 185.

In this paper, we have presented an exact spacetime mapping of two experimental situations with open quantum systems in a regime of loss or gain. The mapping is formulated with the Appendix: The imaginary help of N -point functions of two evolutions of open quantum Gross-Pitaevskii equation gases that obey the Lindblad equation. Our formalism is general as it extends a previous result on quantum field mapping As shown in the main text, this paper essentially focuses on for closed systems [26] to the general case of Markovian open the evolution of N -point functions, which have an equal num- quantum systems with gain and/or loss. For this, we have ber of creation and annihilation operators. In section 4, how- considered an open many-body quantum system consisting of ever, we discussed as a bosonic example the condensate wave a quantum gas in any arbitrary dimension that interacts with function, which is the expectation value of only one creation the environment and evolves in the regime when the Lind- or annihilation operator governed by the imaginary Gross- blad generators feature a loss or a gain. Given that expec- Pitaevskii (iGP) equation. It should be noted that the ana- tation values are unchanged in different pictures of quantum logue of the iGP equation does not exist for fermionic sys- mechanics, we have used the evolution of quantum fields that tems since the expectation value of any field operator would describe the gas dynamics to derive the Heisenberg evolution be zero. This is a consequence of the macroscopic occupation of any arbitrary N -point correlation function of the system. of the lowest energy state being guaranteed for bosons but Our previous result on the quantum field mapping for closed forbiden for fermions due to the Pauli exclusion principle. In quantum systems is rewritten in terms of wave functions, and this Appendix, we discuss the derivation of the iGP equation the extension to open quantum systems is achieved by relat- and the equation that rules the condensate decay rate, i.e., ing onto each other the N -point functions of two different the particle loss rate. First, the derivation of the iGP equa- evolutions of the open quantum system. Such a result is valid tion is carried out directly following the method presented in for both bosonic and fermionic systems. Because the imagi- section 3.2 for a simple example where Oˆ = ψˆ. Secondly, the nary Gross-Pitaevskii equation is a mean-field approximation iGP equation is retrieved from the evolution of the 1-point and a special case of the Lindblad master equation for lossy function, and then the equation that governs the condensate bosonic systems that can provide quantitative results, an il- decay rate is deduced from such an evolution. lustrative example of the mapping has been provided based on such an equation. That example includes two achievable A.1- Direct derivation experiments which are the modulation of the local dissipation rate and the modulation of the interaction strength in Suppose we have Oˆ = ψˆ. Let us expand ultracold gases of bosons confined in harmonic traps. The separately the two terms in the right-hand result suggests that a modulation of the dissipation rate can side of Eq. (22). Then the first one reads

Z 2 Z h ˆ ˆ i D D 0 ˆ ˆ 0† ~ 2 0 ˆ 0 1 D 0 D 00 h ˆ ˆ 0† ˆ 00† 0 00 ˆ 00 ˆ 0 i E Ψ, H = d r Ψ, Ψ − ∇r0 + V (r , t) Ψ + d r d r Ψ, Ψ Ψ U(r , r , t)Ψ Ψ − 2M 2 − − (A-1) D 2 Z E = − ~ ∇2 + V (r, t) Ψ(ˆ r) + dDr0 Ψˆ †(r0)U(r0, r, t)Ψ(ˆ r0)Ψ(ˆ r) = hh(r)Ψ(ˆ r)i. 2M r 11 We have used commutator identities, the boson canon- )condensate wave function. Thus from (A-5) we finally get ical commutators as well as the relation [A,ˆ Bˆ] = − ∂φ(r, t) 2 [A,ˆ Aˆ†] ∂B/∂ˆ Aˆ†, or equally the − branch of the rule (26). i = − ~ ∇2 + V (r, t) − ~ ∂t 2M r The second term of Eq. (22) yields Z + dDr0 φ∗(r0, t)U(r0, r, t)φ(r0, t) (A-7) Z 0 ˆ ˆ D 0 γ(r , t) ˆ ˆ 0† ˆ 0 Tr ΨLρˆ = − d r Tr Ψ Ψ Ψ ρˆ γ(r, t) 2 − i~ φ(r, t). 0 0 2 +ρ ˆΨˆ †Ψˆ 0 − 2Ψˆ 0ρˆΨˆ † In the case of hard-sphere contact potential, we have 0 (A-2) 0 D 0 Z γ(r , t) 0 U(r , r, t) = g(t)δ (r − r ). Thus = − dDr0 Tr [Ψˆ , Ψˆ †]Ψˆ 0ρˆ 2 ∂φ(r, t) 2 0 ~ 2 Z γ(r , t) 0 i~ = − ∇r + V (r, t) + dDr0 Tr Ψˆ †[Ψˆ , Ψˆ 0]ˆρ . ∂t 2M (A-8) 2 γ(r, t) + g(t) |φ(r, t)|2 − i φ(r, t), ~ 2 Using the boson canonical commutator as well as the rule ˆ ˆ which is a more general form of the Gross-Pitaevskii equation hOi = Tr Oρˆ , we get with imaginary potential as given in Eq. (45) of the main text. A similar derivation was provided in Ref. [22], see its supple- Z 0 γ(r , t)D 0 E mental material. By combining Eq. (A-8) and its conjugate, Tr Ψˆ Lˆρˆ = − dDr0 [Ψˆ , Ψˆ †]Ψˆ 0 2 it is possible to derive the condensate decay rate. However we Z γ(r0, t) show here that the decay rate can be obtained directly using = − dDr0 δD(r − r0) hΨˆ 0i (A-3) 2 our evolution of a 1-point function, which means that the iGP γ(r, t) can be inversely derived from the 1-point function. = − hΨˆ i. 2 A.2- Derivation from a 1-point function Summing together both terms, we therefore obtain ˆ ˆ † 0 ˆ Suppose we now have O = Ψ (r1) Ψ(r1), i.e., similar to a 2 local density operator, and F is similar to a one-body local ∂ D ~ 2 i hΨˆ i = − ∇ + V (r, t) Ψ(ˆ r) density. The time evolution of a one-point function can then ~∂t 2M r Z be given by Eq. (38) for N = 1, so we have D 0 ˆ 0† 0 ˆ ˆ 0E + d r Ψ U(r , r, t)ΨΨ ∂ 0 D 0 ←− −→ E i hΨˆ †Ψˆ i = Ψˆ † − h (r0 ) + h (r ) Ψˆ (A-4) ~∂t 1 1 1 1 1 1 γ(r, t) 0 (A-9) − i hΨˆ i 0 ~ γ(r1, t) + γ(r1, t) ˆ † ˆ 2 −i~ hΨ1 Ψ1i. −→ γ(r, t) 2 ≡ h h (r)Ψ(ˆ r)i − i hΨ(ˆ r)i. ~ 2 Using this evolution equation for expectation values, we can obtain the imaginary Gross-Pitaevskii equation and even the In this equation, the correlation function hΨˆ i represents the decay rate equation. condensate wave function. A.2.1- Evolution in the mean-field regime: the iGP equation Mean-field illustration. In the mean-field regime, expectation values can readily be factorized, and we have Consider the evolution equation of the 1-point function in Eq. (A-9) in a regime where the mean-field approximation is ∂ 2 valid. Then we can factorize the expectation values of all op- i hΨˆ i = − ~ ∇2 + V (r, t) hΨˆ i ~∂t 2M r erators. Because the multiplication of expectation values is commutative and considering the evolution of hΨˆ 1i ≡ φ(r1, t) Z 0 0 D 0 ˆ † 0 ˆ ˆ 0 (A-5) ˆ † ∗ + d r hΨ iU(r , r, t)hΨihΨ i and hΨ1 i ≡ φ (r1, t), we get the following set of two equations: γ(r, t) ˆ − i~ hΨi. ∂hΨˆ i γ(r , t) 2 i 1 =h(r )hΨˆ i − i 1 hΨˆ i ~ ∂t 1 1 ~ 2 1 0 (A-10) We set for the mean-field wave functions to be † 0 ∂hΨˆ i 0 γ(r , t) 0 i 1 = − h(r0 )hΨˆ †i − i 1 hΨˆ †i. ~ ∂t 1 1 ~ 2 1 Ψ(ˆ r) ≈ hΨ(ˆ r)i(t) = φ(r, t), (A-6) In that equation, we have Ψˆ †(r) ≈ hΨˆ †(r)i(t) = φ∗(r, t), 2 ~ 2 h(r) = − ∇r + V (r, t) which are expectation values of time independent operators 2M Z (A-11) evaluated at time t, and not the expectation values of time- + dDr00 hψˆ†(r00)iU(r00, r, t)hψˆ(r00)i. evolving operators. The quantity φ(r, t) is actually the (quasi-

12 By taking the complex conjugate of the second equation in the [4] D. Hellweg, L. Cacciapuoti, M. Kottke, T. Schulte, set (A-10), we realize that the two equations are identical. In K. Sengstock, W. Ertmer, and J.J. Arlt, Measurement the case of hard-sphere contact potential, we readily retrieve of the Spatial Correlation Function of Phase Fluctuat- the iGP equation (A-8). ing Bose-Einstein Condensates, Phys. Rev. Lett. 91, 010406 (2003).

A.2.2- Evolution in the equal position limit: the decay [5] S. Manz, R. Bücker, T. Betz, Ch. Koller, S. Hoffer- rate berth, I.E. Mazets, A. Imambekov, E. Demler, A. Per- In order to get the particle loss rate from Eq. (A-9), consider rin, J. Schmiedmayer, and T. Schumm, Two-point den- the special case when the creation and annihilation operators sity correlations of quasicondensates in free expansion, 0 Phys. Rev. A 81, 031610(R) (2010). act at the same position, i.e., r1 = r1 ≡ r. Then we get the dissipative and von Neumann terms to be −i~γ(r, t) n(r, t) D ←− −→ E [6] T. Schweigler, V. Kasper, S. Erne, I. mazets, B. Rauer, ˆ † ˆ and Ψ − h (r) + h (r) Ψ , respectively, where n(r, t) = F. Cataldini, T. Langen, T. Gasenzer, J. Berges, and D E Ψˆ †(r)Ψ(ˆ r) . In the Hamiltonian part, both the single- and J. Schmiedmayer, Experimental characterization of a t quantum many-body system via higher-order correla- two-particle interactions terms vanish, and thus the above tions, Nature 545, 323 (2017). equation becomes [7] J. Schwinger, On the Green’s functions of quantized ∂ 2 D −→ E∗ D −→ E i n(r, t) = ~ Ψˆ ∇2Ψˆ † − Ψˆ † ∇2Ψˆ fields. I, Proc. Natl Acad. Sci. USA 37, 452 (1951). ~∂t 2M r r (A-12) − i~γ(r, t) n(r, t). [8] J. Schwinger, On the Green’s functions of quantized fields. II, Proc. Natl Acad. Sci. USA 37, 455 (1951). −→ ←− It should be noted that h − h is not a null operator because of the quantum pressure resulting from the kinetic term. [9] A. Giraud and J. Serreau, Decoherence and Thermaliza- tion of a Pure Quantum State in Quantum Field The- ory, Phys. Rev. Lett. 104, 230405 (2010). Mean-field illustration. If we invoke mean-field approximation in the above equation, it yields the continuity equation [10] Z. Denis and S. Wimberger, Two-Time Correlation Functions in Dissipative and Interacting Bose-Hubbard ∂ 2 i |φ (r, t) |2 = ~ φ∇2φ∗ − φ∗∇2φ − i γ(r, t) |φ(r, t)|2. Chains, Condens. Matter 3, 2 (2018). ~∂t 2M r r ~ (A-13) [11] C.-L. Hung, X. Zhang, L.-C. Ha, S.-K. Tung, N. Gemelke, and C. Chin, Extracting density-density cor- Summing over all possible r finally gives the relation: relations from in situ images of atomic quantum gases, New J. Phys. 13, 075019 (2011). ∂N Z = − dDr γ(r, t)|φ(r, t)|2. (A-14) ∂t [12] A. Steffens, M. Friesdorf, T. Langen, B. Rauer, T. Schweigler, R. Hübener, J. Schmiedmayer, C.A. Riofró, Obviously, the decay of atom number is due to the loss. In and J. Eisert, Towards experimental quantum-field to- the case when the loss rate is constant, we get a purely ex- mography with ultracold atoms, Nat. Commun. 6, 7663 ponential decay given by N(t) = N exp(−γt). It should be 0 (2014). noted that the mean-field approximation is not necessary for the derivation of the above equation of atom number decay. [13] R.G. Dall, A.G. Manning, S.S. Hodgman, W. RuG- Indeed, directly integrating Eq. (A-12) over space r leads to way, K.V. Kheruntsyan, and A.G. Truscott, Ideal n- the same result. body correlations with massive particles, Nat. Phys. 9, 341 (2013).

References [14] D.J. Wineland, and D. Leibfried, Quantum information processing and metrology with trapped ions, Laser Phys. [1] I. Kukuljan, S. Sotiriadis, and G. Takacs, Correlation Lett. 8, 175 (2011). Functions of the Quantum Sine-Gordon Model in and out of Equilibrium, Phys. Rev. Lett. 121, 110402 (2018). [15] P. Schindler, et al., Experimental repetitive quantum er- ror correction, Science 332, 1059 (2011). [2] S. Weinberg, The Quantum Theory of Fields (Cam- bridge University Press, Cambridge, England, 1995). [16] I. Bloch, J. Dalibard, and S. Nascimbene, Quantum simulations with ultracold quantum gases, Nat. Phys. [3] V. Guarrera, P. W¨urtz, A. Ewerbeck, A. Vogler, G. 8, 267 (2012). Barontini, and H. Ott, Observation of Local Tempo- ral Correlations in Trapped Quantum Gases, Phys. Rev. [17] J.I. Cirac and P. Zoller, Goals and opportunities in Lett. 107, 160403 (2011). quantum simulation, Nat. Phys. 8, 264 (2012).

13 [18] P. W¨urtz,T. Langen, T. Gericke, A. Koglbauer, and [34] H.P. Breuer and F. Petruccione, The Theory of Open H. Ott, Experimental Demonstration of Single-Site Quantum Systems, Oxford University Press (2002). Addressability in a Two-Dimensional Optical Lattice, Phys. Rev. Lett. 103, 080404 (2003). [35] F. Trimborn, D. Witthaut, H. Hennig, G. Kordas, T. Geisel, and S. Wimberger, Decay of a Bose-Einstein [19] A. Grigoriu, H. Rabitz, G. Turinici, Controllability of condensate in a dissipative lattice - the mean-ﬁeld ap- open quantum systems, hal-00696546v1 (2012). proximation and beyond, Eur. Phys. J. D 63, 63 (2011).

[20] C.P. Koch, Controlling open quantum systems: tools, [36] D. Witthaut, F. Trimborn, H. Hennig, G. Kordas, T. achievements, and limitations, J. Phys.: Condens. Mat- Geisel, and S. Wimberger, Beyond mean-field dynamics ter 28, 213001 (2016). in open Bose-Hubbard chains, Phys. Rev. A 83, 063608 [21] H.Z. Shen, S. Xu, S. Yi, and X.X. Yi, Controllable dis- (2011). sipation of a qubit coupled to an engineering reservoir, [37] J.M. Torres, Quantum master equation with balanced Phys. Rev. A 98, 062106 (2018). gain and loss, Phys. Rev. A 89, 052133 (2014). [22] G. Barontini, R. Labouvie, F. Stubenrauch, A. Vogler, V. Guarrera, and H. Ott, Controlling the Dynamics of [38] D. Dast, D. Haag, H. Cartarius, and G. Wunner, an Open Many-Body Quantum System with Localized Closed-form solution of Lindblad master equations with- Dissipation, Phys. Rev. Lett. 110, 035302 (2013). out gain, Phys. Rev. A 90, 052120 (2014). [23] H. Fröml,A. Chiocchetta, C. Kollath, and S. Diehl, [39] D. Poletti, J.-S. Bernier, A. Georges, and C. Kol- Fluctuation-Induced Quantum Zeno Effect, Phys. Rev. lath, Interaction-Induced Impeding of Decoherence and Lett. 122, 040402 (2019). Anomalous Diffusion, Phys. Rev. Lett. 109, 045302 (2012). [24] R. Labouvie, B. Santra, S. Heun, S. Wimberger, and H. Ott, Negative Differential Conductivity in an Interact- [40] C. Garginer and P. Zoller, Quantum noise: a handbook ing Quantum Gas, Phys. Rev. Lett. 115, 050601 (2015). of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics, Springer [25] R. Labouvie, B. Santra, S. Heun, and H. Ott, Bistabil- Science & Business Media (2004). ity in a Driven-Dissipative Superfluid, Phys. Rev. Lett. 116, 235302 (2016). [41] H. Haken, Laser Theory, Encyclopedia of Physics [26] E. Wamba, A. Pelster, and J.R. Anglin, Exact quan- (Berlin, Springer, 1970) tum field mappings between different experiments on [42] E.J.K.P. Nandani, R.A. Römer, S. Tan, and X.-W. quantum gases, Phys. Rev. A 94, 043628 (2016). Guan, Higher-order local and non-local correlations for [27] M. Naraschewski and R.J. Glauber, Spatial coherence 1D strongly interacting Bose gas, New J. Phys. 18, and density correlations of trapped Bose gases, Phys. 055014 (2016). Rev. A 59, 4595 (1999). [43] L. Piroli and P. Calabrese, Local correlations in the at- [28] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Seng- tractive one-dimensional Bose gas: From Bethe ansatz stock, A. Sanpera, G.V. Shlyapnikov, and M. Lewen- to the Gross-Pitaevskii equation, Phys. Rev. A 94, stein, Dark Solitons in Bose-Einstein Condensates, 053620 (2016). Phys. Rev. Lett. 83, 5198 (1999). [44] C. Henkel, T.-O. Sauer, and N.P. Proukakis, Cross-over [29] G. Lindblad, On the Generators of Quantum Dynamical to quasi-condensation: mean-field theories and beyond, Semigroups, Commun. Math. Phys. 48, 119 (1976). J. Phys. B: At. Mol. Opt. Phys. 50, 114002 (2017). [30] T. Prosen, Third quantization: a general method to [45] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fesh- solve master equations for quadratic open Fermi sys- bach resonances in ultracold gases, Rev. Mod. Phys. 82, tems, New J. Phys. 10, 043026 (2008). 1225 (2010). [31] E. Braaten, H.-W. Hammer, and G.P. Lepage, Lindblad equation for the inelastic loss of ultracold atoms, Phys. [46] P. Makotyn, C.E. Klauss, D.L. Goldberger, E.A. Cor- Rev. A 95, 012708 (2017). nell, and D.S. Jin, Universal dynamics of a degenerate unitary Bose gas, Nat. Phys. 10, 116 (2014). [32] G. Kordas, D. Witthaut, P. Buonsante, A. Vezzani, R. Burioni, A.I. Karanikas, and S. Wimberger, The dis- [47] T. Gericke, A scanning electron microscope for ultra- sipative Bose-Hubbard model: Methods and Examples, cold quantum gases, PhD thesis, Johannes Gutenberg- Eur. Phys. J. Special Topics 224, 2127 (2015). UniversitätMainz (2010). [33] U. Weiss, Quantum Dissipative Systems, 3rd Ed. Series [48] T. Gericke, P. Würtz,D. Reitz, T. Langen, and H. Ott, in Modern Condensed Matter Physics: Vol. 13, World High-resolution scanning electron microscopy of an ul- Scientific (2008). tracold quantum gas, Nat. Phys. 4, 949 (2008).

14 [49] A. Mohamadou, E. Wamba, S.Y. Doka, T.B. Ekogo and T.C. Kofané, Generation of matter-wave solitons of the Gross-Pitaevskii equation with a time-dependent com- plicated potential, Phys. Rev. A 84, 023602 (2011). [50] D.A. Zezyulin, I.V. Barashenkov, and V.V. Konotop, Stationary through-flows in a Bose-Einstein condensate with a PT-symmetric impurity, Phys. Rev. A 94, 063649 (2016). [51] D. Dries, S.E. Pollack, J.M. Hitchcock, and R.G. Hulet, Dissipative transport of a Bose-Einstein condensate, Phys. Rev. A 82, 033603 (2010). [52] J. Armijo, T. Jacqmin, K.V. Kheruntsyan, and I. Bou- choule, Probing Three-Body Correlations in a Quan- tum Gas Using the Measurement of the Third Moment of Density Fluctuations, Phys. Rev. Lett. 105, 230402 (2010). [53] D.M. Gangardt and G.V. Shlyapnikov, Local correlations in a strongly interacting one-dimensional Bose gas, New J. Phys. 5, 79 (2003). [54] M. Radonjić,W. Kopylov, A. Balaz, and A. Pelster, Interplay of coherent and dissipative dynamics in condensates of light, New J. Phys. 20, 055014 (2018). [55] E. Stein, F. Vewinger, and A. Pelster, Collective modes of a photon Bose-Einstein condensate with thermo-optic interaction, New J. Phys. 21, 103044 (2019).